The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background

Abstract

In this paper, we derive the early-time asymptotics for fixed-frequency solutions $\phi _\ell$ to the wave equation $\Box _g \phi _\ell =0$ on a fixed Schwarzschild background ($M>0$) arising from the no incoming radiation condition on ${\mathscr {I}}^-$ and polynomially decaying data, $r\phi _\ell \sim t^{-1}$ as $t\rightarrow -\infty$, on either a timelike boundary of constant area radius $r>2M$ (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of $\partial _v(r\phi _\ell )$ along outgoing null hypersurfaces near spacelike infinity $i^0$ contains logarithmic terms at order $r^{-3-\ell }\log r$. In contrast, in case (II), we obtain that the asymptotic expansion of $\partial _v(r\phi _\ell )$ near spacelike infinity $i^0$ contains logarithmic terms already at order $r^{-3}\log r$ (unless $\ell =1$). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity $i^+$ that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate a logarithmically modified Price’s law for each $\ell$-mode. On the other hand, the data of case (II) lead to much stronger deviations from Price’s law. In particular, we conjecture that compactly supported scattering data on ${\mathscr {H}}^-$ and ${\mathscr {I}}^-$ lead to solutions that exhibit the same late-time asymptotics on ${\mathscr {I}}^+$ for each $\ell$: $r\phi _\ell |_{{\mathscr {I}}^+}\sim u^{-2}$ as $u\rightarrow \infty$.

1 Introduction

1.1 Background and Motivation

In this paper, we study the early-time asymptotics, i.e. asymptotics near spatial infinity $i^0$, of solutions, localised on a single angular frequency $\ell =L$, to the wave equation

$$\begin{aligned} \Box _{g_M}\phi _{\ell =L}=0 \end{aligned}$$

(1.1)

on the exterior of a fixed Schwarzschild (or a more general spherically symmetric) background $({\mathscr {M}}_M,g_M)$ under certain assumptions on data near past infinity. The most important of these assumptions is the no incoming radiation condition on ${\mathscr {I}}^-$, stating that the flux of the radiation field on past null infinity vanishes at late advanced times. In addition, we will assume polynomially decaying (boundary) data on either a past-complete timelike hypersurface, or a past-complete null hypersurface.

1.1.1 The Spherically Symmetric Mode

We initiated the study of such data in [12], where we constructed spherically symmetric solutions arising from the no incoming radiation condition, as a condition on data on ${\mathscr {I}}^-$, and polynomially decaying boundary data on a timelike hypersurface $\Gamma$ terminating at $i^-$ (or polynomially decaying characteristic initial data on an ingoing null hypersurface ${\mathscr {C}}_{\mathrm {in}}$ terminating at ${\mathscr {I}}^-$).

The choice for these data, in turn, was motivated by an argument due to D. Christodoulou [4] (based on the monumental proof of the stability of the Minkowski spacetime [5]), which showed that the assumption of Sachs peeling [21, 22] and, thus, of (conformally) smooth null infinity [19] is incompatible with the no incoming radiation condition and the prediction of the quadrupole formula for N infalling masses from $i^-$. The latter predicts that the rate of change of gravitational energy along ${\mathscr {I}}^+$ is given by $\sim -1/|u|^4$ near $i^0$. Indeed, modelling gravitational radiation by scalar radiation, we showed in [12] that the data described above lead to solutions which not only agree with the prediction of the quadrupole approximation (namely that $r^2(\partial _u\phi )^2|_{{\mathscr {I}}^+}\sim |u|^{-4}$ near $i^0$), but also have logarithmic terms in the asymptotic expansion of the spherically symmetric mode $\partial _v(r\phi _0)$ as ${\mathscr {I}}^+$ is approached, thus contradicting the statement of Sachs peeling that such expansions are analytic in 1/r. More precisely, we obtained for the spherically symmetric mode $\phi _0$ that if the limit

$$\begin{aligned} \lim _{{\mathscr {C}}_{\mathrm {in}},u\rightarrow -\infty }|u|r\phi _0:=\Phi ^- \end{aligned}$$

(1.2)

on initial data is non-zero (or, in the timelike case, if a similar condition on $\Gamma$ holds), then it is, in fact, a conserved quantity along ${\mathscr {I}}^-$, and, for sufficiently large negative values of u, one obtains on each outgoing null hypersurface of constant u the asymptotic expansion

$$\begin{aligned} \partial _v(r\phi _0)(u,v)=-2M\Phi ^- \frac{\log r-\log |u|}{r^3}+{\mathscr {O}}(r^{-3}). \end{aligned}$$

(1.3)

In wide parts of the literature, it has been (and still is) assumed that physically relevant spacetimes do possess a smooth null infinity and that, therefore, logarithms as in (1.3) do not appear. The result of [12], in line with [5], thus further puts this assumption in doubt. Furthermore, we showed in [13] that the failure of peeling manifested by the early-time asymptotics (1.3) translates into logarithmic late-time asymptotics near $i^+$, providing evidence for the physical measurability of the failure of null infinity to be smooth. We will return to the discussion of late-time asymptotics in section 1.3.

For more background on the history and relevance of peeling and smooth null infinity, we refer the reader to the introduction of [12].

Finally, we note that the results from [12] were, in fact, obtained for the non-linear Einstein-Scalar field system ($G_{\mu \nu }[g]=T_{\mu \nu }^{sf}[\phi _0]$) under spherical symmetry and then, a fortiori, carried over to the linear case ($G_{\mu \nu }[g]=0$, $\Box _g \phi _0=0$).

1.1.2 Higher $\ell$-Modes

Ultimately, we would like to develop an understanding of the situation for the Einstein vacuum equations without symmetry assumptions (for which the spherically symmetric Einstein-Scalar field system only served as a toy model) in order to close the circle to Christodoulou’s original argument [4], which was an argument pertaining to gravitational, not scalar, radiation. In particular, we would like to understand the prediction of the quadrupole approximation, namely that the rate of gravitational energy loss along ${\mathscr {I}}^+$ is given by $-1/|u|^4$ as $u\rightarrow -\infty$, dynamically, i.e. arising from suitable scattering data, rather than imposing it on ${\mathscr {I}}^+$ as was done in [4]. In view of the multipole structure of gravitational radiation, it thus seems to be necessary to first understand the answer to the following question:

What are the early-time asymptotics for higher $\ell$-modes of solutions to the wave equation $\Box _g \phi =0$ on a fixed Schwarzschild background, arising from the no incoming radiation condition, i.e., what is the analogue of (1.3) for $\ell >0$?

We shall provide a detailed answer to this question in this paper. Let us already paraphrase two special cases of the main statements (which are summarised in section 1.2). Statement 1) below corresponds to Theorems 1.3, 1.4, and Statement 2) corresponds to Theorem 1.5.

Fig. 1

Schematic depiction of the data setup considered in 1): We consider polynomially decaying data on a spherically symmetric timelike hypersurface $\Gamma$, and vanishing data on ${\mathscr {I}}^-$. The latter condition is to be thought of as the no incoming radiation condition

1) Consider solutions $\phi _\ell$ to (1.1) arising from polynomially decaying data ${r^{\ell +1}}\phi _\ell \sim |t|^{-1}$ as $t\rightarrow -\infty$ on a spherically symmetric timelike hypersurface $\Gamma$ and the no incoming radiation condition on ${\mathscr {I}}^-$. (See Figure 1.) Then, schematically, $r\phi _\ell |_{{\mathscr {I}}^+}\sim |u|^{-\ell -1}$ along ${\mathscr {I}}^+$ as $u\rightarrow -\infty$, and the asymptotic expansion of $\partial _v(r\phi _\ell )$ along outgoing null hypersurfaces of constant u near spacelike infinity $i^0$ reads:

$$\begin{aligned} \partial _v(r\phi _\ell )=\frac{f_0(u)}{r^2}+\dots +\frac{f_\ell (u)}{r^{2+\ell }}+C\frac{\log r}{r^{3+\ell }}+\dots , \end{aligned}$$

(1.4)

where C is a non-vanishing constant.

Fig. 2

Schematic depiction of the data setup considered in 2): We consider polynomially decaying data on a spherically symmetric ingoing null hypersurface ${\mathscr {C}}_{\mathrm {in}}$, and vanishing data on the part of ${\mathscr {I}}^-$ that lies to the future of ${\mathscr {C}}_{\mathrm {in}}$. The latter condition is to be thought of as the no incoming radiation condition

2) Alternatively, consider solutions $\phi _\ell$ to (1.1) arising from polynomially decaying data $r\phi _\ell \sim |u|^{-1}$ as $u\rightarrow -\infty$ on a null hypersurface ${\mathscr {C}}_{\mathrm {in}}$ and the no incoming radiation condition. (See Figure 2.) Then, schematically, $r\phi _\ell |_{{\mathscr {I}}^+}\sim |u|^{-\min (\ell +1,2)}$ as $u\rightarrow -\infty$, and the asymptotic expansion of $\partial _v(r\phi _\ell )$ along outgoing null hypersurfaces of constant u near spacelike infinity $i^0$ reads:

$$\begin{aligned} \partial _v(r\phi _\ell )=\frac{f_0(u)}{r^2}+C\frac{\log r}{r^{3}}+\dots , \end{aligned}$$

(1.5)

unless $\ell =1$, in which case we instead have that

$$\begin{aligned} \partial _v(r\phi _\ell )=\frac{f_0(u)}{r^2}+\frac{f_1(u)}{r^3}+C\frac{\log r}{r^{4}}+\dots . \end{aligned}$$

(1.6)

In both cases, C is a generically non-vanishing constant.

By incorporating an $r^\ell$-weight into the boundary data assumption (namely $r^{\ell +1}\phi _\ell |_{\Gamma }\sim |t|^{-1}$), we phrased statement 1) in such a way as to be independent of the behaviour of the area radius r on $\Gamma$: Independently of whether r is constant along $\Gamma$ or divergent (e.g. $r|_{\Gamma }\sim |t|$), the $|t|^{-1}$-decay of $r^{\ell +1}\phi$ on $\Gamma$ translates into $|u|^{-\ell -1}$ decay of $r\phi _\ell$ near ${\mathscr {I}}^-$, causing the logarithmic term in (1.4) to appear $\ell$ orders later than in (1.5).

The difference between (1.5) and (1.6), on the other hand, is a manifestation of certain cancellations that happen if $r\phi _\ell \sim |u|^{-\ell }$ on ${\mathscr {C}}_{\mathrm {in}}$. Similar cancellations are responsible for $r\phi _\ell$ decaying faster on ${\mathscr {I}}^+$ than on ${\mathscr {C}}_{\mathrm {in}}$ in case 2). These cancellations, together with the precise and more general versions of the above statements, will be discussed in detail in section 1.2 below, see also Remark 1.4.

Let us finally remark that, even though higher $\ell$-modes thus decay slower than the spherically symmetric mode near spacelike infinity, we still expect the leading-order asymptotics near future timelike infinity $i^+$ to be dominated by the spherically symmetric mode in the two data setups described above, see also [13] and [3]. However, in the case of smooth compactly supported scattering data on ${\mathscr {I}}^-$ and the past event horizon ${\mathscr {H}}^-$, it turns out that all $\ell$-modes can be expected to have the same decay along ${\mathscr {I}}^+$ as $i^+$ is approached. We will discuss this in detail in section 1.3, see already Figures 3–5.

1.2 Summary of the Main Results

We now give a summary of the main theorems obtained in this paper. They are all stated with respect to Eddington–Finkelstein double null coordinates (u, v) ($\partial _vr=1-\frac{2M}{r}=-\partial _ur$). Let’s first focus on solutions to (1.1) supported on a single $\ell =1$-frequency.

1.2.1 The Case $\ell =1$

Let $\Gamma \subset {\mathscr {M}}_M$ be a spherically symmetric, past-complete timelike hypersurface of constant area radius function $r=R>2M$.¹ Let $\ell =1$ and $|m|\le 1$, and prescribe on $\Gamma$ smooth boundary data for $\phi _{\ell =1}=\phi _1 \cdot Y_{1m}$ that satisfy, as $u\rightarrow -\infty$,

$$\begin{aligned} \left| r^2\phi _1|_{\Gamma }-\frac{C_\Gamma }{|u|}\right| ={\mathscr {O}}_5(|u|^{-1-\varepsilon }) \end{aligned}$$

(1.7)

for some constant $C_\Gamma$ and for some $\varepsilon \in (0,1)$. Moreover, prescribe in a limiting sense that

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _1)(u,v)=0, \quad n=0,\dots , 5 \end{aligned}$$

(1.8)

for all $v\in {\mathbb {R}}$. We interpret this as the condition of no incoming radiation from ${\mathscr {I}}^-$. We then prove the following theorem, in its rough form (see Theorem 5.1 for the precise version):

Theorem 1.1

Given smooth boundary data satisfying (1.7), there exists a unique smooth (finite-energy) solution to (1.1) (restricted to the (1, m)-angular frequency) in the domain of dependence of $\Gamma \cup {\mathscr {I}}^-$ that restricts correctly to these data and satisfies (1.8). Moreover, this solution satisfies along any spherically symmetric ingoing null hypersurface:

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _1)(u,v)&=0, \end{aligned}$$

(1.9)

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1))(u,v)&\equiv I_{\ell =1}^{\mathrm {past}}[\phi ], \end{aligned}$$

(1.10)

where $I_{\ell =1}^{\mathrm {past}}[\phi ]$ is a constant which is non-vanishing as long as $C_\Gamma$ is non-vanishing and R/2M is sufficiently large, and we further have that

$$\begin{aligned} \left| r^2\partial _u(r^2\partial _u(r\phi _1))(u,v) - I_{\ell =1}^{\mathrm {past}}[\phi _1]\right| ={\mathscr {O}}(\max (r^{-1}, |u|^{-\varepsilon })). \end{aligned}$$

(1.11)

In particular, $r\phi _1$ decays like $u^{-2}$ towards ${\mathscr {I}}^-$.

The next theorem translates these results into logarithmic asymptotics along outgoing null hypersurfaces in a neighbourhood of spacelike infinity. Let ${\mathscr {C}}_{\mathrm {in}}$ be a spherically symmetric, past-complete ingoing null hypersurface (e.g. $v=1$). Prescribe on ${\mathscr {C}}_{\mathrm {in}}$ smooth data for $\phi _{\ell =1}=\phi _1 \cdot Y_{1m}$ that satisfy

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _1)&=C_{\mathrm {in}}^{(1)}, \end{aligned}$$

(1.12)

$$\begin{aligned} \left| r^2\partial _u(r^2\partial _u(r\phi _1))-C_{\mathrm {in}}^{(2)}\right|&={\mathscr {O}}(|u|^{-\varepsilon }), \end{aligned}$$

(1.13)

for some constants $C_{\mathrm {in}}^{(1)}$, $C_{\mathrm {in}}^{(2)}$ and for some $\varepsilon \in (0,1)$. Moreover, prescribe equation (1.8) to hold in a limiting sense to the future of ${\mathscr {C}}_{\mathrm {in}}$ for $n=0,1,2$. Then we have (see Theorem 4.1 for the precise version):

Theorem 1.2

Given smooth data satisfying (1.12) and (1.13), there exists a unique smooth solution to (1.1) (restricted to the (1, m)-angular frequency) in the domain of dependence of ${\mathscr {C}}_{\mathrm {in}}\cup {\mathscr {I}}^-$ that restricts correctly to these data and satisfies (1.8). Moreover, this solution satisfies, for sufficiently large negative values of u, the following asymptotics as ${\mathscr {I}}^+$ is approached along any outgoing spherically symmetric null hypersurface:

$$\begin{aligned} \begin{aligned} r^2\partial _v(r\phi _1)(u,v)=&-C_{\mathrm {in}}^{(1)}-2\int _{-\infty }^uF(u')\mathop {}\!\mathrm {d}u'-\frac{2MC_{\mathrm {in}}^{(1)}-2M\int _{-\infty }^uF(u')\mathop {}\!\mathrm {d}u'}{r}\\&-2M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\frac{\log r-\log |u|}{r^2}+{\mathscr {O}}(r^{-2}), \end{aligned} \end{aligned}$$

(1.14)

where F(u) is given by the limit of the radiation field $r\phi _1$ on ${\mathscr {I}}^+$,

$$\begin{aligned} F(u):=\lim _{v\rightarrow \infty }r\phi _1(u,v)=\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{6|u|^2}+{\mathscr {O}}(|u|^{-2-\varepsilon }). \end{aligned}$$

(1.15)

The asymptotics of $r\phi _1$ near ${\mathscr {I}}^+$ can be obtained by integrating $\partial _v(r\phi _1)$ from ${\mathscr {I}}^+$ and combining (1.14) and (1.15). In particular, if $M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\ne 0$, then peeling fails at future null infinity.

Theorem 1.2 applies to the solution of Theorem 1.1, with $C_{\mathrm {in}}^{(1)}=0$ and $C_{\mathrm {in}}^{(2)}=I_{\ell =1}^{\mathrm {past}}[\phi ]$.

Remark 1.1

Let us already make the following observation: We recall from section 1.1.1 that, in the spherically symmetric case ($\ell =0$) studied in [12], the initial u-decay of $r\phi _0$ was transported all the way to ${\mathscr {I}}^+$, that is, we had that $\lim _{v\rightarrow \infty }r\phi _0(u,v)\sim |u|^{-1}$. This fact was closely related to the approximate conservation law satisfied by $\partial _u(r\phi _0)$.² For $\ell =1$, we see that this is no longer the case: The initial $|u|^{-1}$-decay of $r\phi _1$ translates into $|u|^{-2}$-decay on ${\mathscr {I}}^+$.

This improvement in the u-decay on ${\mathscr {I}}^+$ can be traced back to certain cancellations that happen if the |u|-decay of the data comes with a specific power: In fact, notice from (1.15) that if $C_{\mathrm {in}}^{(1)}=0$, the u-decay of $r\phi _1$ on ${\mathscr {I}}^+$ sees no improvement over its initial decay. We will understand these cancellations in more generality in the theorems below, see already equation (1.27) of Theorem 1.5 and the Remark 1.4. See also §4.4.3 for a schematic explanation of these cancellations.

1.2.2 The Case of General $\ell \ge 0$

Let $\ell =L\in {\mathbb {N}}_0$ and $|m|\le L$, let $\Gamma$ be as in §1.2.1, and prescribe on $\Gamma$ smooth boundary data for $\phi _{\ell =L}=\phi _L \cdot Y_{Lm}$ that satisfy, as u tends to $-\infty$,

$$\begin{aligned} \left| r^{L+1}\phi _L|_{\Gamma }-\frac{C_\Gamma }{|u|}\right| ={\mathscr {O}}_{L+4}(|u|^{-1-\varepsilon }) \end{aligned}$$

(1.16)

for some constant $C_\Gamma$ and some $\varepsilon \in (0,1)$, and prescribe again, in a limiting sense, that for all $v\in {\mathbb {R}}$:

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _L(u,v))=0,\quad n=0,\dots ,L+4. \end{aligned}$$

(1.17)

Then we have (see Theorem 8.1 for the precise version):

Theorem 1.3

Given smooth boundary data satisfying (1.16), there exists a unique smooth solution to (1.1) (restricted to the (L, m)-angular frequency) in the domain of dependence of $\Gamma \cup {\mathscr {I}}^-$ that restricts correctly to these data and satisfies (1.17). Moreover, this solution satisfies along any spherically symmetric ingoing null hypersurface:

$$\begin{aligned} \lim _{u\rightarrow -\infty }(r^2\partial _u)^{L-j}(r\phi _L)(u,v)&=0,\qquad \qquad j=0,\dots ,L, \end{aligned}$$

(1.18)

$$\begin{aligned} \lim _{u\rightarrow -\infty }(r^2\partial _u)^{L+1}(r\phi _L)(u,v)&\equiv I_{\ell =L}^{\mathrm {past}}[\phi ],\qquad \qquad \end{aligned}$$

(1.19)

where $I_{\ell =L}^{\mathrm {past}}[\phi ]$ is a constant which is non-vanishing as long as $C_\Gamma$ is non-vanishing and R/2M is sufficiently large, and we further have that

$$\begin{aligned} \left| (r^2\partial _u)^{L+1}(r\phi _1)(u,v) - I_{\ell =L}^{\mathrm {past}}[\phi ]\right| ={\mathscr {O}}(\max (r^{-1},|u|^{-\varepsilon })). \end{aligned}$$

(1.20)

In particular, $r\phi _\ell$ decays like $|u|^{-\ell -1}$ towards ${\mathscr {I}}^-$.

Remark 1.2

Theorem 1.3 also applies to boundary data on more general spherically symmetric timelike hypersurfaces on which r is allowed to tend to infinity. See also Theorems 6.1, 6.2.

Moreover, the proof can also be applied to any inverse polynomial rate for the |u|-decay of the boundary data. In fact, if $r|_{\Gamma }\rightarrow \infty$, one can more generally apply it to growing polynomial rates, $r^{L+1}\phi _L|_{\Gamma }\sim |u|^{-p}$ for some $p<0$, so long as the quantity $r\phi _L|_{\Gamma }$ itself is decaying. This leads to some obvious changes in equations (1.19), (1.20). (Schematically, if $r^{L+1}\phi _L|_{\Gamma }\sim |u|^{-p}$ along $\Gamma$, then $r\phi _L\sim |u|^{-L-p}$ along hypersurfaces of constant v.)

Remark 1.3

Notice that the regularity required for the boundary data (Eq. (1.16)), when restricted to $L=0$, is higher than that of [12]. This is because, for general $L\ge 0$, we need to work with certain energy estimates in order to obtain the sharp decay for transversal derivatives on $\Gamma$, which is not necessary for the $\ell =0$-mode.

As before, the results of Theorem 1.3 translate into logarithmic asymptotics near spacelike infinity: Prescribe on ${\mathscr {C}}_{\mathrm {in}}$ smooth data for $\phi _{\ell =L}=\phi _L \cdot Y_{Lm}$ that satisfy

$$\begin{aligned} \lim _{u\rightarrow -\infty }(r^2\partial _u)^{L-j}(r\phi _L)(u,v)&=0,\qquad \qquad j=0,\dots ,L, \end{aligned}$$

(1.21)

$$\begin{aligned} \left| (r^2\partial _u)^{L+1}(r\phi _1)(u,v) - C^{(L,0)}_{\mathrm {in}}\right|&={\mathscr {O}}(|u|^{-\varepsilon })\qquad \qquad \end{aligned}$$

(1.22)

for some constant $C^{(L,0)}_{\mathrm {in}}$ and some $\varepsilon \in (0,1)$, and further prescribe equation (1.17) to hold in the future of ${\mathscr {C}}_{\mathrm {in}}$ for $n=0,\dots ,L+1$. We prove the following theorem in its rough form (see Theorem 9.1 for the precise version):

Theorem 1.4

Given smooth data satisfying (1.21) and (1.22), there exists a unique smooth solution to (1.1) (restricted to the (L, m)-angular frequency) in the domain of dependence of ${\mathscr {C}}_{\mathrm {in}}\cup {\mathscr {I}}^-$ that restricts correctly to these data and satisfies (1.17). Moreover, this solution satisfies, for sufficiently large negative values of u, the following asymptotics as ${\mathscr {I}}^+$ is approached along any outgoing spherically symmetric null hypersurface:

$$\begin{aligned} \begin{aligned} r^2\partial _v(r\phi _L)(u,v)=\sum _{i=0}^L \frac{f_i^{(L)}(u)}{r^{i}}+\frac{ I_{\ell =L}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ](\log r-\log |u|)}{r^{L+1}}+{\mathscr {O}}(r^{-L-1}) \end{aligned} \end{aligned}$$

(1.23)

where the $f_i^{(L)}$ are smooth functions of u which satisfy $f_i(u)=\frac{\beta _i^{(L)}}{|u|^{L-i}}+{\mathscr {O}}(|u|^{-L+i-\varepsilon })$ for some explicit numerical constants $\beta _i^{(L)}$, and $I_{\ell =L}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ]$ is an explicit constant which can be expressed as a non-vanishing numerical multiple of M and $C^{(L,0)}_{\mathrm {in}}$. In addition, we have that

$$\begin{aligned} \lim _{v\rightarrow \infty } r\phi _L(u,v)=\frac{L!C^{(L,0)}_{\mathrm {in}}}{(2L+1)!|u|^{L+1}}+{\mathscr {O}}(|u|^{-L-1-\varepsilon }). \end{aligned}$$

(1.24)

Now, while Theorem 1.3 generalises Theorem 1.1 in every sense, Theorem 1.4 does not fully generalise Theorem 1.2 since it excludes initial data that satisfy

$$\begin{aligned} \lim _{u\rightarrow -\infty }[r^2\partial _u]^{L-j}(r\phi _L)=C_{\mathrm {in}}^{(L,j+1)} \end{aligned}$$

(1.25)

for $j=0,\dots , L$ and non-vanishing constants $C_{\mathrm {in}}^{(L,j)}$. If only $C_{\mathrm {in}}^{(L,1)}$ is non-vanishing, then, in fact, the above theorem remains valid, albeit with some modifications to the $f_i(u)$ and to the constant $I_{\ell =L}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ]$. More generally, however, we have the following:

Instead of (1.21), (1.22), prescribe on ${\mathscr {C}}_{\mathrm {in}}$ that

$$\begin{aligned} \left| r\phi _L(u,1)-\frac{C_{\mathrm {in}}}{r^p}\right| ={\mathscr {O}}_1( r^{-p-\varepsilon }) \end{aligned}$$

(1.26)

for some $\varepsilon \in (0,1]$, a constant $C_{\mathrm {in}}\ne 0$, and for some $p\in {\mathbb {N}}_0$ ($p=0$ is permitted). Moreover, assume the no incoming radiation condition (1.17) to hold for $n=1,\dots ,L+1$. Then we have (see Theorem 10.1 for the precise version):

Theorem 1.5

Given smooth data satisfying (1.26), there exists a unique smooth solution to (1.1) (restricted to the (L, m)-angular frequency) in the domain of dependence of ${\mathscr {C}}_{\mathrm {in}}\cup {\mathscr {I}}^-$ that restricts correctly to these data and satisfies (1.17). Define $r_0:=|u|-2M\log |u|$. Then the limit of the radiation field satisfies

$$\begin{aligned} \lim _{v\rightarrow \infty }r\phi _L(u,v)=F(u)={\left\{ \begin{array}{ll} {\mathscr {O}}(r_0^{-p-\varepsilon }),&{}\text {if }p\le L \text { and } p\ne 0,\\ C(L,p)\cdot C_{\mathrm {in}} r_0^{-p}+{\mathscr {O}}(r_0^{-p-\varepsilon }), &{}\text {if }p>L \text { or } p=0, \end{array}\right. } \nonumber \\ \end{aligned}$$

(1.27)

for some smooth function F(u) and some non-vanishing numerical constant C(L, p).

Moreover, $\underline{if \, p<L}$, this solution satisfies, for sufficiently large negative values of u, the following asymptotics as ${\mathscr {I}}^+$ is approached along any outgoing spherically symmetric null hypersurface:

$$\begin{aligned} r^2\partial _v(r\phi _L)(u,v)=\sum _{i=0}^{p-1}\frac{f^{(L,p)}_i(u)}{r^i}+\frac{I_{\ell =L}^{\mathrm {future},r^{2+p-L}}[\phi ](\log r-\log |u|)}{r^p}+{\mathscr {O}}\left( \frac{|u|}{r^{p}}\right) , \nonumber \\ \end{aligned}$$

(1.28)

where the $f^{(L,p)}_i$ are smooth functions which satisfy $f^{(L,p)}_i={\mathscr {O}}(r_0^{-p+i+1-\varepsilon })$ if $i<p-1$, and $f^{(L,p)}_{i}=\beta _i^{(L,p)}+{\mathscr {O}}(r_0^{-\varepsilon })$ for some constant $\beta _i^{(L,p)}$ if $i=p-1$. $I_{\ell =L}^{\mathrm {future},r^{2+p-L}}[\phi ]$ is a non-vanishing constant which depends on $p,L,C_{\mathrm {in}}$ and M.

On the other hand, $\underline{if \, p\ge L}$, then

$$\begin{aligned} r^2\partial _v(r\phi _L)(u,v)=\sum _{i=0}^{\max (L,p-1)}\frac{f^{(L,p)}_i(u)}{r^i}+{\mathscr {O}}\left( \frac{\log r}{r^{\max (L+1,p)}}\right) , \end{aligned}$$

(1.29)

where the $f^{(L,p)}_i$ are smooth functions which satisfy $f^{(L,p)}_i={\mathscr {O}}(r_0^{-p+i+1-\varepsilon })$ if $p=L$ and $i<L-1$, and which satisfy $f^{(L,p)}_i=\beta _i^{(L,p)}r_0^{-p+i+1}+{\mathscr {O}}(r_0^{-p+i-\varepsilon })$ for some constants $\beta _i^{(L,p)}$ otherwise (i.e. if $p=L=i$, $p=L=i+1,$ or if $p>L$).

Some remarks are in order.

Remark 1.4

Notice the different behaviour in the cases $p<L$, $p=L$ and $p>L$ in Theorem 1.5. We want to direct the reader’s attention to the following points:

• Equation (1.27) shows that if $0\ne p\le L$, then there is a cancellation and $\lim _{v\rightarrow \infty }(r\phi _L)(u,v)$ decays faster in u than $r\phi _L(u,1)$. See §4.4.3 for a schematic explanation of these cancellations. Such cancellations do not happen if $p=0$ of $p>L$. Moreover, they can be viewed as Minkowskian behaviour, i.e., they can already be seen if $M=0$. In fact, in the course of the proof of Theorem 1.5, we will derive simple and effective expressions for solutions of $\Box _g\phi _L=0$ on Minkowski arising from the no incoming radiation condition and initial data $r\phi _L(u,1)=C/r^p$ (see Proposition 10.4).

• In view of (1.29), we see that “the first logarithmic term” in the expansion of $r^2\partial _v(r\phi _L)$ appears at order $r^{-p-1}\log r$ unless $p=L$, in which case it appears one order later. In particular, it never appears at order $r^{-L-1}\log r$.

Remark 1.5

The proof of Theorem 1.5 can be generalised to positive non-integer p in (1.26) (and even to certain negative p). However, if $p\notin \{1,\dots ,L\}$, we expect no cancellations of the type above to occur. On the other hand, if we assume, for instance, that $r\phi _L(u,1)\sim r^{-p}\log r$ initially, then the same cancellations occur in the range $p\in \{1,\dots L\}$, and one will obtain that $r\phi _L|_{{\mathscr {I}}^+}\sim |u|^{-p}$ if $p\in \{1,\dots , L\}$ and $r\phi _L|_{{\mathscr {I}}^+}\sim |u|^{-p}\log |u|$ otherwise. This observation will be of relevance in future work.

Remark 1.6

All of the above theorems make crucial use of certain approximate conservation laws. These are generalisations of the Minkowskian identities

$$\begin{aligned} \partial _u\left( r^{-2\ell -2}(r^2\partial _v)^{\ell +1}(r\phi _\ell )\right) =0,&\partial _v\left( r^{-2\ell -2}(r^2\partial _u)^{\ell +1}(r\phi _\ell )\right) =0, \end{aligned}$$

and have been used in a very similar context in the recent [3], see also [15]. See already section 3.4 and section 7 for a discussion and derivation of these in the cases $\ell \le 1$, $\ell \ge 0$, respectively. The reason why we stated Theorems 1.4 and 1.5 separately is that the former can be proved in a rather simple way using the second conservation law, i.e. by propagating the initial decay for $(r^2\partial _u)^{\ell +1}(r\phi _\ell )$ in v, whereas, in order to prove Theorem 1.5, we will need to use the conservation law in the u-direction.

Remark 1.7

The constants $I_{\ell =L}^{\mathrm {future},f}[\phi ]$ appearing in the above theorems are modified Newman–Penrose constants. These are closely related to the approximate conservation laws mentioned before. We will discuss this further in the next section.

Remark 1.8

One can generalise all of the above theorems to hold on more general spherically symmetric spacetimes such as the Reissner–Nordström spacetimes in the full physical range of charge parameters $|e|\le M$. In the extremal case $|e|=M$, one can moreover apply the well-known conformal “mirror” isometry to obtain results on the asymptotics near the future event horizon ${\mathscr {H}}^+$, see section 2.2.2 of [12].

1.3 Future Applications: Late-Time Asymptotics and the Role of the Modified Newman–Penrose Constants

The approximate conservation laws mentioned in Remarks 1.6, 1.7 are closely related to the $\ell$-th order Newman–Penrose constants $I_\ell [\phi ]$ defined on future and past null infinity, respectively (see also the original [17, 18], and, more tailored to our context, [2, 3] and section 7 of the present paper). In fact, these $\ell$-th order Newman–Penrose constants play an important role in the study of both early-time asymptotics (near $i^0$) and late-time asymptotics (near $i^+$) of fixed-$\ell$ solutions to the wave equation on Schwarzschild.

While the question of early-time asymptotics has not been investigated much elsewhere, the study of late-time asymptotics has been an active field for decades. The most prominent result in this line of research is the so-called Price’s law [10, 20], see also [14]. Price’s law states that smooth, compactly supported data on a Cauchy hypersurface (i.e. data with trivial early-time asymptotics) for fixed angular frequency solutions $\phi _{\ell =L}=Y_{Lm}\phi _{Lm}$ to the wave equation (1.1) generically lead to the following asymptotics near future timelike infinity $i^+$ (we suppress the m-index in the following):

$$\begin{aligned} r\phi _L|_{{\mathscr {I}}^+}\sim u^{-2-L}, \qquad \phi _L|_{r=\text {constant}}\sim \tau ^{-2L-3}, \qquad \phi _L|_{{\mathscr {H}}^+}\sim v^{-2L-3} \end{aligned}$$

(1.30)

along future null infinity, hypersurfaces of constant r, and the event horizon ${\mathscr {H}}^+$, respectively. This statement has been satisfactorily proved in the recent works [1,2,3], see also [11] and [15]. (For earlier rigorous works on pointwise upper bounds (not asymptotics), see [8, 9] as well as [16].) We also refer the reader to these papers for more general background and motivation for the study of late-time asymptotics.

The question of late-time asymptotics for compactly supported Cauchy data has thus been completely understood. Similar results have been obtained for non-compactly supported data, but in that case, it has typically been assumed that the data are conformally smooth. However, if one’s motivation for studying late-time asymptotics comes from gravitational wave astronomy (i.e. the hope that some devices will eventually be able to measure these asymptotics), then the assumption of smooth compactly supported (or conformally smooth) data on a Cauchy hypersurface becomes questionable – as long as one accepts the general framework of an isolated system. For, if one assumes that the gravitational waves emitting system under consideration has existed for all times, then it will certainly have radiated for all times: Thus, a spacetime describing this system cannot be expected to contain Cauchy hypersurfaces with compact radiation content. On the other hand, the data considered in [12] and the present paper have a clear physical motivation³ and, thus, seem like a more reasonable starting point for the question of physically relevant late-time asymptotics.

Motivated by this, we shall now discuss consequences that our results from section 1.2 have on late-time asymptotics. It turns out that one can gain a simple, intuitive understanding of these in terms of the aforementioned Newman–Penrose constants.

1.3.1 The Timelike Case: A Logarithmically Modified Price’s Law for All $\ell$

Let’s assume that we have a spherically symmetric timelike hypersurface $\Gamma$ that has constant area radius near $i^-$ and terminates at ${\mathscr {H}}^+$. (Note that, if we chose $\Gamma$ to terminate at $i^+$, then we would have to essentially prescribe the late-time asymptotics as boundary data on $\Gamma$. On the other hand, if we choose $\Gamma$ to terminate at ${\mathscr {H}}^+$, then it will turn out that the leading-order late-time asymptotics are completely determined by the data’s behaviour near $i^-$. In particular, they do not depend on the extension of the data towards ${\mathscr {H}}^+$.) Consider first the spherically symmetric mode, and prescribe smooth data for it which, near past timelike infinity $i^-$, behave like $r\phi _0=C|u|^{-1}+{\mathscr {O}}(|u|^{-1-\varepsilon })$, and which smoothly extend to the future event horizon ${\mathscr {H}}^+$; and impose the no incoming radiation condition on ${\mathscr {I}}^-$. Then the results of [12] show⁴ that the past Newman–Penrose constant exists and is conserved along ${\mathscr {I}}^-$:

$$\begin{aligned} I_{\ell =0}^{\mathrm {past}}[\phi ]:=\lim _{u\rightarrow -\infty } r^2\partial _u(r\phi _0)\ne 0. \end{aligned}$$

(1.31)

Moreover, we showed that the finiteness of the past N–P constant, together with the no incoming radiation condition, implies that, even though the future Newman–Penrose constant vanishes ($\lim _{v\rightarrow \infty } r^2 \partial _v(r\phi _0)=0$), a logarithmically modified future Newman–Penrose constant exists and is conserved along ${\mathscr {I}}^+$:

$$\begin{aligned} I_{\ell =0}^{ \mathrm {future},\frac{\log r}{r^3}}[\phi ]:=\lim _{v\rightarrow \infty } \frac{r^3}{\log r} \partial _v(r\phi _0)=-2MI_{\ell =0}^{\mathrm {past}}[\phi ]\ne 0. \end{aligned}$$

(1.32)

In [13], we then applied slight adaptations of the methods of [2] to show that this logarithmically modified Newman–Penrose constant completely determines the leading-order late-time asymptotics near $i^+$:

$$\begin{aligned} r\phi _0(u,\infty )&=\frac{1}{2}I_{\ell =0}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ]\frac{\log u}{u^2}+{\mathscr {O}}(u^{-2}), \end{aligned}$$

(1.33)

$$\begin{aligned} \phi _0(u,v_R(u))&=\frac{1}{2}I_{\ell =0}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ]\frac{\log \tau }{\tau ^3}+{\mathscr {O}}(\tau ^{-3}),\end{aligned}$$

(1.34)

$$\begin{aligned} \phi _0(\infty ,v)&=\frac{1}{2}I_{\ell =0}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ]\frac{\log v}{v^3}+{\mathscr {O}}(v^{-3}), \end{aligned}$$

(1.35)

along ${\mathscr {I}}^+$, hypersurfaces of constant R, and ${\mathscr {H}}^+$, respectively.⁵ In particular, the leading-order late-time behaviour is independent of the extension of the data towards ${\mathscr {H}}^+$ and only depends on the behaviour of the data near $i^-$. We called this a logarithmically modified Price’s law for the $\ell =0$-mode.

Consider now the $\ell =1$-case. If we assume data as in Theorem 1.1 and smoothly extend them to ${\mathscr {H}}^+$, then we obtain that the past N–P constant of order $\ell =1$ exists and is conserved along ${\mathscr {I}}^-$:

$$\begin{aligned} I_{\ell =1}^{\mathrm {past},f}[\phi ]:=\lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1)+Mr\phi _1)=\lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1))\ne 0. \nonumber \\ \end{aligned}$$

(1.36)

It then follows from Theorem 1.2 that the decay encoded in (1.36) (namely, $r\phi _1\sim u^{-2}$), along with the no incoming radiation condition, implies that the future N–P constant vanishes, but that a logarithmically modified future Newman–Penrose constant of order $\ell =1$ exists and is conserved along ${\mathscr {I}}^+$ (see also Theorem 4.2 in §4.4):

$$\begin{aligned} I_{\ell =1}^{ \mathrm {future},\frac{\log r}{r^3}}[\phi ]:=\lim _{v\rightarrow \infty } \frac{r^3}{\log r} \partial _v(r^2\partial _v(r\phi _1)-Mr\phi _1)=4MI_{\ell =1}^{\mathrm {past}}[\phi ]. \end{aligned}$$

(1.37)

One should then be able to combine the results above with those of Angelopoulos–Aretakis–Gajic [3], with adaptations exactly as in [13] (which combined the results of [12] and [1, 2]), in order to obtain near $i^+$:

$$\begin{aligned} r\phi _1(u,\infty )&=C_1\frac{\log u}{u^3}+{\mathscr {O}}(u^{-3}), \end{aligned}$$

(1.38)

$$\begin{aligned} \phi _1(u,v_R(u))&=C_2\frac{\log \tau }{\tau ^5}+{\mathscr {O}}(\tau ^{-5}), \end{aligned}$$

(1.39)

$$\begin{aligned} \phi _1(\infty ,v)&=C_2\frac{\log v}{v^5}+{\mathscr {O}}(v^{-5}), \end{aligned}$$

(1.40)

where $C_1, C_2$ are given by numerical multiples of $I_{\ell =1}^{\log , \mathrm {future}}[\phi ]$. In particular, these constants $C_1,C_2$ should be independent of the extension of the data towards ${\mathscr {H}}^+$. Thus, we would obtain a logarithmically modified Price’s law for $\ell =1$ (cf. §4.4.2).

Fig. 3

Schematic depiction of the situation of §1.3.1: Given smooth data for $r\phi _\ell$ on $\Gamma$ which decay like 1/t near $i^-$, the solution decays like $u^{-\ell -1}$ near ${\mathscr {I}}^-$ by Thm. 1.3 and has finite logarithmically modified N–P constant on ${\mathscr {I}}^+$ by Thm. 1.4. The depicted late-time behaviour near $i^+$ should follow from the methods of [3] and should be independent of the data’s extension towards ${\mathscr {H}}^+$

In fact, we expect the same structure to hold in the case of general $\ell =L$. The data of Theorem 1.3 lead to solutions which have finite L-th order past N–P constant and, by Theorem 1.4 (see also Theorem 9.1), have a finite logarithmically modified L-th order future N–P constant $I_{\ell =L}^{ \mathrm {future},\frac{\log r}{r^3}}[\phi ]$, see §7 for the definition of these. In other words, Theorems 1.3 and 1.4 prove the precise analogues of (1.36) and (1.37) for general $\ell =L$. In view of the remarks above, one should then be able to recover a logarithmically modified Price’s law for each $\ell$ from this. See Figure 3.

What would be more difficult, however, is to show such a statement for fixed, finite regularity of $\phi _L$ instead of assuming smoothness or regularity that is dependent on L. We therefore make the following conjecture:

Conjecture 1

Prescribe data for $\phi$ on $\Gamma$ that have sufficient but fixed, finite regularity and which satisfy $r^\ell \phi _\ell \sim t^{-1}$ as $t\rightarrow -\infty$ for all $\ell$. Moreover, prescribe the no incoming radiation condition on ${\mathscr {I}}^-$. Then there exists an $\ell _0\in {\mathbb {N}}$, increasing with the prescribed regularity of the data, such that, for all $\ell \le \ell _0$, the $\ell$-modes $\phi _\ell$ of the corresponding solution will exhibit the following late-time asymptotics near $i^+$:

$$\begin{aligned} r\phi _\ell |_{{\mathscr {I}}^+}\sim u^{-2-\ell }\log u, \qquad \quad \phi _\ell |_{r=\text {constant}}\sim \tau ^{-2\ell -3}\log \tau , \qquad \quad \phi _\ell |_{{\mathscr {H}}^+}\sim v^{-2\ell -3}\log v. \end{aligned}$$

(1.41)

Moreover, the projection onto higher $\ell >\ell _0$-modes $\phi _{\ell >\ell _0}$ satisfies the upper bounds

$$\begin{aligned} r\phi _{\ell>\ell _0}|_{{\mathscr {I}}^+}&={\mathscr {O}}( u^{-2-\ell _0-\varepsilon }), \quad \phi _{\ell>\ell _0}|_{r=\text {constant}}={\mathscr {O}}( \tau ^{-2\ell _0-3-\varepsilon }), \nonumber \\ \phi _{\ell >\ell _0}|_{{\mathscr {H}}^+}&={\mathscr {O}}( v^{-2\ell _0-3-\varepsilon }) \end{aligned}$$

(1.42)

for some $\varepsilon >0$. If the data are chosen to be smooth, then $\ell _0$ can be chosen to be $\infty$.

See also the comments in §9.5.

A proof of the above conjecture would require revisiting the proof of Thm. 1.3 since, as stated, Thm. 1.3 requires boundary data regularity increasing in angular frequency L. However, if one imposes fixed, finite regularity, it should still be possible to extract weaker decay (compared to that of Thm. 1.3) from the methods of the proof that is consistent with (1.42). On the other hand, once these modifications are understood, one should be able to directly apply the methods of [3], with modifications as in [13], to prove the conjecture.

It would also be interesting to find a definitive answer to the question whether or not the rate (1.42) can be improved without assuming additional regularity.

We finally note that, on the one hand, if the 1/t-decay on initial data is replaced by any integrable decay rate, then the logarithms in (1.41) would disappear and we would expect the usual Price’s law tails. On the other hand, if one considers a timelike hypersurface $\Gamma$ on which $r|_{\Gamma }\rightarrow \infty$ as $u\rightarrow -\infty$, say, $r|_{\Gamma }(u)\sim |u|$, and only imposes $r\phi _\ell |_{\Gamma }\sim |t|^{-1}$, then the expected modifications to Price’s law are much more severe and exactly as in the null case with $p=1$. We will discuss this latter case now.

1.3.2 The Null Case: More Severe Deviations from Price’s Law

In contrast to the timelike case, it turns out that the data considered in Theorem 1.2, which were posed for the $\ell =1$-mode on an ingoing null hypersurface ($r\phi _1|_{{\mathscr {C}}_{\mathrm {in}}}\sim C_{\mathrm {in}}^{(1)}|u|^{-1}$), generally lead to a non-vanishing future Newman–Penrose constant

$$\begin{aligned} I_{\ell =1}^{\mathrm {future}}[\phi ]:=\lim _{v\rightarrow \infty } r^2 \partial _v(r^2\partial _v(r\phi _1)-Mr\phi _1)\ne 0 \end{aligned}$$

(1.43)

if $C_{\mathrm {in}}^{(1)}\ne 0$, cf. Theorem 4.2. In this case, one recovers the following late-time asymptotics (provided that one smoothly extends the data to ${\mathscr {H}}^+$):

$$\begin{aligned} r\phi _1|_{{\mathscr {I}}^+}\sim u^{-2},&\phi _1|_{r=R}\sim \tau ^{-4},&\phi _1|_{{\mathscr {H}}^+}\sim v^{-4}, \end{aligned}$$

(1.44)

with the leading-order asymptotics only depending on $I_{\ell =1}^{\mathrm {future}}[\phi ]$. These late-time asymptotics are one power worse than the Price’s law decay (1.30) for compactly supported data and have also been derived in [3].⁶

In the case of general $\ell \ge 1$, however, the situation is more subtle: The data considered in Theorem 1.5, i.e. $r\phi _L(u,1)\sim |u|^{-p}$, lead, for $p\le L\ne 0$, to solutions where the usual Newman–Penrose constant is infinite, $I_{\ell =L}^\mathrm {future}[\phi ]:=\lim r^2\partial _v\Phi _L=\infty$, where $\Phi _L$ is defined in section 7, Eq. (7.8). Instead, the following $(L-p)$-modified Newman–Penrose constant remains finite and conserved along null infinity (see also Thm. 10.1):

$$\begin{aligned} I_{\ell =L}^{\mathrm {future},r^{2-L+p}}[\phi ]:=\lim _{v\rightarrow \infty } r^{2-L+p}\partial _v\Phi _L\ne 0. \end{aligned}$$

(1.45)

With the decay encoded in (1.45), which is $L-p$ powers worse than in the case of finite unmodified N–P constant, we expect that one can further modify the methods of [3] to then derive late-time asymptotics near $i^+$ which are $L-p+1$ powers slower than the Price’s law decay (1.30) and which do not depend on the data’s extension towards ${\mathscr {H}}^+$ (see Figure 4), provided that the solution is smooth. Cf. the comments in §10.9.

Fig. 4

Schematic depiction of the situation of §1.3.2: Given data for $r\phi _\ell$ on ${\mathscr {C}}_{\mathrm {in} }$ which decay like $1/u^p$ near ${\mathscr {I}}^-$, the solution has finite $(\ell -p)$-modified N–P constant (see (1.45)) on ${\mathscr {I}}^+$ by Thm. 1.5, provided that $p\le \ell$. We also depicted the conjectured late-time behaviour near $i^+$

Analogously to Conjecture 1, we also make the following conjecture for the finite regularity problem:

Conjecture 2

Let $1\le p \in {\mathbb {N}}$, and prescribe data for $\phi$ on ${\mathscr {C}}_{\mathrm {in}}$ that have sufficient but fixed, finite regularity and which satisfy $r\phi _\ell \sim |u|^{-p}$ as $u\rightarrow -\infty$ for all $\ell$. Moreover, prescribe the no incoming radiation condition on ${\mathscr {I}}^-$. Then, for all $\ell \ge p>0$,⁷ the $\ell$-modes $\phi _\ell$ of the corresponding solution will exhibit the following late-time asymptotics near $i^+$ along ${\mathscr {I}}^+$:

$$\begin{aligned} r\phi _\ell |_{{\mathscr {I}}^+}\sim u^{-p-1}. \end{aligned}$$

(1.46)

Moreover, there exists an $\ell _0\in {\mathbb {N}}$, increasing with the prescribed regularity of the data, such that, away from ${\mathscr {I}}^+$, and for some $\varepsilon >0$,

$$\begin{aligned}&\phi _\ell |_{r=\text {constant}}\sim \tau ^{-\ell -p-2},\quad \phi _\ell |_{{\mathscr {H}}^+}\sim v^{-\ell -p-2},\quad \text {for all } \ell \in \{p,\dots ,\ell _0\}, \end{aligned}$$

(1.47)

$$\begin{aligned}&\phi _{\ell>\ell _0}|_{r=\text {constant}}={\mathscr {O}}( \tau ^{-\ell _0-p-2-\varepsilon }),\quad \phi _{\ell >\ell _0}|_{{\mathscr {H}}^+}={\mathscr {O}}( v^{-\ell _0-p-2-\varepsilon }). \end{aligned}$$

(1.48)

If the data are chosen to be smooth, then $\ell _0$ can be chosen to be $\infty$.

Remarkably, if one takes $p=1$ in (1.26), then the asymptotics (1.46), (1.47) for $\ell \ge 1$ would still be a logarithm faster than the ones for $\ell =0$, (1.33)–(1.35), despite the decay of $\partial _v(r\phi _\ell )$ towards spatial infinity being slower for $\ell >0$ than for $\ell =0$.

We note that even if one is willing to assume smoothness, then the modifications to [3] needed to prove (1.47) are different than those of [13], as one now has to deal with a difference in integer powers in decay. (In [13], we treated non-integer modifications in decay.) We expect that one should be able to use time derivatives, rather than time integrals, of our solutions to reduce to the cases treated in [3], and then integrate the asymptotics of these time derivatives from $i^+$ to obtain the asymptotics of the original solution. (This would be the opposite procedure of that described in footnote 6.)

In contrast, the fixed, finite regularity problem, i.e. a proof of Conjecture 2, would require much more elaborate modifications. In fact, since we conjecture the precise late-time asymptotics for all $\ell$ in (1.46), one would now also have to modify the methods of [3] since the procedure outlined in the previous paragraph would, again, require regularity that is increasing in $\ell$. We also want to point out that, since the conjectured asymptotics (1.46) along ${\mathscr {I}}^+$ are independent of $\ell$, an understanding of the fixed, finite regularity problem would be all the more important for applications!

1.3.3 Compactly Supported Scattering Data on ${\mathscr {H}}^-$ and ${\mathscr {I}}^-$

One final natural configuration of data we want to consider is the case of smooth, compactly supported scattering data on ${\mathscr {I}}^-$ and the past event horizon ${\mathscr {H}}^-$. In order to apply our results, we can, without loss of generality, assume that the data on ${\mathscr {H}}^-$ are vanishing (this can be achieved by restricting to sufficiently large negative values of u). Similarly, we can assume, without loss of generality, that the data on ${\mathscr {I}}^-$ are supported in $v_1\le v\le 1$. If we then integrate the wave equation satisfied by the radiation field $r\phi _L$, namely

$$\begin{aligned} \partial _v\partial _u(r\phi _L)=\left( 1-\frac{2M}{r}\right) \left( \frac{-L(L+1)}{r^2}r\phi _L-\frac{2M}{r^3}r\phi _L\right) , \end{aligned}$$

(1.49)

from $v=v_1$ to $v=1$, we obtain that, generically, $\partial _u(r\phi _0)(u,1)\sim r^{-3}$ if $L=0$ and $\partial _u(r\phi _L)(u,1)\sim r^{-2}$ if $L\ge 1$. More precisely, one can derive from (1.49) that if the data on ${\mathscr {I}}^-$ are given by $r\phi _L(-\infty ,v)=:G(v)$, then

$$\begin{aligned} r^2\cdot \partial _u(r\phi _L)(u,1)=-L(L+1)\int _{v_1}^1 G(v)\mathop {}\!\mathrm {d}v+{\mathscr {O}}(r^{-1}), \quad \quad \text {if } L>0,\\ r^3\cdot \partial _u(r\phi _0)(u,1)=-2M\int _{v_1}^1 G(v)\mathop {}\!\mathrm {d}v+{\mathscr {O}}(r^{-1}), \quad \quad \text {if } L=0, \end{aligned}$$

See also §2.2 and §6 of [12] for a detailed discussion of this restricted to the spherically symmetric mode. Thus, since the integrals above are non-vanishing for generic scattering data G, one can show that Theorem 1.5 applies, with (generically) $p=2$ if $L=0$ and with $p=1$ if $L\ge 1$.

The results of Theorem 1.5 then show that if $L=0$, then $\lim _{v\rightarrow \infty }r^3\partial _v(r\phi _0)<\infty$, whereas if $L\ge 1$, then, generically, $I_{\ell =L}^{\mathrm {future},r^{2-L+1}}[\phi ]$ is finite and non-vanishing. Therefore, if $L=0$, one obtains the following late-time asymptotics near $i^+$ [3]:

$$\begin{aligned} r\phi _0|_{{\mathscr {I}}^+}\sim u^{-2}, \quad \quad \phi _0|_{r=R}\sim \tau ^{-3},\quad \quad \phi _0|_{{\mathscr {H}}^+}\sim v^{-3}. \end{aligned}$$

(1.50)

On the other hand, if $L>0$, then, since $p=1$, Conjecture 2 would imply that, generically,

$$\begin{aligned} r\phi _L|_{{\mathscr {I}}^+}\sim u^{-2}, \quad \quad \phi _L|_{r=R}\sim \tau ^{-L-3},\quad \quad \phi _L|_{{\mathscr {H}}^+}\sim v^{-L-3}. \end{aligned}$$

(1.51)

We are thus led to a third conjecture (see Figure 5):

Conjecture 3

Consider compactly supported scattering data on ${\mathscr {H}}^-$ and ${\mathscr {I}}^-$ for (1.1), supported on all angular frequencies, with sufficient but finite regularity. Then there exists an $\ell _0\in {\mathbb {N}}$, increasing with the prescribed regularity of the data, such that, away from ${\mathscr {I}}^+$, and for some $\varepsilon >0$,

$$\begin{aligned}&\phi _\ell |_{r=\text {constant}}\sim \tau ^{-\ell -p-2},\quad \phi _\ell |_{{\mathscr {H}}^+}\sim v^{-\ell -p-2},\quad \text {for all } \ell \in \{0,\dots ,\ell _0\}, \end{aligned}$$

(1.52)

$$\begin{aligned}&\phi _{\ell>\ell _0}|_{r=\text {constant}}={\mathscr {O}}( \tau ^{-\ell _0-p-2-\varepsilon }),\quad \phi _{\ell >\ell _0}|_{{\mathscr {H}}^+}={\mathscr {O}}( v^{-\ell _0-p-2-\varepsilon }). \end{aligned}$$

(1.53)

On the other hand, along future null infinity ${\mathscr {I}}^+$, we have the asymptotic expression

$$\begin{aligned} r\phi |_{{\mathscr {I}}^+}=\sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell C_{\ell m}\cdot Y_{\ell m}(\theta ,\phi )u^{-2}+o(u^{-2}) \end{aligned}$$

(1.54)

for some constants $C_{\ell m}$ which can be computed explicitly from the scattering data on ${\mathscr {I}}^-$ and which are generically non-zero.

Fig. 5

Given compactly supported scattering data for $r\phi _\ell$ on ${\mathscr {H}}^-$ and ${\mathscr {I}}^-$, the solution generically decays like 1/u near ${\mathscr {I}}^-$ (away from the support of the data) unless $\ell =0$. It thus has finite $(\ell -1)$-modified N–P constant (see (1.45)) on ${\mathscr {I}}^+$ by Thm. 1.5. We also depicted the conjectured late-time asymptotics for all $\ell \ge 0$

The asymptotics (1.54) would be in stark contrast to the usual expectation that the asymptotic behaviour on ${\mathscr {I}}^+$, i.e. the physically measurable behaviour, is dominated by low frequencies. It would therefore also be interesting to find the precise form of the constants $C_{\ell m}$ to see how much each frequency contributes. We leave this, as well as the resolution of Conjectures 1–3, to future work.

1.4 Structure and Guide to Reading the Paper

This paper is structured as follows: We first recall the family of Schwarzschild spacetimes and recall some geometric preliminaries in §2. We then recall relevant results on the wave equation on a Schwarzschild background in §3.

The rest of the paper is divided into two parts: In part I, which comprises sections 4–6, we focus solely on the $\ell =1$-case. This part is written with an emphasis on being instructive and providing intuition for the main results and some (but not all) of the methods used to prove them, which might otherwise be camouflaged by the large amount of inductions in the case of general $\ell$. In part II, which comprises sections 7–10, we then develop a more systematic approach for the case of general $\ell$.

Part I is structured as follows: In §4, we treat the case of data on an ingoing null hypersurface and prove Theorem 1.2. In §5, we then treat the case of boundary data on a timelike hypersurface $\Gamma$ of constant area radius and prove Theorem 1.1. We shall explain how to lift the restriction to constant area radii and treat boundary data on more general spherically symmetric hypersurfaces in §6.

Part II is structured as follows: In §7, we derive the higher-order approximate conservation laws for general $\ell$-modes and the associated higher-order Newman–Penrose constants. Equipped with these, we then consider the case of boundary data on a hypersurface $\Gamma$ of constant area radius and prove Theorem 1.3 in §8. The generalisation to more general $\Gamma$ proceeds similarly to the one in §6 and is left to the reader. The last two sections, §9 and §10, again concern data on a null hypersurface. In §9, we consider the fast initial decay implied by Thm. 1.3 and prove Theorem 1.4. Section 10 then generalises these results to slowly decaying data (using different methods) and contains the proof of Theorem 1.5. Various inductive proofs of statements made in part II are deferred to the appendix A.

Depending on the reader’s taste, she can either begin with a thorough reading of part I and then skim through §7–§9 of part II and carefully read §10, which introduces an approach not presented in part I.

Alternatively, she can skip directly to part II and occasionally refer back to part I for details, e.g. on the treatment of boundary data on a timelike hypersurface of varying area radius in §6.

In any case, an effort was made to make each section of the paper as self-contained as possible.

2 Geometric Preliminaries

2.1 The Schwarzschild Spacetime Manifold

The (exterior of the) Schwarzschild family of spacetimes $({\mathscr {M}}_M,g_M)$, $M>0$,⁸ is given by the family of manifolds

$$\begin{aligned} {\mathscr {M}}_M={\mathbb {R}}\times (2M, \infty )\times {\mathbb {S}}^2, \end{aligned}$$

covered by the coordinate chart $(v,r,\theta , \varphi )$, with $v\in {\mathbb {R}}$, $r\in (2M,\infty )$, $\theta \in (0,\pi )$ and $\varphi \in (0,2\pi )$, where $(\theta ,\varphi )$ denote the standard spherical coordinates on ${\mathbb {S}}^2$, and by the family of metrics

$$\begin{aligned} g_M=-D(r)\mathop {}\!\mathrm {d}v^2+2\mathop {}\!\mathrm {d}v\mathop {}\!\mathrm {d}r+r^2(\mathop {}\!\mathrm {d}\theta ^2+\sin ^2\theta \mathop {}\!\mathrm {d}\varphi ^2), \end{aligned}$$

(2.1)

where

$$\begin{aligned} D(r)=1-\frac{2M}{r}. \end{aligned}$$

(2.2)

Upon introducing the tortoise coordinate $r^*$ as

$$\begin{aligned} r^*(r):=R+\int _R^r D^{-1}(r')\mathop {}\!\mathrm {d}r' \end{aligned}$$

(2.3)

for some $R>2M$, and defining

$$\begin{aligned} u:=v-r^*, \end{aligned}$$

(2.4)

one obtains a double null covering $(u,v,\theta ,\varphi )$ of ${\mathscr {M}}_M$, with $u\in (\infty , \infty )$, $v\in (-\infty , \infty )$. In these coordinates, the metric takes the form

$$\begin{aligned} g_M=-4D(r)\mathop {}\!\mathrm {d}u\mathop {}\!\mathrm {d}v+r^2(\mathop {}\!\mathrm {d}\theta ^2+\sin ^2\theta \mathop {}\!\mathrm {d}\varphi ^2). \end{aligned}$$

(2.5)

Throughout the remainder of this paper, we will always work within this (u, v)-coordinate system.

From the definitions (2.3), (2.4), it follows that $\partial _vr=-\partial _ur=D$ and that, for sufficiently large values of r:

$$\begin{aligned} |r-(v-u)+2M\log r|={\mathscr {O}}(1). \end{aligned}$$

(2.6)

The estimate (2.6) will be used frequently throughout the paper.

The vector field $T=\partial _u+\partial _v$ is a Killing vector field, the static Killing vector field of the Schwarzschild spacetime, which equips $({\mathscr {M}}_M,g_M)$ with a time orientation.

While the metric (2.5) in double null coordinates (u, v) becomes singular near $u=\infty$, we see from the form (2.1) that one can smoothly extend $({\mathscr {M}}_M,g_M)$ to and beyond “$u=\infty$” in (v, r)-coordinates. The set “$u=\infty$” is referred to as ${\mathscr {H}}^+$, or future event horizon, and the region beyond it as the black hole region of the Schwarzschild spacetime. Similarly, one can extend $({\mathscr {M}}_M,g_M)$ to and beyond “$v=-\infty$” (denoted ${\mathscr {H}}^-$) by working in coordinates (u, r).

One the other hand, we will often consider functions $f\in C^{\infty }({\mathscr {M}}_M)$ such that the e.g. the limit $\lim _{v\rightarrow \infty }f(u,v,\theta ,\varphi )$ exists and is continuous in $u,\theta$ and $\varphi$. In these cases, we will interpret the limit as living on the abstract set $\{u,v=\infty ,\theta ,\varphi \}$, which we well refer to as future null infinity or ${\mathscr {I}}^+$. Similarly, past null infinity ${\mathscr {I}}^-$ corresponds to the set of points $\{u=-\infty ,v,\theta ,\varphi \}$. One can think of these sets as being attached to ${\mathscr {M}}$ as boundaries, but the differentiable structure of this extension plays no role in this paper. See Figure 6.

We introduce two null foliations of ${\mathscr {M}}_M$: A foliation by ingoing null hypersurfaces

$$\begin{aligned} {\mathscr {C}}_{v=V}={\mathscr {M}}_M\cap \{v=V\}, \end{aligned}$$

(2.7)

and a foliation by outgoing null hypersurfaces

$$\begin{aligned} {\mathscr {C}}_{u=U}={\mathscr {M}}_M\cap \{u=U\}. \end{aligned}$$

(2.8)

We will often just write ${\mathscr {C}}_U$ instead of ${\mathscr {C}}_{u=U}$, and, similarly, ${\mathscr {C}}_{V}$ instead of ${\mathscr {C}}_{v=V}$. It will always be clear from the context whether we refer to ingoing or outgoing null hypersurfaces. Moreover, if $f:{\mathbb {R}}\rightarrow (2M,\infty )$ is a smooth function of u, we shall denote by $\Gamma _f$ the following timelike hypersurface:

$$\begin{aligned} \Gamma _{f}={\mathscr {M}}_M\cap \{v=v_{\Gamma _{f}}(u)\}, \end{aligned}$$

(2.9)

where $v_{\Gamma _{f}}(u)$ is defined via

$$\begin{aligned} v_{\Gamma _{f}}(u)-u=2r^*(f(u)). \end{aligned}$$

In the special case where $f(u)=R>2M$ is a constant, we simply write $\Gamma _f=\Gamma _R$ and $v_{\Gamma _{f}}(u)=v_R(u).$

In the sequel, we will drop the subscript M in ${\mathscr {M}}_M$ and $g_M$, and we will frequently quotient out the spheres for a given spherically symmetric subset of ${\mathscr {M}}$ without writing it (for instance, we will denote the set of all points (u, v) s.t. $(u,v,\theta ,\varphi )\in \Gamma _R$ by $\Gamma _R$, too).

Fig. 6

Depiction of the Schwarzschild manifold ${\mathscr {M}}$. Also depicted is the region $\Delta$ in which we apply the divergence theorem (2.11)

2.2 The Divergence Theorem

Let ${\mathscr {D}}$ be any simply connected subset of ${\mathscr {M}}$ with piecewise smooth boundary $\partial {\mathscr {D}}$. If J is a smooth 1-form, then we have by the divergence theorem:

$$\begin{aligned} \int _{{\mathscr {D}}}\mathrm {div}J=\int _{\partial {\mathscr {D}}}J\cdot n_{\partial {\mathscr {D}}} \end{aligned}$$

(2.10)

Here, $n_{\partial {\mathscr {D}}}$ is the normal to $\partial {\mathscr {D}}$, and integration over the canonical volume form is implied. If $\partial {\mathscr {D}}$ contains null pieces, then there is no canonical choice of volume form or normal on these. In this case, we shall choose the product of volume form and normal in such a way that the divergence theorem (2.10) applies (using Stokes’ Theorem). For instance, if $\Delta$ is the region bounded by $\Gamma _R\cap \{u_1\le u\le u_2\}$, ${\mathscr {C}}_{u_1}\cap \{v_R(u_1)\le v\le v_2\}$, ${\mathscr {C}}_{u_2}\cap \{v_R(u_2)\le v\le v_2\}$ and ${\mathscr {C}}_{v_2}\cap \{u_1\le u\le u_2\}$, then we have

$$\begin{aligned}&\int _{\Gamma _R\cap \{u_1\le u\le u_2\}} r^2\mathop {}\!\mathrm {d}(u+v)\mathop {}\!\mathrm {d}\Omega \,J\cdot (\partial _u-\partial _v) +\int _{{\mathscr {C}}_{u_1}\cap \{v_R(u_1)\le v\le v_2\}} r^2 \mathop {}\!\mathrm {d}v \mathop {}\!\mathrm {d}\Omega \,J\cdot \partial _v\nonumber \\&=\int _{{\mathscr {C}}_{v_2}\cap \{u_1\le u\le u_2\}} r^2\mathop {}\!\mathrm {d}u \mathop {}\!\mathrm {d}\Omega \, J\cdot \partial _u+\int _{{\mathscr {C}}_{u_2}\cap \{v_R(u_2)\le v\le v_2\}} r^2 \mathop {}\!\mathrm {d}v \mathop {}\!\mathrm {d}\Omega \,J\cdot \partial _v\nonumber \\&\quad -\int _{\Delta } 2Dr^2 \mathop {}\!\mathrm {d}u\mathop {}\!\mathrm {d}v\mathop {}\!\mathrm {d}\Omega \, \mathrm {div}J , \end{aligned}$$

(2.11)

where $d\Omega =\sin \theta \mathop {}\!\mathrm {d}\theta \mathop {}\!\mathrm {d}\varphi$ is the volume form of the unit sphere. See Figure 6 for a depiction of this region.

2.3 Frequently Appearing Integrals

We conclude this section with a computation of an integral that makes frequent appearances in this paper:

Lemma 2.1

Let $N, N'\in {\mathbb {N}}$ with $N>N'+1$. Then

$$\begin{aligned} (N-1)\int _{-\infty }^u\frac{r^{N'}}{|u'|^{N}}\mathop {}\!\mathrm {d}u'=\frac{r^{N'}}{|u|^{N-1}} +\sum _{k=1}^{N'}\frac{r^{N'-k}}{|u|^{N-1-k}}\prod _{j=1}^k\frac{N'+1-j}{N-1-j}\left( 1+{\mathscr {O}}(r^{-1})\right) . \nonumber \\ \end{aligned}$$

(2.12)

Proof

The proof is straight-forward, but nevertheless provided in the appendix A.1. $\square$

3 Generalities on the Wave Equation

In this section, we collect some important facts about the wave equation

$$\begin{aligned} \Box _g\phi :=\nabla ^\mu \nabla _\mu \phi =0 \end{aligned}$$

(3.1)

on a Schwarzschild background, where $\nabla$ denotes the Levi–Civita connection of g.

3.1 Existence and Uniqueness

We recall the following two standard existence results:

Proposition 3.1

(Existence for characteristic initial data) Let $f\in C^{\infty }({\mathscr {C}}_{v_1}\cap \{u_1\le u\le u_2\})$ and $h\in C^{\infty }({\mathscr {C}}_{u_1}\cap \{v_1\le v\le v_2\})$ be two smooth functions satisfying the usual corner condition. Then there exists a unique smooth function $\phi :{\mathscr {M}}\cap \{v_1\le v\le v_2,u_1\le u\le u_2\}\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \phi |_{{\mathscr {C}}_{v_1}\cap \{u_1\le u\le u_2\}}=f,&\phi |_{{\mathscr {C}}_{u_1}\cap \{v_1\le v\le v_2\}} =h, \end{aligned}$$

and

$$\begin{aligned} \Box _g\phi =0. \end{aligned}$$

Proposition 3.2

(Existence for mixed characteristic/boundary data) Let $f\in C^{\infty }(\Gamma _R\cap \{u_1\le u\le u_2\})$ and $h\in C^{\infty }({\mathscr {C}}_{u_1}\cap \{v_R(u_1)\le v\le v_2\})$ be two smooth functions satisfying the usual corner condition. Then there exists a unique smooth function $\phi :{\mathscr {M}}\cap \{u_1\le u\le u_2,v_R(u)\le v\le v_2\}\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \phi |_{\Gamma _R\cap \{u_1\le u\le u_2\}}=f,&\phi |_{{\mathscr {C}}_{u_1}\cap \{v_R(u_1)\le v\le v_2\}} =h, \end{aligned}$$

and

$$\begin{aligned} \Box _g\phi =0. \end{aligned}$$

3.2 The Basic Energy Currents

We define, with respect to any coordinate basis, and for any smooth scalar field $f\in C^{\infty }({\mathscr {M}})$, the following energy momentum tensor:

$$\begin{aligned} {\mathbf {T}}_{\mu \nu }[f]:=\partial _\mu f\partial _\nu f-\frac{1}{2} g_{\mu \nu } \partial ^\xi f\partial _\xi f \end{aligned}$$

Moreover, if V is any smooth vector field on ${\mathscr {M}}$, we define the energy current $J^V[f]$ according to

$$\begin{aligned} J^V[\phi ](\cdot ):={\mathbf {T}}[\phi ](V,\cdot ). \end{aligned}$$

With the divergence theorem (2.10) in mind, we compute

$$\begin{aligned} \mathrm {div}J^V[f]=K^V[f]+{\mathscr {E}}^V[f], \end{aligned}$$

(3.2)

where

$$\begin{aligned} K^V[f]&:={\mathbf {T}}^{\mu \nu }\nabla _{\mu } V_{\nu }, \end{aligned}$$

(3.3)

$$\begin{aligned} {\mathscr {E}}^V[f]&:=V(f)\Box _g f. \end{aligned}$$

(3.4)

Note that $K^V[f]$ vanishes if V is Killing (in view of the symmetry of ${\mathbf {T}}$), whereas ${\mathscr {E}}^V[f]$ vanishes if f is a solution to the wave equation. Thus, $K^V[f]$ measures the failure of V to be Killing and ${\mathscr {E}}^V[f]$ measures the failure of f to solve the wave equation.

3.3 Decomposition into Spherical Harmonics

One can decompose any smooth function $f:{\mathscr {M}}\rightarrow {\mathbb {R}}$ into its projections onto spherical harmonics,

$$\begin{aligned} f=\sum _{\ell '=0}^\infty f_{\ell =\ell '}, \end{aligned}$$

such that

$$\begin{aligned} f_{\ell =\ell '}(u,v,\theta , \varphi )=\sum _{m=-\ell '}^{m=\ell '}f_{\ell 'm}(u,v)Y_{\ell 'm}(\theta , \varphi ), \end{aligned}$$

where the $Y_{\ell 'm}$ are the spherical harmonics. These form a complete basis on $L^2({\mathbb {S}}^2)$ of orthogonal eigenfunctions to the spherical Laplacian , with eigenvalues $-\ell '(\ell '+1)$. In particular, in view of the spherical symmetry of the Schwarzschild spacetime, if $\phi$ solves $\Box _g\phi =0$, so does $\phi _{\ell =L}$:

$$\begin{aligned} \Box _g\phi =0 \implies \Box _g\phi _{\ell =L}=0 \end{aligned}$$

for any $L\ge 0$. In the sequel, we will frequently suppress the m-index of $\phi _{\ell m}(u,v)$ and just write $\phi _\ell$ instead.

Finally, we recall the Poincaré inequality on the sphere:

Lemma 3.1

Let $L>0$, and let $f_{\ell \ge L}\in C^2({{\mathbb {S}}^2})$ be supported only on $\ell$-modes with $\ell \ge L$. Then

(3.5)

3.4 The Commuted Wave Equations and the Higher-Order Newman–Penrose Constants

In the double null coordinates (2.5), the wave operator $\Box _g$ acting on any scalar function f takes the form

(3.6)

Hence, if $\phi$ solves the wave equation $\Box _g\phi =0$, then we obtain the following wave equation for the radiation field $r\phi$ (recall that $\partial _vr=D=-\partial _ur$):

(3.7)

Notice that if we restrict to the spherically symmetric mode $r\phi _{\ell =0}$, this gives rise to the approximate conservation law

$$\begin{aligned} \partial _u\partial _v(r\phi _0)=-\frac{2MD}{r^3}r\phi _0. \end{aligned}$$

(3.8)

This equation (3.8) is closely related to the existence of conserved quantities along null infinity, the so-called the Newman–Penrose constants

$$\begin{aligned} I_{\ell =0}^{\mathrm {future}}[\phi ](u)&:=\lim _{v\rightarrow \infty }r^2\partial _v(r\phi _0)(u,v), \end{aligned}$$

(3.9)

$$\begin{aligned} I_{\ell =0}^{\mathrm {past}}[\phi ](v)&:=\lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _0)(u,v), \end{aligned}$$

(3.10)

which, under suitable assumptions on $\phi$, remain conserved along ${\mathscr {I}}^+$, ${\mathscr {I}}^-$, respectively. Equation (3.8) (or rather, the non-linear analogue thereof) played a crucial role in proving our results from [12] and is, in fact, ubiquitous in the studies of asymptotics for the wave equation on Schwarzschild backgrounds, see e.g. [2, 6].

However, for higher $\ell$-modes, the approximate conservation law (3.8) is no longer available, and the RHS of $\partial _u\partial _v(r\phi _{\ell =L})$ has a bad $r^{-2}$-weight. This difficulty appears already in the Minkowski spacetime, i.e. for $M=0$. There, it can be resolved by commuting with $(r^2\partial _v)^\ell$, $(r^2\partial _u)^\ell$, respectively. Indeed, if $M=0$, one has the following precise conservation laws:

$$\begin{aligned} \partial _u(r^{-2L-2}(r^2\partial _v)^{(L+1)}(r\phi _L))=0,\\ \partial _v(r^{-2L-2}(r^2\partial _u)^{(L+1)}(r\phi _L))=0. \end{aligned}$$

One can find generalisations of these conservation laws in Schwarzschild. This is done in §7 of the paper. For now, we believe it to be more instructive to only explain what happens to the $\ell =1$-modes. If we naively commute the wave equation for $\ell =1$, namely

$$\begin{aligned} \partial _u\partial _v(r\phi _1)=-\frac{2D}{r^2}r\phi _1\left( 1+\frac{M}{r}\right) , \end{aligned}$$

(3.11)

with $r^2\partial _v$, then we find

$$\begin{aligned} \partial _u(r^{-2}\partial _v(r^2\partial _v(r\phi _1)))=-10 MD\frac{r^2\partial _v(r\phi _1)}{r^5}-2MD\frac{r\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) . \end{aligned}$$

(3.12)

We see that the top-order term in (3.12) comes with a good $r^{-5}$-weight. Moreover, the problematic $r^{-4}$-weight multiplying $r\phi _1$ can be removed by subtracting $Mr\phi _1$ in the following way:

$$\begin{aligned} \partial _u(r^{-2}\partial _v(r^2\partial _v(r\phi _1)-M r\phi _1))=-12 MD\frac{r^2\partial _v(r\phi _1)}{r^5}-6M^2 D\frac{r\phi _1}{r^5}. \end{aligned}$$

(3.13)

Similarly, for u and v interchanged, we obtain

$$\begin{aligned} \partial _v(r^{-2}\partial _u(r^2\partial _u(r\phi _1)))=-10 MD\frac{r^2\partial _u(r\phi _1)}{r^5}+2MD\frac{r\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) \end{aligned}$$

(3.14)

and

$$\begin{aligned} \partial _v(r^{-2}\partial _u(r^2\partial _u(r\phi _1)+M r\phi _1))=-12 MD\frac{r^2\partial _u(r\phi _1)}{r^5}+6M^2 D\frac{r\phi _1}{r^5}. \end{aligned}$$

(3.15)

The approximate conservation laws (3.13), (3.15) give rise to the following higher-order Newman–Penrose constants:

$$\begin{aligned} I_{\ell =1}^{\mathrm {future}}[\phi ](u)&:=\lim _{v\rightarrow \infty }r^2\partial _v(r^2\partial _v(r\phi _1)-Mr\phi _1)(u,v), \end{aligned}$$

(3.16)

$$\begin{aligned} I_{\ell =1}^{\mathrm {past}}[\phi ](v)&:=\lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1)+Mr\phi _1)(u,v), \end{aligned}$$

(3.17)

which, under suitable assumptions on $\phi$, remain conserved along ${\mathscr {I}}^+$, ${\mathscr {I}}^-$, respectively. Equations (3.13) and (3.15) will play a similar role in the asymptotic analysis of the $\ell =1$-mode as equation (3.8) did in the analysis of [12].

3.5 Notational Conventions

We use the notation that $f\sim g$ (or $f\lesssim g$) if there exists a uniform constant $C>0$ such that $C^{-1}g\le f\le Cg$ (or $f\le C g$). Similarly, we use the convention that $f={\mathscr {O}}(g)$ if there exists a uniform constant $C>0$ such that $|f|\le C g$. If f and g are functions depending on a single variable x, and if $k\in {\mathbb {N}}$, we also say that $f={\mathscr {O}}_k(g)$ if there exist uniform constants $C_j>0$ such that $|\partial _x^j f|\le C_j|\partial _x^j g|$ for all $j\le k$. Finally, we use the usual algebra of constants ($C+D=C=CD\dots$).

4 Part I: The case $\ell =1.$

In this part of the paper, we focus solely on the analysis of the $\ell =1$-modes. The aim of this part is to give some intuition for the decay rates and the methods used to prove them. The confident reader may wish to skip directly to the discussion of general $\ell$ in Part II.

We first treat the case of data on an ingoing null hypersurface and prove Theorem 1.2 in §4. We then treat the case of boundary data on a timelike hypersurface of constant area radius r and prove Theorem 1.1 in §5. Finally, we explain how to generalise to the case of boundary data on timelike hypersurfaces on which r is allowed to vary in §6.

Throughout Part I, $\phi$ will always denote a solution to $\Box _g\phi =0$ which is localised on an $(\ell ,m)$-frequency with $\ell =1$, $|m|\le 1$ fixed. We use the notation from §3.3, that is, we write $\phi =\phi _{\ell =1}=\phi _1(u,v)\cdot Y_{1m}(\theta ,\varphi )$.

5 Data on an Ingoing Null Hypersurface ${\mathscr {C}}_{v=1}$

In this section, we consider solutions $\phi$ arising from polynomially decaying data on an ingoing null hypersurface ${\mathscr {C}}_{v=1}$ and from vanishing data on ${\mathscr {I}}^-$, and we show asymptotic estimates near spatial infinity for these. In particular, this section contains the proof of Theorem 1.2 from the introduction.

5.1 Initial Data Assumptions and the Main Theorem (Theorem 4.1)

Prescribe smooth characteristic/scattering data for the wave equation (1.1) restricted to (1, m) which satisfy on ${\mathscr {C}}_{v=1}$

$$\begin{aligned} r^2\partial _u(r\phi _1)(u,1)&=C_{\mathrm {in}}^{(1)}+{\mathscr {O}}(r^{-1}), \end{aligned}$$

(4.1)

$$\begin{aligned} r^2\partial _u(r^2\partial _u(r\phi _1))(u,1)&=C_{\mathrm {in}}^{(2)}+{\mathscr {O}}(r^{-\eta }) \end{aligned}$$

(4.2)

for some $\eta \in (0,1)$, and which moreover satisfy for all $v\ge 1$:

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _1)(u,v)=0 \end{aligned}$$

(4.3)

for $n=0,1,2$. We interpret this latter assumption as the no incoming radiation condition.

The main result of this section then is:

Theorem 4.1

By standard scattering theory [7], there exists a unique smooth scattering solution $\phi _1\cdot Y_{1m}$ in ${\mathscr {M}}\cap \{v\ge 1\}$ attaining these data. Let $U_0$ be a sufficiently large negative number. Then, for all $(u,v)\in {\mathscr {D}}:=(-\infty ,U_0]\times [1,\infty )$, the outgoing derivative of $r\phi _1$ satisfies, for fixed values of u, the following asymptotic expansion as ${\mathscr {I}}^+$ is approached:

$$\begin{aligned} \begin{aligned} r^2\partial _v(r\phi _1)(u,v)=&-C_{\mathrm {in}}^{(1)}-2\int _{-\infty }^uF(u')\mathop {}\!\mathrm {d}u'-\frac{2MC_{\mathrm {in}}^{(1)}-2M\int _{-\infty }^uF(u')\mathop {}\!\mathrm {d}u'}{r}\\&-2M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\frac{\log r-\log |u|}{r^2}+{\mathscr {O}}(r^{-2}), \end{aligned} \end{aligned}$$

(4.4)

where F(u) is given by the limit of the radiation field $r\phi _1$ on ${\mathscr {I}}^+$

$$\begin{aligned} F(u):=\lim _{v\rightarrow \infty }r\phi _1(u,v)=\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{6|u|^2}+{\mathscr {O}}(|u|^{-2-\eta }). \end{aligned}$$

(4.5)

In particular, if $M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\ne 0$, then peeling fails at future null infinity.

Remark 4.1

The methods of our proof can also directly be applied to data which only have

$$\begin{aligned} r^2\partial _u(r\phi _1)(u,1)=C_{\mathrm {in}}^{(1)}+{\mathscr {O}}_1(r^{-\eta }) \end{aligned}$$

for $\eta \in (0,1)$. In that case, one would, schematically, obtain $\partial _v(r\phi _1)=\frac{f_1(u)}{r^2}+\frac{f_2(u)}{r^3}+{\mathscr {O}}(r^{-3-\eta })$.

In order to prove the theorem, we shall first establish the asymptotics of $r\phi _1$, using equations (3.11) and (3.15), in §4.2, and then establish the asymptotics of $\partial _v(r\phi _1)$, using (3.11) and (3.13), in §4.3. We shall make some important comments in §4.4.

5.2 Asymptotics for $r\phi _1$

We recall from §3.4 the two wave equations

$$\begin{aligned} \partial _u\partial _v(r\phi _1)=-\frac{2D}{r^2}r\phi _1\left( 1+\frac{M}{r}\right) \end{aligned}$$

(4.6)

and

$$\begin{aligned} \partial _v(r^{-2}\partial _u(r^2\partial _u(r\phi _1)))=-10 MD\frac{r^2\partial _u(r\phi _1)}{r^5}+2MD\frac{r\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) . \end{aligned}$$

(4.7)

The reason that we here choose to work with (4.7) rather than (3.15) is that, in view of the no incoming radiation condition, the bad $r^{-4}$-weight multiplying $r\phi _1$ in (4.7) is not a problem (since $r\phi _1$ itself will decay).

Throughout the rest of §4, $U_0$ will be a sufficiently large negative number (the largeness depending only on data), and ${\mathscr {D}}$ will be as in Thm. 4.1.

5.2.1 A Weighted Energy Estimate and Almost-Sharp Decay for $r\phi _1$

We first prove almost-sharp decay using an energy estimate:

Proposition 4.1

Define the following energies:

$$\begin{aligned} E_{q}^{[u_1,u_2]}(v):=\int _{u_1}^{u_2} |u|^q \left( (\partial _u(r\phi _1))^2 +(r\phi _1)^2 \frac{2D}{r^2}\left( 1+\frac{M}{r}\right) \right) (u,v)\mathop {}\!\mathrm {d}u,\\ E_{q}^{[v_1,v_2]}(u):=\int _{v_1}^{v_2}|u|^q \left( (\partial _v(r\phi _1))^2 +(r\phi _1)^2 \frac{2D}{r^2}\left( 1+\frac{M}{r}\right) \right) (u,v)\mathop {}\!\mathrm {d}v. \end{aligned}$$

Then the following energy inequality holds for all $v_2>v_1\ge 1$, $q\ge 0$ and for $0>U_0\ge u_2>u_1$:

$$\begin{aligned} E_{q}^{[u_1,u_2]}(v_2)+E_{q}^{[v_1,v_2]}(u_2)\le E_{q}^{[u_1,u_2]}(v_1)+E_{q}^{[v_1,v_2]}(u_1). \end{aligned}$$

(4.8)

Proof

Multiply the wave equation (4.6) with $2 T(r\phi _1)$ (recall that $T=\partial _u+\partial _v$) to obtain:

$$\begin{aligned} 0=\partial _u\left( (\partial _v(r\phi _1))^2\right) +\partial _v\left( (\partial _u(r\phi _1))^2\right) +T\left( \frac{2D(r\phi _1)^2}{r^2}\left( 1+\frac{M}{r}\right) \right) . \end{aligned}$$

This would already lead to the standard energy estimate, but we can exploit a certain monotonicity to obtain a weighted energy estimate: For this, we multiply the above expression with $|u|^q$ and recall that $u<0$:

$$\begin{aligned} 0&=\partial _v\left( |u|^q (\partial _u(r\phi _1))^2 +\frac{2D|u|^q(r\phi _1)^2 }{r^2}\left( 1+\frac{M}{r}\right) \right) \\&\quad +\partial _u\left( |u|^q (\partial _v(r\phi _1))^2 +\frac{2D|u|^q(r\phi _1)^2 }{r^2}\left( 1+\frac{M}{r}\right) \right) \\&\quad +q|u|^{q-1}\left( (\partial _v(r\phi _1))^2+\frac{2D(r\phi _1)^2}{r^2}\left( 1+\frac{M}{r}\right) \right) . \end{aligned}$$

Finally, integrating this in u and v using the fundamental theorem of calculus gives

$$\begin{aligned} \begin{aligned} E_{q}^{[u_1,u_2]}(v_2)&+E_{q}^{[v_1,v_2]}(u_2)= E_{q}^{[u_1,u_2]}(v_1)+E_{q}^{[v_1,v_2]}(u_1)\\&-\int _{v_1}^{v_2}\int _{u_1}^{u_2}q|u|^{q-1}\left( (\partial _v(r\phi _1))^2+\frac{2D(r\phi _1)^2}{r^2}\left( 1+\frac{M}{r}\right) \right) \mathop {}\!\mathrm {d}u\mathop {}\!\mathrm {d}v. \end{aligned} \end{aligned}$$

(4.9)

$\square$

Remark 4.2

A similar result holds for any fixed angular frequency solution. Moreover, in view of Lemma 3.5, the above proof also works for any $\phi$ supported on angular frequencies $\ell \ge L$, for some $L\ge 1$.

From this weighted $L^2$-estimate, we can already derive almost-sharp pointwise decay:

Corollary 1

There is a constant C depending only on data such that, throughout ${\mathscr {D}}$:

$$\begin{aligned} |r\phi _1(u,v)|\le \frac{C}{|u|},&|\partial _u(r\phi _1)|\le \frac{C}{|u|^2}. \end{aligned}$$

(4.10)

Moreover, we have that, for all $v\ge 1$:

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _1)(u,v)\equiv C_{\mathrm {in}}^{(1)}. \end{aligned}$$

(4.11)

Proof

We consider the energy estimate above with $q=2$ and let $(u,v)\in {\mathscr {D}}$. Then

$$\begin{aligned} \begin{aligned} r\phi _1(u,v)&=\int _{-\infty }^u\partial _u(r\phi _1)(u',v)\mathop {}\!\mathrm {d}u'\\&\le \left( \int _{-\infty }^u |u'|^{-2}\mathop {}\!\mathrm {d}u'\right) ^\frac{1}{2} \left( \int _{-\infty }^u |u'|^{2}(\partial _u(r\phi _1))^2(u,v)\mathop {}\!\mathrm {d}u'\right) ^\frac{1}{2} \\&\le \left( \int _{-\infty }^u |u'|^{-2}\mathop {}\!\mathrm {d}u'\right) ^\frac{1}{2} \left( E_2^{[-\infty ,u]}(1)+\lim _{u'\rightarrow -\infty }E_2^{[1,v]}(u')\right) ^\frac{1}{2}\le \frac{C}{|u|} \end{aligned} \end{aligned}$$

(4.12)

for some constant C solely determined by initial data. Here, we used the no incoming radiation condition (4.3) in the first step, Cauchy–Schwarz in the second step, and the energy estimate in the third step. In the last estimate, we then inserted the initial data assumptions⁹ (4.1) and used that $\lim _{u'\rightarrow -\infty }E_2^{[1,v]}(u')=0$. To show this latter statement, consider first the energy estimate with $q=0$ to obtain a bound of the form $\phi _1\lesssim r^{-\frac{1}{2}}$. Then, insert this bound into (4.6) to obtain $\partial _v(r\phi _1)\lesssim r^{-\frac{1}{2}}$, and repeat the argument with, say, $q=1/2$, and iterate.

Plugging the bound (4.12) into the wave equation (4.6) and integrating from initial data $v=1$, we moreover obtain that

$$\begin{aligned} |\partial _u(r\phi _1)|\le \frac{C}{u^2}, \end{aligned}$$

and that, in fact, the limit of $|u|^2\partial _u(r\phi _1)$ remains constant along ${\mathscr {I}}^-$. $\square$

5.2.2 Asymptotics for $\partial _u(r\phi _1)$ and $r\phi _1$

We now make the decay from Corollary 1 sharp:

Proposition 4.2

There is a constant C depending only on data such that $r\phi _1$ satisfies the following asymptotic expansion throughout ${\mathscr {D}}$:

$$\begin{aligned} \left| r\phi _1(u,v)-\frac{C_{\mathrm {in}}^{(1)}}{r}-\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{6|u|^2}\right| \le \frac{C}{|u|^{2+\eta }}+\frac{C}{r|u|}. \end{aligned}$$

(4.13)

In particular, we have

$$\begin{aligned} \lim _{v\rightarrow \infty }r\phi _1(u,v)=\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{6|u|^2}+{\mathscr {O}}(|u|^{-2-\eta }). \end{aligned}$$

(4.14)

Proof

We integrate the approximate conservation law (4.7) from $v=1$:

$$\begin{aligned} \begin{aligned} r^{-2}\partial _u(r^2\partial _u(r\phi _1))&(u,v)=r^{-2}\partial _u(r^2\partial _u(r\phi _1))(u,1)\\&+\int _1^v \left( \frac{-10MD\partial _u(r\phi _1)}{r^3}+\frac{2MDr\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) \right) (u,v') \mathop {}\!\mathrm {d}v'. \end{aligned} \end{aligned}$$

(4.15)

Using that $\partial _vr=D$ and plugging in the initial data assumption (4.2) as well as the almost sharp bounds obtained in Corollary 1, we obtain

$$\begin{aligned} \partial _u(r^2\partial _u(r\phi _1))(u,v)\lesssim \frac{r^2}{|u|^{4}}, \end{aligned}$$

(4.16)

from which, in turn, we obtain via integrating that

$$\begin{aligned} \left| r^2\partial _u(r\phi _1)-\lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _1)\right| \lesssim \int _{-\infty }^u \frac{r^2}{|u'|^{4}}\mathop {}\!\mathrm {d}u'\lesssim \frac{r^2}{|u|^{3}}, \end{aligned}$$

(4.17)

where the last inequality can be seen by recalling that $r\sim v-u$, or by an application of Lemma 2.1, see also Eq. (4.20) below. Now, by Corollary 1, we have $\lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _1)=C_{\mathrm {in}}^{(1)}.$ Thus, integrating once more in u and using that $r\phi _1$ vanishes on ${\mathscr {I}}^-$, we obtain that

$$\begin{aligned} \left| r\phi _1-C_{\mathrm {in}}^{(1)}r^{-1}\right| \lesssim |u|^{-2}. \end{aligned}$$

This estimate provides us with the leading-order behaviour of $r\phi _1$ in r. To also understand the leading-order u-decay of $r\phi _1$, we insert our improved bounds back into equation (4.15):

$$\begin{aligned} \begin{aligned} r^{-2}\partial _u(r^2\partial _u(r\phi _1))(u,v)=\frac{C_{\mathrm {in}}^{(2)}}{|u|^4} +\int _{1}^v\frac{-10MC_{\mathrm {in}}^{(1)}}{r^5}+\frac{2MC_{\mathrm {in}}^{(1)}}{r^5}\mathop {}\!\mathrm {d}v' +{\mathscr {O}}(|u|^{-4-\eta }). \end{aligned} \end{aligned}$$

(4.18)

Hence, by again converting the v-integration into r-integration using $\partial _vr=D$,

$$\begin{aligned} \partial _u(r^2\partial _u(r\phi _1))=r^2\left( \frac{C_{\mathrm {in}}^{(2)}}{|u|^4}-\frac{2MC_{\mathrm {in}}^{(1)}}{|u|^4}+\frac{2MC_{\mathrm {in}}^{(1)}}{|r|^4}\right) +{\mathscr {O}}(r^2|u|^{-4-\eta }). \end{aligned}$$

(4.19)

Integrating this from past null infinity, we again encounter the integral $\int _{-\infty }^u\frac{r^2}{|u'|^4}\mathop {}\!\mathrm {d}u'$, which we compute via Lemma 2.1 with $N'=2$ and $N=4$.

$$\begin{aligned} \begin{aligned} \int \frac{r^2}{|u|^4}\mathop {}\!\mathrm {d}u =\frac{1}{3}\sum _{k=0}^2\frac{r^k}{|u|^{k+1}}+{\mathscr {O}}(|u|^{-2}). \end{aligned} \end{aligned}$$

(4.20)

We therefore obtain the following estimate for $\partial _u(r\phi _1)$:

$$\begin{aligned} \partial _u(r\phi _1)(u,v)=\frac{C_{\mathrm {in}}^{(1)}}{r^2}+\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{3}\left( \frac{1}{|u|^3}+\frac{1}{|u|^2r}+\frac{1}{|u|r^2}\right) +{\mathscr {O}}(r^{-3}+|u|^{-3-\eta }). \nonumber \\ \end{aligned}$$

(4.21)

In particular, we thus get that

$$\begin{aligned} \lim _{v\rightarrow \infty }\partial _u(r\phi _1)(u,v)=\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{3|u|^3}+{\mathscr {O}}(|u|^{-3-\eta }). \end{aligned}$$

(4.22)

Integrating once more in u finishes the proof of the proposition. $\square$

5.3 Asymptotics for $\partial _v(r\phi _1)$ and Proof of Thm. 4.1

Equipped with an asymptotic expression for $r\phi _1$, we can now compute the asymptotics of $\partial _v(r\phi _1)$. We first derive the leading-order asymptotics of $\partial _v(r\phi _1)$ up to order ${\mathscr {O}}(r^{-3})$, using only the wave equation (4.6), and then determine the next-to-leading-order asymptotics up to ${\mathscr {O}}(r^{-4}\log r)$ using the commuted equation (3.12).

5.3.1 Leading-Order Asymptotics of $\partial _v(r\phi _1)$

Plugging the asymptotics (4.13) of $r\phi _1$ into the wave equation (4.6) and integrating the latter from past null infinity, we obtain

$$\begin{aligned} \partial _v(r\phi _1)(u,v)=-\frac{C_{\mathrm {in}}^{(1)}}{r^2}+{\mathscr {O}}(r^{-2}|u|^{-1}). \end{aligned}$$

(4.23)

In order to find the ${\mathscr {O}}(r^{-2}|u|^{-1})$-term, we commute the wave equation with $r^2$,

$$\begin{aligned} \partial _u(r^2\partial _v(r\phi _1))=-2Dr\partial _v(r\phi _1)-2D\left( r\phi _1+\frac{M}{r}r\phi _1\right) , \end{aligned}$$

(4.24)

to find, upon integrating, that

$$\begin{aligned} r^2\partial _v(r\phi _1)(u,v)= & {} -C_{\mathrm {in}}^{(1)}-\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{3|u|}\nonumber \\&+{\mathscr {O}}\left( \frac{\log (v-u)-\log |u|}{v}+\frac{1}{|u|^{1+\eta }}+\frac{1}{r}\right) , \end{aligned}$$

(4.25)

where we used Eq. (2.6) and the fact that

$$\begin{aligned} \int _{-\infty }^u\frac{1}{r(u',v)|u'|}\mathop {}\!\mathrm {d}u'\sim \int _{-\infty }^u\frac{1}{(v-u')|u'|}\mathop {}\!\mathrm {d}u'=\frac{\log (v-u)-\log |u|}{v}. \end{aligned}$$

(4.26)

In fact, the ${\mathscr {O}}(\log r)$-terms in (4.25) do not appear: By writing $r\phi _1$ as¹⁰$r\phi _1=\lim _{{\mathscr {I}}^+}r\phi _1-\int _v^\infty \partial _v(r\phi _1)$ in Eq. (4.24), we can improve the asymptotic estimate (4.25) to

$$\begin{aligned} r^2\partial _v(r\phi _1)(u,v)=-C_{\mathrm {in}}^{(1)}-2\int _{-\infty }^u\lim _{v\rightarrow \infty }r\phi _1(u',v)\mathop {}\!\mathrm {d}u'+{\mathscr {O}}(r^{-1}). \end{aligned}$$

This cancellation is related to the one that gives rise to the approximate conservation law (3.13). In the above, we used (see also Eq. (4.49) of [12]) that

$$\begin{aligned} \int _{-\infty }^u \frac{\log (v-u')-\log |u'|}{vr}\mathop {}\!\mathrm {d}u'\lesssim \int _{-\infty }^u \frac{\log (v-u')-\log |u'|}{v(v-u')}\mathop {}\!\mathrm {d}u'\le \frac{\pi ^2}{6}\frac{1}{v-u}\lesssim \frac{1}{r}. \nonumber \\ \end{aligned}$$

(4.27)

We summarise our findings in

Proposition 4.3

We have the following asymptotics throughout ${\mathscr {D}}$:

$$\begin{aligned} \partial _v(r\phi _1)(u,v)&=\frac{\lim _{{\mathscr {I}}^+}r^2\partial _v(r\phi _1)(u)}{r^2}+{\mathscr {O}}(r^{-3}), \end{aligned}$$

(4.28)

$$\begin{aligned} r\phi _1(u,v)&=\lim _{{\mathscr {I}}^+}r\phi _1(u)-\frac{\lim _{{\mathscr {I}}^+}r^2\partial _v(r\phi _1)(u)}{r}+{\mathscr {O}}(r^{-2}), \end{aligned}$$

(4.29)

where $\lim _{{\mathscr {I}}^+}r\phi _1(u)$ is given by (4.14), and where

$$\begin{aligned} \lim _{{\mathscr {I}}^+}r^2\partial _v(r\phi _1)(u)= & {} -C_{\mathrm {in}}^{(1)}-2\int _{-\infty }^u\lim _{{\mathscr {I}}^+}r\phi _1(u') \nonumber \\= & {} -C_{\mathrm {in}}^{(1)}-\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{3|u|}+{\mathscr {O}}(|u|^{-1-\eta }). \end{aligned}$$

(4.30)

5.3.2 Next-to-Leading-Order Asymptotics for $\partial _v(r\phi _1)$ (Proof of Thm. 4.1)

Proof of Theorem 4.1

Equipped with the leading-order asymptotics for $\partial _v(r\phi _1)$ and $r\phi _1$, we now find the asymptotic behaviour of $\partial _v(r^2\partial _v(r\phi _1))$ using the commuted wave equation

$$\begin{aligned} \partial _u(r^{-2}\partial _v(r^2\partial _v(r\phi _1)))=-10 MD\frac{r^2\partial _v(r\phi _1)}{r^5}-2MD\frac{r\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) . \end{aligned}$$

(4.31)

By the no incoming radiation condition (4.3) and the fundamental theorem of calculus, we have

$$\begin{aligned} \begin{aligned} r^{-2}\partial _v(r^2\partial _v(r\phi _1))&(u,v) =\int _{-\infty }^u \frac{-10MDr^2\partial _v(r\phi _1)}{r^5}-\frac{2MDr\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) \mathop {}\!\mathrm {d}u'. \end{aligned} \end{aligned}$$

(4.32)

Plugging the asymptotics from Prop. 4.3 into the above, we obtain that

$$\begin{aligned} \begin{aligned}&r^{-2}\partial _v(r^2\partial _v(r\phi _1))(u,v)\\&=\int _{-\infty }^u -\frac{8MD\lim _{{\mathscr {I}}^+}r^2\partial _v(r\phi _1)(u')}{r^5}-\frac{2MD\lim _{{\mathscr {I}}^+}r\phi _1(u')}{r^4} \mathop {}\!\mathrm {d}u'+{\mathscr {O}}(r^{-5}). \end{aligned} \end{aligned}$$

(4.33)

Evaluating the integrals in a similar way to (4.26), we thus find

$$\begin{aligned} r^2\partial _v(r^2\partial _v(r\phi _1))&=2MC_{\mathrm {in}}^{(1)}-M\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{3|u|}\nonumber \\&\quad +{\mathscr {O}}\left( \frac{\log (1-v/u)}{v}+\frac{1}{|u|^{1+\eta }}+\frac{1}{r}\right) . \end{aligned}$$

(4.34)

Notice that the ${\mathscr {O}}$-terms in (4.34) all integrate to ${\mathscr {O}}(1/r)$ when multiplied by 1/r (cf. (4.27)).

To find the next-to-leading-order logarithmic terms, we commute the approximate conservation law (4.31) with $r^4$:

$$\begin{aligned} \partial _u(r^2\partial _v(r^2\partial _v(r\phi _1)))&=-\frac{4D}{r} r^2\partial _v(r^2\partial _v(r\phi _1)) \nonumber \\&-\frac{10MD}{r}r^2\partial _v(r\phi _1)-2MDr\phi _1\left( 1+\frac{4M}{r}\right) . \end{aligned}$$

Integrating this from past null infinity and plugging in (as in (4.33)) the asymptotics for $r^2\partial _v(r^2\partial _v(r\phi _1))$, $r^2\partial _v(r\phi _1)$ and $r\phi _1$ from (4.34) and Prop. 4.3, respectively, we find:

$$\begin{aligned}&r^{2}\partial _v(r^2\partial _v(r\phi _1))(u,v)\nonumber \\&=2MC_{\mathrm {in}}^{(1)}+\int _{-\infty }^u \frac{12MD}{r}\frac{C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)}}{3|u'|} -2M\lim _{{\mathscr {I}}^+}r\phi _1(u') \mathop {}\!\mathrm {d}u'+{\mathscr {O}}(r^{-1}) \nonumber \\&=2MC_{\mathrm {in}}^{(1)}-2M\int _{-\infty }^u\lim _{{\mathscr {I}}^+}r\phi _1\mathop {}\!\mathrm {d}u'+4M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\frac{\log (v-u)-\log |u|}{v}+{\mathscr {O}}(r^{-1}). \end{aligned}$$

(4.35)

We can now fix u and integrate the above in v from ${\mathscr {I}}^+$ to obtain for $\partial _v(r\phi _1)$:

$$\begin{aligned} \begin{aligned} r^2\partial _v(r\phi _1)(u,v)=&\lim _{{\mathscr {I}}^+}r^2\partial _v(r\phi _1)(u)-\frac{\lim _{{\mathscr {I}}^+ }r^2\partial _v(r^2\partial _v(r\phi _1))(u)}{r}\\ -&2M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\frac{\log (v-u)-\log |u|}{r^2}+{\mathscr {O}}(r^{-2}), \end{aligned} \end{aligned}$$

(4.36)

where

$$\begin{aligned} \lim _{{\mathscr {I}}^+ }r^2\partial _v(r^2\partial _v(r\phi _1))(u)=2MC_{\mathrm {in}}^{(1)}-2M\int _{-\infty }^u\lim _{{\mathscr {I}}^+}r\phi _1\mathop {}\!\mathrm {d}u'. \end{aligned}$$

This concludes the proof of Theorem 4.1. $\square$

5.4 Comments

5.4.1 The Newman–Penrose Constant $I_{\ell =1}^{\mathrm {future}}[\phi ]$

It is instructive to also write down the asymptotics of the quantity related to the higher-order Newman–Penrose constant $I_{\ell =1}^{\mathrm {future}}[\phi ]$ (recall the definition (3.16)):

Theorem 4.2

Let $U_0$ be a sufficiently large negative number. Then, throughout ${\mathscr {D}}=(-\infty ,U_0]\times [1,\infty )$, the outgoing derivative of the combination $r^2\partial _v(r\phi _1)-Mr\phi _1$ satisfies, for fixed values of u, the following asymptotic expansion as ${\mathscr {I}}^+$ is approached:

$$\begin{aligned} \begin{aligned} r^2\partial _v(r^2\partial _v(r\phi _1)-Mr\phi _1)=3MC_{\mathrm {in}}^{(1)}+4M(C_{\mathrm {in}}^{(2)}-2MC_{\mathrm {in}}^{(1)})\frac{\log r-\log |u|}{r}+{\mathscr {O}}(r^{-1}). \end{aligned} \end{aligned}$$

(4.37)

In particular, $I_{\ell =1}^{\mathrm {future}}[\phi ](u)\equiv 3MC_{\mathrm {in}}^{(1)}$ is conserved along ${\mathscr {I}}^+$.

5.4.2 The Case $C_{\mathrm {in}}^{(1)}=0$: A Logarithmically Modified Price’s Law

Notice that if $C_{\mathrm {in}}^{(1)}=0$, then $I_{\ell =1}^{\mathrm {future}}[\phi ]=0$. However, one can still define a conserved quantity along future null infinity in this case, namely

$$\begin{aligned} I_{\ell =1}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ](u):=\lim _{v\rightarrow \infty }\frac{r^3}{\log r}\partial _v(r^2\partial _v(r\phi _1)-Mr\phi _1)(u,v), \end{aligned}$$

(4.38)

which, in our case, is given by $4MC_{\mathrm {in}}^{(2)}$. In particular, by using similar methods to the ones from [13], which combined the results of [12] and [2], one can thus obtain that the late time asymptotics of the $\ell =1$-mode, if one smoothly extends the data to ${\mathscr {H}}^+$, have logarithmic corrections at leading order. In particular, one can obtain that $r\phi _1(u,\infty )=Cu^{-3}\log u+{\mathscr {O}}(u^{-3})$ along ${\mathscr {I}}^+$, and that $\partial _v\phi _1(\infty ,v)=C'v^{-5}\log v+{\mathscr {O}}(v^{-5})$ along the event horizon ${\mathscr {H}}^+$, where the constants C and $C'$ can be expressed explicitly in terms of $C_{\mathrm {in}}^{(2)}$.

In order to show this, one needs to combine the results of the present paper with those of the recent [3] and make modifications to [3] similar to those in [13], see the discussion of §1.3.

5.4.3 Discussion of the Cancellations of Remark 1.4 and the Case of General $\ell$

Recall the cancellations discussed for general $\ell$ in Remark 1.4. Let us here give some intuition for them, restricting, of course, to the case $\ell =1$.

Theorem 4.1 shows that, if $r\phi _1\sim 1/|u|$ initially, this translates to $r\phi |_{{\mathscr {I}}^+}\sim u^{-2}$ on null infinity. We found this "cancellation" somewhat tacitly, namely by transporting decay for the commuted quantity $r^2\partial _u(r^2\partial _u(r\phi _1))$ along ${\mathscr {I}}^-$. It is maybe easiest to explain why this approach produces no cancellations for $p\in (0,1)$ or $p\in (1,2)$: If $p\in (1,2)$, then the estimate (4.16) becomes worse, not better, since the initial data term of (4.15) now decays slower. On the other hand, if $p\in (0,1)$, then (4.17) fails, as the limit $\lim r^2\partial _u(r\phi _1)$ diverges. In fact, this shows that the proof of the present section fails for $p<1$.

There also is a more direct way of understanding the cancellation for $p=1$: In view of the estimate (4.23), we have that, schematically,

$$\begin{aligned} r\phi (u,v)= & {} r\phi (u,1)+\int _1^v \partial _v(r\phi )\mathop {}\!\mathrm {d}v'=r\phi (u,1)+\int _1^v \frac{-C_{\mathrm {in}}^{(1)}}{r^2(u,v')}\mathop {}\!\mathrm {d}v' \\= & {} \frac{C_{\mathrm {in}}^{(1)}}{|u|}+\frac{C_{\mathrm {in}}^{(1)}}{r}-\frac{C_{\mathrm {in}}^{(1)}}{|u|}=\frac{C_{\mathrm {in}}^{(1)}}{r}, \end{aligned}$$

where we used that $r(u,1)\sim |u|$. From this point of view, it is clear that such cancellations only happen if $r\phi _1\sim 1/|u|^p$ for $p=1$. Our more systematic approach of §10, in which we analyse general $\ell$-modes, will understand the cancellations of Remark 1.4 in a generalised form of the above computation. Indeed, in §10, we will avoid using the conservation law in the v-direction entirely, and instead only use the conservation law in the u-direction: Instead of propagating decay for $(r^2\partial _u)^{\ell +1}(r\phi _\ell )$ in v and then integrating this $\ell +1$ times from ${\mathscr {I}}^-$ , we will directly obtain an estimate for $(r^2\partial _v)^{\ell +1}(r\phi _\ell )$ by integrating from ${\mathscr {I}}^-$ in u, and then integrate this estimate $\ell$ times from $v=1$, carefully analysing at each step the initial data contributions. In particular, this approach will also allow for slower decay in the initial data. See already §10.3 for a more detailed overview of the approach for general $\ell$.

6 Boundary Data on a Timelike Hypersurface $\Gamma _R$

Having obtained asymptotic estimates for solutions arising from polynomially decaying initial data on an ingoing null hypersurface in the previous section, we now want to obtain similar estimates for solutions arising from polynomially decaying boundary data on a timelike hypersurface $\Gamma _R$. The main result of this section is the proof of Theorem 1.1.

In contrast to the previous section, we here need to construct our solutions at the same time as we prove estimates on them.

We use the notation from §3.3, that is, we write $\phi =\phi _{\ell =1}=\phi _1(u,v)\cdot Y_{1m}(\theta ,\varphi )$.

6.1 Overview of the Ideas and Structure of the Section

Let us briefly recall the approach that we followed in our treatment of the $\ell =0$-mode in [12]: Given polynomially decaying boundary data on $\Gamma _R$, we first considered a sequence of compactly supported boundary data that would approach the original boundary data. This allowed us to use the method of continuity, i.e. bootstrap arguments. We then assumed decay for $r\phi _0$, and improved it by essentially integrating the wave equation (3.8) first in u and then in v (from $\Gamma _R$) and exploiting 2M/R as a "small" parameter. In fact, we also showed that one can avoid exploiting smallness in 2M/R using a Grönwall argument.

If we want to follow a similar approach for $\ell =1$, it is not sufficient to consider the uncommuted wave equation (3.11) in view of its non-integrable $r^{-2}$-weight. Instead, it seems more appropriate to use the approximate conservation law (3.13) and bootstrap decay on the combination

$$\begin{aligned} \Phi :=r^2\partial _v(r\phi _1)-Mr\phi _1. \end{aligned}$$

The first and main difficulty then becomes apparent: $\Phi |_{\Gamma _R}$ is not given by boundary data (we prescribe boundary data tangent to $\Gamma$). One way of overcoming this difficulty is to exploit certain cancellations in the wave equation; this however requires one to have knowledge on the T-derivative of $r\phi _1$. Alternatively, one can estimate $r^2\partial _v(r\phi _1)|_{\Gamma _R}$ using an energy estimate which only uses "a square root" of the bootstrapped decay of $r^2\partial _v(r\phi _1)$. We will make use of both of these approaches, the former for lower-order derivatives $r^2\partial _vT^n(r\phi _1)$ (where we have room to make assumptions on $T^{n+1}(r\phi _1)$), and the latter for the top-order derivative $r^2\partial _vT^N(r\phi _1)$, $n<N$. In fact, using only the latter approach is sufficient, but we find it instructive to also include the former as it since it highlights the importance of commuting with T. In the more systematic approach of the discussion of general $\ell$ in §8, we will, however, exclusively use the latter approach.

Equipped with a boundary estimate on $\Phi$, we can then hope to close the bootstrap argument by simply integrating (3.13) first in u and then in v, and exploiting 2M/R as a small parameter. In fact, as in the $\ell =0$-case, one can avoid this smallness assumption. The only additional subtlety here is that, in order to estimate the RHS of (3.13), we need to control $r\phi _1$ and $\partial _v(r\phi _1)$, which is not directly provided by a bootstrap assumption on the combination $\Phi$. We will deal with this by estimating $r\phi _1$ against the integral over $\partial _v(r\phi _1)$ from $\Gamma _R$, and either just exploiting smallness in 2M/R or using a more elaborate Grönwall argument.

Structure We first state our initial boundary data assumptions for $\phi _1$, as well as the main theorem, in §5.2.1. Then, in order to gain access to the method of continuity, we smoothly cut-off the boundary data in §5.2.2. These will lead to finite solutions $\phi _1^{(k)}$ in the sense of Proposition 3.2. Using bootstrap methods as outlined above, we can then estimate $r^2\partial _vT^n(r\phi _1^{(k)})$ and $T^n(r\phi _1^{(k)})$ in §5.3.

In order to later show that Theorem 4.1 can be applied (i.e. to show that the limit $\lim _{u\rightarrow -\infty }(r^2\partial _u)^2(r\phi _1)(u,v)$ exists), we will also need to show some auxiliary estimates on the differences $r^2\partial _vT^n(r\phi _1^{(k)}-|u|T(r\phi _1^{(k)})$. This is done in §5.4.

In §5.5, we finally show that the finite solutions $\phi _1^{(k)}$ tend to a limiting solution and show that Theorem 4.1 can be applied to it, thus proving Theorem 1.1. We make some closing comments in §5.6.

6.2 The Setup

6.2.1 The Initial/Boundary Data and the Main Theorem (Theorem 5.1)

Throughout the rest of this section, we shall assume that $R>2M$ is a constant. In particular, $T=\partial _u+\partial _v$ will be tangent to $\Gamma _R$. We then prescribe smooth boundary data ${\hat{\phi }}_1$ on $\Gamma _R={\mathscr {M}}_M\cap \{v=v_R(u)\}$ that satisfy, for $u\le U_0<0$ and $|U_0|$ sufficiently large, the upper bounds

$$\begin{aligned} \left| T^n(r{\hat{\phi }}_1)\right|&\le \frac{n! C_{\mathrm {in}}^\Gamma }{R|u|^{n+1}},&n=0,1,\dots ,N+1, \end{aligned}$$

(5.1)

$$\begin{aligned} \left| T^n\left( r{\hat{\phi }}_1-|u|T(r{\hat{\phi }}_1)\right) \right|&\le \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{R|u|^{n+1+\varepsilon }},&n=0,\dots ,N'+1 \end{aligned}$$

(5.2)

for some positive constants $C_{\mathrm {in}}^\Gamma$, $C_{\mathrm {in},\varepsilon }^\Gamma$, $\varepsilon \in (0,1)$ and $N,N'\ge 0$ integers, and which also satisfy the following lower bound:

$$\begin{aligned} \left| T(r{\hat{\phi }}_1)\right| \ge \frac{C_{\mathrm {in}}^\Gamma }{2R|u|^{2}}>0. \end{aligned}$$

(5.3)

Moreover, we demand, in a limiting sense, that, for all v,

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _1)(u,v)=0, \quad n=1,\dots , N+1. \end{aligned}$$

(5.4)

Then the main result of this section is

Theorem 5.1

Let $R>2M$ be a constant. Then there exists a unique solution $\phi _1$ to Eq. (3.11) in ${\mathscr {D}}_{\Gamma _R}:={\mathscr {M}}\cap \{v\ge v_R(u)\}$ that restricts correctly to ${\hat{\phi }}_1$ on $\Gamma _R$, $\phi _1|_{\Gamma _R}={\hat{\phi }}_1$, and that satisfies (5.4). Moreover, if $U_0$ is a sufficiently large negative number, then there exists a constant $C=C(2M/R,C_{\mathrm {in}}^\Gamma )$, depending only on data, such that $\phi _1$ obeys the following bounds throughout ${\mathscr {D}}_{\Gamma _R}\cap \{u\le U_0\}$:

$$\begin{aligned} \left| r^2\partial _vT^n(r\phi _1)(u,v)\right|&\le \frac{C}{|u|^{n+1}},&n=0,\dots ,N, \end{aligned}$$

(5.5)

$$\begin{aligned} \left| T^n(r\phi _1)(u,v)\right|&\le \frac{C}{|u|^{n+1}}\max \left( r^{-1},|u|^{-1}\right) ,&n=0,\dots ,N-1. \end{aligned}$$

(5.6)

Finally, if $N\ge 4$ and $N'\ge 2$, then we have, along any ingoing null hypersurface ${\mathscr {C}}_v$, that

$$\begin{aligned} r^2\partial _u(r\phi _1)(u,v)&={\mathscr {O}}(r^{-1}), \end{aligned}$$

(5.7)

$$\begin{aligned} r^2\partial _u(r^2\partial _u(r\phi _1))(u,v)&={\tilde{C}}+{\mathscr {O}}(r^{-1}+|u|^{-\varepsilon }), \end{aligned}$$

(5.8)

where ${\tilde{C}}$ is a constant that is non-vanishing if R/2M is sufficiently large. In particular, $\phi _1$ satisfies the assumptions of Theorem 4.1 with $C_{\mathrm {in}}^{(1)}=0$, $C_{\mathrm {in}}^{(2)}={\tilde{C}}$ and $\varepsilon =\eta$.

Remark 5.1

Let us already draw the reader’s attention to the fact that the data above lead to solutions with $C_{\mathrm {in}}^{(1)}=0$ (cf. (4.1)). In view of the comments in §4.4.2, this suggests that the data considered here lead to a logarithmically modified Price’s law near $i^+$.

Remark 5.2

Instead of considering data with ${\hat{\phi }}_1\sim |u|^{-1}$, we can also consider data with ${\hat{\phi }}_1\sim |u|^{-p}$ for $p>0$ and derive a similar result with some obvious modifications.

Remark 5.3

It may be instructive for the reader to keep the following solution to (3.11) in the case $M=0$ in mind:

$$\begin{aligned} \partial _u\partial _v\left( \frac{1}{2|u|^2}+\frac{1}{|u|r}\right) =-\frac{2}{r^2}\left( \frac{1}{2|u|^2}+\frac{1}{|u|r}\right) . \end{aligned}$$

(5.9)

6.2.2 Cutting of the Data and Replacing ${\mathscr {I}}^-$ with ${\mathscr {C}}_{u=-k}$

As mentioned before, in order to appeal to bootstrap arguments, we need to work in compact regions. We therefore need to cut the boundary data off and then recover the original data using a limiting argument. Let $(\chi _k(u))_{k\in {\mathbb {N}}}$ be a sequence of positive smooth cut-off functions such that

$$\begin{aligned} \chi _k={\left\{ \begin{array}{ll} 1,&{} u\ge - k+1,\\ 0,&{}u\le -k, \end{array}\right. } \end{aligned}$$

and cut off the highest-order derivative: $\chi _k \cdot T^{N+1}{\hat{\phi }}$. We then have

$$\begin{aligned} \int _{-\infty }^u \chi _k T^{N+1}{\hat{\phi }}_1=\chi _k T^{N}{\hat{\phi }}_1-\int _{-\infty }^{u}(T\chi _k)( T^{N}{\hat{\phi }}_1 )=\chi _k T^{N}{\hat{\phi }}_1+\theta _k\cdot {\mathscr {O}}(k^{-N-1}), \end{aligned}$$

where $\theta _k$ equals 1 on $\{u\ge -k\}$ and 0 elsewhere. Similarly, we obtain inductively that

$$\begin{aligned} \underbrace{\int \dots \int }_{n\, \mathrm { times}} \chi _k T^{N+1}{\hat{\phi }}_1=\chi _k T^{N+1-n}{\hat{\phi }}_1+\theta _k\cdot {\mathscr {O}}(k^{-N-2+n}). \end{aligned}$$

In particular, if we denote $\overbrace{\int \dots \int }^{N+1\text { times}} \chi _k T^{N+1}{\hat{\phi }}_1$ as ${\hat{\phi }}_1^{(k)}$, then the bounds (5.1), (5.2) imply, for sufficiently large negative values of u and for some constant $C'_{\mathrm {in}}=C'_{\mathrm {in}}(N,N')$:

$$\begin{aligned} \left| T^n\left( r{\hat{\phi }}^{(k)}_1\right) \right|&\le \frac{n! C_{\mathrm {in}}^\Gamma }{R|u|^{n+1}},&n=0,1,\dots ,N+1, \end{aligned}$$

(5.10)

$$\begin{aligned} \left| T^n\left( r{\hat{\phi }}^{(k)}_1-|u|T\left( r{\hat{\phi }}^{(k)}_1\right) \right) \right|&\le \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{R|u|^{n+1+\varepsilon }}+C'_{\mathrm {in}}\theta _k\cdot \frac{C_{\mathrm {in}}^\Gamma }{Rk^{n+1}},&n=0,1,\dots ,N'+1 \end{aligned}$$

(5.11)

Notice that, in the second line above, we lose some decay due to the $\theta _k$-term arising from the cut-off. Since we will take the limit $k\rightarrow \infty$ in the end, this only poses a minor difficulty.

Throughout the next two sections (§5.3 and §5.4), we shall assume initial/boundary data satisfying the estimates (5.10) and (5.11) and moreover satisfying

$$\begin{aligned} \phi _1(u=-k,v)=0 \end{aligned}$$

(5.12)

for all $v\ge v_R(-k)$. We shall denote the unique solutions to these initial/boundary value problems as $\phi _1^{(k)}$. For the next two sections, we shall drop the superscript (k), only to reinstate it in §5.5, where we will show that the solutions $\phi _1^{(k)}$ tend towards a limiting solution.

6.3 Estimates for $\partial _vT^n (r\phi _1)$ and $T^n(r\phi _1)$

Let $U_0$ be a sufficiently large negative number, and let ${\hat{\phi }}_1$ be smooth data on $\Gamma _R$, supported on $\Gamma _R\cap \{-k< u\}$ and satisfying (5.10). By Prop. 3.2, there exists a unique smooth solution $\phi _1$ throughout ${\mathscr {D}}_{\Gamma _R}\cap \{-k\le u\}$ such that $\phi _1(-k,v)=0$ for all $v\ge v_R(-k)$ and such that $\phi _1|_{\Gamma _R}={\hat{\phi }}_1$. We will now derive the following uniform-in-k estimates on this solution $\phi _1$:

Proposition 5.1

Let $\phi _1$ be the solution as described above, and let $N\ge 1$. Then, if $|U_0|$ is sufficiently large, there exists a constant $C=C(2M/R,C_{\mathrm {in}}^\Gamma )$ (in particular, this constant does not depend on k), which can be chosen to be independent of R/2M for large enough R/2M, such that the following estimates hold throughout ${\mathscr {D}}_{\Gamma _R}\cap \{-k\le u\le U_0\}$:

$$\begin{aligned} \left| r^2\partial _vT^n (r\phi _1)(u,v)\right|&\le \frac{C}{|u|^{n+1}},&n=0,1,\dots ,N, \end{aligned}$$

(5.13)

$$\begin{aligned} \left| T^n (r\phi _1)(u,v)\right|&\le \frac{C}{|u|^{n+1}}\max \left( r^{-1},|u|^{-1}\right) ,&n=0,1,\dots ,N-1. \end{aligned}$$

(5.14)

Proof

The proof is divided into the sections §5.3.1–§5.3.5. In §5.3.1–§5.3.4, we present a bootstrap argument and exploit 2M/R as a small parameter to improve the bootstrap assumptions. An overview over this bootstrap argument will be given in §5.3.1.

We will then explain how to lift the smallness assumption on 2M/R by partially replacing the bootstrap argument with a Grönwall-type argument in §5.3.5.

6.3.1 The Bootstrap Assumptions

Let $\{C_{\mathrm {BS}}^{(n)}, n=1,\dots ,N\}$ and $\{C_{\mathrm {BS},\phi }^{(m)},m=0,\dots ,N-1\}$ be two sets of sufficiently large positive constants. We shall make the following bootstrap assumptions on $\phi _1$:

$$\begin{aligned} \left| r^2\partial _vT^n (r\phi _1)(u,v)\right| \le \frac{C_{\mathrm {BS}}^{(n)}}{|u|^{n+1}} \end{aligned}$$

(BS(n))

for $n=1,\dots ,N$, and

$$\begin{aligned} \left| T^m (r\phi _1)(u,v)\right| \le \frac{C_{\mathrm {BS},\phi }^{(m)}}{|u|^{m+1}}\max \left( r^{-1},|u|^{-1}\right) \end{aligned}$$

(BS'(m))

for $m=0,\dots ,N-1$.

We now define $\Delta$ to be the subset of all $(u,v)\in X:= \{(u,v)|-k< u\le U_0, v_R(u)< v\}$ such that, for all $(u',v')\in X$ with $u'\le u$ and $v'\le v$, (BS(n)) and (BS’(m)) hold for all $n=1,\dots ,N$, $m=0,\dots ,N-1$, respectively.

By compactness and continuity, $\Delta$ is non-empty if the constants $C_{\mathrm {BS}}^{(n)},C_{\mathrm {BS},\phi }^{(m)}$ are chosen sufficiently large. Moreover, $\Delta$ is trivially closed in X. We shall show that $\Delta$ is also open by improving each of the bootstrap assumptions within $\Delta$.

We shall first improve the bootstrap assumptions for the lower-order T-derivatives ($n\le N-2$) by explicitly exploiting the precise behaviour for higher T-derivatives in §5.3.2 in order for the reader to get a clear intuition for the origin of the assumed rates. In §5.3.3, we will then improve the bootstrap assumption away from the top-order derivative ($n\le N-1$), where we no longer have the sharp decay for $T^N(r\phi _1)$ available. Finally, in §5.3.4, we will improve the bootstrap assumptions for the top-order derivatives.

Since the approach of §5.3.4 applies to derivatives of any order, the reader can in principle skip §5.3.2–§5.3.3, which are included for pedagogical reasons, and go directly to §5.3.4. In fact, §5.3.4 only requires the bootstrap assumptions (BS(n)) (and not (BS’(m))). In particular, when going through §5.3.2–§5.3.3, the reader can focus on the arguments without having to pay close attention to the bootstrap constants.

6.3.2 Closing Away from the Top-Order Derivatives $j\le N-2$

The idea is to exploit the fact that, for $M=0$, $\phi _1=1/r^2$ is a stationary solution. In particular, we expect $\partial _v(r^2\phi _1)$ to have some cancellations (see (5.17)), and $r^2\partial _v(r\phi _1)$ to remain approximately conserved in u and v (see (5.19)). (We remind the reader of the example solution (5.9).)

Proposition 5.2

Let $0\le j\le N-2$. Then, for sufficiently large values of R/2M and $|U_0|$, and if the ratios $C_{\mathrm {BS},\phi }^{(j)}/C_{\mathrm {BS},\phi }^{(j+1)}$, $C_{\mathrm {BS}}^{(j)}/C_{\mathrm {BS},\phi }^{(j+1)}$ are chosen large enough, we have throughout $\Delta$ that, in fact,

$$\begin{aligned} \left| r^2\partial _vT^j (r\phi _1)(u,v)\right|&\le \frac{1}{2} \frac{C_{\mathrm {BS}}^{(j)}}{|u|^{j+1}}, \end{aligned}$$

(5.15)

$$\begin{aligned} \left| T^j (r\phi _1)(u,v)\right|&\le \frac{1}{2} \frac{C_{\mathrm {BS},\phi }^{(j)}}{|u|^{j+1}}\max \left( r^{-1},|u|^{-1}\right) . \end{aligned}$$

(5.16)

Proof

Fix $j\le N-2$ and assume (BS’(m)) for $m=j,j+1$. Motivated by the comment above, we compute

$$\begin{aligned} \partial _u(r^{-2}\partial _v(r^2\phi _1))=\frac{DT(r\phi _1)}{r^2}-\frac{8MD}{r^4}r\phi _1. \end{aligned}$$

(5.17)

Commuting with $T^j$, plugging in the bootstrap assumptions, and integrating (5.17) from $u=-k$, we find (recall that $\partial _ur=-D$):

$$\begin{aligned} \left| r^{-2}T^j\partial _v(r^2\phi _1)(u,v)\right| \le \int _{r_v(-k)}^{r_v(u)} \frac{C_{\mathrm {BS},\phi }^{(j+1)}}{r^3|u|^{j+2}}+\frac{8MC_{\mathrm {BS},\phi }^{(j)}}{r^5|u|^{j+1}}\mathop {}\!\mathrm {d}r\le \frac{C_{\mathrm {BS},\phi }^{(j+1)}}{2r^2|u|^{j+2}}+\frac{2MC_{\mathrm {BS},\phi }^{(j)}}{r^4|u|^{j+1}}. \end{aligned}$$

(5.18)

Here, we denoted $r_v(u)$ as the unique r such that $r^*(r)=v-u$. Now, we similarly compute

$$\begin{aligned} \partial _u(r^2\partial _v(r\phi _1))=-2D\partial _v(r^2\phi _1)-6DM\frac{r\phi _1}{r}. \end{aligned}$$

(5.19)

Commuting again with $T^j$, plugging in the bound (5.18) for $T^j\partial _v(r^2\phi _1)$ from above, and integrating (5.19) in u, we then find:

$$\begin{aligned} \left| r^2\partial _vT^j(r\phi _1)(u,v)\right|&\le \int _{r_v(-k)}^{r_v(u)}\frac{C_{\mathrm {BS},\phi }^{(j+1)}}{|u|^{j+2}}\mathop {}\!\mathrm {d}u+\int _{r_v(-k)}^{r_v(u)}\frac{4MC_{\mathrm {BS},\phi }^{(j)}}{r^2|u|^{j+1}}+\frac{6MC_{\mathrm {BS},\phi }^{(j)}}{r^2|u|^{j+1}}\mathop {}\!\mathrm {d}r \nonumber \\&\le \frac{C_{\mathrm {BS},\phi }^{(j+1)}}{(j+1)|u|^{j+1}}+\frac{10MC_{\mathrm {BS},\phi }^{(j)}}{r|u|^{j+1}}. \end{aligned}$$

(5.20)

For large enough R and $C_{\mathrm {BS}}^{(j)}/C_{\mathrm {BS},\phi }^{(j+1)}$, this proves the first part of the proposition.

Moreover, inserting (5.20) back into (5.18) and writing $\partial _v(r^2\phi _1)=r\partial _v(r\phi _1)+Dr\phi _1$, we obtain

$$\begin{aligned} \begin{aligned} D\left| T^j(r\phi _1)\right|&\le \left| T^j\partial _v(r^2\phi _1)\right| +\frac{1}{r}\left| r^2T^j\partial _v(r\phi _1)\right| \\&\le \frac{C_{\mathrm {BS},\phi }^{(j+1)}}{2|u|^{j+2}}+\frac{2MC_{\mathrm {BS},\phi }^{(j)}}{r^2|u|^{j+1}}+\frac{C_{\mathrm {BS},\phi }^{(j+1)}}{(j+1)r|u|^{j+1}}+\frac{10MC_{\mathrm {BS},\phi }^{(j)}}{r^2|u|^{j+1}}. \end{aligned} \end{aligned}$$

(5.21)

This proves the second part of the proposition for large enough R and $C_{\mathrm {BS},\phi }^{(j)}/C_{\mathrm {BS},\phi }^{(j+1)}$. $\square$

6.3.3 Closing Away from the Top-Order Derivatives $j= N-1$

In the previous proof, we crucially needed the sharp decay of $T^{j+1}(r\phi _1)$, which we no longer have access to if $j+1=N$. We therefore proceed differently now. We will use the approximate conservation law (3.13). In fact, since we still have sharp decay for $T^j(r\phi _1)$, it will suffice to consider (the T-commuted)

$$\begin{aligned} \partial _u(r^{-2}\partial _v(r^2\partial _v(r\phi _1)))=-10 MD\frac{r^2\partial _v(r\phi _1)}{r^5}-2MD\frac{r\phi _1}{r^4}\left( 1+\frac{4M}{r}\right) , \end{aligned}$$

(5.22)

since, as long as we have the extra r-decay of $T^j(r\phi _1)$, the bad $r^{-4}$-weight multiplying $T^j(r\phi _1)$ poses no problem.

Proposition 5.3

Let $0\le j\le N-1$. Then, for sufficiently large values of R/2M and $|U_0|$, and if $C_{\mathrm {BS}}^{(j)}$ and $C_{\mathrm {BS},\phi }^{(j)}$ are chosen large enough, we have throughout $\Delta$ that, in fact,

$$\begin{aligned} \left| r^2\partial _vT^j (r\phi _1)(u,v)\right|&\le \frac{1}{2} \frac{C_{\mathrm {BS}}^{(j)}}{|u|^{j+1}}, \end{aligned}$$

(5.23)

$$\begin{aligned} \left| T^j (r\phi _1)(u,v)\right|&\le \frac{1}{2} \frac{C_{\mathrm {BS},\phi }^{(j)}}{r|u|^{j+1}}\max \left( r^{-1},|u|^{-1}\right) . \end{aligned}$$

(5.24)

Proof

Fix $j\le N-1$ and assume (BS(n)) for $n=j,j+1$ and (BS’(m)) for $m=j$. The idea is to integrate (5.22) twice, first from $u=-k$ and then from $\Gamma _R$. In doing so, we will pick up the boundary term $r^2T^j(r\phi _1)|_{\Gamma _R}$, which is not given by data. We will therefore estimate this boundary term by using the (T-commuted) Eq. (5.17): First, note that, by integrating the bound (BS(n)) for $n=j+1$ from $\Gamma _R$, we have

$$\begin{aligned} \left| T^{j+1}(r\phi _1)(u,v)\right| \le \frac{(j+1)!C_{\mathrm {in}}^\Gamma }{R|u|^{j+1}}+\frac{C_{\mathrm {BS}}^{(j+1)}}{R|u|^{j+1}}. \end{aligned}$$

(5.25)

Hence, by integrating equation (5.17) from $u=-k$, we obtain

$$\begin{aligned} \begin{aligned} \left| r^{-2}T^j\partial _v(r^2\phi _1)(u,v)\right| \le \int _{r_v(-k)}^{r_v(u)} \frac{(j+1)!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j+1)}}{r^2R|u|^{j+2}}+\frac{8MC_{\mathrm {BS},\phi }^{(j)}}{r^5|u|^{j+1}}\mathop {}\!\mathrm {d}r, \end{aligned} \end{aligned}$$

and, consequentially,

$$\begin{aligned} \left| r^2T^j\partial _v(r\phi _1)+Dr\cdot T^j(r\phi _1)\right| \le r^2\frac{(j+1)!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j+1)}}{R|u|^{j+2}}+\frac{2MC_{\mathrm {BS},\phi }^{(j)}}{r|u|^{j+1}}. \end{aligned}$$

Evaluating this bound on $\Gamma _R$ and applying the triangle inequality gives

$$\begin{aligned} \left| r^2T^j\partial _v(r\phi _1)|_{\Gamma _R}\right| \le \left( 1-\frac{2M}{R}\right) \frac{j!C_{\mathrm {in}}^\Gamma }{|u|^{j+1}}+\frac{2MC_{\mathrm {BS},\phi }^{(j)}}{R|u|^{j+1}}+R\frac{(j+1)!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j+1)}}{|u|^{j+2}}. \nonumber \\ \end{aligned}$$

(5.26)

Notice that, for sufficiently large $|U_0|$, the last term becomes subleading. Moreover, if 2M/R is suitably small, the first term, in fact, dominates.

Equipped with this estimate for the boundary term, we can now integrate (the $T^j$-commuted) approximate conservation law (5.22), first from $u=-k$:

$$\begin{aligned}&r^{-2}\partial _v(r^2\partial _vT^j(r\phi _1))(u,v)\\&=\int _{-k}^u -10 MD\frac{r^2\partial _vT^j(r\phi _1)}{r^5}-2MD\frac{T^j(r\phi _1)}{r^4}\left( 1+\frac{4M}{r}\right) \mathop {}\!\mathrm {d}u' \\&\le \int _{-r_v(k)}^{r_v(u)} \frac{10MC_{\mathrm {BS}}^{(j)}}{r^5|u|^{j+1}}+\frac{2MC_{\mathrm {BS},\phi }^{(j)}}{r^5|u|^{j+1}}\left( 1+\frac{4M}{r}\right) \mathop {}\!\mathrm {d}r. \end{aligned}$$

We thus obtain:

$$\begin{aligned} \left| \partial _v(r^2\partial _vT^j(r\phi _1))\right| \le \frac{10MC_{\mathrm {BS}}^{(j)}}{4r^2|u|^{j+1}}+\frac{MC_{\mathrm {BS},\phi }^{(j)}}{2r^2|u|^{j+1}}\left( 1+\frac{16M}{5R}\right) . \end{aligned}$$

(5.27)

Finally, integrating (5.27) from $\Gamma _R$, and estimating the boundary term via (5.26), we obtain

$$\begin{aligned} \begin{aligned} \left| r^2\partial _vT^j(r\phi _1)\right| \le&D(R)\frac{j!C_{\mathrm {in}}^\Gamma }{|u|^{j+1}}+\frac{2MC_{\mathrm {BS},\phi }^{(j)}}{R|u|^{j+1}}+R\frac{(j+1)!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j+1)}}{|u|^{j+2}}\\ +&D^{-1}(R)\left( \frac{10MC_{\mathrm {BS}}^{(j)}}{4R|u|^{j+1}}+\frac{MC_{\mathrm {BS},\phi }^{(j)}}{2R|u|^{j+1}}\left( 1+\frac{16M}{5R}\right) \right) . \end{aligned} \end{aligned}$$

(5.28)

(The factor $D^{-1}$ comes from substituting $\mathrm {d}v$ with $\mathrm {d}r$ in the integral.) If $U_0$ and R/2M are sufficiently large, and if $C_{\mathrm {BS}}^{(j)}$ is chosen suitably large relative to $j!C_{\mathrm {in}}^\Gamma$, then the RHS can be shown to be smaller than $\frac{C_{\mathrm {BS}}^{(j)}}{2|u|^{j+1}}$. We thus recover the first statement of the proposition.

In order to show the second statement, we exploit the fact that we have also obtained an estimate on $r^2\partial _v(r^2 \partial _vT^j(r\phi _1))$ and use the following identity (which follows directly from the wave equation (3.11)):

$$\begin{aligned} -\frac{2D}{r^2}T^j(r\phi _1)\left( 1+\frac{M}{r}\right) =T^j\partial _v\partial _u(r\phi _1)=-\partial _v^2T^j(r\phi _1)+\partial _vT^{j+1}(r\phi _1). \nonumber \\ \end{aligned}$$

(5.29)

The last term of the equation above is controlled by the bootstrap assumption (BS(n)) for $n=j+1$. For the other term, we can write:

$$\begin{aligned} \partial _v^2T^j(r\phi _1)=\frac{1}{r^2}\partial _v\left( r^2\partial _vT^j(r\phi _1)\right) -\frac{2D}{r}\partial _vT^{j}(r\phi _1). \end{aligned}$$

We therefore can estimate $T^j(r\phi _1)$ as follows:

$$\begin{aligned} 2D\left( 1+\frac{M}{r}\right) \left| T^j(r\phi _1)\right|\le & {} \left| \partial _v\left( r^2\partial _vT^j(r\phi _1)\right) \right| \nonumber \\&+\frac{2D}{r}\left| r^2\partial _vT^{j}(r\phi _1)\right| +\left| r^2\partial _vT^{j+1}(r\phi _1)\right| . \end{aligned}$$

(5.30)

Finally, plugging in the estimates (5.27), (5.28), as well as the assumption (BS(n)) for $n=j+1$, into the estimate (5.30) shows that if $U_0$ and R/2M are sufficiently large, and if $C_{\mathrm {BS},\phi }^{(j)}$ is chosen suitably large relative to $j!C_{\mathrm {in}}^\Gamma /2$, then the RHS is smaller than $\frac{C_{\mathrm {BS},\phi }^{(j)}}{2r|u|^{j+1}}$, thus proving the proposition.¹¹$\square$

6.3.4 Closing the Top-Order Derivative $j=N$

In the previous proof, we used the sharp u-decay of $T^{j+1}(r\phi _1)$ (see (5.26)), combined with equation (5.17), to estimate the boundary term $r^2\partial _vT^j(r\phi _1)|_{\Gamma _R}$ on $\Gamma _R$. At the highest order in derivatives, we can no longer do this. Instead, we will estimate the boundary term using an energy estimate. This energy estimate will be wasteful in terms of r-decay, but sharp in terms of u-decay and, therefore, useful on $\Gamma _R$. Moreover, it only requires a "square root" of the bootstrap estimate on $r^2\partial _vT^j(r\phi _1)$ and, thus, allows for improvement.

Another difference to the previous section will be that, since we can no longer assume the sharp decay of $T^{j+1}(r\phi _1)$, we will have to work with the approximate conservation law (3.13) instead of (5.22). (Recall that the former has a better r-weight multiplying $T^j(r\phi _1)$.) This will give us an estimate on $T^j\Phi =r^2\partial _vT^j(r\phi _1)-MT^j(r\phi _1)$. As mentioned in the introduction to this section, we will simply exploit the largeness in R/2M to estimate $r^2\partial _vT^j(r\phi _1)$ in terms of $T^j\Phi$.

Proposition 5.4

Let $0\le j\le N$. Then, for sufficiently large values of R/2M and $|U_0|$, and if $C_{\mathrm {BS}}^{(j)}$ is chosen large enough, we have throughout $\Delta$ that, in fact,

$$\begin{aligned} \left| r^2\partial _vT^j (r\phi _1)(u,v)\right|&\le \frac{1}{2} \frac{C_{\mathrm {BS}}^{(j)}}{|u|^{j+1}}. \end{aligned}$$

(5.31)

Proof

Fix $j\le N$ and assume (BS(n)) for $n=j$.

Recall the definition (3.2). Since T is Killing and $T^j\phi _{\ell =1}$ solves the wave equation, we have

$$\begin{aligned} \mathrm {div}J^T[T^j\phi _{\ell =1}]=0. \end{aligned}$$

(5.32)

We want to apply the divergence theorem to this identity. We recall the notation $\phi _{\ell =1}=\phi _1\cdot Y_{1m}$, and denote by the covariant derivative on the unit sphere. We compute

Let now $(u,v)\in \Delta$. Then, applying the divergence theorem as in (2.11) to (5.32), we obtain

$$\begin{aligned} \begin{aligned}&\int _{{\mathscr {C}}_{v}\cap \{-k\le u'\le u\}} r^2\mathop {}\!\mathrm {d}u'\, \mathop {}\!\mathrm {d}\Omega \, J^T[T^j\phi _{\ell =1}]\cdot \partial _u\\ \le&\int _{\Gamma _R\cap \{-k\le u'\le u\}} r^2 \mathop {}\!\mathrm {d}(u'+v')\mathop {}\!\mathrm {d}\Omega \,J^T[T^j\phi _{\ell =1}]\cdot (\partial _u-\partial _v). \end{aligned} \end{aligned}$$

(5.33)

Doing the integrals over the sphere, using that $u+v=2u+r^*(R)$ along $\Gamma _R$, and plugging in the expressions for the fluxes from above, we obtain

$$\begin{aligned} \begin{aligned}&\int _{-k}^u\left( r^2(\partial _uT^j\phi _{1})^2+D|T^j\phi _{1}|^2\right) (u',v)\mathop {}\!\mathrm {d}u' \\ \le&\int _{\Gamma _R\cap \{-k\le u'\le u\}} \left( 2r^2 T^{j+1}\phi _{1}\cdot (T-2\partial _v)T^j \phi _{1}\right) \left( u',u'+r^*(R)\right) \mathop {}\!\mathrm {d}u'. \end{aligned} \end{aligned}$$

(5.34)

Observe that we can estimate the right-hand side of (5.34) by using the boundary data assumption (5.10) for $T^{j+1}\phi _1$ and the bootstrap assumption (BS(n)) with $n=j$ for $\partial _vT^j\phi _1$.

On the other hand, the left-hand side of (5.34) controls $\sqrt{r}T^j\phi$, as can be seen by applying first the fundamental theorem of calculus and then the Cauchy–Schwarz inequality:

$$\begin{aligned} \left| T^j\phi _1(u,v)\right| \le \left( \int _{-k}^u \frac{1}{r^2(u',v)}\mathop {}\!\mathrm {d}u'\right) ^\frac{1}{2}\left( \int _{-k}^u r^2\left( \partial _uT^j\phi _1\right) ^2(u',v)\mathop {}\!\mathrm {d}u'\right) ^\frac{1}{2}. \end{aligned}$$

Applying the energy identity (5.34) to the above estimate gives

$$\begin{aligned} Dr(T^j\phi _1)^2&\le \int _{\Gamma _R} 2|T^{j+1}(r\phi _1)|\left( |T^{j+1}(r\phi _1)|+2|\partial _vT^j(r\phi _1)|+\frac{2D}{r} |T^j(r\phi _1)|\right) \mathop {}\!\mathrm {d}u'\\&\le \int _{\Gamma _R} \frac{2(j+1)!C_{\mathrm {in}}^\Gamma }{R|u'|^{j+2}}\left( \frac{2(j+1)!C_{\mathrm {in}}^\Gamma }{R|u'|^{j+2}}+\frac{2C_{\mathrm {BS}}^{(j)}}{R^2|u'|^{j+1}}+\frac{2D}{R} \frac{2j!C_{\mathrm {in}}^\Gamma }{R|u'|^{j+1}}\right) \mathop {}\!\mathrm {d}u'\\&\le \frac{1}{R^3|u|^{2j+2}}\underbrace{\left( 4D(j!)^2(C_{\mathrm {in}}^\Gamma )^2+2j!C_{\mathrm {in}}^\Gamma C_{\mathrm {BS}}^{(j)}\right) }_{:=A}+{\mathscr {O}}\left( \frac{1}{|u|^{2j+3}R^2}\right) . \end{aligned}$$

Plugging this bound¹² into the wave equation (3.11) and integrating (3.11) from $u=-k$ results in the following bound on the boundary term $\partial _vT^j(r\phi _1)|_{\Gamma _R}$:

$$\begin{aligned} \left| \partial _vT^j(r\phi _1)|_{\Gamma _R}\right| \le \int \frac{2\sqrt{AD}}{R^{\frac{3}{2}}|u|^{j+1}}\frac{1}{r^{3/2}}\left( 1+\frac{M}{r}\right) \mathop {}\!\mathrm {d}u\le \frac{4\sqrt{A}}{R^2|u|^{j+1}}\frac{1+\frac{M}{3R}}{\sqrt{1-\frac{2M}{R}}}. \nonumber \\ \end{aligned}$$

(5.35)

In fact, we see that the estimate on the boundary term closes by itself!

Having obtained a bound on the boundary term, we can proceed as in the previous proof. We insert the bootstrap estimate (BS(n)) for $n=j$ and the estimate

$$\begin{aligned} |T^j(r\phi _1)|\le \frac{j!C_{\mathrm {in}}^\Gamma }{R|u|^{j+1}}+\frac{C_{\mathrm {BS}}^{(j)}}{R|u|^{j+1}} \end{aligned}$$

(5.36)

implied by it into the approximate conservation law (3.13) to find (recall $\Phi =r^2\partial _v(r\phi _1)-Mr\phi _1$):

$$\begin{aligned} \begin{aligned} |r^{-2}\partial _vT^j\Phi (u,v)|&\le \int _{-k}^{u} \frac{6M^2D}{r^5}\frac{j!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j)}}{R|u'|^{j+1}}+\frac{12MD}{r^5}\frac{C_{\mathrm {BS}}^{(j)}}{|u'|^{j+1}}\mathop {}\!\mathrm {d}u'\\&\le \frac{3M^2}{2r^4}\frac{j!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j)}}{R|u|^{j+1}}+\frac{3MC_{\mathrm {BS}}^{(j)}}{r^4|u|^{j+1}}. \end{aligned} \end{aligned}$$

(5.37)

Multiplying the above estimate by $r^2$, integrating from $\Gamma _R$ and using the bound (5.35) to estimate the boundary term, we thus obtain

$$\begin{aligned} \begin{aligned} |T^j\Phi |\le \frac{4\sqrt{A}}{|u|^{j+1}}\frac{1+\frac{M}{3R}}{\sqrt{1-\frac{2M}{R}}}+\frac{Mj!C_{\mathrm {in}}^\Gamma }{R|u|^{j+1}}+\frac{1}{1-\frac{2M}{R}}\left( \frac{3M^2}{2}\frac{j!C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}^{(j)}}{R^2|u|^{j+1}}+\frac{3MC_{\mathrm {BS}}^{(j)}}{R|u|^{j+1}}\right) . \end{aligned} \end{aligned}$$

(5.38)

Importantly, $C_{\mathrm {BS}}^{(j)}$ in the above estimate is either multiplied by decaying R-weights, or appears sublinearly inside a square root (in A).

We can now combine (5.38) with (5.36) and write

$$\begin{aligned} |r^2\partial _vT^j(r\phi _1)|\le |T^j\Phi |+M|T^j(r\phi _1)| \end{aligned}$$

(5.39)

to close the bootstrap assumption, provided that R and $C_{\mathrm {BS}}^{(j)}$ are chosen large enough. $\square$

In order to close the entire bootstrap argument, one can now first apply Proposition 5.4 to $j=N$, then apply Proposition 5.3 to $j=N-1$ and, finally, apply Proposition 5.2 to all $j\le N-2$. One thus obtains that $\Delta$ is open and hence closes the bootstrap argument. In particular, we have established the proof of Proposition 5.1.

More systematically, one could instead make only the bootstrap assumptions (BS(n)) (without assuming (BS’(m))), apply Proposition 5.4 to all $j\le N$ in order to close the bootstrap argument, and then use the identity (5.29) in order to obtain the remaining estimates for $T^j(r\phi _1)$. This will be the approach followed in section 8. $\square$

6.3.5 Removing the Smallness Assumption on 2M/R.

In the proofs of the previous sections §5.3.2–§5.3.4, we exclusively followed continuity methods, which required us to exploit 2M/R as a small parameter at various steps. It turns out that one can partially replace the continuity argument with a Grönwall argument to remove all smallness assumptions on 2M/R. Let us briefly sketch how this works.

Let $\phi$ denote the finite solution as described in the beginning of §5.3. First, we remark that the proof of Proposition 5.4 shows that one can obtain an estimate of the form

$$\begin{aligned} |\partial _vT^{j}(r\phi _1)|_{\Gamma _R}| \le \frac{C}{\sqrt{R}}|u|^{-j-1} \end{aligned}$$

(5.40)

without requiring largeness in R (this can be obtained by assuming a bootstrap estimate on $r^2\partial _vT^j(r\phi _1)|_{\Gamma _R}$ and improving it using the energy estimate, cf. (5.35)).

Equipped with this boundary term estimate, one can then obtain an estimate on $r^2\partial _vT^j(r\phi )$ throughout ${\mathscr {D}}_{\Gamma _R}\cap \{u\ge -k\}$ as follows: Let (u, v) in ${\mathscr {D}}_{\Gamma _R}\cap \{u\ge -k\}$. For simpler notation, set $j=0$. Then, by the fundamental theorem of calculus,

$$\begin{aligned} \left| \Phi (u,v)-\Phi (u,v_{R}(u))\right|&\le \int _{v_{R}(u)}^v r^2(u,v')\int _{-k}^u \left| 12MD\frac{r^2\partial _v(r\phi _1)}{r^5}-6M^2 D\frac{r\phi _1}{r^5} \right| \nonumber \\&\quad (u',v')\mathop {}\!\mathrm {d}u'\mathop {}\!\mathrm {d}v'. \end{aligned}$$

(5.41)

Recalling the definition of $\Phi =r^2\partial _v(r\phi _1)-Mr\phi _1$, estimating the $Mr\phi _1$-term against the integral over $\partial _v(r\phi _1)$ from $\Gamma$, and applying Tonelli in the inequality above, we obtain:

$$\begin{aligned}&|r^2\partial _v(r\phi _1)(u,v)|\le \left| r^2\partial _v(r\phi _1)(u,v_{R}(u))\right| +\int _{v_{R}(u)}^vM \frac{|r^2\partial _v(r\phi _1)|}{r^2}(u,v')\mathop {}\!\mathrm {d}v'\nonumber \\&\quad + \int _{-k}^u\int _{v_{R}(u)}^v r^2(u,v') \left| 12MD\frac{r^2\partial _v(r\phi _1)}{r^5}-6M^2 D\frac{r\phi _1}{r^5} \right| (u',v')\mathop {}\!\mathrm {d}v'\mathop {}\!\mathrm {d}u'. \nonumber \\ \end{aligned}$$

(5.42)

We now also write the $r\phi _1$-term in the double integral as an integral over $\partial _v(r\phi _1)$ from $\Gamma$ and pull out the relevant suprema:

$$\begin{aligned}&|r^2\partial _v(r\phi _1)(u,v)|\le \left| r^2\partial _v(r\phi _1)(u,v_{R}(u))\right| +\int _{v_{R}(u)}^v M\frac{|r^2\partial _v(r\phi _1)|}{r^2}(u,v')\mathop {}\!\mathrm {d}v'\nonumber \\&\quad + C\int _{-k}^u\frac{\sup _{v'\in [v_{R}(u),v]} |r^2\partial _v(r\phi _1)(u',v')|}{r^2(u',v_{\Gamma _R}(u))} + \frac{ \sup _{v'\in [v_{R}(u'),v]}|r^2\partial _v(r\phi _1)(u',v')|}{r^2(u',v_{R}(u))} \mathop {}\!\mathrm {d}u'. \nonumber \\ \end{aligned}$$

(5.43)

We already control the boundary term on the RHS. If we now fix u and just regard the last two integrals on the RHS as some monotonically increasing function of v, we can apply Grönwall’s inequality in the v-direction to obtain that

$$\begin{aligned} |r^2\partial _v(r\phi _1)(u,v)|&\le C\left( \left| r^2\partial _v(r\phi _1)(u,v_{R}(u))\right| \right. \nonumber \\&\left. + \int _{-k}^u \frac{ \sup _{v'\in [v_{R}(u'),v]}|r^2\partial _v(r\phi _1)(u',v')|}{r^2(u',v_{\Gamma _R}(u))} \mathop {}\!\mathrm {d}u'\right) .\nonumber \\ \end{aligned}$$

(5.44)

Finally, we take the supremum in v, $\sup _{v'\in [v_R(u),v]}$, on the RHS and apply another Grönwall inequality, this time in u. This then shows that

$$\begin{aligned} \sup _{v'\in [v_{R}(u),v]}|r^2\partial _v(r\phi _1)(u,v')|\le C\left| r^2\partial _v(r\phi _1)(u,v_{R}(u))\right| , \end{aligned}$$

(5.45)

and thus shows (5.5) for $n=0$.

Clearly, this approach requires no smallness assumption on 2M/R other than $R>2M$. Nevertheless, in hopes of simplifying the presentation, we will keep exploiting 2M/R as a small parameter throughout the remainder of the paper (i.e. §5.4 and §8). However, as the argument above shows, these smallness assumptions can always be lifted if one only wants to show upper bounds. The only times where we really need 2M/R as a small parameter is when we show lower bounds on $r\phi _1$ etc., see already §5.5.2.

6.4 Estimates for $\partial _vT^n(r\phi _1-|u|T(r\phi _1))$

The results obtained thus far are sufficient to show the first two estimates of Theorem 5.1. In fact, not much modification is needed to also show certain lower bounds. However, something different needs to be done in order to establish the existence of the limit $\lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1))$ (i.e. to prove Eq. (5.8)). A crucial ingredient for this is to prove decay estimates for the differences $T^j(\partial _v(r\phi _1)-|u|\partial _vT(r\phi _1))$ (the reader may wish to already have a look at §5.5.2 to understand the role played by these quantities).

Therefore, let from now on $\phi _1$ be as described in the beginning of §5.3, but with the additional assumption that also the lower bound (5.11) holds on the boundary data. We will now establish the following uniform decay estimates:

Proposition 5.5

Let $\phi _1$ be the solution described above, and let $1\le N'\le N+2$. Then, if $|U_0|$ is sufficiently large, there exists a constant $C=C(2M/R,C_{\mathrm {in}}^\Gamma \,C_{\mathrm {in},\varepsilon }^\Gamma )$ (in particular, this constant does not depend on k), which can be chosen independent of R for large enough R, such that the following estimates hold throughout ${\mathscr {D}}_{\Gamma _R}\cap \{-k\le u\le U_0\}$:

$$\begin{aligned} \left| r^2T^n\partial _v\left( r\phi _1-|u|T(r\phi _1)\right) \right| \le \frac{C}{|u|^{n+1+\varepsilon }}+\theta _k\cdot \frac{C}{k^{n+1}},&n=0,\dots ,N'. \end{aligned}$$

(5.46)

Proof

The proof will be very similar to the proof of Proposition 5.1. We will again treat 2M/R as a small parameter, keeping in mind that this restriction can lifted as in §5.3.5. $\square$

6.4.1 The Bootstrap Assumptions

Let $\{C_{\mathrm {BS},\varepsilon }^{(n)},0=1,\dots ,N'\}$ be a set of sufficiently large positive constants, and let $\Delta$ be defined as in 5.3.1, with the additional requirement that also

$$\begin{aligned} \left| r^2T^n\partial _v\left( r\phi _1-|u|T(r\phi _1)\right) \right| \le \frac{C_{\mathrm {BS},\varepsilon }^{(n)}}{|u|^{n+1+\varepsilon }}+\theta _k\cdot \frac{C_{\mathrm {BS},\varepsilon }^{(n)}}{k^{n+1}} \end{aligned}$$

(BS''(n))

holds for $n=0,\dots ,N$’. We shall improve these estimates in the following. Note that we only assume estimates on the $\partial _v$-derivatives, so we can just use the method of the proof of Proposition 5.4 with some adaptations. The crucial observation is that, while the differences $T^n(\phi _1-|u|T\phi _1)$ do not solve the wave equation, the error term is of the form $\partial _vT^{n+1}(r\phi _1)$, over which we already have sharp control by Proposition 5.1.

6.4.2 Improving the Bootstrap Assumptions

Proposition 5.6

Let $j\in \{0,\dots ,N'\}$ for $N'\le N+2$. Then, for sufficiently large values of R/2M and $|U_0|$, and if $C_{\mathrm {BS},\varepsilon }^{(j)}$ is chosen large enough, we have throughout $\Delta$ that, in fact,

$$\begin{aligned} \left| r^2T^j\partial _v\left( r\phi _1-|u|T(r\phi _1)\right) \right| \le \frac{1}{2}\frac{C_{\mathrm {BS},\varepsilon }^{(j)}}{|u|^{j+1+\varepsilon }}+\frac{1}{2} \theta _k\cdot \frac{C_{\mathrm {BS},\varepsilon }^{(j)}}{k^{n+1}}. \end{aligned}$$

(5.47)

Proof

We shall only need to assume (BS”(n)) for $n=j$ and, in addition, the results of Proposition 5.1. We shall follow the structure of the proof of Proposition 5.4.

First, we require an estimate of $r^2T^j\partial _v(r\phi _1-|u|T(r\phi _1))$ on the boundary $\Gamma _R$. Recall that, in the previous proof, we obtained such an estimate by using an energy estimate to obtain a bound on $\sqrt{r}T^j\phi _1$ with sharp decay in u, and by then using the wave equation to convert this into a bound for $\partial _vT^j(r\phi _1)$ on $\Gamma _R$. Proceeding along the same lines for the differences under consideration, we are led to consider the current $J^T[T^j(\phi _1-|u|T\phi _1)]$. The divergence of this current is no longer vanishing. Instead, we have¹³

$$\begin{aligned} \begin{aligned} \mathrm {div}J^T\left[ T^j\left( \phi _1-|u|T\phi _1\right) \right]&=\Box _g\left( T^j\left( \phi _1-|u|T\phi _1\right) \right) \cdot T\left( T^j\left( \phi _1-|u|T\phi _1\right) \right) \\&=-\frac{1}{Dr}\partial _vT^{j+1}(r\phi _1)\cdot \frac{1}{r}T^{j+1}\left( r\phi _1-|u|T(r\phi _1)\right) , \end{aligned} \end{aligned}$$

(5.48)

where we used the formula (3.6) for $\Box _g$. Using the estimates from Proposition 5.1 and the fact that $j+2\le N'+2\le N$, we can thus bound $\mathrm {div}J^T$ as follows:

$$\begin{aligned} \left| \mathrm {div}J^T\left[ T^j\left( \phi _1-|u|T\phi _1\right) \right] \right| \le 2\cdot \frac{C^2}{R\cdot r^{4}|u|^{2j+4}}. \end{aligned}$$

(5.49)

Applying the divergence theorem, in the form of (2.11), to the current $J^T$, and doing the integrals over the sphere, we then arrive at (compare to Eq. (5.34))

$$\begin{aligned} \begin{aligned}&\int _{-k}^u r^2\left( \partial _uT^j(\phi _1-|u'|T\phi _1)\right) ^2\mathop {}\!\mathrm {d}u' \\ \le&\int _{\Gamma _R\cap \{-k\le u'\le u\}} 2r^2 T^{j+1}(\phi _1-|u'|T\phi _1)\cdot (2\partial _v-T)T^j(\phi _1-|u'|T\phi _1) \mathop {}\!\mathrm {d}u'\\ +&\int _{-k}^u\int _{v_R(u')}^v r^2\left| \mathrm {div}J^T\left[ T^j(\phi _1-|u'|T\phi _1)\right] \right| \mathop {}\!\mathrm {d}u'\mathop {}\!\mathrm {d}v'. \end{aligned} \end{aligned}$$

(5.50)

We have already estimated the terms inside the bulk term in (5.49). Indeed, we can see that the contribution to the RHS of the estimate (5.50) above is subleading:

$$\begin{aligned} \int _{-k}^u\int _{v_R(u)}^v r^2\left| \mathrm {div}J^T[T^j(\phi _1-|u'|T\phi _1)]\right| \mathop {}\!\mathrm {d}u'\mathop {}\!\mathrm {d}v'\le \frac{{\tilde{C}}}{R^2|u|^{2j+3}} \end{aligned}$$

(5.51)

for some constant ${\tilde{C}}$. On the other hand, we can estimate the boundary term in (5.50) by plugging in the boundary data assumptions (5.11) for $n=j,j+1$ and the bootstrap assumption (BS”(n)) for $n=j$. This gives:

$$\begin{aligned}&\int _{\Gamma _R\cap \{-k\le u'\le u\}} 2r^2 T^{j+1}(\phi _1-|u'|T\phi _1)\cdot (2\partial _v-T)T^j(\phi _1-|u'|T\phi _1) \mathop {}\!\mathrm {d}u'\\ \le&\int _{\Gamma _R\cap \{-k\le u'\le u\}}2\left( \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{R|u'|^{2+j+\varepsilon }}+\frac{C_{\mathrm {in}}^\Gamma }{Rk^{j+2}} \right) \\&\cdot \left( \frac{2}{R^2}\left( \frac{C_{\mathrm {BS},\varepsilon }^{(j)}}{|u'|^{j+1+\varepsilon }}+\frac{C_{\mathrm {BS},\varepsilon }^{(j)}}{k^{j+1}}\right) +\left( \frac{2D}{R}+\frac{1}{|u'|}\right) \left( \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{R|u'|^{1+j+\varepsilon }}+\frac{C_{\mathrm {in}}^\Gamma }{Rk^{j+1}} \right) \right) \mathop {}\!\mathrm {d}u' \\ \le&\frac{8}{R^3}\max (C_{\mathrm {in},\varepsilon }^\Gamma ,C_{\mathrm {in}}^\Gamma )\left( C_{\mathrm {BS},\varepsilon }^{(j)}+\max (C_{\mathrm {in},\varepsilon }^\Gamma ,C_{\mathrm {in}}^\Gamma )\right) \left( \frac{1}{k^{j+1}}+\frac{1}{|u|^{j+1+\varepsilon }} \right) ^2\cdot (1+{\mathscr {O}}(|u|^{-1}). \end{aligned}$$

We thus find, using the fundamental theorem of calculus, Cauchy–Schwarz and the energy estimate (5.50) above, that

$$\begin{aligned} \sqrt{Dr}|T^j(\phi _1-|u|T\phi _1)|\le \frac{\sqrt{A'}}{R^{\frac{3}{2}}}\left( \frac{1}{k^{j+1}}+\frac{1}{|u|^{j+1+\varepsilon }}\right) (1+{\mathscr {O}}(|u|^{-\frac{1}{2}})), \end{aligned}$$

(5.52)

where $A'$ is a constant which, importantly, only depends linearly on $C_{\mathrm {BS},\varepsilon }^{(j)}$. We now use the wave equation (3.11) in order to derive an estimate for the $\partial _v$-derivative on the boundary. We compute from (3.11) that

$$\begin{aligned} \begin{aligned} \partial _u\partial _v(T^j(r\phi _1&-|u| T(r\phi _1)))=-\frac{2D}{r^2}\left( 1+\frac{M}{r}\right) (T^j(r\phi _1-|u| T(r\phi _1)))+\partial _vT^{j+1}(r\phi _1). \end{aligned} \end{aligned}$$

(5.53)

Note that we control the error term $\partial _vT^{j+1}(r\phi _1)$ by Proposition 5.1; in fact, it is subleading in terms of u-decay. Integrating (5.53) from $u=-k$, and plugging in the estimate (5.52), we find that (see also (5.35))

$$\begin{aligned} |\partial _vT^j(r\phi _1-|u|T(r\phi _1))|\le \frac{4\sqrt{A'}}{R^2}\frac{1+\frac{M}{3R}}{\sqrt{1-\frac{2M}{R}}}\left( \frac{1}{k^{j+1}}+\frac{1}{|u|^{j+1+\varepsilon }}\right) (1+{\mathscr {O}}(|u|^{-\frac{1}{2}})). \nonumber \\ \end{aligned}$$

(5.54)

We have thus established an estimate on the boundary term. Now, in order to improve the bootstrap assumption, we want to appeal to the approximate conservation law (3.13). We compute that

$$\begin{aligned} \begin{aligned}&\partial _u\left( r^{-2}\partial _vT^j(\Phi -|u|T\Phi )\right) =r^{-2}\partial _vT^{j+1}\Phi \\&-\frac{6M^2D}{r^5}(T^j(r\phi _1-|u| T(r\phi _1)))-\frac{12MD}{r^5}r^2\partial _vT^j(r\phi _1-|u| T(r\phi _1)). \end{aligned} \end{aligned}$$

(5.55)

Again, we control the error term $r^{-2}\partial _vT^{j+1}\Phi$ by Proposition 5.1; in fact, it has more u-decay than the other terms:

$$\begin{aligned} |r^{-2}\partial _vT^{j+1}\Phi |\le \frac{C}{r^4|u|^{j+2}}. \end{aligned}$$

Converting some of the additional |u|-decay present in $\partial _vT^{j+1}\Phi$ into r-decay,

$$\begin{aligned} |r^{-2}\partial _vT^{j+1}\Phi |\le \frac{C}{r^{5-\delta }|u|^{j+1+\delta }}, \end{aligned}$$

for some suitable $1>\delta >\varepsilon$, and repeating the computations leading to (5.38), we thus find

$$\begin{aligned}&|T^j(\Phi -|u|T\Phi )|\le \frac{4\sqrt{A'}}{R^2}\frac{1+\frac{M}{3R}}{\sqrt{1-\frac{2M}{R}}}\left( \frac{1}{k^{j+1}}+\frac{1}{|u|^{j+1+\varepsilon }}\right) +\frac{MC_{\mathrm {in},\varepsilon }^\Gamma }{R|u|^{j+1+\varepsilon }}+ \frac{MC_{\mathrm {in}}^\Gamma }{Rk^{j+1}}\nonumber \\&\quad +\frac{1}{1-\frac{2M}{R}}\left( \frac{3M^2}{2}\frac{C_{\mathrm {in},\varepsilon }^\Gamma +C_{\mathrm {BS},\varepsilon }^{(j)}}{R^2}+\frac{3MC_{\mathrm {BS},\varepsilon }^{(j)}}{R}\right) \left( \frac{1}{k^{j+1}}+\frac{1}{|u|^{j+1+\varepsilon }}\right) +{\mathscr {O}}(|u|^{-{j-1-\delta }}). \nonumber \\ \end{aligned}$$

(5.56)

Importantly, $C_{\mathrm {BS},\varepsilon }^{(j)}$ in the above estimate appears either multiplied by decaying R-weights or sublinearly inside a square root (in $A'$). Therefore, if R/2M and $C_{\mathrm {BS},\varepsilon }^{(j)}$ are sufficiently large, we can improve the bootstrap assumption (and thus prove the proposition) by again writing

$$\begin{aligned} \left| r^2 T^j\partial _v\left( r\phi _1-|u|T(r\phi _1)\right) \right| \le \left| T^j\left( \Phi -|u|T\Phi \right) \right| +M\left| T^j\left( r\phi _1-|u|T(r\phi _1)\right) \right| . \nonumber \\ \end{aligned}$$

(5.57)

$\square$

This concludes the proof of Proposition 5.5.

6.5 Proof of Thm. 5.1

Recall from §5.2.2 the definition of the sequence of solutions $\phi _1^{(k)}$, each arising from data satisfying (5.10), (5.11) and (5.12). We have shown sharp, uniform-in-k decay for these solutions in Propositions 5.1 and 5.5.

We will now smoothly extend these solutions to the zero solution for $u\le -k$ and show that they converge uniformly to a pointwise limit $\phi _1$ as $k\rightarrow -\infty$, which therefore still satisfies the uniform bounds of Propositions 5.1 and 5.5.

6.5.1 Sending ${\mathscr {C}}_{u=-k}\rightarrow {\mathscr {I}}^-$

Proposition 5.7

Let $\{\phi _1^{(k)}\}_{k\in {\mathbb {N}}}$ be the sequence of solutions described in §5.2.2 extended with the zero solution for $u\le -k$. This sequence $\{\phi _1^{(k)}\}_{k\in {\mathbb {N}}}$ tends to a uniform limit $\phi _1$ as $k\rightarrow \infty$,

$$\begin{aligned} \lim _{k\rightarrow \infty }||\phi _1^{(k)}-\phi _1 ||_{C^N({\mathscr {D}}_{\Gamma _R})}=0. \end{aligned}$$

(5.58)

In fact, this limiting solution is the unique smooth solution that restricts correctly to the data of §5.2.1, and it satisfies, throughout ${\mathscr {D}}_{\Gamma _R}\cap \{u\le U_0\}$, and for sufficiently large negative values of $U_0$, the following bounds for some constant $C=C(2M/R,C_{\mathrm {in}}^\Gamma ,C_{\mathrm {in},\varepsilon }^\Gamma )$ which can be chosen independent of R for large enough R:

$$\begin{aligned}&\left| r^2\partial _vT^n (r\phi _1)(u,v)\right| \le \frac{C}{|u|^{n+1}},&n=0,1,\dots ,N, \end{aligned}$$

(5.59)

$$\begin{aligned}&\left| T^n (r\phi _1)(u,v)\right| \le \frac{C}{|u|^{n+1}}\max \left( r^{-1},|u|^{-1}\right) ,&n=0,1,\dots ,N-1. \end{aligned}$$

(5.60)

Moreover, if $N'\le N+2$, we also have

$$\begin{aligned} \left| r^2T^n\partial _v(r\phi _1-|u|T(r\phi _1))\right| \le \frac{C}{|u|^{n+1+\varepsilon }},&n=0,1,\dots ,N'. \end{aligned}$$

(5.61)

Proof

We show that the sequence is Cauchy. Let $\delta >0$ arbitrary. We need to show that there exists $K\in {\mathbb {N}}$ such that

$$\begin{aligned} ||\phi _1^{(n)}-\phi _1^{(k)}||_{C^N({\mathscr {D}}_{\Gamma _R})}<\delta \end{aligned}$$

(5.62)

for all $n,k>K$. This is done by splitting ${\mathscr {D}}_{\Gamma _R}$ into three regions: $u\le -n$, $-n\le u\le -k+1$ and $-k+1\le u$, where we assumed without loss of generality that $n>k$.

In the first region, $u\le -n$, both solutions vanish, so there is nothing to show.

Notice that, by linearity, the difference $\Delta \phi _1:=\phi _1^{(n)}-\phi _1^{(k)}$ is itself a solution to the wave equation (3.11), with vanishing data on $u=-n$ and compactly supported boundary data on $\Gamma _R\cap \{-n\le u\le -k+1\}$. We can therefore simply apply the results of Proposition 5.1 to $\Delta \phi _1$ in the second region, $-n\le u\le -k+1$, and obtain, for some constant $C_1$, that

$$\begin{aligned} ||\phi _1^{(n)}-\phi _1^{(k)}||_{C^N({\mathscr {D}}_{\Gamma _R}\cap \{-n\le u\le -k+1\})}\le \frac{C_1}{k}. \end{aligned}$$

(5.63)

In the third region¹⁴, $-k+1\le u$, we apply the energy estimate (5.34) to the difference $\Delta \phi _1$:

$$\begin{aligned} \begin{aligned}&\int _{-n}^u r^2(\partial _uT^j\Delta \phi _{1})^2+D|T^j\Delta \phi _{1}|^2\mathop {}\!\mathrm {d}u' \\&\quad \le \int _{\Gamma _R\cap \{-n\le u'\le -k+1 \}} 2r^2 T^{j+1}\Delta \phi _{1}\cdot (2\partial _v-T)T^j \Delta \phi _{1} \mathop {}\!\mathrm {d}u'. \end{aligned} \end{aligned}$$

Here, we used that the boundary data for $\Delta \phi _1$ are compactly supported in $u\le -k+1$. We can now estimate the integral over $\Gamma _R$ by plugging in the boundary data assumptions for the $T^j\Delta \phi _1$-terms and by plugging in the previously obtained estimate (5.63) for the terms $\partial _vT^j \Delta \phi _1$. We thus find that

$$\begin{aligned} |T^j\Delta \phi (u,v)|\le \left( \int _{-n}^u \frac{1}{r^2}\mathop {}\!\mathrm {d}u'\right) ^{\frac{1}{2}}\left( \int _{-n}^u r^2(\partial _uT^j\Delta \phi _1(u,v))^2\mathop {}\!\mathrm {d}u' \right) ^{\frac{1}{2}}\le \frac{1}{\sqrt{r}} \frac{C}{k^{j+1}} \end{aligned}$$

for some constant $C>0$. From these estimates on $T^j\Delta \phi _1$, we can obtain estimates on $\partial _vT^j\Delta \phi _1$ by simply integrating (3.11) from $u=-k+1$ (where $\partial _vT^j \Delta \phi _1\lesssim k^{-j-1}$). This shows that there exists a constant $C_2$ such that

$$\begin{aligned} ||\phi _1^{(n)}-\phi _1^{(k)}||_{C^N({\mathscr {D}}_{\Gamma _R}\cap \{u\ge -k+1\})}\le \frac{C_2}{k}. \end{aligned}$$

(5.64)

Combining (5.63) and (5.64) shows that (5.62) holds for all $n>k>K$ provided that $K>\frac{C_1+C_2}{2\delta }$.

We have thus established the uniform convergence of the sequence $\{\phi _1^{(k)}\}$. In view of the uniformity of the convergence, the bounds from Propositions 5.1 and 5.5 carry over to the limiting solution, thus proving the estimates (5.59)–(5.61). Moreover, the methods of the proof show that this is the unique solution that has vanishing energy flux on ${\mathscr {I}}^-$ and satisfies the assumptions of §5.2.1. This concludes the proof. $\square$

6.5.2 The Limit $\lim _{u\rightarrow -\infty ,v=\text {constant}}r^2\partial _u\left( r^2\partial _u(r\phi )\right)$

Finally, we establish that the limiting solution constructed above satisfies (5.8). For this, we will also need to assume the lower bound (5.3) on data.

Proposition 5.8

Consider the solution of Proposition 5.7, and assume in addition that $N\ge 4$ and $N'\ge 2$, as well as the lower bound on data (5.3). Then the following limit exists, is independent of v, and is non-vanishing so long as $C_{\mathrm {in}}^\Gamma$ is non-vanishing and R/2M is sufficiently large:

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u\left( r^2\partial _u(r\phi _1(u,v))\right) ={{\tilde{C}}}\ne 0. \end{aligned}$$

(5.65)

Moreover, we have that $\lim _{u\rightarrow -\infty }r^2\partial _u(r\phi _1)(u,v)=0$ and

$$\begin{aligned} r^2\partial _u\left( r^2\partial _u(r\phi _1(u,v))\right) -\lim _{u\rightarrow -\infty }r^2\partial _u\left( r^2\partial _u(r\phi _1(u,v))\right) ={\mathscr {O}}(\max (|u|^{-\varepsilon },r^{-1})). \nonumber \\ \end{aligned}$$

(5.66)

Proof

We first establish the existence of the limit

$$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^2 r^2\partial _vT(r\phi _1)(u,v)=:{\mathscr {L}}(v) \end{aligned}$$

(5.67)

by computing

$$\begin{aligned} \begin{aligned} \partial _u(|u|^2 r^2\partial _vT(r\phi _1)) =\underbrace{-2|u|r^2\partial _vT(r\phi _1)+|u|^2r^2\partial _vT^2(r\phi _1)}_{={\mathscr {O}}(|u|^{-1-\varepsilon })}-\underbrace{|u|^2\partial _v(r^2\partial _vT(r\phi _1))}_{={\mathscr {O}}(r^{-2})}, \end{aligned} \end{aligned}$$

(5.68)

where we bounded the first two terms using (5.61), and the third term by plugging in the bounds (5.59), (5.60) into the approximate conservation law (3.13) and integrating in u (see also (5.27)). In fact, the bound on the last term also shows that ${\mathscr {L}}(v)$ is independent of v, ${\mathscr {L}}(v)\equiv {\mathscr {L}}$. We have thus shown that

$$\begin{aligned} |u|^2 r^2\partial _vT(r\phi _1)-\lim _{u\rightarrow -\infty }|u|^2 r^2\partial _vT(r\phi _1)={\mathscr {O}}(\max (|u|^{-\varepsilon },r^{-1})). \end{aligned}$$

(5.69)

We now show that ${\mathscr {L}}$ is non-vanishing by using the lower bound (5.3): We have from (the T-commuted) equation (5.17) that

$$\begin{aligned} r^{-2}\partial _vT(r^2\phi _1)=\int \frac{D T^2(r\phi _1)}{r^2}-\frac{8MD}{r^4}T(r\phi _1)\mathop {}\!\mathrm {d}u \le \frac{MC}{r^4|u|^2}+{\mathscr {O}}(|u|^{-3}r^{-2}). \end{aligned}$$

Evaluating the above on $\Gamma _R$ gives

$$\begin{aligned} \left| R^2\partial _vT(r\phi _1)+D R^2 T\phi _1|_{\Gamma _R}\right| \le \frac{MC}{R|u|^2}+{\mathscr {O}}(|u|^{-3}R^{-2}), \end{aligned}$$

which, if R is chosen large enough (recall that the constant C in the estimates above can be chosen independently of R if R is large enough), can be chosen to be less than $\frac{C_{\mathrm {in}}^\Gamma }{4|u|^2}$. This results in the following lower bound on $\Gamma _R$:

$$\begin{aligned} \left| r^2\partial _vT(r\phi _1)|_{\Gamma _R}\right| \ge \frac{C_{\mathrm {in}}^\Gamma }{4|u|^2}. \end{aligned}$$

(5.70)

Using once more the estimate on $|u|^2\partial _v(r^2\partial _vT(r\phi _1))\sim r^{-2}$ and integrating it from $\Gamma _R$ shows that, if R is chosen suitably large, we in fact have

$$\begin{aligned} \left| r^2\partial _vT(r\phi _1)|_{\Gamma _R}\right| \ge \frac{C_{\mathrm {in}}^\Gamma }{8|u|^2}. \end{aligned}$$

(5.71)

In summary, we have thus established that $\lim _{u\rightarrow -\infty }|u|^2 r^2\partial _vT(r\phi _1)(u,v)={\mathscr {L}}(v)\equiv {\mathscr {L}}\ne 0.$

Finally, in order to relate ${\mathscr {L}}$ to the limit of $r^2\partial _u(r^2\partial _u(r\phi _1))$ in question, we write

$$\begin{aligned} r^2\partial _u(r^2\partial _u(r\phi _1))= & {} r^4 T^2(r\phi _1)-r^2\partial _v(r^2 T(r\phi _1))-r^2\partial _u(r^2\partial _v(r\phi _1))\nonumber \\= & {} r^4 T^2(r\phi _1)-r^4\partial _vT(r\phi _1)-2D r^3 T(r\phi _1)\nonumber \\&+2Dr^2\partial _v(r^2\phi _1)+6MD r^2\phi _1, \end{aligned}$$

(5.72)

where we used Eq. (5.19) in the last line. Notice that the last term decays like $|u|^{-1}$, whereas the other terms do not decay. We now express the limits of each of the other terms in terms of ${\mathscr {L}}$.

Computing $\lim _{u\rightarrow -\infty } |u|^{j+1}rT^j(r\phi _1)$. Observe that (5.61) implies that

$$\begin{aligned}&\lim _{u\rightarrow -\infty } |u|^2r^2\partial _vT(r\phi _1)=\frac{1}{2}\lim _{u\rightarrow -\infty } |u|^3r^2\partial _vT^2(r\phi _1) \nonumber \\&\quad =\frac{1}{6} \lim _{u\rightarrow -\infty }|u|^4r^2\partial _vT^3(r\phi _1) = \dots =\frac{1}{(j+1)!}\lim _{u\rightarrow -\infty } |u|^{2+j}r^2\partial _vT^{j+1}(r\phi _1) \end{aligned}$$

(5.73)

for all $j\le N'$. Now, use the wave equation (3.11) to write

$$\begin{aligned} -2D T^j(r\phi _1)\left( 1+\frac{M}{r}\right)&=r^2\partial _vT^{j+1}(r\phi _1)-r^2\partial _v^2 T^{j}(r\phi _1)\\&=r^2\partial _vT^{j+1}(r\phi _1)+2Dr\partial _vT^j(r\phi _1)-\partial _v(r^2\partial _vT^j(r\phi _1)). \end{aligned}$$

The last term decays faster than the others, and we conclude that

$$\begin{aligned} \begin{aligned} -2\lim _{u\rightarrow -\infty } |u|^{j+1}r T^j(r\phi _1)&=\lim _{u\rightarrow -\infty }|u|^{j+2}r^2\partial _vT^{j+1}(r\phi _1) \\&\quad +2\lim _{u\rightarrow -\infty }|u|^{j+1}r^2\partial _vT^j(r\phi _1). \end{aligned} \end{aligned}$$

(5.74)

Plugging (5.73) into the expression above, and setting $j=1, 2$, respectively, we thus obtain

$$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^2r T(r\phi _1)&=-\frac{1}{2}(2{\mathscr {L}}+2{\mathscr {L}})=-2{\mathscr {L}}, \end{aligned}$$

(5.75)

$$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^3 r T^2(r\phi _1)&=-\frac{1}{2} (6{\mathscr {L}}+2\cdot 2{\mathscr {L}})=-5{\mathscr {L}}. \end{aligned}$$

(5.76)

Computing $\lim _{u\rightarrow -\infty }r^2\partial _v(r^2\phi _1)$. In order to compute the limit of $r^2\partial _v(r^2\phi _1)$, we use equation (5.17) to write

$$\begin{aligned} r^2\partial _v(r^2\phi _1)&=r^4\int \frac{DT(r\phi _1)}{r^2}-\frac{8MD(r\phi _1)}{r^4}\mathop {}\!\mathrm {d}u\\&=r^4\int \frac{\lim _{u\rightarrow -\infty } |u|^2 r T(r\phi _1)}{r^3|u|^2}\mathop {}\!\mathrm {d}u+{\mathscr {O}}(|u|^{-1}+r|u|^{-1-\varepsilon }), \end{aligned}$$

from which we conclude, using (2.6), that

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _v(r^2\phi _1)=\frac{1}{4} \lim _{u\rightarrow -\infty }|u|^2r T(r\phi _1)=-\frac{1}{2}{\mathscr {L}}. \end{aligned}$$

(5.77)

Finally, inserting the identities (5.75), (5.76) and (5.77) back into (5.72), we find that

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1))=-5{\mathscr {L}}-{\mathscr {L}}+2\cdot 2{\mathscr {L}}-\frac{2}{2}{\mathscr {L}}=-3{\mathscr {L}}. \end{aligned}$$

(5.78)

This proves equation (5.65). Estimate (5.66) as well as the vanishing of $\lim _{u\rightarrow -\infty }r^2\partial _u(r\phi )$ follow similarly. $\square$

Combining the previous two propositions, Propositions 5.7 and 5.8, proves Theorem 5.1.

6.6 A Comment on the Stationary Solution

We have already remarked in §4.4 that we expect the vanishing of $\lim _{u\rightarrow -\infty } r^2\partial _u(r\phi _1)$ to lead to late-time asymptotics with logarithmic terms appearing at leading order if the data on $\Gamma _R$ are smoothly extended to ${\mathscr {H}}^+$ (for instance, we would have $r\phi _1|_{{\mathscr {I}}^+}(u)\sim \frac{\log u}{u^3}$ as $u\rightarrow \infty$). In other words, we expect our choice of polynomially decaying boundary data to lead to a logarithmically modified Price’s law for $\ell =1$.

Note that the limit $\lim _{u\rightarrow -\infty } r^2\partial _u(r\phi _1)$ would not vanish if we included the stationary solution, that is to say: if we added a constant to our initial data. Using the structure of equations¹⁵ (5.17), (5.19) presented in the proof of Proposition 5.2, or other methods, it is indeed not difficult to see that the stationary solution behaves like $\partial _v(r\phi _1)\sim -\partial _u(r\phi _1)\sim r^{-2}$.

On the other hand, we see that if we prescribe decaying data on $\Gamma _R$, then the solution will behave, roughly speaking, like the stationary solution multiplied by that decay. Now, since the stationary solution for higher $\ell$-modes will decay faster in $\ell$, $r\phi _\ell \sim r^{-\ell }$ (see eqns. (5.79), (5.80)), we thus expect that if we prescribe decaying boundary data for higher $\ell$-modes, then the corresponding solution will decay increasingly faster towards ${\mathscr {I}}^-$, and $(r^2\partial _u)^j(r\phi _\ell )$ will vanish to higher and higher orders. We will build on this intuition and make it precise in §8.

7 Boundary Data on a Timelike Hypersurface $\Gamma _f$

In the previous section, we showed how to construct solutions and prove sharp decay in the case of polynomially decaying boundary data on hypersurfaces of constant $r=R$. We now outline how to generalise to spherically symmetric hypersurfaces on which r is allowed to vary. In fact, not much modification will be needed.

7.1 The Setup

For the sake of notational simplicity, we restrict our attention to spherically symmetric hypersurfaces $\Gamma _{f}\subset {\mathscr {M}}$ that have timelike generators that are given by¹⁶

$$\begin{aligned} T^{(s)}=\partial _u+\frac{1}{1+|u|^{-s}}\partial _v=T-\frac{|u|^{-s}}{1+|u|^{-s}}\partial _v,&s>0. \end{aligned}$$

(6.1)

Notice that we normalised $T^{(s)}$ such that $T^{(s)}u=1$.

Since the cases $s>1$ and $s\le 1$ are quite different, we shall treat them separately. Let’s first consider the case $s>1$:

7.2 The Case Where $r|_{\Gamma _f}$ Attains a Finite Limit ($s>1$):

7.2.1 Initial/Boundary Data Assumptions and the Main Theorem

Let $\Gamma _f$ be as described above, and let $s>1$. We then prescribe smooth boundary data ${\hat{\phi }}_1$ for $\phi _1$ on $\Gamma _f$ which satisfy, for $u\le U_0< 0$ and $|U_0|$ sufficiently large:

$$\begin{aligned} \left| (T^{(s)})^n(r{\hat{\phi }}_1)\right|&\le \frac{n! C_{\mathrm {in}}^\Gamma }{r|u|^{n+1}},&n=0,1,\dots ,N+1, \end{aligned}$$

(6.2)

$$\begin{aligned} \left| (T^{(s)})^n(r{\hat{\phi }}_1-|u|T^{(s)}(r{\hat{\phi }}_1))\right|&\le \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{r|u|^{n+1+\varepsilon }},&n=0\dots ,N'+1 \end{aligned}$$

(6.3)

for some positive constants $C_{\mathrm {in}}^\Gamma$, $C_{\mathrm {in},\varepsilon }^\Gamma$, $\varepsilon \in (0,1)$ and for $N,N'>1$ positive integers. Moreover, we demand, in a limiting sense, that, for all v

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _1)(u,v)=0,\quad n=1,\dots N+1. \end{aligned}$$

(6.4)

Then we obtain, as in the previous section:

Theorem 6.1

Let $N\ge 4$ and $N'\ge 2$. Then there exists a unique solution $\phi _1$ to Eq. (3.11) in ${\mathscr {D}}_{\Gamma _f}={\mathscr {M}}\cap \{v\ge v_{\Gamma _f}(u)\}$ which restricts correctly to ${\hat{\phi }}_1$ on $\Gamma _f$, $\phi _1|_{\Gamma _f}={\hat{\phi }}_1$, and which satisfies (6.4).

Moreover, the estimates from Theorem 5.1 apply to this solution, with ${{\tilde{C}}}\ne 0$ being non-zero provided that a lower bound on data is specified and that R/2M is sufficiently large.

7.2.2 Outline of the Proof

As the proof only requires small modifications to the proof of Theorem 5.1, we will only give an outline. There are two closely related ways one can go about this: One can either work with the generators of $\Gamma _f$, i.e. replace all T’s from the proof of Theorem 5.1 with $T^{(s)}$’s, and estimate the resulting error terms (which would always exhibit faster decay than the other terms) – this was the approach of [12]. Alternatively, one can continue working with T and exploit the fact that the difference of, say, $T(r\phi _1)-T^{(s)}(r\phi _1)=\frac{|u|^{-s}}{1+|u|^{-s}}\partial _v(r\phi _1)$ decays faster than either of the terms on the left-hand side as long as $s>1$. Thus, an estimate on $T(r\phi _1)$ immediately gives control on $T^{(s)}(r\phi _1)$ and vice versa.¹⁷ We shall follow the second approach:

Proof of Theorem 6.1

First, we cut off the data as in section 5.2.2. We then introduce the set of bootstrap assumptions as in section 5.3.1 (with the only modification that the set X defined below Eq. (BS’(m)) now contains all $v\ge v_{\Gamma _f}(u)$). The proof of Proposition 5.2 remains unchanged. The proof of Proposition 5.3 requires the modification that, now, it isn’t $T^j(r\phi _1)$ which on $\Gamma _f$ is given by data, but $(T^{(s)})^j(r\phi _1)$. However, this can easily be dealt with by writing

$$\begin{aligned} T(r\phi _1)=T^{(s)}(r\phi _1)+\frac{|u|^{-s}}{1+|u|^{-s}}\partial _v(r\phi _1) \end{aligned}$$

(6.5)

(and similarly for $T^j$), and then plugging in the bootstrap assumption for $\partial _v(r\phi _1)$, using the fact that, because $s>1$, the $\partial _v(r\phi _1)$-term has more u-decay than the $T^{(s)}(r\phi _1)$-term. It can thus be absorbed into the latter for large enough $|U_0|$.

Let’s now move to the proof of Proposition 5.4. Applying the divergence theorem gives (we denote the induced volume element on $\Gamma _f$ by $r^2\mathop {}\!\mathrm {d}t_{\Gamma _f}\mathop {}\!\mathrm {d}\Omega$)

$$\begin{aligned} \begin{aligned}&\int _{{\mathscr {C}}_{v}\cap \{-k\le u'\le u\}} r^2\mathop {}\!\mathrm {d}u' \mathop {}\!\mathrm {d}\Omega \, J^T[T^j\phi _{\ell =1}]\cdot \partial _u\\&\quad \le \int _{\Gamma _f\cap \{-k\le u'\le u\}} r^2 \mathop {}\!\mathrm {d}t_{\Gamma _f}\mathop {}\!\mathrm {d}\Omega \,J^T[T^j\phi _{\ell =1}]\cdot \left( \partial _u-\partial _v+\frac{|u'|^{-s}}{1+|u'|^{-s}}\partial _v\right) , \end{aligned} \end{aligned}$$

(6.6)

which implies (cf. (5.34))

$$\begin{aligned}&\int _{-k}^u r^2(\partial _uT^j\phi _{1})^2+D|T^j\phi _{1}|^2\mathop {}\!\mathrm {d}u' \nonumber \\&\quad \lesssim \int \limits _{\Gamma _f\cap \{-k\le u'\le u\}} r^2 \left( T^{j+1}\phi _{1}\cdot (T-2\partial _v)T^j \phi _{1}\right. \nonumber \\&\quad \left. +\frac{|u'|^{-s}}{1+|u'|^{-s}}\left( (T^j \partial _v\phi _1)^2+\frac{2D}{r^2}(T^j\phi _1)^2\right) \right) \mathop {}\!\mathrm {d}u'. \end{aligned}$$

(6.7)

As before, we can now write $T^j(r\phi _1)|_{\Gamma _f}=(T^{(s)})^{j}(r\phi _1)|_{\Gamma _f}+{\mathscr {O}}(|u|^{-j-s})$ to find that

$$\begin{aligned} \int _{-k}^u r^2(\partial _uT^j\phi _{1})^2+D|T^j\phi _{1}|^2\mathop {}\!\mathrm {d}u'\lesssim & {} \int \limits _{\Gamma _f\cap \{-k\le u'\le u\}} -2r^2 \left( (T^{(s)})^{j+1}\phi _{1}\cdot 2\partial _vT \phi _{1} \right) \\&+{\mathscr {O}}(|u'|^{-2j-3-s})\mathop {}\!\mathrm {d}u'. \end{aligned}$$

From here, we arrive at the analogue of the estimate (5.38). We can thus prove the analogue of Proposition 5.4.

In a similar fashion, one can then follow the steps of sections 5.4 and 5.5 to conclude the proof of Theorem 6.1. $\square$

7.3 The Case Where $r|_{\Gamma _f}$ Diverges ($s\le 1$):

There are two main differences in the case $s\le 1$. On the one hand, if we write, as above,

$$\begin{aligned} T(r\phi _1)=\underbrace{T^{(s)}(r\phi _1)}_{\sim |u|^{-2}r^{-1}}+\frac{|u|^{-s}}{1+|u|^{-s}}\underbrace{\partial _v(r\phi _1)}_{\sim |u|^{-1}r^{-2}}, \end{aligned}$$

(6.8)

then we immediately see that, on $\Gamma _f$, where $r\sim |u|^{1-s}$ if $s\ne 1$, both terms on the RHS can be expected to have the same decay.¹⁸ This means that we have to be more careful in estimating the boundary terms in the energy estimate. On the other hand, since now $r|_{\Gamma }$ tends to infinity, it will be much more straight-forward to show the existence of the limit $\lim _{u\rightarrow -\infty }|u|^2r^2\partial _vT(r\phi _1)$.

7.3.1 Initial/Boundary Data Assumptions and the Main Theorem

Let $\Gamma _f$ be as described in section 6.1, and let $s\le 1$. We prescribe smooth boundary data ${\hat{\phi }}_1$ for $\phi _1$ on $\Gamma _f$ which satisfy, for $u\le U_0< 0$ and $|U_0|$ sufficiently large:

$$\begin{aligned} \left| (T^{(s)})^n(r^2{\hat{\phi }}_1)- \frac{n!C_{\mathrm {in}}^\Gamma }{|u|^{n+1}}\right| ={\mathscr {O}}(|u|^{-n-1-\varepsilon }),&n=0,\dots ,5 \end{aligned}$$

(6.9)

for some positive constant $C_{\mathrm {in}}^\Gamma$. Moreover, we demand, in a limiting sense, that, for all v

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _1)(u,v)=0\quad n=1,\dots 5. \end{aligned}$$

(6.10)

Then we obtain, as in the previous section:

Theorem 6.2

There exists a unique solution $\phi _1$ to Eq. (3.11) in ${\mathscr {D}}_{\Gamma _f}={\mathscr {M}}\cap \{v\ge v_{\Gamma _f}(u)\}$ which restricts correctly to ${\hat{\phi }}_1$ on $\Gamma _f$, $\phi _1|_{\Gamma _f}={\hat{\phi }}_1$, and which satisfies (6.10).

Moreover, if $U_0$ is a sufficiently large negative number, then there exists a constant $C=C(C_{\mathrm {in}}^\Gamma )$ (depending only on data) such that $\phi _1$ obeys the following bounds throughout ${\mathscr {D}}_{\Gamma _f}\cap \{u\le U_0\}$:

$$\begin{aligned}&|r^2\partial _vT^n(r\phi _1)(u,v)| \le \frac{C}{|u|^{n+1}},&n=0,\dots ,4, \end{aligned}$$

(6.11)

$$\begin{aligned}&|T^n(r\phi _1)(u,v)| \le \frac{C}{|u|^{n+1}}\max \left( r^{-1},|u|^{-1}\right) ,&n=0,\dots ,3. \end{aligned}$$

(6.12)

Finally, along any ingoing null hypersurface ${\mathscr {C}}_v$, we have

$$\begin{aligned} r^2\partial _u(r\phi _1)(u,v)&={\mathscr {O}}(|u|^{-1}), \end{aligned}$$

(6.13)

$$\begin{aligned} r^2\partial _u(r^2\partial _u(r\phi _1)))(u,v)&={\left\{ \begin{array}{ll} {\tilde{C}}+{\mathscr {O}}(|u|^{-\varepsilon '}),&{} \text {if }s<1,\\ {\tilde{C}}+{\mathscr {O}}(\log ^{-1} |u|),&{}\text {if }s=1, \end{array}\right. } \end{aligned}$$

(6.14)

where ${\tilde{C}}=3C_{\mathrm {in}}^\Gamma$ is determined explicitly by initial data, and $\varepsilon '=\min (\varepsilon , s, 1-s)$.

Remark 6.1

We remark that, in the case $s=1$, the fact that the ${\mathscr {O}}(\log ^{-1} |u|)$-term in (6.14) is non-integrable means that Theorem 4.1 cannot be applied directly. Since this is a very specific issue, we make no attempts to fix it in this presentation.

7.3.2 Outline of the Proof

Proof

As before, only a sketch of the proof will be provided.

We cut the data off as before. Let us first show (6.11) for $n=0$:

Proof of (6.11) for $\mathbf {n=0}$: We follow the proof of Proposition 5.4. We first need to acquire an estimate for $\partial _v(r\phi _1)$ on $\Gamma _f$. We assume as a bootstrap assumption that

$$\begin{aligned} |r^2\partial _v(r\phi _1)|\le \frac{C_{\mathrm {BS}}}{|u|} \end{aligned}$$

(6.15)

for a suitable constant $C_{\mathrm {BS}}$. We recall from the energy estimate (6.7):

$$\begin{aligned} \begin{aligned}&\int _{-k}^u r^2(\partial _u\phi _{1})^2+D|\phi _{1}|^2\mathop {}\!\mathrm {d}u' \\&\quad \lesssim \int \limits _{\Gamma _f\cap \{-k\le u'\le u\}} r^2 \left( T\phi _{1}\cdot (T-2\partial _v) \phi _{1}+\frac{|u'|^{-s}}{1+|u'|^{-s}}\left( (\partial _v\phi _1)^2+\frac{2D}{r^2}\phi _1^2\right) \right) \mathop {}\!\mathrm {d}u' \end{aligned} \end{aligned}$$

(6.16)

Note that the $(\partial _v\phi _1)^2$-terms in the above are potentially dangerous since they could lead to a $C_{\mathrm {BS}}^2$-term, which would make it impossible to improve the bootstrap assumption. However, upon writing again

$$\begin{aligned} T\phi _1=T^{(s)}\phi _1 +\frac{|u|^{-s}}{1+|u|^{-s}}\partial _v\phi _1, \end{aligned}$$

(6.17)

we find that that the $(\partial _v\phi _1)^2$-terms, in fact, appear with a benign sign:

$$\begin{aligned}&\int _{\Gamma _f} r^2 \left( T\phi _{1}\cdot (T-2\partial _v) \phi _{1}+\frac{|u'|^{-s}}{1+|u'|^{-s}}\left( (\partial _v\phi _1)^2+\frac{2D}{r^2}\phi _1^2\right) \right) \mathop {}\!\mathrm {d}u'\\&\quad =\int _{\Gamma _f} r^2 \left( (T\phi _1)^2-\frac{|u'|^{-s}}{1+|u'|^{-s}}(\partial _v\phi _1)^2-2\partial _v\phi _1T^{(s)}\phi _1+\frac{|u'|^{-s}}{1+|u'|^{-s}}\frac{2D}{r^2}\phi _1^2\right) \mathop {}\!\mathrm {d}u'\\&\quad \le \int _{\Gamma _f} r^2 \left( (T^{(s)}\phi _1)^2+2|\partial _v\phi _1||T^{(s)}\phi _1|\frac{1+2|u'|^{-s}}{1+|u'|^{-s}}+\frac{|u'|^{-s}}{1+|u'|^{-s}}\frac{2D}{r^2}\phi _1^2\right) \\&+r^2 \frac{|u'|^{-2s}}{(1+|u'|^{-s})^2} (\partial _v\phi _1)^2\mathop {}\!\mathrm {d}u'\\&\quad \lesssim \int _{\Gamma _f} r^2 \left( \frac{(C_{\mathrm {in}}^\Gamma )^2}{r^4|u'|^4}+\frac{(C_{\mathrm {in}}^\Gamma )^2}{r^6|u'|^{2+2s}} +\frac{C_{\mathrm {in}}^\Gamma (C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}})}{r^5|u'|^2}\left( \frac{1}{|u'|}+\frac{1}{r|u'|^{s}}\right) +\frac{(C_{\mathrm {in}}^\Gamma )^2}{r^6|u'|^{2+s}}\right) \\&\qquad +r^2 \frac{(C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}})^2}{r^6|u'|^{2+2s}} \mathop {}\!\mathrm {d}u' , \end{aligned}$$

where we used the boundary data assumption and the bootstrap assumption in the last estimate. Using now the fact that $|u|^{1-s}\lesssim r$ if $s\ne 1$ (or $\log |u|\lesssim r$ if $s=1$), as well as the fundamental theorem of calculus and the Cauchy–Schwarz inequality, combined with the energy estimate (6.16), we obtain that

$$\begin{aligned} r\phi _1^2\lesssim \frac{C_{\mathrm {in}}^\Gamma (C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}})}{r^3|_{\Gamma _f}|u|^2} \end{aligned}$$

(6.18)

Importantly, $C_{\mathrm {BS}}$ does not appear quadratically in the above estimate. Plugging this bound into the wave equation (3.11) and integrating in u, we obtain that

$$\begin{aligned} \left| r^2\partial _v(r\phi _1)|_{\Gamma _f}\right| \lesssim \frac{\sqrt{C_{\mathrm {BS}}^2+C_{\mathrm {BS}}C_{\mathrm {in}}^\Gamma }}{|u|^2}. \end{aligned}$$

(6.19)

Having obtained a bound for the boundary term, we can now, as in the proof of Proposition 5.4, use the approximate conservation law (3.13) to close the bootstrap argument for $\partial _v(r\phi _1)$. Indeed, we can obtain a bound for $\Phi$ (similarly to how we obtained (5.38)) and then use the fact that, by integrating the bootstrap assumption from $\Gamma _f$, we have

$$\begin{aligned} |r\phi _1|\le \frac{C_{\mathrm {in}}^\Gamma +C_{\mathrm {BS}}}{r|_{\Gamma _f}|u|}. \end{aligned}$$

(6.20)

In view of $\log |u|\lesssim r|_{\Gamma _f}$, this decays faster than $r^2\partial _v(r\phi _1)$. Therefore, the bound for $\Phi$ immediately translates into a bound for $r^2\partial _v(r\phi _1)$. This closes the bootstrap argument.

Proof of (6.11) for $\mathbf {n>0}$: Having proved (6.11) for $n=0$, we now outline the proof for $n>0$. In fact, the only thing that changes is that, in the energy equality (6.16), we now need to express $T^j\phi _1$ in terms of $(T^{(s)})^j\phi _1$ for $j>1$, which leads to more "error" terms. For instance, we have

$$\begin{aligned} T^2\phi _1= & {} (T^{(s)})^2\phi _1+2\frac{|u|^{-s}}{1+|u|^{-s}}\partial _vT\phi _1+T\left( \frac{|u|^{-s}}{1+|u|^{-s}}\right) \partial _v\phi _1\nonumber \\&-\frac{|u|^{-2s}}{(1+|u|^{-s})^2}\partial _v^2\phi _1. \end{aligned}$$

(6.21)

We have already obtained estimates for the last two terms. Moreover, we can estimate the first term above from the boundary data assumptions, and the second one via a bootstrap assumption on $\partial _vT(r\phi _1)$. Plugging these estimates into (6.7) for $j=2$ then improves the bootstrap assumption.

We leave the cases $j>2$ to the reader. (Notice that when e.g. expressing $T^4\phi _1$ in terms of $(T^{(s)})^4 \phi _1$, there will also be, for instance, a term containing $\partial _v^4\phi _1$. We can estimate this by simply commuting the wave equation twice with $\partial _v$ and appealing to the proof of (6.11) for $n=0$. The other terms can be dealt with similarly.)

Proof of (6.12): We can obtain the estimates (6.12) for $n\le 3$ by using the wave equation as in (5.29) and the already obtained bounds (6.11).

Proof of (6.13) The proof of (6.13) is straight-forward. We simply write:

$$\begin{aligned} r^2\partial _u(r\phi _1)=r^2T(r\phi _1)-r^2\partial _v(r\phi _1). \end{aligned}$$

(6.22)

Proof of (6.14): Finally, we prove (6.14). As in the proof of Proposition 5.8, we will first compute the limit $\lim _{u\rightarrow -\infty }|u|^2r^2\partial _vT(r\phi _1)$.

In view of the approximate conservation law (3.13) and the fact that $r|_{\Gamma _f}$ tends to infinity, we have that

$$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^2r^2\partial _vT(r\phi _1)(u,v)=\lim _{u\rightarrow -\infty }|u|^2r^2\partial _vT(r\phi _1)(u,v_{\Gamma _f}(u)). \end{aligned}$$

(6.23)

We estimate $\partial _vT(r\phi _1)|_{\Gamma _f}$ as follows. Integrating the T-commuted (5.17), we obtain that

$$\begin{aligned} r^2\partial _vT(r\phi _1)+r^2T \phi _1={\mathscr {O}}\left( \frac{r}{|u|^3}+\frac{1}{r|u|^2}\right) , \end{aligned}$$

(6.24)

from which we read off that

$$\begin{aligned} |u|^2r^2\partial _vT(r\phi _1)(u,v)=-C_{\mathrm {in}}^\Gamma +{\left\{ \begin{array}{ll} {\mathscr {O}}\left( \frac{1}{\log |u|}\right) ,&{}s=1,\\ {\mathscr {O}}\left( \frac{1}{|u|^{s}}+\frac{1}{|u|^{1-s}}+\frac{1}{|u|^{\varepsilon }}\right) ,&{} s<1, \end{array}\right. } \end{aligned}$$

(6.25)

and, in particular, that

$$\begin{aligned} {\mathscr {L}}:=\lim _{u\rightarrow -\infty }|u|^2r^2\partial _vT(r\phi _1)(u,v)=-C_{\mathrm {in}}^\Gamma . \end{aligned}$$

(6.26)

Here, we used that

$$\begin{aligned} r^2T\phi _1=T(r^2\phi _1)=T^{(s)}(r^2\phi _1)+\frac{|u|^{-s}}{1+|u|^{-s}}\partial _v(r^2\phi _1) \end{aligned}$$

and the fact that, in view of (5.17), the second term above decays faster than the first.

Similarly, we find that

$$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^{j+1}r^2\partial _vT^j(r\phi _1)(u,v)=-j!C_{\mathrm {in}}^\Gamma \end{aligned}$$

(6.27)

for $j\le 3$. We can now compute, exactly as in the proof of Proposition 5.8, the expressions (5.75), (5.76) and (5.77), from which it follows, using the identity (5.72), that

$$\begin{aligned} \lim _{u\rightarrow -\infty }r^2\partial _u(r^2\partial _u(r\phi _1))(u,v)=-3{\mathscr {L}}=3C_{\mathrm {in}}^\Gamma . \end{aligned}$$

(6.28)

This concludes the proof. $\square$

8 Part II: Generalising to all $\ell \ge 0$.

Having understood the case $\ell =1$ in detail in the previous sections §4–§6, we now want to analyse the general case. As explained in §1.4, this second part of the paper can be understood mostly independently of part I.

As a zeroth step, we need to establish higher $\ell$-analogues of the approximate conservation laws (3.13), (3.15). This is achieved in §7. We then treat the case of timelike boundary data in §8, restricting the presentation however to cases of hypersurfaces of constant area radius. Then, we treat the case of characteristic initial data in §9 and §10.

The sections §8 and §9 are similar in spirit to §5 and §4, respectively. (The reasons for the reversed order of the sections are solely of presentational, not of mathematical nature.) On the other hand, §10 follows a different mathematical structure than §4: While the methods of §4 and §9 can only treat data that decay like $r\phi _\ell \lesssim |u|^{-\ell }$, the approach of §10 allows us to also treat slowly decaying (and even growing) initial data.

9 The Higher-Order Newman–Penrose Constants

In this section, we derive higher-order conservation laws and define the Newman–Penrose constants associated with them.

9.1 Generalising the Approximate Conservation Law (3.13)

In order to generalise the approximate conservation law in u (3.13), we first require a general formula for commutations of the wave equation with $[r^2\partial _v]^N$:

Proposition 7.1

Let $\phi$ be a smooth solution to $\Box _g\phi =0$, and let $N\ge 0$. Then $\phi$ satisfies

(7.1)

where the constants $a_j^N, b_j^N$ and $c_j^N$ are given explicitly by

$$\begin{aligned} a_j^N&=(2^j-1)\left( {\begin{array}{c}N\\ j\end{array}}\right) +(2^{j+2}-2)\left( {\begin{array}{c}N+1\\ j+2\end{array}}\right) , \end{aligned}$$

(7.2)

$$\begin{aligned} b_j^N&=\left( {\begin{array}{c}N\\ j\end{array}}\right) , \end{aligned}$$

(7.3)

$$\begin{aligned} c_j^N&=2^j\left( {\begin{array}{c}N\\ j\end{array}}\right) +2^{j+2}\left( {\begin{array}{c}N+1\\ j+2\end{array}}\right) , \end{aligned}$$

(7.4)

and we use the convention that $\left( {\begin{array}{c}N\\ j\end{array}}\right) =0$ if $j>N$.

Proof

A proof is given in the appendix A.2. $\square$

Notice that, in particular, $a_0^N=N(N+1)$ and $b_0^N=1$. Hence, if we restrict to solutions supported on the $\ell =L$-angular frequencies, and consider (7.1) for $N=L$, there will be a cancellation in the highest-order derivatives. One can then iteratively subtract (7.1) for $N<L$, multiplied with a suitable constant, to obtain an approximate conservation law. This is done in

Corollary 2

Let $\phi =\sum _{|m|\le L}\phi _{Lm}\cdot Y_{Lm}$ be a smooth solution to $\Box _g\phi =0$ supported on the angular frequencies $\ell =L\ge 0$. In what follows, we shall suppress the m-index. Let $N\ge 0$, and define, for $j\le N$,

$$\begin{aligned} {\tilde{a}}_j^{N,L}:=a_j^N-b_j^N\cdot L(L+1), \end{aligned}$$

(7.5)

and let $\{x_i^{(L)}\}_{0\le i\le L}$ be a set of constants with $x_0^{(L)}=1$. Then $\phi$ satisfies

$$\begin{aligned}&\partial _u\partial _v\left( [r^2\partial _v]^L(r\phi _L)+\sum _{i=1}^L(2M)^ix_i^{(L)}[r^2\partial _v]^{L-i}(r\phi _L)\right) \nonumber \\&\quad =-\frac{2LD}{r}\partial _v[r^2\partial _v]^L(r\phi _L)-\sum _{i=1}^L(2M)^ix_i^{(L)}\frac{2(L-i)D}{r}\partial _v[r^2\partial _v]^{L-i}(r\phi _L)\nonumber \\&\quad +\sum _{j=0}^L\frac{D}{r^2}(2M)^j[r^2\partial _v]^{L-j}(r\phi _L)\sum _{i=0}^j\left( x_i^{(L)}{\tilde{a}}_{j-i}^{L-i,L}-x_i^{(L)} c_{j-i}^{L-i}\cdot \frac{2M}{r} \right) . \end{aligned}$$

(7.6)

Proof

This is a straight-forward computation.$\square$

Definition 7.1

(The generalised higher-order future Newman–Penrose constant) Let $\phi$ be as in Corollary 2, and define the constants $x_i^{(L)}$ for $1\le i\le L$ as follows:

$$\begin{aligned} x_i^{(L)}:=-\dfrac{1}{{\tilde{a}}_0^{L-i,L}}\sum _{k=0}^{i-1}{\tilde{a}}_{i-k}^{L-k,L}x^{(L)}_k. \end{aligned}$$

(7.7)

This is well-defined since ${\tilde{a}}_0^{L-i,L}\ne 0$ for $i>0$ and since $x^{(L)}_0=1$. We further denote

$$\begin{aligned} \Phi _{L}:=[r^2\partial _v]^L(r\phi _{L})+\sum _{i=1}^L(2M)^ix_i^{(L)}[r^2\partial _v]^{L-i}\phi _{L}, \end{aligned}$$

(7.8)

and define, for any smooth function $f(r)=o(r^3)$, the generalised higher-order Newman–Penrose constant according to

$$\begin{aligned} I_{\ell =L}^{\mathrm {future},f}[\phi ](v):=\lim _{v\rightarrow \infty }f\partial _v\Phi _L(u,v). \end{aligned}$$

(7.9)

Corollary 3

The quantity $\Phi _{L}$ defined above satisfies the following approximate conservation law:

$$\begin{aligned} \partial _u(r^{-2L}\partial _v\Phi _L)= & {} \sum _{j=0}^L\frac{D}{r^{3+2L}}(2M)^{j+1}[r^2\partial _v]^{L-j}(r\phi _{L})\nonumber \\&\left( 2(j+1)x^{(L)}_{j+1}-\sum _{i=0}^j x_i^{(L)} c_{j-i}^{L-i}\right) . \end{aligned}$$

(7.10)

Here, we used the notation that $x_i^{(L)}=0$ for $i\ge L$.

In particular, under suitable assumptions on $\phi$, the generalised higher-order N–P constant defined above is conserved along ${\mathscr {I}}^+$:

$$\begin{aligned} I_{\ell =L}^{\mathrm {future},f}[\phi ](v) \equiv I_{\ell =L}^{\mathrm {future},f}[\phi ] . \end{aligned}$$

(7.11)

Remark 7.1

It is helpful to keep in mind that the quantity $\partial _v\Phi _L$ can always be written as

$$\begin{aligned} \partial _v\Phi _L=\partial _v(r^2p_1(r^2p_2\partial _v(\dots r^2p_L\partial _v(r\phi _L)\dots ))), \end{aligned}$$

(7.12)

where the $p_i$ are polynomials in 1/r. Intuitively, this indicates that one should typically be able to recover an estimate for $r\phi _L$ from $\Phi _L$ by simply integrating L times. However, we will never need to make use of the form (7.12) in this paper.

9.2 Generalising the Approximate Conservation Law (3.15)

We follow a similar procedure to derive approximate conservation laws in the $\partial _v$-direction. We have

Proposition 7.2

Let $\phi$ be a smooth solution to $\Box _g\phi =0$, and let $N\ge 0$. Then $\phi$ satisfies

(7.13)

where ${\underline{a}}_j^N=(-1)^ja_j^N$, ${\underline{b}}_j^N=(-1)^jb_j^N$ and ${\underline{c}}_j^N=(-1)^jc_j^N$.

Proof

The proof follows along the same steps as the one of Proposition 7.1. See the appendix A.2 for details. $\square$

Definition 7.2

(The generalised higher-order past Newman–Penrose constant) Let $\phi$ be as in Corollary 2, let $\tilde{{\underline{a}}}_j^{N,L}:=(-1)^j{\tilde{a}}_j^{N,L}$, let $x^{(L)}_0=1$, and define, for $1\le i\le L$,

$$\begin{aligned} {\underline{x}}_i^{(L)}=-\dfrac{1}{\tilde{{\underline{a}}}_0^{L-i,L}}\sum _{k=0}^{i-1}\tilde{{\underline{a}}}_{i-k}^{L-k,L}{\underline{x}}^{(L)}_k. \end{aligned}$$

(7.14)

We then denote

$$\begin{aligned} {\underline{\Phi }}_{L}:=[r^2\partial _u]^L(r\phi _{L})+\sum _{i=1}^L(2M)^i{\underline{x}}_i^{(L)}[r^2\partial _u]^{L-i}(r\phi _{L}), \end{aligned}$$

(7.15)

and, moreover, define for any smooth function $f(r)=o(r^3)$ the generalised higher-order Newman–Penrose constant according to

$$\begin{aligned} I_{\ell =L}^{\mathrm {past},f}[\phi ](u):=\lim _{u\rightarrow -\infty }f\partial _u{\underline{\Phi }}_L(u,v). \end{aligned}$$

(7.16)

Corollary 4

The quantity ${\underline{\Phi }}_{L}$ defined above satisfies the following approximate conservation law:

$$\begin{aligned} \partial _v(r^{-2L}\partial _u{\underline{\Phi }}_L) =\sum _{j=0}^L\frac{D}{r^{3+2L}}(2M)^{j+1}[r^2\partial _u]^{L-j}(r\phi _{L})\left( -2(j+1){\underline{x}}^{(L)}_{j+1}-\sum _{i=0}^j {\underline{x}}_i^{(L)} {\underline{c}}_{j-i}^{L-i}\right) . \nonumber \\ \end{aligned}$$

(7.17)

Here, we used the notation that ${\underline{x}}_i^{(L)}=0$ for $i\ge L$.

In particular, under suitable assumptions on ${\underline{\Phi }}$, the generalised higher-order N–P constant defined above is conserved along ${\mathscr {I}}^-$:

$$\begin{aligned} I_{\ell =L}^{\mathrm {past},f}[\phi ](u) \equiv I_{\ell =L}^{\mathrm {past},f}[\phi ] . \end{aligned}$$

(7.18)

10 Boundary Data on a Timelike Hypersurface $\Gamma _R$

Equipped with the approximate conservation laws (7.10), (7.17), we now generalise the results of §5. More precisely, we construct higher $\ell$-mode solutions to (1.1) (and derive estimates for them) that arise from polynomially decaying boundary data on a timelike hypersurface $\Gamma _R$ of constant area radius $r=R$ and the no incoming radiation condition. In particular, the present section contains the proof of Theorem 1.3. The generalisation to boundary data on hypersurfaces on which r is allowed to vary then proceeds as in §6 and is left to the reader.

Throughout the rest of this section, we shall assume that $R>2M$ is a constant and that $\phi$ is a solution to (1.1) supported on a single angular frequency (L, m), with $|m|\le L$ and $L\ge 0$. In the usual abuse of notation of §3.3, we omit the m-index, that is, we write $\phi =\phi _{Lm}Y_{Lm}=\phi _L Y_{Lm}$.

10.1 Initial/Boundary Data Assumptions

We prescribe smooth boundary data ${\hat{\phi }}_L$ on $\Gamma _R={\mathscr {M}}_M\cap \{v=v_R(u)\}$ that satisfy, for $u\le U_0<0$ and $|U_0|$ sufficiently large, the following upper bounds:

$$\begin{aligned} \left| T^n(r{\hat{\phi }}_L)\right|&\le \frac{n! C_{\mathrm {in}}^\Gamma }{R^L|u|^{n+1}},&n=0,1,\dots ,N+1, \end{aligned}$$

(8.1)

$$\begin{aligned} \left| T^n(r{\hat{\phi }}_L-|u|T(r{\hat{\phi }}_L))\right|&\le \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{R^L|u|^{n+1+\varepsilon }},&n=0,\dots ,N'+1 \end{aligned}$$

(8.2)

for some positive constants $C_{\mathrm {in}}^\Gamma$, $C_{\mathrm {in},\varepsilon }^\Gamma$, $\varepsilon \in (0,1)$ and $N,N'\ge 0$ integers, and which also satisfy the lower bound

$$\begin{aligned} \left| T(r{\hat{\phi }}_L)\right| \ge \frac{C_{\mathrm {in}}^\Gamma }{2R^L|u|^{n+1}}>0. \end{aligned}$$

(8.3)

Moreover, we demand, in a limiting sense, that, for all v,

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _L)(u,v)=0,\quad n=1,\dots N+1. \end{aligned}$$

(8.4)

This latter condition is to be thought of as the no incoming radiation condition.

10.2 The Main Theorem (Theorem 8.1)

The main result of this section is

Theorem 8.1

Let $R>2M$. Then there exists a unique solution $\phi _L\cdot Y_{Lm}$ to Eq. (1.1) in ${\mathscr {D}}_{\Gamma _R}={\mathscr {M}}\cap \{v\ge v_R(u)\}$ that restricts correctly to ${\hat{\phi }}_L \cdot Y_{Lm}$ on $\Gamma _R$, $\phi _L|_{\Gamma _R}={\hat{\phi }}_L$, and that satisfies (8.4).

Moreover, if $U_0$ is a sufficiently large negative number, then there exists a constant $C=C(2M/R,C_{\mathrm {in}}^\Gamma ,L)$, depending only on data, such that $\phi _L$ obeys the following bounds throughout ${\mathscr {D}}_{\Gamma _R}\cap \{u\le U_0\}$:

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi _L)\right| \le \frac{C}{|u|^{i+1}}\min (r,|u|)^{-\min {({\tilde{j}},N-i)}} \end{aligned}$$

(8.5)

for all $j=-1,\dots , L$ and for all $i=0,\dots , N$, and ${\tilde{j}}:=\max (j,0)$.

Finally, if $N-2\ge N'\ge L+1$, then we have along any ingoing null hypersurface ${\mathscr {C}}_v$:

$$\begin{aligned}&[r^2\partial _u]^{L-j}(r\phi _L)(u,v)={\mathscr {O}}(r^{-1-j}),\qquad \quad j=0,\dots , L, \end{aligned}$$

(8.6)

$$\begin{aligned}&[r^2\partial _u]^{L+1}(r\phi _L)(u,v)={\tilde{C}}+{\mathscr {O}}(r^{-1}+|u|^{-\varepsilon }) \end{aligned}$$

(8.7)

for some constant ${\tilde{C}}$ which can be shown to be non-vanishing if R/2M is sufficiently large.

Remark 8.1

A similar result holds true for more general timelike hypersurfaces $\Gamma _f$ (on which, in particular, r is allowed to tend to infinity) as discussed in §6. We leave the proof to the interested reader.

10.3 Overview of the proof

We shall first give an overview over the proof of Theorem 8.1.

• I In a first step, we construct a sequence of smooth compactly supported data ${\hat{\phi }}^{(k)}_L$ as in §5.2.2, which lead to solutions $\phi _L^{(k)}$ in the sense of Prop. 3.2. The purpose of this is that we will then be able to use the method of continuity (i.e. bootstrap arguments) on these finite solutions $\phi _L^{(k)}$ .

• II We then assume (in the form of a bootstrap assumption) that the estimate $\left| r^2\partial _v(r\phi ^{(k)}_L)|_{\Gamma _R}\right| \le \frac{C_{\mathrm {BS}}}{R^{L-1}|u|}$ holds on $\Gamma _R$. An application of an energy estimate will imply that $\left| r^2\partial _v(r\phi ^{(k)}_L)\right| \le C'(\mathrm {data})\cdot \frac{\sqrt{ C_{\mathrm {BS}}}}{R^L|u|}$ and, thus, improve this assumption. From this, we then inductively integrate equation (7.1) to obtain estimates for the boundary terms $\left| [r^2\partial _v]^{L-j}(r\phi ^{(k)}_L)|_{\Gamma _R}\right|$, $j=0,\dots L$. The same estimates hold upon commuting with $T^i$.

• III In a third step, we assume decay on $[r^2\partial _v]^{L}(r\phi ^{(k)}_L)$ and integrate the approximate conservation law (7.10) in u and in v (the integration in v from $\Gamma _R$ outwards is why we need the estimates on the boundary terms from step II) to improve this decay, exploiting 2M/R as a small parameter. (We recall from §5.3.5 that any smallness assumptions on 2M/R can be recovered by replacing the bootstrap argument with a Grönwall argument.) Integrating this estimate for $[r^2\partial _v]^{L}(r\phi ^{(k)}_L)$ then j times from $\Gamma _R$ and also commuting with T establishes the following estimates:

  $$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi ^{(k)}_L)\right| \le \frac{C}{|u|^{i+1}}R^{-\max {(j,0)}} \end{aligned}$$

  (8.8)

  for $i=0,\dots , N$, $j=-1,0,\dots , L$, and for $C\ne C(k)$ a constant.

• IV In a fourth step, we adapt the methods of steps II and III as in §5.4 to obtain estimates on the boundary terms $[r^2\partial _v]^{L-j}T^i (r\phi ^{(k)}_L-|u| T(r\phi ^{(k)}_L))|_{\Gamma _R}$ and, from these, establish the auxiliary estimates (modulo corrections arising from the cut-off terms, cf. (8.31)):

  $$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i\left( r\phi ^{(k)}_L-|u|T(r\phi ^{(k)}_L)\right) \right| \le \frac{C}{|u|^{i+1+\varepsilon }}R^{-\max (j,0)} \end{aligned}$$

  (8.9)

  for $i=0,\dots ,N'$ and $j=-1,0, \dots , L$.

• V In a fifth step, we show, as in §5.5, that the solutions $\phi _L^{(k)}$ tend uniformly to a limiting solution $\phi _L$, which still satisfies the estimates (8.8) and (8.9) above.

• VI In a sixth step, we use the estimate (8.8), together with the identities

  $$\begin{aligned}&\sum _{j=0}^N\left( a_j^N-L(L+1)b_j^N-c_j^N\frac{2M}{r}\right) [r^2\partial _v]^{N-j}T^i(r\phi _L) \nonumber \\&\quad =[r^2\partial _v]^{N+1} T^{i+1}(r\phi _L) +\frac{2D(N+1)}{r}[r^2\partial _v]^{N+1}T^i(r\phi _L)\nonumber \\&\quad -\frac{1}{r^2}[r^2\partial _v]^{N+2}T^i(r\phi _L) \end{aligned}$$

  (8.10)

  implied by (7.1), to obtain the improved estimates (8.5), i.e. to convert the R-weights of (8.8) into r-weights, using an "upwards-downwards induction".

• VII In a seventh step, we use equation (8.10) to obtain a lower bound for $[r^2\partial _v]^{L}T(r\phi _L)|_{\Gamma }$ on $\Gamma$, provided that the lower bound (8.3) for $T(r\phi _L)|_{\Gamma }$ on data is specified. Using the estimate (8.8) with $j=-1$, we can then obtain a global lower bound for $[r^2\partial _v]^{L}T(r\phi _L)$, provided that R/2M is sufficiently large. Furthermore, and independently of this lower bound, we can use the auxiliary estimate (8.9) to show that the following limit exists and is independent of v:

  $$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^2[r^2\partial _v]^LT(r\phi _L)(u,v)=:{\mathscr {L}}. \end{aligned}$$

  (8.11)

  By the lower bounds obtained before, this limit is non-vanishing.

• VIII Finally, we prove (8.6) and (8.7) by writing

  $$\begin{aligned}{}[r^2\partial _u]^{L+1-j}(r\phi _L)=[r^2T-r^2\partial _v]^{L+1-j}(r\phi _L), \end{aligned}$$

  (8.12)

  and by expressing each term in the expansion of the above expression in terms of ${\mathscr {L}}$, using the relations (8.9) and (8.10).

10.4 Proof of Theorem 8.1

We now prove Theorem 8.1, following the structure outlined above. The proof will be self-contained, with the exceptions of steps IV and V, for which we will refer to §5 for details.

Proof

Throughout this proof, C shall denote a constant that depends only on $C_{\mathrm {in}}^\Gamma ,C_{\mathrm {in},\varepsilon }^\Gamma$, $2M/R, M, N, N', L$ (and, in particular, not on k) and can be bounded independently of R for sufficiently large R. Moreover, C is allowed to vary from line to line. We will also assume $U_0$ to be sufficiently large, where this largeness, again, only depends on data, i.e. on C.

10.4.1 Step I: Cutting off the data

We let $(\chi _k(u))_{k\in {\mathbb {N}}}$ be a sequence of smooth cut-off functions such that

$$\begin{aligned} \chi _k(u)={\left\{ \begin{array}{ll}1,&{}u\ge -k+1, \\ 0,&{}u\le -k, \end{array}\right. } \end{aligned}$$

and cut-off the the highest-order derivative: $\chi _kT^{N+1}{\hat{\phi }}_L$. We then define, as in §5.2.2, ${\hat{\phi }}_L^{(k)}$ to be the $N+1$-th T-integral of $\chi _kT^{N+1}{\hat{\phi }}_L$ from $-\infty$. Then ${\hat{\phi }}_L^{(k)}$ satisfies the following bounds:

$$\begin{aligned} \left| T^n(r{\hat{\phi }}^{(k)}_L)\right|&\le \frac{n! C_{\mathrm {in}}^\Gamma }{R^L|u|^{n+1}},&n=0,1,\dots ,N+1, \end{aligned}$$

(8.13)

$$\begin{aligned} \left| T^n\left( r{\hat{\phi }}^{(j)}_L-|u|T(r{\hat{\phi }}^{(k)}_L)\right) \right|&\le \frac{C_{\mathrm {in},\varepsilon }^\Gamma }{R^L|u|^{n+1+\varepsilon }}+C\theta _k\cdot \frac{C_{\mathrm {in}}^\Gamma }{Rk^{n+1}},&n=0,1,\dots ,N'+1, \end{aligned}$$

(8.14)

where $\theta _k$ equals 1 if $u\ge -k$, and 0 otherwise.

The boundary data ${\hat{\phi }}^{(k)}_L$, combined with the no incoming radiation condition (8.4), lead to unique solutions $\phi _L^{(k)}$, which vanish identically for $u\le -k$, and which solve the finite initial/boundary value problem (in the sense of Prop. 3.2) where ${\hat{\phi }}^{(k)}_L$ is specified on $\Gamma _R$, and where $r\phi _L^{(k)}=0$ on $\{u=-k\}$.

In steps II–IV below, we will show uniform-in-k estimates on these solutions $\phi _L^{(k)}$, temporarily dropping the superscript (k) and denoting them simply by $\phi _L$. We will re-instate this superscript in step V, where we will show that the solutions $\phi _L^{(k)}$ tend to a limiting solution as $k\rightarrow -\infty$.

10.4.2 Step II: Estimates on the boundary terms

Claim 1

Let $U_0$ be a sufficiently large negative number. Then there exist constants $B^{(i)}$ such that

$$\begin{aligned} \left| r^2\partial _vT^i(r\phi _L)\right| (u,v_{\Gamma _R}(u))\le \frac{B^{(i)}}{R^{L-1}|u|^{i+1}} \end{aligned}$$

(8.15)

for $i=0,\dots , N$ and for all $u\le U_0$.

Proof

We fix $i\le N$, and assume (8.15) with $B^{(i)}$ sufficiently large as a bootstrap assumption.

Recall the definition of the energy current (3.2). We apply the divergence theorem in the form (2.11) to the identity (we abuse notation and omit the $Y_{Lm}$)

$$\begin{aligned} \mathrm {div}J^T[T^i\phi _L]=0 \end{aligned}$$

(8.16)

as in equation (5.34) in order to obtain

$$\begin{aligned} \begin{aligned}&\int _{-k}^u r^2 \left( \partial _uT^i \phi _1\right) ^2(u',v)\mathop {}\!\mathrm {d}u'\\&\quad \le \int _{\Gamma _R\cap \{-k\le u' \le u\}}2r^2 \left( \left| T^{i+1}\phi _L\right| ^2+2\left| T^{i+1}\phi _L\cdot \partial _vT^i\phi _L\right| \right) (u', v_{\Gamma _R}(u'))\mathop {}\!\mathrm {d}u'. \end{aligned}\end{aligned}$$

(8.17)

We estimate, on $\Gamma _R$:

$$\begin{aligned}\left| r\partial _vT^i\phi _L\right| =\left| \partial _vT^i(r\phi _L)-DT^i\phi _L\right| \le \frac{B^{(i)}}{R^{2+L-1}|u|^{i+1}}+\frac{(i+1)!C_{\mathrm {in}}^\Gamma }{R^{L+1}|u|^{i+1}}.\end{aligned}$$

(In the above, we used the bootstrap assumption (8.15) and the boundary data assumption (8.13).) We thus obtain that

$$\begin{aligned}&\int _{-k}^u r^2 \left( \partial _uT^j \phi _1\right) ^2(u',v)\mathop {}\!\mathrm {d}u'\\&\quad \le \int _{\Gamma _R\cap \{-k\le u' \le u\}}2R^2 \cdot \left( \left( \frac{(i+2)!C_{\mathrm {in}}^\Gamma }{R^{L+1}|u|^{i+2}}\right) ^2\right. +\left. 2\frac{(i+2)!C_{\mathrm {in}}^\Gamma \left( B^{(i)}+(i+1)!C_{\mathrm {in}}^\Gamma \right) }{R^{2L+1}|u|^{2i+3}}\right) \\&\qquad \quad (u', v_{\Gamma _R}(u'))\mathop {}\!\mathrm {d}u'\\&\quad \le C\cdot \frac{B^{(i)}}{R^{2L+1}|u|^{2i+2}} \end{aligned}$$

for some constant C as described in the beginning of the proof. We now apply the fundamental theorem of calculus and the Cauchy–Schwarz inequality to obtain

$$\begin{aligned} T^i\phi _L(u,v)=\int _{-k}^u \partial _uT^i\phi _{L}(u',v)\mathop {}\!\mathrm {d}u'\le \frac{1}{\sqrt{Dr}}\cdot \frac{\sqrt{CB^{(i)}}}{R^{L+\frac{1}{2}}|u|^{i+1}}. \end{aligned}$$

Inserting this bound into (7.1) with $N=0$, and integrating the latter from $u=-k$, we find that

$$\begin{aligned}&\left| \partial _vT^i(r\phi _L)\right| (u,v_{\Gamma _R}(u)) \nonumber \\&\quad \le \int _{-k}^u\frac{\sqrt{CB^{(i)}}}{R^{L+\frac{1}{2}}|u'|^{i+1}}\left( L(L+1)+\frac{2M}{r}\right) \frac{D}{r^{\frac{3}{2}}}\mathop {}\!\mathrm {d}u'\le \frac{C\sqrt{B^{(i)}}}{R^{L+1}|u|^{i+1}}. \end{aligned}$$

(8.18)

This improves the bootstrap assumption (8.15), provided that $B^{(i)}$ is chosen sufficiently large. $\square$

Claim 2

Let $U_0$ be a sufficiently large negative number. Then there exists a constant C such that

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}(r\phi _L)\right| (u,v_{\Gamma _R}(u))\le \frac{C}{R^{j}|u|^{i+1}} \end{aligned}$$

(8.19)

for $i=0, \dots , N$ and $j=0,\dots , L-1$, and for all $u\le U_0$.

Proof

In the proof of the previous claim, we have in fact shown that (cf. (8.18))

$$\begin{aligned} \left| \partial _vT^i(r\phi _L)\right| (u,v)\le \frac{C}{R^{L+\frac{1}{2}}\sqrt{r}|u|^{i+1}} \end{aligned}$$

(8.20)

for all $v\ge v_{\Gamma _R}(u)$. Let us assume inductively that

$$\begin{aligned} \left| \partial _v[r^2\partial _v]^n T^i(r\phi _L)\right| (u,v)\le \frac{Cr^{-\frac{1}{2}+n}}{R^{L+\frac{1}{2}}|u|^{i+1}} \end{aligned}$$

(8.21)

for some fixed $n<\max (L-2,1)$ and for all $u\le U_0, v\ge v_{\Gamma _R}(u)$, noting that we have already established the case $n=0$. We then insert this inductive assumption into (7.1) with $N=n+1$ and integrate the latter in u to find

$$\begin{aligned}&\left| r^{-2(n+1)}\partial _v[r^2\partial _v]^{n+1} T^i(r\phi _L)\right| (u,v) \\&\quad \le \int _{-k}^u \sum _{j=0}^{n+1}\frac{D}{r^{2(n+1)+2}}\cdot \frac{Cr^{\frac{3}{2}+n}}{R^{L+\frac{1}{2}}|u'|^{i+1}}\mathop {}\!\mathrm {d}u'\le \frac{1}{r^{2(n+1)}}\cdot \frac{C r^{\frac{1}{2}+n}}{R^{L+\frac{1}{2}}|u|^{i+1}}; \end{aligned}$$

so (8.21) holds for $n+1$ as well. Evaluating on $\Gamma _R$ completes the proof. $\square$

10.4.3 Step III: The main estimates

Claim 3

Let $U_0$ be a sufficiently large negative number. There exists a constant C such that

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi _L)\right| (u,v)\le \frac{C}{|u|^{i+1}}R^{-\max {(j,0)}} \end{aligned}$$

(8.22)

for $i=0,\dots , N$, $j=-1,0,\dots , L$, and for all $u\le U_0$, $v\ge v_{\Gamma _R}(u)$.

Proof

In order to simplify the presentation, we will additionally assume that R/2M is sufficiently large. This largeness assumption can be lifted by replacing bootstrap argument below by a Grönwall argument as in §5.3.5.

Let us fix $i\le N$. We make the following bootstrap assumption:

$$\begin{aligned} \left| [r^2\partial _v]^{L}T^i(r\phi _L)\right| (u,v)\le \frac{C_{\mathrm {BS}}^{(i)}}{|u|^{i+1}}, \end{aligned}$$

(8.23)

where $C_{\mathrm {BS}}^{(i)}$ is a constant to be specified later. Notice that, by integrating this up to L times from $\Gamma$, estimating at each step the boundary term by (8.19), this implies

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi _L)\right| (u,v)\le \frac{C+C_{\mathrm {BS}}^{(i)}}{R^j|u|^{i+1}} \end{aligned}$$

(8.24)

for $j=0,\dots , L$. In particular, if R and $C_{\mathrm {BS}}^{(i)}$ are chosen sufficiently large, then we have

$$\begin{aligned} \left| [r^2\partial _v]^{L}T^i(r\phi _L)\right| (u,v)\le 2\cdot |\Phi _L|(u,v), \end{aligned}$$

(8.25)

where we recall the definition (7.8) of $\Phi _L$. We now plug the bounds (8.24) into the approximate conservation law (7.10) and integrate the latter in u from $u=-k$ to obtain that

$$\begin{aligned} \left| r^{-2L}\partial _vT^i\Phi _L\right| (u,v)\le \frac{C\cdot C_{\mathrm {BS}}^{(i)}}{r^{2L+2}|u|^{i+1}}. \end{aligned}$$

(8.26)

We then integrate this bound in v from $\Gamma$, estimating the boundary term $T^i\Phi _L|_{\Gamma _R}$ using (8.15), to obtain that

$$\begin{aligned} \left| T^i\Phi _L\right| (u,v)\le \frac{C}{|u|^{i+1}}+\frac{C\cdot C_{\mathrm {BS}}^{(i)}}{R|u|^{i+1}}. \end{aligned}$$

(8.27)

Finally, we can choose R and $C_{\mathrm {BS}}^{(i)}$ large enough such that

$$\begin{aligned} |[r^2\partial _v]^{L}T^i(r\phi _L)|\le |2T^i\Phi _L |\le \frac{C_{\mathrm {BS}}^{(i)}}{2|u|^{i+1}}. \end{aligned}$$

(8.28)

This improves the bootstrap assumption (8.23) and thus proves (8.22) for $j=0$. The result for $j>0$ then follows in view of the estimates (8.24), and the result for $j=-1$ follows from (8.26). $\square$

10.4.4 Step IV: The auxiliary estimates

As in the $\ell =1$-case (cf. §5.4), we will need some auxiliary estimates in order to later be able to show that certain quantities attain limits on ${\mathscr {I}}^-$. These auxiliary estimates will be estimates on the differences $[r^2\partial _v]^{L}T^i \left( r\phi ^{(k)}_L-|u| T(r\phi ^{(k)}_L)\right)$. As in step II, we first need estimates on the boundary terms:

Claim 4

Let $U_0$ be a sufficiently large negative number, and let $N'\le N-2$. Then there exists a constant C such that

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i (r\phi _L-|u| T(r\phi _L))\right| (u,v_{\Gamma _R}(u))\le \frac{C}{R^{j}|u|^{i+1+ \varepsilon }}+\theta _k\cdot \frac{C}{R^{j}k^{i+1}} \nonumber \\ \end{aligned}$$

(8.29)

for $i=0,\dots ,N'$ and $j=0, \dots , L$, and for all $u\le U_0$.

Proof

The proof for the case $j=L-1$ is similar to that of Claim 1: We first assume (8.29) (with $j=L-1$) as a bootstrap assumption. The main difference to the proof of Claim 1 is that we then apply the divergence theorem to

$$\begin{aligned} \mathrm {div}J^T[T^i(\phi _L-|u|T\phi _L)]&=\Box _g(T^i(\phi _L-|u|T\phi _L))\cdot T(T^i(\phi _L-|u|T\phi _L))\\&=-\frac{1}{Dr}\partial _vT^{i+1}(r\phi _L)\cdot \frac{1}{r}T^{i+1}(r\phi _L-|u|T(r\phi _L)), \end{aligned}$$

rather than to $\mathrm {div}J^T[T^i\phi _L]=0$. (Here, we used the expression (3.6) for $\Box _g$.) This gives rise to a non-trivial bulk term. However, the estimates we established in Claim 3 provide sufficient bounds for this term.

Using the fundamental theorem of calculus and the Cauchy–Schwarz inequality, one then obtains an estimate on (cf. (5.52))

$$\begin{aligned} \sqrt{r}T^i(\phi _L-|u|T\phi _L)\le \frac{C}{R^{\frac{2L+1}{2}}}\left( \frac{1}{|u|^{i+1+\varepsilon }}+\frac{1}{k^{i+1}}\right) . \end{aligned}$$

In order to translate this into an estimate on $\partial _vT^i(r\phi _L-|u|T(r\phi _L))$, we consider the wave equation satisfied by $\partial _u\partial _vT^i(r\phi _L-|u|T(r\phi _L))$,

$$\begin{aligned} \partial _u\partial _vT^i(r\phi _L-|u|T(r\phi _L))= & {} -\frac{D}{r^2}\left( L(L+1)+\frac{2M}{r}\right) T^i(r\phi _L-|u|T(r\phi _L)) \nonumber \\&+\overbrace{\partial _vT^{i+1}(r\phi _L)}^{\le Cr^{-L-1}|u|^{-i-2}}, \end{aligned}$$

(8.30)

where we again note that the error term $\partial _vT^{i+1}(r\phi _L)$ can be bounded by the previous estimates (Claim 3), and integrate in u. This improves the bootstrap assumption.

The general case $j\le L-1$ then follows as in the proof of Claim 2, noting that, when considering the wave equations for $\partial _u\partial _v[r^2\partial _v]^j T^i(r\phi _L-|u|T(r\phi _L))$, the error terms compared to (7.1) will always be given by $\partial _v[r^2\partial _v]^jT^{i+1}(r\phi _L)$, which we already control by Claim 3. See also the proof of Proposition 5.5 for more details. $\square$

Having obtained estimates on the boundary terms, we can now prove:

Claim 5

Let $U_0$ be a sufficiently large negative number, and let $N'\le N-2$. Then there exists a constant C such that

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi ^{(k)}_L-|u|T(r\phi ^{(k)}_L))\right| (u,v)\le \frac{C}{R^{ \max (j,0)}}\left( \frac{1}{|u|^{i+1+\varepsilon }}+\frac{1}{k^{i+1}}\right) \end{aligned}$$

(8.31)

for $i=0,\dots ,N'$, $j=-1,0, \dots , L$, and for all $u\le U_0$, $v\ge v_{\Gamma _R}(u)$.

Proof

The proof is similar to that of Claim 3, with the main modifications being that we now use Claim 4 in order to estimate the boundary terms. Furthermore, instead of the approximate conservation law (7.10), we now consider the equations

$$\begin{aligned}&\partial _u(r^{-2L}\partial _vT^i(\Phi _L-|u|T \Phi _L))=\overbrace{r^{-2L}\partial _vT^{i+1}\Phi _L}^{\le Cr^{-2L-2}|u|^{-i-2}} \nonumber \\&\quad +\sum _{j=0}^L\frac{D(2M)^{j+1}}{r^{2L+3}}[r^2\partial _v]^{L-j}T^i(r\phi _L-|u|T(r\phi _L)) \left( 2(j+1)x^{(L)}_{j+1}-\sum _{i=0}^j x_i^{(L)} c_{j-i}^{L-i}\right) , \nonumber \\ \end{aligned}$$

(8.32)

in which we again control the error terms $r^{-2L}\partial _vT^{i+1}\Phi _L$ by Claim 3. Indeed, they decay faster near $\Gamma _R$ (that is, they have more u-decay).¹⁹ See the proof of Proposition 5.5 for more details. $\square$

10.4.5 Step V: Taking the limit $k\rightarrow \infty$

So far, we have proved uniform-in-k estimates on the sequence of solutions $\phi _L^{(k)}$ (whose elements vanish on $u\le -k$) constructed in Step I. We now show that these solutions converge uniformly to another solution $\phi _L$:

Claim 6

The sequence $\{\phi _L^{(k)}\}_{k\in {\mathbb {N}}}$ tends to a uniform limit $\phi _L$ as $k\rightarrow \infty$,

$$\begin{aligned} \lim _{k\rightarrow \infty }||\phi _L^{(k)}-\phi _L ||_{C^N({\mathscr {D}}_{\Gamma _R})}=0. \end{aligned}$$

(8.33)

In fact, this limiting solution is the unique smooth solution that restricts correctly to the data of §8.1, and it satisfies for all $u\le U_0$ and $v\ge v_{\Gamma _R}(u)$, for sufficiently large negative values of $U_0$, the following bounds for some constant C:

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi ^{(k)}_L)\right| (u,v)\le \frac{C}{|u|^{i+1}}R^{-\max {(j,0)}} \end{aligned}$$

(8.34)

for $i=0,\dots , N$ and $j=-1,0,\dots , L$. Furthermore, if $N'\le N-2$, then we also have

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi ^{(k)}_L-|u|T(r\phi ^{(k)}_L))\right| (u,v)\le \frac{C}{|u|^{i+1+\varepsilon }}R^{ -\max (j,0)}. \end{aligned}$$

(8.35)

for $i=0,\dots ,N'$ and $j=-1,0, \dots , L$.

Proof

The proof proceeds, mutatis mutandis, as the proof of Proposition 5.7. $\square$

10.4.6 Step VI: Proving sharp decay for $[r^2\partial _v]^{L-j} T^i(r\phi _L)$ (Proof of (8.5))

Claim 7

Let $U_0$ be a sufficiently large negative number. There exists a constant C such that the solution $\phi _L$ from Claim 6 satisfies

$$\begin{aligned} \left| [r^2\partial _v]^{L-j}T^i(r\phi _L)\right| (u,v)\le \frac{C}{|u|^{i+1}}\min (r,|u|)^{-\min {({\tilde{j}},N-i)}} \end{aligned}$$

(8.36)

for all $j=-1,\dots , L$, $i=0,\dots , N$, and for all $u\le U_0$, $v\ge v_{\Gamma _R}(u)$. Here, ${\tilde{j}}:=\max (j,0)$.

Proof

We will prove this inductively by showing the following lemma:

Lemma 8.1

Let $n<L$. Then there exists a constant C such that

$$\begin{aligned} \left| [r^2\partial _v]^{n-j}T^i(r\phi _L)\right| (u,v)\le \frac{C}{|u|^{i+1}}\min (r, |u|)^{-\min (j+1, N-i)} \end{aligned}$$

(8.37)

for all $j=0,\dots , n$, for all $i=0,\dots N$, and for all $u\le U_0$, $v\ge v_{\Gamma _R}(u)$.

Indeed, once this lemma is shown for $n=L-1$, then (8.36) follows in view of Claim 3 (which already provides the sharp estimates for $n=L, L+1$). $\square$

Proof of Lemma 8.1

We first show (8.37) for $n=0$. We derive from (7.1) with $N=0$ that

$$\begin{aligned}&-\frac{D}{r^2}T^i(r\phi _L)\left( L(L+1)+\frac{2M}{r}\right) =-\partial _v^2T^i(r\phi _L)+\partial _vT^{i+1}(r\phi _L)\nonumber \\&\quad =-\frac{1}{r^2}\partial _v(r^2\partial _vT^i(r\phi _L))+\frac{2D}{r}\partial _vT^i(r\phi _L)+\partial _vT^{i+1}(r\phi _L). \end{aligned}$$

(8.38)

We can assume, without loss of generality, that $i<N$, as (8.37) follows directly from (8.22) if $i=N$. If $i<N$, then we can insert the estimates from (8.22) into (8.38) to find that

$$\begin{aligned} |T^i(r\phi _L)|\le \frac{C}{r^2|u|^{i+1}}+\frac{C}{r|u|^{i+1}}+\frac{C}{|u|^{i+2}}. \end{aligned}$$

(8.39)

This establishes (8.37) for $n=0$.

Let us now assume that (8.37) holds for some fixed $n<L-1$. We shall show that it then also holds for $n+1$. We derive from (7.1) the following generalisation of (8.38):

$$\begin{aligned}&D\left( a_0^{n+1}-b_0^{n+1}L(L+1)-c_0^{n+1}\frac{2M}{r}\right) [r^2\partial _v]^{n+1}T^i(r\phi _L)\nonumber \\&=-\sum _{j=0}^n(2M)^{j+1}D\left( a_{j+1}^{n+1}-b_{j+1}^{n+1}L(L+1)-c_{j+1}^{n+1}\frac{2M}{r}\right) [r^2\partial _v]^{n-j}T^i(r\phi _L)\nonumber \\&\quad +[r^2\partial _v]^{n+2} T^{i+1}(r\phi _L)+\frac{2D(n+2)}{r}[r^2\partial _v]^{n+2}T^i(r\phi _L)-\frac{1}{r^2}[r^2\partial _v]^{n+3}T^i(r\phi _L). \nonumber \\ \end{aligned}$$

(8.40)

Notice that, since $n+1<L$, the difference $a_0^{n+1}-b_0^{n+1}L(L+1)$ is non-zero. Therefore, estimating the terms in the second line of (8.40) using the induction assumption, and the terms in the third line of (8.40) using (8.22) (keeping in mind that $n+3\le L+1$ and assuming as before that $i<N$), we obtain

$$\begin{aligned}&\left| [r^2\partial _v]^{n+1}T^i(r\phi _L) \right| \le \frac{C}{|u|^{i+1}}\min (r, |u|)^{-\min (1,N-i)}\nonumber \\&\quad +\frac{C}{|u|^{i+2}}+\frac{C}{r|u|^{i+1}}+\frac{C}{r^2|u|^{i+1}}\le \frac{C}{|u|^{i+1}}\min (r, |u|)^{-\min (1,N-i)}. \end{aligned}$$

(8.41)

This establishes (8.37) for $n+1$, restricted to $j=0$. For $j>0$, we use another induction, this time going down in derivatives:

Sublemma 1

There exists a constant C such that

$$\begin{aligned} \left| [r^2\partial _v]^{n+1-j}T^i(r\phi _L)\right| (u,v)\le \frac{C}{|u|^{i+1}}\min (r,|u|)^{-\min (1+j, N-i)} \end{aligned}$$

(8.42)

for all $j=0,\dots , n+1$, and for all $u\le U_0$, $v\ge v_{\Gamma _R}(u)$.

Proof of Sublemma 1

We have already established (8.42) for $j=0$. Let us now assume that it holds for $j\le n$ fixed. We shall show that (8.42) then also holds for $j+1$. We can restrict to $i< N-(j+2)$ since the result would be trivial otherwise. Using (8.40), we obtain the estimate (writing $n+1-(j+1)=n-j$):

$$\begin{aligned}&|[r^2\partial _v]^{n-j}T^i(r\phi _L)|\le \sum _{k=0}^{n-j-1}\frac{C}{|u|^{i+1}}\min (r,|u|)^{-\min (k+2+j, N-i)}\nonumber \\&\quad +\frac{C}{|u|^{i+2}}\min (r,|u|)^{-\min (j+1, N-(i+1))}+\frac{C}{|u|^{i+1}r}\min (r,|u|)^{-\min (j+1, N-i))},\nonumber \\ \end{aligned}$$

(8.43)

where we used the induction assumption (8.37) to estimate the first term on the RHS. This establishes (8.42) for $j+1$ and, thus, proves the sublemma. $\square$

Sublemma 1 proves (8.37) for $n+1$ and, hence, completes the proof of Lemma 8.1. $\square$

10.4.7 Step VII: The limit ${\mathscr {L}}=\lim _{u\rightarrow -\infty }|u|^2[r^2\partial _v]^LT(r\phi _L)$

Throughout the rest of the proof, we shall assume that $N-2\ge N'\ge L+1$, and that $\phi _L$ denotes the solution from Claim 6.

Claim 8

The limit ${\mathscr {L}}(v):=\lim _{u\rightarrow -\infty }|u|^2[r^2\partial _v]^LT(r\phi _L)(u,v)$ exists and is independent of v. Moreover, along any ingoing null hypersurfaces of constant v,

$$\begin{aligned} |u|^2[r^2\partial _v]^LT(r\phi _L)(u,v)-{\mathscr {L}}={\mathscr {O}}(r^{-1}+ |u|^{-\varepsilon }). \end{aligned}$$

(8.44)

In fact, if the lower bound (8.3) is assumed, and if R/2M is chosen large enough, then ${\mathscr {L}}\ne 0$.

Proof

We show that the limit exists by computing

$$\begin{aligned}&\partial _u(|u|^2[r^2\partial _v]^{L}T(r\phi _L))\nonumber \\&\quad =\underbrace{-2|u| [r^2\partial _v]^{L}T(r\phi _L)+|u|^2[r^2\partial _v]^{L}T^2(r\phi _L)}_{\le C|u|^{-1-\varepsilon }}-\underbrace{|u|^2\partial _v[r^2\partial _v]^{L}T(r\phi _L)}_{\le Cr^{-2}}. \nonumber \\ \end{aligned}$$

(8.45)

The first two terms together can be bounded by $|u|^{-1-\varepsilon }$ in view of estimate (8.35) from Claim 6. The third term can be bounded by $r^{-2}$ in view of estimate (8.36) from Claim 7. This establishes the existence of the limit ${\mathscr {L}}(v)$. The independence of v follows directly from the bound on the third term.

It is left to show that this limit is not zero. For this, we will also need to establish a lower bound for $|u|^2[r^2\partial _v]^LT(r\phi _L)$ on $\Gamma _R$. First, we observe that, in view of the identity (8.38), we have, on $\Gamma _R$, that

$$\begin{aligned} \left| R^L\cdot T(r\phi _L)(L(L+1))+2R^{L-1}[r^2\partial _v]T(r\phi _L)\right| \le \frac{C}{|u|^{3}}+\frac{C}{R|u|^2}. \end{aligned}$$

(8.46)

Thus, if R and $|U_0|$ are chosen sufficiently large, we have, as a consequence of the lower bound (8.3):

$$\begin{aligned} \left| r^2\partial _vT(r\phi _L)\right| (u,v_{\Gamma _R}(u))\ge \frac{C_{\mathrm {in}}^\Gamma }{4R^{L-1}|u|^2} \end{aligned}$$

(8.47)

Similarly, we can now show inductively, using (8.40) instead of (8.38), that, say,

$$\begin{aligned} \left| [r^2\partial _v]^{L-j} T(r\phi _L)\right| (u,v_{\Gamma _R}(u))\ge \frac{C_{\mathrm {in}}^\Gamma }{2^{L-j+1}R^{j}|u|^2}. \end{aligned}$$

(8.48)

for $j=0,\dots , L-1$, provided that R is chosen sufficiently large.²⁰

Using the lower bound above for $j=0$ and then integrating the bound (8.36) for $j=-1$ from $\Gamma _R$, one obtains, provided that R is chosen sufficiently large,

$$\begin{aligned} \left| [r^2\partial _v]^{L} T(r\phi _L)\right| (u,v)\ge \frac{C_{\mathrm {in}}^\Gamma }{2^{L+2}|u|^2} \end{aligned}$$

(8.49)

for all $v\ge v_{\Gamma _R}(u), u\le U_0$. This shows that ${\mathscr {L}}\ne 0$ and thus completes the proof. (Notice that this approach only works for the highest-order v-derivative (i.e. for $j=0$).) $\square$

We now show that various different limits can be computed from this limit.

Lemma 8.2

The following limits exist and satisfy the relations

$$\begin{aligned} \lim _{u\rightarrow -\infty }|u|^{1+i}r^j[r^2\partial _v]^{L-j}T^i(r\phi _L)={\mathscr {L}}^{(i,j)} \end{aligned}$$

(8.50)

for $i=0,\dots , N'+1$ and $j=0,\dots , L$, provided that $i+j\le N'+1$, where ${\mathscr {L}}^{(i,j)}$ are rational multiples of ${\mathscr {L}}$.

Proof

We will first show (8.50) for $j=0$. Indeed, we have ${\mathscr {L}}^{(0,0)}={\mathscr {L}}^{(1,0)}={\mathscr {L}}$, and, in view of the estimates (8.35) from Claim 6, we have the relations

$$\begin{aligned} \lim _{u\rightarrow -\infty } |u|^2[r^2\partial _v]^L T(r\phi _L)&=\frac{1}{2}\lim _{u\rightarrow -\infty } |u|^3[r^2\partial _v]^L T^2(r\phi _L)\nonumber \\&=\frac{1}{6} \lim _{u\rightarrow -\infty }|u|^4[r^2\partial _v]^L T^3(r\phi _L)\nonumber \\&=\dots =\frac{1}{i!}\lim _{u\rightarrow -\infty } |u|^{1+i}[r^2\partial _v]^L T^i(r\phi _L), \end{aligned}$$

(8.51)

provided that $i\le N'+1$. We thus have established that, for all $i\le N'+1$:

$$\begin{aligned} {\mathscr {L}}^{(i,0)}=i!{\mathscr {L}}. \end{aligned}$$

(8.52)

We now prove (8.50) inductively in j. Without loss of generality, we may assume that $L>0$. We assume that we have already established (8.50) for (i, j), for some fixed $j<L$ and for all $i\le N'+1-j$. We then show that (8.50) also holds for $(i,j+1)$, provided that $i+j+1\le N'+1$: Indeed, if $i+j\le N'$, we can appeal to equation (8.40) (with $n+1$ replaced by $L-j-1$) to obtain

$$\begin{aligned}&(a_0^{L-j-1}-b_0^{L-j-1}L(L+1))\lim _{u\rightarrow -\infty }|u|^{2+i}r^{j+1}[r^2\partial _v]^{L-j-1}T^i(r\phi _L)\nonumber \\&\quad =\lim _{u\rightarrow -\infty }|u|^{2+i+1}r^j[r^2\partial _v]^{L-j}T^{i+1}(r\phi _L)+2(L-j)\nonumber \\&\qquad \lim _{u\rightarrow -\infty }|u|^{2+i}r^j[r^2\partial _v]^{L-j}T^i(r\phi _L), \end{aligned}$$

(8.53)

where we used (8.36) to estimate the lower-order derivatives $[r^2\partial _v]^{L-j-2-k}$ for $k\ge 0$. We have thus established that

$$\begin{aligned} {\mathscr {L}}^{(i,j+1)}=\frac{1}{a_0^{L-j-1}-b_0^{L-j-1}L(L+1)}\cdot \left( {\mathscr {L}}^{(i+1,j)}+2(L-j){\mathscr {L}}^{(i,j)}\right) \end{aligned}$$

(8.54)

for all $0\le i\le N'+1-(j+1)$, $0\le j<L$. (Notice that, in this range of indices, $a_0^{L-j-1}-b_0^{L-j-1}L(L+1)\ne 0$.) We leave it to the reader to derive explicit expressions for ${\mathscr {L}}^{(i,j)}$ from the recurrence relations above. $\square$

10.4.8 Step VIII: The limit $\lim _{u\rightarrow -\infty }[r^2\partial _u]^{L+1}(r\phi _L)$ (Proof of (8.6) and (8.7))

Claim 9

The limit $\lim _{u\rightarrow -\infty }[r^2\partial _u]^{L+1}(r\phi _L)={{\tilde{C}}}\ne 0$ exists and can be computed explicitly in terms of the constants ${\mathscr {L}}^{(i,j)}$. Moreover, we have along any ingoing null hypersurface of constant v:

$$\begin{aligned}&[r^2\partial _u]^{L-j}(r\phi _L)(u,v)={\mathscr {O}}(\min (r,|u|)^{-1- j}),\qquad \quad j=0,\dots , L, \end{aligned}$$

(8.55)

$$\begin{aligned}&[r^2\partial _u]^{L+1}(r\phi _L)(u,v)={\tilde{C}}+{\mathscr {O}}(r^{-1}+|u|^{-\varepsilon }). \end{aligned}$$

(8.56)

Proof

We will prove this by writing $\partial _u=T-\partial _v$ and using the following lemma:

Lemma 8.3

Let f be a smooth function, and let $n\in {\mathbb {N}}$. Then

$$\begin{aligned} (r^2\partial _u)^n f=(r^2T-r^2\partial _v)^nf&=\sum _{k=0}^n (-1)^{n-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \sum _{i=0}^{k-1}\alpha ^{(n,k)}_i(r+{\mathscr {O}}(1))^i(r^2 T)^{k-i}\right) \nonumber \\&\quad [r^2\partial _v]^{n-k}f \end{aligned}$$

(8.57)

for some constants $\alpha ^{(n,k)}_i$.

Proof

A proof is provided in the Appendix A.3. $\square$

Indeed, applying this lemma with $n=L-j$ for $j\ge 0$ immediately proves (8.55) upon inserting the bounds (8.36) from Claim 7. On the other hand, applying the lemma with $n=L+1$ and recalling Lemma 8.2, we find

$$\begin{aligned} \lim _{u\rightarrow -\infty }[r^2\partial _u]^{L+1}(r\phi _L)=\sum _{k=0}^{L+1}(-1)^{L+1-k}\left( {\begin{array}{c}L+1\\ k\end{array}}\right) \sum _{i=0}^{k-1}\alpha ^{(L+1,k)}_i {\mathscr {L}}^{(k-i,L+1-k)}={{\tilde{C}}} \nonumber \\ \end{aligned}$$

(8.58)

since we assumed that $N'\ge L+1$. Equation (8.56) then follows similarly.

Finally, we need to show that ${{\tilde{C}}}\ne 0$. Instead of explicitly computing the sum above, we proceed by contradiction: Suppose that (8.56) holds with ${{\tilde{C}}}= 0$. Then, in view of (8.55), we have that, say, on $v=1$, $r\phi _L(u,1)={\mathscr {O}}(|u|^{-L-1-\varepsilon })$. Inductively inserting this estimate into (7.10) and integrating in u (cf. Proposition 10.1), this implies that $[r^2\partial _v]^L(r\phi _L)(u,v)={\mathscr {O}}(|u|^{-1-\varepsilon })$, which would imply that ${\mathscr {L}}=0$, a contradiction. $\square$

This concludes the proof of Theorem 8.1. $\square$

11 Data on an Ingoing Null Hypersurface ${\mathscr {C}}_{v=1}$ I

Having obtained an understanding of solutions arising from timelike boundary data in the previous section, we now aim to understand solutions arising from polynomially decaying initial data on an ingoing null hypersurface. We will, in the present section, focus on initial data which decay as predicted by Theorem 8.1 of the previous section. While the present section completely generalises the methods of §4 and contains a proof of Theorem 1.4, it requires fast initial decay on the data (depending on $\ell$). The case of more slowly decaying data will thus need to be treated differently and is discussed in §10.

Throughout this section, we shall again assume that $\phi$ is a solution to (1.1), supported on a single angular frequency (L, m) with $|m|\le L$. In the usual abuse of notation, we omit the m-index, that is, we write $\phi =\phi _{Lm}\cdot Y_{Lm}=\phi _L\cdot Y_{Lm}$.

11.1 Initial Data Assumptions

Prescribe smooth characteristic/scattering data for (1.1), restricted to the angular frequency (L, m), that satisfy on ${\mathscr {C}}_{v=1}$

$$\begin{aligned}&\lim _{u\rightarrow -\infty }[r^2\partial _u]^{L+1-i}(r\phi _L)(u,1)=C_{\mathrm {in}}^{(L,i)},\qquad \quad i=1,\dots ,L, \end{aligned}$$

(9.1)

$$\begin{aligned}&[r^2\partial _u]^{L+1}(r\phi _L)(u,1)=C_{\mathrm {in}}^{(L,0)}+{\mathscr {O}}(r^{-\varepsilon }) \end{aligned}$$

(9.2)

for some $\varepsilon \in (0,1)$, where the $C_{\mathrm {in}}^{(L,i)}$ are constants, and which moreover satisfy for all $v\ge 1$:

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _L)(u,v)=0 \end{aligned}$$

(9.3)

for $n=0,\dots , L+1$. We interpret this latter assumption as the no incoming radiation condition.

11.2 The Main Theorem (Theorem 9.1)

Motivated by the previous Theorem 8.1, we will only consider the case where $C_{\mathrm {in}}^{(L,i)}=0$ for $i>0$ for now. The other cases, and further generalisations that do not require conformal regularity on the initial data, will be treated in §10, as they have to be dealt with in a different way. Let us mention, however, that the proof of the present section still works if additionally $C_{\mathrm {in}}^{(L,1)}\ne 0$.

Theorem 9.1

By standard scattering theory [7], there exists a unique smooth scattering solution $\phi _L\cdot Y_{Lm}$ in ${\mathscr {M}}\cap \{v\ge 1\}$ attaining the data of §9.1.

Let $U_0$ be a sufficiently large negative number. Assume moreover that $C_{\mathrm {in}}^{(L,i)}=0$ for all $i=1,\dots , L$ and that $C_{\mathrm {in}}^{(L,0)}\ne 0$. Then, for all $u\le U_0$, the limit of the radiation field on future null infinity is given by

$$\begin{aligned} \lim _{v\rightarrow \infty }(r\phi _L)(u,v)=\frac{L!C_{\mathrm {in}}^{(L,0)}}{(2L+1)!|u|^{L+1}}+{\mathscr {O}}(|u|^{-L-1-\varepsilon }), \end{aligned}$$

(9.4)

and, throughout ${\mathscr {D}}=(-\infty ,U_0]\times [1,\infty )$, the outgoing derivative of $r\phi _L$ satisfies, for fixed values of u, the following asymptotic expansion as ${\mathscr {I}}^+$ is approached:

$$\begin{aligned} \partial _v(r\phi _L)(u,v)=\sum _{i=0}^L \frac{f_i^{(L)}(u)}{r^{2+i}}+\frac{ (-1)^LM\cdot B^*}{(L+1)!}\frac{\log r-\log |u|}{r^{3+L}}+{\mathscr {O}}(r^{-3-L}). \end{aligned}$$

(9.5)

Here, the $f^{(L)}_i(u)$ are smooth functions of u which satisfy $f^{(L)}_i(u)=\dfrac{(-1)^i\beta _{L-1-i}^{(L)}}{i!|u|^{L-i}}+{\mathscr {O}}(|u|^{-L+i-\varepsilon })$ for $i<L$ and $f^{(L)}_i(u)=\dfrac{2M(-1)^L{{\tilde{a}}}^{L,L}_1\beta _1^{(L)}}{L!|u|}+{\mathscr {O}}(|u|^{-1-\varepsilon })$ for $i=L$, and $B^*=2(2x_1^{(L)}-c_0^L)\beta _0^{(L)}$. The constants $\beta _i^{(L)}$ are given explicitly by the formulae²¹

$$\begin{aligned} \beta _i^{(L)}=\frac{L!C_{\mathrm {in}}^{(L,0)}}{(2L+1)!}\cdot \frac{i!}{L!}\prod _{k=0}^{L-i-1} ( a_0^k -b_0^kL(L+1))=(-1)^{L-i}\frac{(2L-i)!C_{\mathrm {in}}^{(L,0)}}{(2L+1)!}. \end{aligned}$$

(9.6)

Moreover, the quantity $\Phi _L$ defined in Eq. (7.8) has the expansion

$$\begin{aligned} \partial _v\Phi _L=\frac{ M\cdot B^*(\log r-\log |u|)}{r^{3}}+{\mathscr {O}}(r^{-3}), \end{aligned}$$

(9.7)

and, in particular, the logarithmically modified Newman–Penrose constant is finite and conserved:

$$\begin{aligned} I_{\ell =L}^{\mathrm {future},\frac{\log r}{r^3}}[\phi ](u):=\lim _{v\rightarrow \infty }\frac{r^3}{\log r}\partial _v\Phi _L(u,v)=M\cdot B^*\ne 0. \end{aligned}$$

(9.8)

Remark 9.1

The first two statements of the above theorem, (9.4) and (9.5), still apply if one lifts the restriction $C_{\mathrm {in}}^{(L,1)}=0$, albeit with different constants and with different $f_i^{(L)}(u)$. See also Remark 9.2. In particular, we again have a cancellation if $r\phi _L\sim 1/|u|^L$ initially: The initial $|u|^{-L}$-decay translates into $|u|^{-L-1}$-decay on ${\mathscr {I}}^+$. Equations (9.7) and (9.8), on the other hand change, change: The leading-order decay behaviour of $\partial _v\Phi _L$ is now given by $\sim \frac{1}{r^2}$, and, in particular, the usual Newman–Penrose constant

$$\begin{aligned} I_{\ell =L}^{\mathrm {future}}[\phi ]:=\lim _{v\rightarrow \infty }r^2\partial _v\Phi _L(u,v) \end{aligned}$$

(9.9)

will be finite, generically non-vanishing, and conserved along future null infinity. If one also allows $C_{\mathrm {in}}^{(L,i)}\ne 0$ for $i>1$, then the modifications to Theorem 9.1 will be more severe, see already Theorem 10.1.

11.2.1 Overview of the proof

We will prove the theorem in two steps. First, we will obtain an asymptotic estimate for $r\phi _L$ (which will, in particular, imply (9.4)) by integrating (7.13) in v from data and then integrating the result $L+1$ times in u from ${\mathscr {I}}^-$.

Then, we will use this estimate to get the leading-order decay of $\partial _v(r\phi _L)$ by integrating equation (7.1) with $N=0$. Once this is achieved, we will inductively obtain leading-order asymptotics for $(r^2\partial _v)^n(r\phi _L)$ using the corresponding (7.1), from which we can, in turn, deduce higher-order asymptotics for $\partial _v(r\phi _L)$. This will prove (9.5). Equation (9.7) then follows in a similar fashion from the approximate conservation law (7.10).

11.3 Asymptotics for $r\phi _L$

Proposition 9.1

There exists a constant C depending only on data such that $r\phi _L$ satisfies the following asymptotic expansion throughout ${\mathscr {D}}$:

$$\begin{aligned} \left| r\phi _L(u,v)-\frac{L!C_{\mathrm {in}}^{(L,0)}}{(2L+1)!|u|^{L+1}}\right| \le \frac{C}{|u|^{L+1+\varepsilon }}+\frac{C}{r|u|^L}. \end{aligned}$$

(9.10)

In particular, we have

$$\begin{aligned} \lim _{v\rightarrow \infty }r\phi _L(u,v)=\frac{L!}{(2L+1)!|u|^{L+1}}+{\mathscr {O}}(|u|^{-L-1-\varepsilon }). \end{aligned}$$

(9.11)

Proof

By applying the weighted energy estimate of Proposition 4.1 (whose proof still works for higher $\ell$-modes), we obtain the decay estimate (cf. Corollary 1):

$$\begin{aligned} |r\phi _L(u,v)|\le C|u|^{-L-1} \end{aligned}$$

(9.12)

for some constant C depending only on initial data.

By inserting this estimate into Eq. (7.13) with $N=0$ and integrating the latter from $v=1$, we then obtain that $|\partial _u(r\phi _L)|\le C|u|^{-L-2}$. Similarly, by inductively integrating Eq. (7.13) for higher $N\le L$ from $v=1$, we find that

$$\begin{aligned} |(r^2\partial _u)^n(r\phi )|\le C\frac{r^{2n}}{|u|^{L+1+n}}. \end{aligned}$$

(9.13)

We can plug these estimates into (7.13) with $N=L$,

$$\begin{aligned}&r^{2L}\partial _v(r^{-2L}\partial _u[r^2\partial _u]^L(r\phi _L))=-{\underline{c}}_0^L\frac{2MD}{r^3}\overbrace{[r^2\partial _u]^L(r\phi _L)}^{\lesssim r^{2L}|u|^{-2L-1}}\nonumber \\&\quad +\sum _{j=1}^N\frac{D}{r^2}(2M)^j\underbrace{[r^2\partial _u]^{L-j}(r\phi _L)}_{\lesssim r^{2L-2}|u|^{-2L}}\cdot \left( {\underline{a}}_j^L-{\underline{b}}_j^L-{\underline{c}}_j^L\frac{2M}{r} \right) , \end{aligned}$$

(9.14)

and integrate in v from $v=1$ to find that

$$\begin{aligned} \left| \partial _u[r^2\partial _u]^{L}(r\phi _L)(u,v)-C_{\mathrm {in}}^{(L,0)}\frac{r^{2L}}{|u|^{2L+2}}\right| \le C\frac{r^{2L}}{|u|^{2L+2+\varepsilon }}. \end{aligned}$$

(9.15)

Essentially, we can now integrate (9.15) $L+1$ times from ${\mathscr {I}}^-$ to improve the bootstrap assumption. For this, we first apply Lemma 2.1 with $N'=2L$, $N=2L+2$ to (9.15) (and divide by $r^2$) to obtain that

$$\begin{aligned} \left| \partial _u[r^2\partial _u]^{L-1}(r\phi _L)(u,v)-\frac{C_{\mathrm {in}}^{(L,0)}}{2L+1}\left( \frac{r^{2(L-1)}}{|u|^{2L+1}}+ \sum _{k=1}^{2L}\frac{r^{2(L-1)-k}}{|u|^{2L+1-k}}\right) \right| \le C\frac{r^{2(L-1)}}{|u|^{2L+1+\varepsilon }}. \nonumber \\ \end{aligned}$$

(9.16)

Here, we used that the boundary term vanishes,²²

$$\begin{aligned} \lim _{u\rightarrow -\infty }[r^2\partial _u]^L(r\phi _L)(u,v)=0. \end{aligned}$$

Notice that the terms inside the sum $\sum _{k=1}^{2L}$ decay faster near ${\mathscr {I}}^+$ than the $\frac{r^{2(L-1)-k}}{|u|^{2L+1-k}}$-term inside (9.16). Therefore, inductively applying the above procedure L more times²³, one obtains

$$\begin{aligned} \left| r\phi _L(u,v)-\frac{1}{(2L+1)\cdots (L+1)}\frac{C_{\mathrm {in}}^{(L,0)}}{|u|^{L+1}}\right| \le \frac{C}{|u|^{L+1+\varepsilon }}+\frac{C}{r|u|^L}. \end{aligned}$$

(9.17)

This proves the proposition. $\square$

Remark 9.2

We stated in Remark 9.1 that parts of Theorem 9.1 still apply if one assumes that also $C_{\mathrm {in}}^{(L,1)}=\lim _{u\rightarrow -\infty }[r^2\partial _u]^L(r\phi _L)\ne 0$. Let us explain the modifications to the proof above needed to see this: First, one needs to replace the $|u|^{-L-1}$-decay in (9.12) with $|u|^{-L}$-decay. This then leads to the RHS’s of (9.13) and (9.14) to decay one power in u slower. However, this still produces the same leading-order decay of $\partial _u[r^2\partial _u]^L(r\phi _L)$ in (9.15). Upon integrating this from ${\mathscr {I}}^-$, one picks up the limit $C_{\mathrm {in}}^{(L,1)}=\lim _{u\rightarrow -\infty }[r^2\partial _u]^L(r\phi _L)$ and obtains an asymptotic estimate for $[r^2\partial _u]^L(r\phi _L)$ near ${\mathscr {I}}^-$. One can then, as was done in §4, insert this asymptotic estimate for $[r^2\partial _u]^L(r\phi _L)$ back into (9.14) and proceed with the rest of the proof as above. In view of the boundary term coming from $\lim [r^2\partial _u]^L(r\phi _L)$, one then obtains, instead of (9.10),

$$\begin{aligned} \left| r\phi _L(u,v)-\frac{L!C_{\mathrm {in}}^{(L,0)}+C'(C_{\mathrm {in}}^{(L,1)})}{(2L+1)!|u|^{L+1}}-\frac{C''(C_{\mathrm {in}}^{(L,1)})}{r^L}\right| \le \frac{C}{|u|^{L+1+\varepsilon }} \end{aligned}$$

(9.18)

for some $C'(C_{\mathrm {in}}^{(L,1)})$, $C''(C_{\mathrm {in}}^{(L,1)})$ which depend only on M, L and $C_{\mathrm {in}}^{(L,1)}$.

The proof of the asymptotics for $\partial _v(r\phi _L)$, presented in the next section, remains largely unchanged. See also §4 for details.

11.4 Asymptotics for $\partial _v(r\phi _L)$ and Proof of Theorem 9.1

Proof of Theorem 9.1

Having obtained the asymptotics for $r\phi _L$ along ${\mathscr {I}}^+$, we can now compute the asymptotics of $\partial _v(r\phi _L)$. For the sake of notational simplicity, we restrict to $L\ne 0$ for now, the case $L=0$ is recovered in (9.31).

We first compute the leading-order asymptotics by integrating the wave equation (7.1) with $N=0$,

$$\begin{aligned} \partial _u\partial _v(r\phi _L)=-\frac{D}{r^2}\left( L(L+1)+\frac{2M}{r}\right) r\phi _L \end{aligned}$$

(9.19)

from past null infinity (where $\partial _v(r\phi _L)$ vanishes by assumption (9.3)) and by plugging in the estimate (9.10). This yields, after also commuting (9.19) with $r^2$,

$$\begin{aligned} \left| r^2\partial _v(r\phi _L)(u,v)+\frac{(L+1)!}{(2L+1)!}\frac{C_{\mathrm {in}}^{(L,0)}}{|u|^L} \right| \le \frac{C}{|u|^{L+\varepsilon }}+\frac{C}{r|u|^{L-1}}, \end{aligned}$$

(9.20)

where, from now on, C will be a constant which depends only on data and which is allowed to vary from line to line. More precisely, by writing $r\phi _L(u,v)=r\phi _L(u,\infty )-\int _{v}^\infty \partial _v(r\phi )(u,v)\mathop {}\!\mathrm {d}v'$ in (9.19), we can write

$$\begin{aligned} \left| r^2\partial _v(r\phi _L)(u,v)+L(L+1)\int _{-\infty }^u \lim _{v\rightarrow \infty }(r\phi )(u',v)\mathop {}\!\mathrm {d}u' \right| \le \frac{C}{r|u|^{L-1}}. \end{aligned}$$

(9.21)

Let us now make the following induction assumption. Let $n\ge 1$. Then we assume that for all $L\ge i\ge n$:

$$\begin{aligned} \left| [r^2\partial _v]^{L-i}(r\phi _L)-\frac{\beta _i^{(L)}}{|u|^{i+1}}\right| \le \frac{C}{|u|^{i+1+\varepsilon }}+\frac{C}{r|u|^i} \end{aligned}$$

(9.22)

for some non-vanishing constants $\beta _i^{(L)}\in {\mathbb {Q}}$. Since we have already established that this holds true for $n=L$ with $\beta _{L}^{(L)}=\frac{L!C_{\mathrm {in}}^{(L,0)}}{(2L+1)!}$, it suffices to show that (9.22) also holds for $n-1\ge 1$, provided that it holds for n. Therefore, we now consider (7.1) with $N=L-n\ge 0$:

$$\begin{aligned}&\partial _u\left( r^{-2(L-n)}\partial _v[r^2\partial _v]^{L-n}(r\phi _L)\right) \nonumber \\&=\frac{1}{r^{2(L-n)}}\sum _{i=n}^L\frac{D(2M)^{i-n}}{r^2}[r^2\partial _v]^{L-i}(r\phi _L)\left( a_{i-n}^{L-n}-b_{i-n}^{L-n}L(L+1)-c_{i-n}^{L-n}\frac{2M}{r}\right) . \end{aligned}$$

(9.23)

Plugging in the induction assumption (9.22) for the terms on the RHS and then integrating (9.23) gives that $\partial _v[r^2\partial _v]^{L-n}(r\phi _L)$ is of order ${\mathscr {O}}(r^{-2}|u|^{-n})$. Moreover, commuting now (9.23) with $r^{2(L-(n-1))}$, we obtain that

$$\begin{aligned}&\partial _u(r^2\partial _v[r^2\partial _v]^{L-n}(r\phi _L))=\overbrace{-\frac{2(L-(n-1))D}{r}\cdot r^2\partial _v[r^2\partial _v]^{L-n}(r\phi _L)}^{\lesssim r^{-1}|u|^{-n}}\nonumber \\&\quad +\sum _{i= n}^L D(2M)^{i-n}\underbrace{[r^2\partial _v]^{L-i}(r\phi _L)}_{\lesssim |u|^{-n-1}}\left( a_{i-n}^{L-n}-b_{i-n}^{L-n}L(L+1)-c_{i-n}^{L-n}\frac{2M}{r}\right) , \end{aligned}$$

(9.24)

from which, in turn, we recover, by again integrating from ${\mathscr {I}}^-$, that

$$\begin{aligned} \left| [r^2\partial _v]^{L-(n-1)}(r\phi _L)-\frac{\beta _{n-1}^{(L)}}{|u|^{n}}\right| \le \frac{C}{|u|^{n+\varepsilon }}+\frac{C}{r|u|^{n-1}}, \end{aligned}$$

(9.25)

with $\beta _{n-1}^{(L)}$ given by

$$\begin{aligned} n\beta _{n-1}^{(L)}={\beta _n^{(L)}}\left( a_0^{L-n}-b_0^{L-n}L(L+1)\right) \ne 0. \end{aligned}$$

(9.26)

This proves (9.22) for all $n\ge 1$ and, thus, that (9.25) holds for all $n\ge 2$. In fact, it is easy to see that (9.25) also holds for $n=1$, with the $r^{-1}|u|^{-n+1}$-term on the RHS replaced by $\log (1-v/u)/v$ (cf. (4.26)).

In order to get a similar estimate to (9.25) for $n=0$, we recall the crucial cancellation in (7.10) for $N=L$ (namely $a_0^L-b_0^LL(L+1)=0$). We are thus led to consider, in a very similar fashion to the above, the equation

$$\begin{aligned} \partial _u(r^{-2L}\partial _v[r^2\partial _v]^{L}(r\phi _L))= & {} -\frac{1}{r^{2L}}\frac{D}{r^2}\cdot c_0^L\frac{2M}{r}[r^2\partial _v]^L(r\phi _L)\nonumber \\&\quad +\frac{1}{r^{2L}}\sum _{i=1}^L\frac{D(2M)^i}{r^2}[r^2\partial _v]^{L-i}(r\phi _L)\nonumber \\&\quad \left( a_i^{L}-b_i^{L}L(L+1)-c_i^{L}\frac{2M}{r}\right) . \end{aligned}$$

(9.27)

The first term on the RHS is bounded by $Cr^{-2L-3}|u|^{-1}$, whereas the other terms in the sum are bounded by $C{r^{-2L-2}|u|^{-2}}$. More precisely, we have

$$\begin{aligned}&\partial _u(r^{-2L}\partial _v[r^2\partial _v]^{L}(r\phi _L))=-\frac{2MD}{r^{2L+3}|u|} c_0^L\beta _0^{(L)}\nonumber \\&\quad +\frac{2MD}{r^{2L+2}|u|^2}\left( a_1^L-b_1^L L(L+1)\right) \beta _1^{(L)}\nonumber \\&\quad +{\mathscr {O}}\left( r^{-2L-2}|u|^{-2-\varepsilon }+r^{2L-3}\frac{\log 1-\frac{v}{u}}{v}\right) . \end{aligned}$$

(9.28)

Integrating this from ${\mathscr {I}}^-$ then yields that

$$\begin{aligned}{}[r^2\partial _v]^{L+1}(r\phi _L)(u,v)&=\frac{2M}{|u|}\left( a_1^L-b_1^L L(L+1)\right) \beta _1^{(L)}\nonumber \\&\quad +{\mathscr {O}}\left( |u|^{-1-\varepsilon }+\frac{1}{r} +\frac{\log 1-\frac{v}{u}}{v}\right) , \end{aligned}$$

(9.29)

where, in the two asymptotic equalities above, we made use of the integral estimates (4.26) and (4.27). In order to find the logarithmic next-to leading order asymptotics, we insert the estimates above into the approximate conservation law (7.10):

$$\begin{aligned} \partial _u(r^2\partial _v\Phi _L)&=(2L+2)Dr \partial _v\Phi _L +\sum _{j=0}^L\frac{D}{r}(2M)^{j+1}[r^2\partial _v]^{L-j}(r\phi _{L})\nonumber \\&\left( 2(j+1)x^{(L)}_{j+1}-\sum _{i=0}^j x_i^{(L)} c_{j-i}^{L-i}\right) . \end{aligned}$$

(9.30)

From this, we then obtain, in a similar way to how we proved the estimates above, that

$$\begin{aligned} r^2\partial _v\Phi _L(u,v)=2M(2x_1^{(L)}-c_0^L)\beta _0^{(L)}\frac{\log (v-u)-\log |u|}{v}+{\mathscr {O}}(r^{-1}). \end{aligned}$$

(9.31)

Notice that the difference $2x_1^{(L)}-c_0^L$ is non-vanishing for all $L(\ge 0)$ since $x_1^{(L)}=-\frac{L}{2}$ and $c_0^L=1+2L(L+1)$.

Comparing equations (9.29) and (9.31) then gives us the next-to-leading-order asymptotics for $[r^2\partial _v]^{L+1}(r\phi _L)$ since, for $i>0$, the terms $[r^2\partial _v]^{L-i}(r\phi _L)$ contained in $\Phi _L$ do not contain logarithmic terms at next-to-leading order, which can be seen by integrating (9.29) i times from ${\mathscr {I}}^+$.

Finally, the statement (9.5) follows by simply integrating these asymptotics L times from ${\mathscr {I}}^+$ and using (9.22) for the arising boundary terms on ${\mathscr {I}}^+$.

This concludes the proof of Theorem 9.1.$\square$

11.5 Comments

11.5.1 A Logarithmically Modified Price’s Law at All Orders

We expect Theorem 9.1 (and, in particular, Eq. (9.8)) to imply a logarithmically modified Price’s law for the $\ell =L$-mode (see also the remarks in §4.4). However, Theorem 9.1 only applies if $C_{\mathrm {in}}^{(L,i)}=0$ for all $i>0$. This assumption, in turn, is motivated by the results of the previous §8 (Eq. (8.6) from Theorem 8.1). Therefore, although the general situation ($C_{\mathrm {in}}^{(L,i)}\ne 0$) might (and will) be quite different, we can expect that the data considered in §8, i.e. data on a timelike boundary data which decay like $r\phi _\ell \sim |t|^{-1}$ near $i^-$ and which are smoothly extended to ${\mathscr {H}}^+$, lead to a logarithmically modified Price’s law, for each $\ell$. To be concrete, the expected decay rates would then be

$$\begin{aligned} r\phi _L|_{{\mathscr {I}}^+}\sim u^{-L-2}\log u,\quad \phi _L|_{{\mathscr {H}}^+}\sim v^{-2L-3}\log v \end{aligned}$$

(9.32)

near $i^+$. Moreover, the leading-order asymptotics should be independent of the extension of the data towards ${\mathscr {H}}^+$. As has been discussed in §1.3, the proof of the above expectation should follow by combining the results of this paper with those of [3], similarly to how [13] combined the results of [12] with those of [1, 2], so long as sufficient regularity (depending on L) is assumed. The fixed-regularity problem, on the other hand, seems much more difficult, cf. Conjecture 1.

11.5.2 The Case $C_{\mathrm {in}}^{(L,i)}\ne 0$

Notably, the proof presented in this section cannot be directly applied to the case $C_{\mathrm {in}}^{(L,i)}\ne 0$ for $i>1$, since one would encounter several difficulties related to the quantities $(r^2\partial _u)^i(r\phi _L)$. (Notice already that the limits $\lim _{u\rightarrow -\infty }[r^2\partial _u]^{i}(r\phi _L)$ grow like $v^{i-1}$ for $i=1,\cdots , L$, and like $v^{L-1}$ for $i=L+1$.) Furthermore, working with the quantity $[r^2\partial _u]^{L+1}(r\phi _L)$ requires strong conformal regularity assumptions. In the next section, we shall therefore obtain asymptotics for much more general data by working only with the quantities $[r^2\partial _v]^i(r\phi _L)$ and not using the approximate conservation law in v (7.17) at all.

12 Data on an Ingoing Null Hypersurface ${\mathscr {C}}_{v=1}$ II

In this final section, we present a different approach towards obtaining the early-time asymptotics of $\partial _v(r\phi _L)$ of solutions $\phi _L$ arising from polynomially decaying initial data on a null hypersurface ${\mathscr {C}}_{v=1}$, without requiring any conformal regularity and/or fast decay on initial data. In particular, this section contains the proof of Theorem 1.5 from the introduction and can also treat the cases $C_{\mathrm {in}}^{(L,i)}\ne 0$ from the previous section.

Throughout this section, we shall again assume that $\phi$ is a solution to (1.1), supported on a single angular frequency (L, m) with $|m|\le L$. In the usual abuse of notation, we omit the m-index, that is, we write $\phi =\phi _{Lm}\cdot Y_{Lm}=\phi _L\cdot Y_{Lm}$.

12.1 Initial Data Assumptions

Prescribe smooth characteristic/scattering data for (1.1), restricted to the angular frequency (L, m), that satisfy on ${\mathscr {C}}_{v=1}$

$$\begin{aligned} \left| r\phi _L(u,1)-\frac{C_{\mathrm {in}}}{r^p}\right| ={\mathscr {O}}_1( r^{-p-\varepsilon }) \end{aligned}$$

(10.1)

for some $\varepsilon \in (0,1]$, a constant $C_{\mathrm {in}}>0$ and for some $p\in {\mathbb {N}}_0$, and which moreover satisfy

$$\begin{aligned} \lim _{u\rightarrow -\infty }\partial _v^n(r\phi _L)(u,v)=0 \end{aligned}$$

(10.2)

for all $v\ge 1$ and for all $n=1,\dots , L+1$.

Notice that if $p=1$ in (10.1), then this includes the cases $C_{\mathrm {in}}^{(L,i)}\ne 0$ from §9.

12.2 The Main Theorem (Theorem 10.1)

Theorem 10.1

By standard scattering theory [7], there exists a unique smooth scattering solution $\phi _L\cdot Y_{Lm}$ in ${\mathscr {M}}\cap \{v\ge 1\}$ attaining the data of §10.1.

Let $U_0$ be a sufficiently large negative number, let ${\mathscr {D}}=(-\infty ,U_0]\times [1,\infty )$, and let $r_0:=r(u,1)=|u|-2M\log |u|+{\mathscr {O}}(1)$. Then the following statements hold for all $(u,v)\in {\mathscr {D}}$:

a) We have that:

$$\begin{aligned} \lim _{v\rightarrow \infty }r\phi _L(u,v)=F(u)={\left\{ \begin{array}{ll} {\mathscr {O}}(r_0^{-p-\varepsilon }),&{}\text {if }p\le L\text { and } p\ne 0\\ {C_0\cdot }C_{\mathrm {in}}r_0^{-p}+{\mathscr {O}}(r_0^{-p-\varepsilon }), &{}\text {if }p>L \text { or } p=0, \end{array}\right. } \nonumber \\ \end{aligned}$$

(10.3)

for some smooth function F(u) and some explicit, non-vanishing constant $C_0=C_0(L,p)$.

b) Moreover, the outgoing derivative of the radiation field $\partial _v(r\phi _L)$ satisfies the following asymptotic expansion $\underline{if \,p<L}$:

$$\begin{aligned} r^2\partial _v(r\phi _L)(u,v)= & {} \sum _{i=0}^{p-1}\frac{f^{(L,p)}_i(u)}{r^i}+\frac{M\cdot C_1(\log r-\log |u|)+C_2 r_0}{r^p} \nonumber \\&+{\mathscr {O}}\left( \frac{|u|^{1-\varepsilon }}{r^{p}}\right) , \end{aligned}$$

(10.4)

where the $f^{(L,p)}_i$ are smooth functions of order $f^{(L,p)}_i={\mathscr {O}}(r_0^{-p+i+1-\varepsilon })$ if $i<p-1$, or of order $f^{(L,p)}_{i}=C_3^{L,p,i}+{\mathscr {O}}(r_0^{-\varepsilon })$ if $i=p-1$.

On the other hand, $\underline{if \,p\ge L}$, then

$$\begin{aligned} r^2\partial _v(r\phi _L)(u,v)=\sum _{i=0}^{L}\frac{f^{(L,p)}_i(u)}{r^i}+{\mathscr {O}}\left( \frac{\log r}{r^{L+1}}\right) , \end{aligned}$$

(10.5)

where the $f^{(L,p)}_i$ are smooth functions of order $f^{(L,p)}_i={\mathscr {O}}(r_0^{-p+i+1-\varepsilon })$ if $p=L$ and $i<L-1$, and which are given by $f^{(L,p)}_i=C_3^{L,p,i}r_0^{-p+i+1}+{\mathscr {O}}(r_0^{-p+i-\varepsilon })$ otherwise (i.e. if $p=L=i$, if $p=L=i+1,$ or if $p>L$).

In each case, we have explicit, non-vanishing expressions for the constants $C_1,C_2,C_3^{L,p,i}$ that depend only on $L,p,i,C_{\mathrm {in}}$ (and not on M).

c) Finally, $\underline{if \,p\le L}$, then the following limit exists, is non-vanishing, and is independent of u:

$$\begin{aligned} \lim _{v\rightarrow \infty }r^{2+p-L}\partial _v\Phi _L(u,v)=I^{\mathrm {future},r^{2+p-L}}_{\ell =L}[\phi ]\ne 0. \end{aligned}$$

(10.6)

$\underline{If \,p=L+1}$, then the following limit exists, is non-vanishing, and is independent of u:

$$\begin{aligned} \lim _{v\rightarrow \infty }\frac{r^3}{\log r}\partial _v\Phi _L(u,v)=I^{\mathrm {future},\frac{\log r}{r^3}}_{\ell =L}[\phi ]\ne 0. \end{aligned}$$

(10.7)

$\underline{If \,p>L+1}$, then $\partial _v\Phi _L={\mathscr {O}}(r^{-3})$, and all modified Newman–Penrose constants vanish on ${\mathscr {I}}^+$.

In each case, we have explicit expressions for the constants $I^{\mathrm {future},r^{2+L-p}}_{\ell =L}[\phi ],I^{\mathrm {future},\frac{\log r}{r^3}}_{\ell =L}[\phi ]$. These depend only on $L,p,M,C_{\mathrm {in}}$.

All explicit expressions for constants are listed in the proof of this theorem on page 97.

Remark 10.1

Notice that we often expressed u-decay in terms of $r_0$ rather than u in order to compactly express logarithmic contributions: For instance, if $\varepsilon =1$, and if we express decay directly in terms of u, then we have an additional ${\mathscr {O}}(|u|^{-p-1}\log |u|)$-contribution in the second line of (10.3).

Remark 10.2

The faster decay in (10.3) for $p\le L$ can be traced back to certain cancellations. These already happen for $M=0$. In fact, we will, in the course of the proof, derive effective expressions for exact solutions to the wave equation on Minkowski which have data $r\phi _L(u,1)=C_{\mathrm {in}}/r^p$ and satisfy the no incoming radiation condition (10.2). (See Proposition 10.4.)

Remark 10.3

If also the next-to-leading-order behaviour on initial data is specified in (10.1), we can upgrade the ${\mathscr {O}}$-symbols in (10.3) etc. to precise asymptotics, see Corollary 5.

Remark 10.4

With a bit more effort, one can extend the analysis of the proof (using for instance time integrals as was done in [12]) to show that (10.5) can be improved to

$$\begin{aligned} r^2\partial _v(r\phi _L)(u,v)=\sum _{i=0}^{\max {(L,p-1)}}\frac{f^{(L,p)}_i(u)}{r^i}+{\mathscr {O}}\left( \frac{\log r}{r^{\max (L+1,p)}}\right) . \end{aligned}$$

Notice that "the first logarithmic term" of the expansion of $r^2\partial _v(r\phi _L)$ never appears at order $r^{-L}\log r$: It either appears at order $r^{-L-i}\log r$ or at order $r^{-L+i}\log r$, with $i>0$. In this sense, there is a cancellation happening at $L=p$. In particular, if $p=1$, then the expansion of $\partial _v(r\phi _L)$ contains a logarithmic term at order $r^{-3}\log r$ for all $L\ne 1$ (including $L=0$), whereas the first logarithmic term for $L=1$ only appears at order $r^{-4}\log r$.

Remark 10.5

Using the methods of the proof, one can show a very similar result if one assumes more generally that $0\le p\in {\mathbb {R}}$. (In fact, one should also be able to consider a certain range of positive p!) In this case, however, the cancellation (10.3) in general no longer appears; it seems to be a special property of $p\in \{1,\dots , L\}$.

12.3 Overview of the Proof

In contrast to the proof of §9, we will, in this section, only use the approximate conservation law (7.10) and obtain an asymptotic estimate for $[r^2\partial _v]^i(r\phi _L)$ directly from data. For this, we will first need to compute $[r^2\partial _v]^N(r\phi _L)$ for $N\le L$ on data, i.e. on $v=1$, by inductively integrating the relevant equation (7.1). This is done in Proposition 10.1 in §10.4. We then make a bootstrap assumption on the decay of $[r^2\partial _v]^L(r\phi _L)$ and improve it using (7.10). Once we have obtained a sharp estimate on $[r^2\partial _v]^L(r\phi _L)$ in this way (Proposition 10.2 in §10.5), we can then inductively integrate from $v=1$ to obtain a sharp estimate for $[r^2\partial _v]^{L-i}(r\phi _L)$ (Proposition 10.3 in §10.6). In doing so, we pick up an "initial data term" with each integration. These data terms will all be of the same order, so there might be cancellations. We will understand these cancellations in Proposition 10.4 in §10.7. The results are then summarised in Corollary 5. Finally, the proof of Theorem 10.1 is given in §10.8.

The disadvantage of this more direct approach to the asymptotics of $\partial _v$-derivatives of $r\phi _L$ is that we gain no direct information on the asymptotics of $\partial _u$-derivatives. On the other hand, this should also be seen as an advantage since this approach requires no assumption on the conformal regularity of the initial data on $v=1$.

12.4 Computing Transversal Derivatives on Data

Inserting the initial data bound (10.1) into the wave equation (7.1) with $N=0$, and integrating from $u=-\infty$, where $\partial _v(r\phi _L)$ vanishes by the no incoming radiation condition (10.2), we obtain that on $v=1$

$$\begin{aligned} \left| \partial _v(r\phi _L)+\frac{L(L+1)C_{\mathrm {in}}}{(2+p-1)r^{1+p}}\right| \le C r^{-1-p-\varepsilon } \end{aligned}$$

(10.8)

for some constant C. In turn, inserting this estimate into (7.1) with $N=1$, one obtains an estimate for $[r^2\partial _v]^2(r\phi _L)$. Proceeding inductively, one obtains the following

Proposition 10.1

Let $\phi _L$ be as in Theorem 10.1. Then we have on $v=1$ that

$$\begin{aligned}{}[r^2\partial _v]^{N+1}(r\phi _L)&=C_{[r^2\partial _v]^{N+1}}r^{N+1-p}+{\mathscr {O}}(r^{N+1-p-\varepsilon }),&\text {for }N=0,\dots , L-1, \end{aligned}$$

(10.9)

$$\begin{aligned} ^{L+1}(r\phi _L)&=C_{[r^2\partial _v]^{L+1}}r^{L-p}+{\mathscr {O}}(r^{L-p-\varepsilon }).&\end{aligned}$$

(10.10)

Here, we defined the constants

$$\begin{aligned} C_{[r^2\partial _v]^{N+1}}:=&\frac{p!C_{\mathrm {in}}}{(N+1+p)!}\cdot \prod _{i=0}^N(a_0^i-b_0^iL(L+1)),\quad N=0,\dots , L-1, \end{aligned}$$

(10.11)

$$\begin{aligned} C_{[r^2\partial _v]^{L+1}}:=&\frac{2M}{L+2+p}\left( -c_0^L C_{[r^2\partial _v]^{L}}+(a_1^L-b_1^LL(L+1)C_{[r^2\partial _v]^{L-1}}\right) . \end{aligned}$$

(10.12)

Proof

Inductively integrate equation (7.1). $\square$

12.5 Precise Leading-Order Behaviour of $[r^2\partial _v]^L(r\phi _L)$

Equipped with the initial data estimates (10.9), we now prove

Proposition 10.2

Let $\phi _L$ be as in Theorem 10.1. Then we have

$$\begin{aligned}&[r^2\partial _v]^L(r\phi _L)(u,v)\nonumber \\&\quad =[r^2\partial _v]^L(r\phi _L)(u,1)+2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-\min (p,L))!(L+1+p)!}{(2L+2)!}\nonumber \\&\qquad \cdot {\left\{ \begin{array}{ll} \frac{r^{L-1-p}}{(L-1-p)} (1+{\mathscr {O}}(\frac{|u|}{r}+\frac{1}{r^\varepsilon })),&{}L>p+1,\\ \log r-\log |u| +{\mathscr {O}}(1),&{} L=p+1,\\ {\mathscr {O}}(|u|^{L-1-p}) ,&{}L<p+1.\end{array}\right. } \end{aligned}$$

(10.13)

If $L=p$, then we can write more precisely:

$$\begin{aligned}{}[r^2\partial _v]^L(r\phi _L)(u,v)&=[r^2\partial _v]^L(r\phi _L)(u,1)\nonumber \\&\quad + 2MC_{[r^2\partial _v]^{L}}\left( \frac{(2x_1^{(L)}-c_0^L)}{2L+2}-x_1^{(L)}\right) r_0^{-1}+{\mathscr {O}}(|u|^{-1-\varepsilon }). \end{aligned}$$

(10.14)

Notice that if $L<p+1$, then the second line of (10.13) decays faster than the first line, whereas if $L\ge p+1$, the second line determines the leading-order r-behaviour.

In principle, we can also compute the case $L<p$ more precisely, but since it has already been dealt with in §9, we choose not to. Suffice it to say that if $L<p$, then there will also be a logarithmic term at order $(\log r-\log |u)/r^{p+1-L}$, and if $L=p$, there will be a logarithmic term at order $(\log r-\log |u)/r^{p+2-L}$. Cf. Remark 10.4.

Proof

We will prove the proposition by first making a bootstrap assumption to obtain a preliminary estimate on $[r^2\partial _v]^L(r\phi _L)$ (see (10.22)), and then using this preliminary estimate to obtain the sharp leading-order decay. $\square$

12.5.1 A Preliminary Estimate

We can deduce from the energy estimate from Proposition 4.1 that $|r\phi _L|\le C |u|^{-p}$ for some constant C solely determined by initial data. Cf. Corollary 1. (This is the reason why we also assumed decay on the first derivative in (10.1).) By repeating the calculations done in §10.4, we can then derive from $|r\phi _L|\le C |u|^{-p}$ the estimates

$$\begin{aligned}{}[r^2\partial _v]^{N+1}(r\phi _L)(u,v)\le C r^{N+1}|u|^{-p}+{\mathscr {O}}(r^{N+1-p-\varepsilon }) \end{aligned}$$

(10.15)

for $N=0,\dots , L-1$ and another constant C.

Consider now the set X of all $V> 1$ such that the bootstrap assumption

$$\begin{aligned} \left| [r^2\partial _v]^L(r\phi _L)\right| (u,v)\le C_{\mathrm {BS}} \max (r^{L-p},|u|^{L-p}) \end{aligned}$$

(10.16)

holds for all $1\le v\le V$ and for some suitably chosen constant $C_{\mathrm {BS}}$. The $\max$ above distinguishes between the cases of growth ($L-p>0$) and decay ($L-p<0$). For easier readability, we will suppose for the next few lines that $L\ge p$. This assumption will be removed in (10.22). In view of the estimate (10.15), this set is non-empty provided that $C_{\mathrm {BS}}$ is sufficiently large, and it suffices to improve the assumption (10.16) within X to deduce that $X=(1,\infty )$. Indeed, if we assume that $\sup _X v$ is finite and improve estimate (10.16) within X, then (10.15) shows that $\sup _X v+\delta$ would still be in X for sufficiently small $\delta$ by continuity. (Here, we used that the RHS of (10.15) can be written as $C(V)\cdot \max (r^{N+1-p},|u|^{N+1-p})$ for some continuous function C(V).)

Let us therefore improve the bound (10.16) inside X: First, note that (10.16) implies that

$$\begin{aligned} \left| [r^2\partial _v]^{N}(r\phi _L)\right| (u,v)\lesssim C_{\mathrm {BS}} \max (r^{N-p},|u|^{N-p}). \end{aligned}$$

(10.17)

for $N=0,\dots ,L$, where we also used (10.9). Recall now equation (7.10):

$$\begin{aligned} \partial _u(r^{-2L}\partial _v\Phi _L) =\sum _{j=0}^L\frac{D}{r^{3+2L}}(2M)^{j+1}[r^2\partial _v]^{L-j}(r\phi _{L})\left( 2(j+1)x^{(L)}_{j+1}-\sum _{i=0}^j x_i^{(L)} c_{j-i}^{L-i}\right) . \nonumber \\ \end{aligned}$$

(10.18)

As a consequence of (10.17) and the bootstrap assumption (10.16), the RHS is bounded by $r^{-L-3-p}$:

$$\begin{aligned} \left| \partial _u(r^{-2L}\partial _v\Phi _L)\right| \lesssim \frac{2MD}{r^{L+3+p}}\cdot C_{\mathrm {BS}}+{\mathscr {O}}(r^{-L-4-p})\lesssim \frac{C_{\mathrm {BS}}}{r^{L+3+p}}. \end{aligned}$$

(10.19)

We integrate this bound in u from ${\mathscr {I}}^-$, where $r^{-2L}\partial _v\Phi _L=0$ by the no incoming radiation condition (10.2). This yields

$$\begin{aligned} \left| r^{-2L}\partial _v\Phi _L\right| \lesssim \frac{C_{\mathrm {BS}}}{r^{L+2+p}}. \end{aligned}$$

(10.20)

We now recall the definition of $\Phi _L$ from (7.8) and estimate the difference $\partial _v\Phi _L-\partial _v[r^2\partial _v]^L(r\phi _L)$ using once more the bootstrap assumption, resulting in the bound

$$\begin{aligned} \left| \partial _v[r^2\partial _v]^L(r\phi _L)\right| \lesssim C_{\mathrm {BS}}r^{L-2-p}. \end{aligned}$$

(10.21)

Finally, we integrate the bound above from $v=1$,

$$\begin{aligned}&\left| [r^2\partial _v]^L(r\phi _L)(u,v)-[r^2\partial _v]^L(r\phi _L)(u,1)\right| \nonumber \\&\quad \lesssim {\left\{ \begin{array}{ll} C_{\mathrm {BS}}r^{L-1-p},&{} L> 1+p,\\ C_{\mathrm {BS}}(\log r-\log |u|), &{} L=1+p,\\ C_{\mathrm {BS}}|u|^{L-1-p}, &{} L<1+p,\end{array}\right. } \end{aligned}$$

(10.22)

which, combined with the initial data bound (10.9), improves the bootstrap assumption. (The third case in (10.22) follows from considerations similar to the above.) However, we can already read off from (10.22) that, unless $L<1+p$, it is actually the RHS of (10.22) that determines the leading-order r-behaviour of $[r^2\partial _v]^L(r\phi _L)$, whereas the data term on the LHS only determines the leading-order u-behaviour. We will understand the precise behaviour of the RHS in the next section.

12.5.2 Precise Leading-Order Behaviour of $[r^2\partial _v]^L(r\phi _L)$

We again restrict to $p\le L$ for simpler notation, the only major difference if $p>L$ is explained in Remark 10.6. We will also assume for simplicity that $\varepsilon <1$, leaving the case $\varepsilon =1$ to the reader.

In order to find the precise leading-order behaviour of $[r^2\partial _v]^L(r\phi _L)$, we repeat the previous steps, with the difference that we now use the improved estimate²⁴

$$\begin{aligned} \left| [r^2\partial _v]^L(r\phi _L)-C_{[r^2\partial _v]^{L}}|u|^{L-p} \right|&\le C r^{L-1-p}(\delta _{L,p+1}(\log r-\log |u|)+1)\nonumber \\&+C|u|^{L-p-\varepsilon }, \end{aligned}$$

(10.23)

implied by (10.22) and (10.9), instead of the preliminary estimate (10.16). Similarly, we improve the estimate (10.17) to

$$\begin{aligned} \left| [r^2\partial _v]^N(r\phi _L) \right| \le C |u|^{N-p}+Cr^{N-1-p}(\log r-\log |u|+1) \end{aligned}$$

(10.24)

for all $N=0,\dots , L-1$. Inserting these two bounds into (7.10), we obtain

$$\begin{aligned} \partial _u(r^{-2L}\partial _v\Phi _L)= & {} \frac{2MD}{r^{2L+3}}(2x_1^{(L)}-x_0^{(L)}c_0^L)\cdot C_{[r^2\partial _v]^{L}}|u|^{L-p}\nonumber \\&+{\mathscr {O}}\left( r^{-L-4-p}(\log r-\log |u|+1)+r^{-2L-3}|u|^{L-p-\varepsilon }\right) . \nonumber \\ \end{aligned}$$

(10.25)

Integrating the above in u gives

$$\begin{aligned} r^{-2L}\partial _v\Phi _L=\int _{-\infty }^u \frac{2MD}{r^{2L+3}}(2x_1^{(L)}-x_0^{(L)}c_0^L)\cdot C_{[r^2\partial _v]^{L}}|u'|^{L-p}\mathop {}\!\mathrm {d}u'+{\mathscr {O}}(r^{L-2-p-\varepsilon }). \end{aligned}$$

(10.26)

On the LHS of (10.26), we have

$$\begin{aligned} r^{-2L}\partial _v\Phi _L&=r^{-2L}\partial _v[r^2\partial _v]^L(r\phi _L) +2Mx_1^{(L)}\frac{C_{[r^2\partial _v]^{L}}|u|^{L-p}}{r^{2L+2}} \nonumber \\&\quad +{\mathscr {O}}\left( \frac{r^{L-1-p}+\log r +|u|^{L-p-\varepsilon }}{r^{2L+2}}\right) . \end{aligned}$$

(10.27)

In order to estimate the RHS of (10.26), we recall that $x_0^{(L)}=1$, and compute the integral using the following

Lemma 10.1

Let $N, N'\in {\mathbb {N}}$ with $N>N'+1$. Then

$$\begin{aligned} (N-1)\int _{-\infty }^u\frac{|u'|^{N'}}{|r|^{N}}\mathop {}\!\mathrm {d}u'= \sum _{k=0}^{N'}\frac{|u|^{N'-k}}{|r|^{N-1-k}}\prod _{j=1}^k\frac{N'+1-j}{N-1-j}+{\mathscr {O}}(r^{N'-N}).\nonumber \\ \end{aligned}$$

(10.28)

Proof

The proof proceeds almost identically to the proof of Lemma 2.1. Alternatively, one can also compute the integral directly by writing $|u'|=r+v+{\mathscr {O}}(\log r)$. This latter approach is also useful for $N'\notin {\mathbb {N}}$. $\square$

Remark 10.6

When considering the case $p>L$, then the lemma above slightly changes (i.e. for $N'<0$). While it is trivial to obtain the $|u|^{L-1-p}$-decay claimed in (10.13), one can also obtain a more precise statement: In fact, if $N'<0$, then the above integral is precisely the one that gave rise to the logarithmic terms in the previous sections §4 and §9 (see, for instance, Eq. (4.26)). In particular, if $N>0>N'$, the integrals of (10.28) will lead to logarithmic terms at order $(\log r-\log |u|)/r^{N-N'-1}$.

Applying Lemma 10.1 (with $N'=L-p$ and $N=2L-3$) to (10.26), we obtain

$$\begin{aligned} r^{-2L}\partial _v\Phi _L&=2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!}{(2L+2)\cdots (L+2+p)}\cdot \frac{1}{r^{L+2+p}}\nonumber \\&\quad \left( 1+{\mathscr {O}}\left( \frac{|u|}{r}+\frac{1}{r^{\varepsilon }}\right) \right) \end{aligned}$$

(10.29)

where we used that $|u|^q/r^q\le 1$ for any $q>0$. Finally, using (10.27) to write $\partial _v\Phi _L\sim \partial _v[r^2\partial _v]^L(r\phi _L)$, and integrating from $v=1$, we obtain, $\underline{if \,L>1+p}$,

$$\begin{aligned}&[r^2\partial _v]^L(r\phi _L)(u,v)\nonumber \\&\quad = [r^2\partial _v]^L(r\phi _L)(u,1)+2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!(L+1+p)!}{(L-1-p)(2L+2)!}\nonumber \\&\qquad \cdot \, r^{L-1-p} \left( 1+{\mathscr {O}}\left( \frac{|u|}{r}+\frac{1}{r^{\varepsilon }}\right) \right) . \end{aligned}$$

(10.30)

On the other hand, $\underline{if \,L=1+p}$, then we obtain

$$\begin{aligned}&[r^2\partial _v]^L(r\phi _L)(u,v)\nonumber \\&\quad = [r^2\partial _v]^L(r\phi _L)(u,1)+2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!(L+1+p)!}{(2L+2)!}\nonumber \\&\qquad \cdot \,(\log r-\log |u|) +{\mathscr {O}}(1) \end{aligned}$$

(10.31)

where we used that $\log r(u,1)=\log |u|+{\mathscr {O}}(|u|^{-1})$.

The case $\underline{L<1+p}$ follows in much the same way. The only difference is that one now also needs to take the second term on the RHS of (10.27) into account since it will give a contribution of the same order as the $r^{-2L}\partial _v\Phi _L$-term. For the latter, one can derive an estimate similar to (10.29). This concludes the proof of Proposition 10.2. $\square$

12.6 Precise Leading-Order Behaviour of $[r^2\partial _v]^{L-i}(r\phi _L)$

Proposition 10.3

Let $\phi _L$ be as in Theorem 10.1, and let $0\le j\le L$. Then we have $\underline{for \,j<L-1-p:}$

$$\begin{aligned}&[r^2\partial _v]^{L-j}(r\phi _L)(u,v)\nonumber \\&\quad =\mathrm {data}_{L-j}+2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-2-p-j)!(L-p)(L+1+p)!}{(2L+2)!}\nonumber \\&\qquad \cdot \, r^{L-1-p-j}\left( 1+{\mathscr {O}}\left( \frac{|u|}{r}+\frac{1}{r^{\varepsilon }}\right) \right) \end{aligned}$$

(10.32)

On the other hand, $\underline{if \,j=L-1-p}$, we have

$$\begin{aligned}&[r^2\partial _v]^{p+1}(r\phi _L)(u,v)=\mathrm {data}_{p+1}\nonumber \\&\quad +2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)(L+1+p)!}{(2L+2)!}\cdot (\log r-\log |u|) +{\mathscr {O}}(1). \nonumber \\ \end{aligned}$$

(10.33)

Finally, $\underline{if \,j>L-1-p}$, we have

$$\begin{aligned}{}[r^2\partial _v]^{L-j}(r\phi _L)(u,v)=\mathrm {data}_{L-j}+{\mathscr {O}}(|u|^{-j+(L-1-p)}). \end{aligned}$$

(10.34)

Moreover, if also $p\le L-1$, then $[r^2\partial _v]^{L-j}(r\phi _L)$ possesses an asymptotic expansion in powers of 1/r up to $r^{-j+L-p}$, with a logarithmic term appearing at order

$$\begin{aligned} 2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!(L+p+1)!}{(L-p-1)!(2L+2)!}\frac{(-1)^{j-(L-1-p)}}{(j-(L-1-p))!}\cdot \frac{\log r-\log |u|}{r^{j-(L-1-p)}}. \nonumber \\ \end{aligned}$$

(10.35)

In the above, the expression $\mathrm {data}_{L-j}$ is shorthand for

$$\begin{aligned}&\mathrm {data}_{L-j}:=[r^2\partial _v]^{L-j}(r\phi _L)(u,1)\nonumber \\&\quad +\sum _{i=1}^j \underbrace{\int _{r(u,1)}^{r(u,v)}\frac{1}{Dr_{(i)}^2}\dots \int _{r(u,1)}^{r_{(2)}}\frac{1}{Dr_{(1)}^2}\mathop {}\!\mathrm {d}r_{(1)}\cdots \mathop {}\!\mathrm {d}r_{(i)}}_{i \text {integrals}}\cdot [r^2\partial _v]^{L-j+i}(r\phi _L)(u,1). \nonumber \\ \end{aligned}$$

(10.36)

Remark 10.7

Notice that (10.35) only holds for $p\le L-1$. Indeed, we already know from the results of §9 (or §4 for $L=1$) that if $p=L$, then the first logarithmic term in the expansion of $[r^2\partial _v]^{L-j}(r\phi _L)$ will appear at order $\dfrac{\log r-\log |u|}{r^{j-(L-2-p)}}$. For $p>L$, in contrast, one can show the first logarithmic term will appear at $\dfrac{\log r-\log |u|}{r^{j-(L-1-p)}}$, although we won’t show this here. (Again, we have in fact already shown this for $p=L+1$ in §9.) In this sense, there is a cancellation happening at $p=L$.

Proof

We simply need to integrate (10.30) (or, in general, the bound (10.13)) j times from $v=1$, using at each step the initial data bounds (10.9). If $j<L-1-p$, one obtains inductively that

$$\begin{aligned}&[r^2\partial _v]^{L-j}(r\phi _L)(u,v)\nonumber \\&\quad = [r^2\partial _v]^{L-j}(r\phi _L)(u,1)+\sum _{i=1}^j \int _{r(u,1)}^{r(u,v)}\frac{1}{Dr_{(i)}^2}\int _{r(u,1)}^{r_{(i)}}\frac{1}{Dr_{(i-1)}^2}\dots \nonumber \\&\qquad \int _{r(u,1)}^{r_{(2)}}\frac{1}{Dr_{(1)}^2} \mathop {}\!\mathrm {d}r_{(1)}\cdots \mathop {}\!\mathrm {d}r_{i-1}\mathop {}\!\mathrm {d}r_{(i)}[r^2\partial _v]^{L-j+i}(r\phi _L)(u,1)\nonumber \\&\qquad +2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!(L+1+p)!\cdot r^{L-1-p-j}}{(L-1-p)\cdots (L-1-p-j)\cdot (2L+2)!}\nonumber \\&\qquad \cdot \left( 1+{\mathscr {O}}\left( \frac{|u|}{r}+\frac{1}{r^{\varepsilon }}\right) \right) . \end{aligned}$$

(10.37)

Here, the terms in the second line come from integrating the initial data contributions (divided by $r^2$) in the estimates for $[r^2\partial _v]^{L-j+i}(r\phi _L)$. Notice that all these terms have the same leading-order u-decay, so there might be cancellations between them. We will return to this in §10.7. For now, we simply leave them as they are and write them as $\mathrm {data}_{L-j}$.

From (10.37), one deduces that if $j=L-1-p$:

$$\begin{aligned}&[r^2\partial _v]^{p+1}(r\phi _L)(u,v)=\mathrm {data}_{p+1}\nonumber \\&\quad +2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!(L+1+p)!}{(L-1-p)!(2L+2)!}\cdot (\log r-\log |u|) +{\mathscr {O}}(1). \nonumber \\ \end{aligned}$$

(10.38)

Assuming that $L-1-p\ge 0$, we finally integrate (10.38) again from $v=1$ (and write $j':=j-(L-1-p)$) to obtain that, for all $L\ge j\ge L-1-p$:

$$\begin{aligned}&[r^2\partial _v]^{L-j}(r\phi _L)(u,v)=\mathrm {data}_{L-j}\nonumber \\&\quad +2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!(L+p+1)!}{(L-p-1)!(2L+2)!}\frac{(-1)^{j'}}{j'!}\cdot \left( \frac{\log r-\log |u|}{r^{j'}}\right) \nonumber \\&\quad +{\mathscr {O}}(|u|^{-j'}), \end{aligned}$$

(10.39)

where we inductively used that, for any $q>0$,

$$\begin{aligned} (q-1)\int _{r(u,1)}^{r(u,v)}\frac{\log r-\log |u|}{r^q}\mathop {}\!\mathrm {d}r=-\frac{\log r-\log |u|}{r^{q-1}}+\frac{1}{|u|^{q-1}}-\frac{1}{r^{q-1}}. \end{aligned}$$

Notice that, for $j>L-1-p$, in contrast to (10.37) and (10.38), the leading-order r-decay of $[r^2\partial _v]^{p+1-j}(r\phi _L)$ is no longer determined by the second line of (10.39), but by the first line, namely the initial data terms. (If $j=L-p$, the second line still provides the next-to-leading-order behaviour in r.) To nevertheless prove the fourth claim (10.35) of the proposition, one can simply obtain an analogue of (10.39) by integrating the estimate (10.38) j times from future null infinity, rather than from $v=1$. This concludes the proof. $\square$

Setting $j=L$ in the above, we get

$$\begin{aligned}&r\phi _L(u,v)=r\phi _L(u,1)+\sum _{i=1}^L \int _{r(u,1)}^{r(u,v)}\frac{1}{Dr_{(i)}^2}\dots \int _{r(u,1)}^{r_{(2)}}\frac{1}{Dr_{(1)}^2}\mathop {}\!\mathrm {d}r_{(1)}\cdots \nonumber \\&\quad \mathop {}\!\mathrm {d}r_{(i)}[r^2\partial _v]^{i}(r\phi _L)(u,1) +{\mathscr {O}}\left( \frac{\log r-\log |u|}{r^{p+1}}\right) +{\mathscr {O}}\left( \frac{1}{|u|^{p+1}}\right) . \end{aligned}$$

(10.40)

In view of (10.9), the above estimate shows that $r\phi _L=C|u|^{-p}+{\mathscr {O}}(r^{-1}|u|^{-p+1})$ for some constant C, however, this constant C might potentially be zero. Indeed, we already know that this is what happens in the case $L=1=p$ (discussed in §4), to which, in fact, (10.40) applies. (Recall that we showed that $r\phi _1\sim 1/r+1/|u|^2$ if $r\phi _1\sim 1/|u|$ initially.) We discuss these potential cancellations now.

12.7 Cancellations in the Initial Data Contributions

We now analyse the $v=1$-contributions $\mathrm {data}_{L-j}$ in the first (and second) line(s) of (10.37)–(10.39) in more detail. Define $r(u,1)=r_0(u)$. We will prove the following

Proposition 10.4

Let $0\le j\le L$. The expression $\mathrm {data}_{L-j}$ defined in (10.36) evaluates to²⁵

$$\begin{aligned} \mathrm {data}_{L-j}=C_{\mathrm {in}}r_0^{L-p-j}\sum _{n=0}^j S_{L,p,j,n}\left( \frac{r(u,1)}{r(u,v)}\right) ^n+{\mathscr {O}}(r_0^{L-p-j-\varepsilon }), \end{aligned}$$

(10.41)

where the $S_{L,p,j,n}$ are constants that are computed explicitly in Eq. (10.53). They never vanish if $p> L$. However, if $p\le L$, then they vanish if and only if $L-p+n+1\le j\le L$.

Remark 10.8

The computations required for the proof of the above are completely Minkowskian. This is to be understood in the sense that the M-dependence of (10.41) is entirely contained in the ${\mathscr {O}}(r_0^{L-p-j-\varepsilon })$ term. In fact, the above proposition provides us with exact solutions to the linear wave equation on Minkowski that arise from initial data $r\phi _L(u,1)=C_{\mathrm {in}}r_0^{-p}$ and the no incoming radiation condition (10.2).

Proof

We first require an expression for the integrals in $\mathrm {data}_{L-j}$. For this, we prove

Lemma 10.2

Let $k\in {\mathbb {N}}$. Then

$$\begin{aligned} \underbrace{\int _{r(u,1)}^{r(u,v)}\frac{1}{r_{(k)}^2}\dots \int _{r(u,1)}^{r_{(2)}}\frac{1}{r_{(1)}^2}\mathop {}\!\mathrm {d}r_{(1)}\cdots \mathop {}\!\mathrm {d}r_{(k)}}_{k\text { integrals}}=\frac{1}{k!}\left( \frac{1}{r(u,1)}-\frac{1}{r(u,v)}\right) ^k. \end{aligned}$$

(10.42)

Proof

The proof is deferred to the Appendix A.4. $\square$

Equipped with this lemma, we can write the data contributions in the estimates of Proposition 10.3, namely

$$\begin{aligned} \mathrm {data}_{L-j}=\sum _{i=0}^j \int _{r(u,1)}^{r(u,v)}\frac{1}{Dr_{(i)}^2}\dots \int _{r(u,1)}^{r_{(2)}}\frac{1}{Dr_{(1)}^2}\mathop {}\!\mathrm {d}r_{(1)}\cdots \mathop {}\!\mathrm {d}r_{(i)}[r^2\partial _v]^{L-j+i}(r\phi _L)(u,1), \nonumber \\ \end{aligned}$$

(10.43)

as follows, writing from now on $r=r(u,v)$ and $r_0=r(u,1)$ and estimating the $D^{-1}$-terms against $1+{\mathscr {O}}(r_0^{-1})$:

$$\begin{aligned} \mathrm {data}_{L-j}=\sum _{i=0}^j \frac{1}{i!}\left( \frac{1}{r_0}-\frac{1}{r}\right) ^i [r^2\partial _v]^{L-j+i}(r\phi _L)(u,1)\cdot \left( 1+{\mathscr {O}}(r_0^{-1})\right) . \nonumber \\ \end{aligned}$$

(10.44)

We now insert the estimates (10.9) to write this as

$$\begin{aligned} \mathrm {data}_{L-j}= & {} \sum _{i=0}^j \frac{p!\prod _{k=0}^{L-j+i-1}(a_0^k-L(L+1))}{i!(L+p-j+i)!}\left( \frac{1}{r_0}-\frac{1}{r}\right) ^i \nonumber \\&\cdot \,C_{\mathrm {in}}r_0^{L-p-j+i}\left( 1+{\mathscr {O}}(r_0^{-\varepsilon })\right) , \end{aligned}$$

(10.45)

where we used that $b_0^k=1$. By noting that

$$\begin{aligned} a_0^k-L(L+1)=k(k+1)-L(L+1)=-(L+k+1)(L-k), \end{aligned}$$

we can further express the product as

$$\begin{aligned} \prod _{k=0}^{L-j+i-1}(a_0^k-L(L+1))= & {} (-1)^{L-j+i}\frac{(2L-j+i)!L!}{L!(j-i)!}\nonumber \\&=(-1)^{L-j+i}\frac{(2L-j+i)!}{(j-i)!}. \end{aligned}$$

(10.46)

This yields

$$\begin{aligned} \mathrm {data}_{L-j}= & {} \sum _{i=0}^j \frac{ (-1)^{L-j+i}p!(2L-j+i)!}{i!(L+p-j+i)!(j-i)!}\left( \frac{1}{r_0}-\frac{1}{r}\right) ^i \nonumber \\&\cdot \,C_{\mathrm {in}}r_0^{L-p-j+i}\left( 1+{\mathscr {O}}(r_0^{-\varepsilon })\right) . \end{aligned}$$

(10.47)

A cancellation at leading-order, i.e. at order $r_0^{L-p-j}$, takes place if the sum

$$\begin{aligned} \sum _{i=0}^j (-1)^{L-j+i} \frac{p!(2L-j+i)!}{i!(L+p-j+i)!(j-i)!}=:(-1)^{L-j}p!\cdot \mathrm {sum}(L,p,j) \nonumber \\ \end{aligned}$$

(10.48)

vanishes. To understand when this happens, we prove the following

Lemma 10.3

Let $0\le j\le L$. If $p>L$ or $p=0$, then $\mathrm {sum}(L,p,j)$ never vanishes. If $0<p\le L$, then $\mathrm {sum}(L,p,j)$ vanishes if and only if $j\in \{L-p+1,\dots ,L\}$.

More precisely, if $p>L$, then

$$\begin{aligned} \mathrm {sum}(L,p,j)=\overbrace{\int _0^1\int _0^{x_{p-L}}\cdots \int _{0}^{x_{2}}}^{p-L \text { integrals}}x_{1}^{2L-j}\frac{(1-x_{1})^j}{j!}\mathop {}\!\mathrm {d}x_{1}\cdots \, dx_{p-L-1}\mathop {}\!\mathrm {d}x_{p-L},\nonumber \\ \end{aligned}$$

(10.49)

which is manifestly positive. On the other hand, if $p\le L$, then

$$\begin{aligned} \mathrm {sum}(L,p,j)=(-1)^j\left( {\begin{array}{c}L-p\\ L-p-j\end{array}}\right) \frac{(2L-j)!}{(L+p)!}, \end{aligned}$$

(10.50)

where we use the convention that $\left( {\begin{array}{c}n\\ k\end{array}}\right) =0$ if $k<0\le n$.

In fact, equation (10.50) also applies to $p>L$ if we define in the standard way $\left( {\begin{array}{c}L-p\\ L-p-j\end{array}}\right) :=(-1)^{j} \left( {\begin{array}{c}p-L+j-1\\ j\end{array}}\right)$.

Proof

The proof is deferred to the appendix A.5. Notice, however, that one can make certain soft statements without having to do any computations. For instance, if we consider the case $p=1$ and suppose there are no cancellations for $j=L$, then we would obtain from (10.40) an estimate of the form $r\phi _L=C/|u|+{\mathscr {O}}(1/r+1/|u|^2)$. Inserting this into the wave equation (7.1) with $N=0$ would then give that $\partial _v(r\phi _L)\sim \log r/r^2$, a contradiction to the estimate (10.39) for $j=L-1$. Thus, there has to be a cancellation at $j=1$; in other words, $\mathrm {sum}(L,1,1)=0$. However, we here choose to calculate the sums explicitly. $\square$

Lemma 10.3 provides us with an understanding of cancellations at leading-order, i.e. at order $r_0^{L-p-j}$. Similarly, we can understand cancellations at higher order in (10.47), say at order $r_0^{L-p-j+n}r^{-n}$, by considering the corresponding sum

$$\begin{aligned} \sum _{i=0}^j(-1)^n \left( {\begin{array}{c}i\\ n\end{array}}\right) (-1)^{L-j+i} \frac{p!(2L-j+i)!}{i!(L+p-j+i)!(j-i)!}=: (-1)^{L-j}p!\cdot \mathrm {sum}(L,p,j,n). \end{aligned}$$

(10.51)

Understanding this sum is straight-forward: We have

$$\begin{aligned} \mathrm {sum}(L,p,j,n)=&\sum _{i=0}^j(-1)^n\left( {\begin{array}{c}i\\ n\end{array}}\right) (-1)^{i} \frac{(2L-j+i)!}{i!(L+p-j+i)!(j-i)!}\\ =\frac{1}{n!}&\sum _{i=n}^j (-1)^{i-n} \frac{(2L-j+i)!}{(i-n)!(L+p-j+i)!(j-i)!}\\ =\frac{1}{n!}&\sum _{i=0}^{j-n} (-1)^{i} \frac{(2L-j+i+n)!}{i!(L+p-j+i+n)!(j-i-n)!}, \end{aligned}$$

and thus

$$\begin{aligned} \mathrm {sum}(L,p,j,n)=\frac{1}{n!}\mathrm {sum}(L,p,j-n), \end{aligned}$$

(10.52)

where we set $\mathrm {sum}(L,p,j)=0$ if $j<0$. In particular, in view of Lemma 10.3 above, we obtain that if $j\ge n$ and $p> L$, no cancellations occur. On the other hand, if $j\ge n$ and $p\le L$, then cancellations occur if and only if $L-p+n+1\le j\le L$ and $j-n\ge 1$.

This concludes the proof of Proposition 10.4, with the constants $S_{L,p,j,n}$ being given by

$$\begin{aligned} S_{L,p,j,n}=(-1)^{L-j}\frac{p!}{n!}\mathrm {sum}(L,p,j-n), \end{aligned}$$

(10.53)

where $\mathrm {sum}(L,p,j-n)$ is computed explicitly in Lemma 10.3. $\square$

12.8 Summary and Proof of Theorem 10.1

We can roughly (and schematically) summarise the results obtained so far as

$$\begin{aligned}{}[r^2\partial _v]^{L-j}(r\phi _L)(u,v)\sim {\left\{ \begin{array}{ll} |u|^{L-p-j},&{} \text { if } L<p \text { or } j=L-p\ge 0,\\ |u|^{L-p-j}(|u|^{-\varepsilon }+|u|r^{-1}) ,&{} \text { if } L\ge p \text { and } j\ge L-p+1,\\ \log r-\log |u| +|u|, &{} \text { if } L> p \text { and } j=L-p-1,\\ r^{L-1-p-j}+|u|^{L-p-j}, &{} \text { if } L> p \text { and } 0\le j<L-p-1. \end{array}\right. } \nonumber \\ \end{aligned}$$

(10.54)

The first case follows from estimate (10.34) from Proposition 10.3 and the fact that there are no cancellations in the data term $\mathrm {data}_{L-j}$ in view of Proposition 10.4. The leading-order behaviour is thus entirely determined by the data.

In contrast, the second case follows from (10.34) from Proposition 10.3 and the fact that there are cancellations in the data term $\mathrm {data}_{L-j}$ in view of Proposition 10.4. Notice moreover that if $\varepsilon <1$, then the leading-order behaviour will only have contributions from the data. (To see this, one needs to repeat the calculations of Proposition 10.1, taking into account also the subleading terms.) If $\varepsilon =1$, then there will, in addition, be contributions from the ${\mathscr {O}}$-terms in (10.34). Note that, if desired, all of these contributions can be computed explicitly by following the steps above but without discarding the subleading terms.

The third case follows from (10.33) from Proposition 10.3, with the |u|-term coming again from the data contribution $\mathrm {data}_{L-j}$, which contains no cancellations in view of Proposition 10.4.

The fourth case follows in the same way from (10.32) from Proposition 10.3, with the $|u|^{L-p-j}$-term coming again from the data contribution $\mathrm {data}_{L-j}$, which contains no cancellations in view of Proposition 10.4.

More precisely, we have the following

Corollary 5

Let $\phi _L$ and ${\mathscr {D}}$ be as in Theorem 10.1, and recall that $r_0:=r(u,1)=|u|-2M\log |u|+{\mathscr {O}}(1)$, as well as the constants $S_{L,p,j,n}$ defined in (10.53).

• 1.) If $\underline{L<p and j\ge 0}$, or if $\underline{j=L-p\ge 0}$, then we have throughout ${\mathscr {D}}$:

  $$\begin{aligned}{}[r^2\partial _v]^{L-j}(r\phi _L)=C_{\mathrm {in}}S_{L,p,j,0}\cdot r_0^{L-p-j}(1+{\mathscr {O}}(|u|r^{-1}+|u|^{-\varepsilon })). \end{aligned}$$

  (10.55)

• 2.) If $\underline{L\ge p and j\ge L-p+1}$, then we have throughout ${\mathscr {D}}$:

  $$\begin{aligned}{}[r^2\partial _v]^{L-j}(r\phi _L)={\mathscr {O}} (r_0^{L-p-j-\varepsilon })+{\mathscr {O}}(r^{-1}|u|^{L-p-j+1}). \end{aligned}$$

  (10.56)

  Indeed, if we suppose instead of (10.1) that

  $$\begin{aligned} |r\phi -C_{\mathrm {in}}r^{-p}+C_{\mathrm {in}}{}_{,\varepsilon } r^{-p-\varepsilon }|\le C r^{-p-\varepsilon '} \end{aligned}$$

  (10.57)

  for some constants $C, C_{\mathrm {in}}{}_{,\varepsilon }$ and for some $0<\varepsilon \le 1<\varepsilon '$, then we have

  $$\begin{aligned}{}[r^2\partial _v]^{L-j}(r\phi _L)={{\tilde{C}}} \cdot r_0^{L-p-j-\varepsilon } +{\mathscr {O}}(|u|^{L-p-j-1})+{\mathscr {O}}(r^{-1}|u|^{L-p-j+1}) \nonumber \\ \end{aligned}$$

  (10.58)

  for some constant ${{\tilde{C}}}={{\tilde{C}}}(L,p,j,\varepsilon ,M,C_{\mathrm {in}},C_{\mathrm {in}}{}_{,\varepsilon })$ which we can compute explicitly.

• 3.) If $\underline{L\ge p and j=L-p-1}$, then we have throughout ${\mathscr {D}}$:

  $$\begin{aligned}{}[r^2\partial _v]^{p+1}(r\phi _L)&=C'_1 \cdot (\log r-\log |u|)+C_{\mathrm {in}}S_{L,p,L-p-1,0}\cdot r_0 \nonumber \\&+{\mathscr {O}}(|u|^{1-\varepsilon }), \end{aligned}$$

  (10.59)

  with the constant $C'_1$ being given by

  $$\begin{aligned} C'_1=(-1)^L\cdot 2M(2x_1^{(L)}-c_0^L)\cdot \frac{p!(L-p)(L+1+p)}{(2L+2)(2L+1)}\cdot C_{\mathrm {in}}. \end{aligned}$$

  (10.60)

• 4.) If $\underline{L>p and 0\le j\le L-p-2}$, then we have throughout ${\mathscr {D}}$:

  $$\begin{aligned}&[r^2\partial _v]^{L-j}(r\phi _L)=C'_2 \cdot r^{L-1-p-j}+C_{\mathrm {in}}S_{L,p,j,0} \cdot r_0^{L-p-j} +{\mathscr {O}}(|u|^{L-p-j-\varepsilon })\nonumber \\&+{\mathscr {O}}(r^{L-2-p-j}|u|), \end{aligned}$$

  (10.61)

  with the constant $C'_2$ being given by $C'_2=(L-2-p-j)!\cdot C'_1$.

Proof

The proof is obtained by combining the results of Propositions 10.1–10.4 in the manner described above (below (10.54)). Notice that we expressed the constant $C_{[r^2\partial _v]^{L}}$ appearing in (10.32) and (10.33) (and defined in (10.11)) as $C_{[r^2\partial _v]^{L}}=(-1)^L \frac{p!(2L)!}{(L+p)!}\cdot C_{\mathrm {in}}$, which follows from (10.46). $\square$

12.8.1 Proof of Theorem 10.1

Proof of Theorem 10.1

Part a) of the theorem follows directly from Corollary 5, with $C_0$ given by $C_0=S_{L,p,L,0}$.

The first part of b) follows by dividing (10.59) by $r^2$, integrating from ${\mathscr {I}}^+$, and repeating the procedure $p-1$ terms. The boundary terms $\lim _{v\rightarrow \infty }[r^2\partial _v]^{L-p-i}(u,v)$ we pick up with each integration are estimated via (10.55) or (10.56), thus giving rise to the functions $f_i^{(L,p)}$ and their leading-order behaviour. This proves (10.4), with the constants $C_1, C_2$ given by

$$\begin{aligned} MC_1=\frac{(-1)^p}{p!}C_1',&C_2=\frac{(-1)^p}{p!} S_{L,p,L-p-1,0}\cdot C_{\mathrm {in}}. \end{aligned}$$

(10.62)

The second part of b) follows similarly: We take (10.55) with $j=0$ and integrate $L-1$ times from ${\mathscr {I}}^+$, using at each step either (10.55) or (10.56) to estimate the boundary terms on ${\mathscr {I}}^+$. This alone only gives an expansion of $r^2\partial _v(r\phi _L)$ up to $r^{1-L}$. The higher-order behaviour can be obtained by also taking into account the estimate for $\partial _v[r^2\partial _v]^L(r\phi _L)$ implied by the estimates (10.27) and (10.29). One can obtain expressions for the constants $C_3^{L,p,i}$ in much the same way as for $C_1,C_2$, using also (10.27) and (10.29) for $i=L$.

Finally, part c) of Theorem 10.1 follows by (in the case $p=L+1$ a slightly modified version of) (10.29). We have for $p\le L$ (recall $C_{[r^2\partial _v]^{L}}=(-1)^L \frac{p!(2L)!}{(L+p)!}\cdot C_{\mathrm {in}}$):

$$\begin{aligned} I_{\ell =L}^{\mathrm {future}, r^{2+p-L}}[\phi ]=2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L)\frac{(L-p)!}{(2L+2)\cdots (L+2+p)}, \end{aligned}$$

(10.63)

and for $p=L+1$:

$$\begin{aligned} I_{\ell =L}^{\mathrm {future}, \frac{\log r}{r^3}}[\phi ]=2MC_{[r^2\partial _v]^{L}}(2x_1^{(L)}-c_0^L). \end{aligned}$$

(10.64)

This concludes the proof of Theorem 10.1. $\square$

12.9 Comments: More Severe Modifications to Price’s Law

We have already discussed in detail in §9.5 that we expect to obtain a logarithmically modified Price’s law for each $\ell$ provided that one smoothly extends the data to the event horizon and that $p=L+1$ in (10.1), which, in turn, is the decay predicted by the results of §8. On the other hand, in view of equation (10.6), one can expect that the modification to Price’s law is much more severe for $p\le L>0$: Indeed, we expect that if $p\le L$, then one obtains asymptotics near $i^+$ that are $L-p+1$ powers worse than in the case of smooth compactly supported data, i.e., we expect that

$$\begin{aligned} r\phi _L|_{{\mathscr {I}}^+}\sim u^{-L-2+(L-p+1)}=u^{-1-p},\quad \phi _L|_{{\mathscr {H}}^+}\sim v^{-2L-3+(L-p+1)}=v^{-L-p-2} \nonumber \\ \end{aligned}$$

(10.65)

near $i^+$. The reader should compare this to the behaviour of the $\ell =0$-mode for $p=1$,

$$\begin{aligned} r\phi _0|_{{\mathscr {I}}^+}\sim u^{-2}\log u,\quad \phi _0|_{{\mathscr {H}}^+}=v^{-3}\log v, \end{aligned}$$

(10.66)

which was proved in [13].

We again refer the reader to §1.3 and Conjecture 2 therein for a more detailed discussion.

Appendix

This appendix contains various proofs which have been omitted in the main body of the paper.

1.1 A.1 Proof of Lemma 2.1

Proof

The result clearly holds $N=2$. Let us now assume that (2.12) holds for a fixed N and for all $N'<N-1$. Then it also holds for $N+1$ and for any $N'<N$. Indeed,

$$\begin{aligned} N\int _{-\infty }^u\frac{r^{N'}}{|u'|^{N+1}}\mathop {}\!\mathrm {d}u'=\int _{-\infty }^u \partial _u\left( \frac{r^{N'}}{|u'|^{N}} \right) +\frac{N'r^{N'-1}}{|u'|^{N}}\left( 1-\frac{2M}{r}\right) \mathop {}\!\mathrm {d}u', \end{aligned}$$

(A.1)

and we can apply the induction assumption to the second term on the RHS to obtain the result (after some standard shifting of indices). $\square$

1.2 A.2 Proofs of Propositions 7.1 and 7.2

Proof of Proposition 7.1

We prove (7.1) by induction, noting that it is true for $N=0$ with $a_0^0=0$, $b_0^0=1=c_0^0$. Assume now that (7.1) holds for a fixed N. We have

$$\begin{aligned} \partial _v\partial _u[r^2\partial _v]^{N+1}(r\phi )= & {} \partial _v\left( r^2\partial _u\partial _v[r^2\partial _v]^N(r\phi )\right) +\partial _v\left( -2r^2\cdot \frac{D}{r}\partial _v[r^2\partial _v]^N(r\phi )\right) \nonumber \\= & {} \partial _v\left( r^2\partial _u\partial _v[r^2\partial _v]^N(r\phi )\right) -\frac{2D}{r}\partial _v[r^2\partial _v]^{N+1}(r\phi )+\left( 1-\frac{4M}{r}\right) \nonumber \\&\frac{2D}{r^2}[r^2\partial _v]^{N+1}(r\phi ). \end{aligned}$$

(A.2)

Using the induction hypothesis, we compute the first term on the RHS according to

(A.3)

where we defined $a_j^N, b_j^N, c_j^N:=0$ for $j>N$.

Plugging the above equation back into (A.2), we thus obtain

(A.4)

with

$$\begin{aligned} a_0^{N+1}=a_0^N+2(N+1),&b_0^{N+1}=b_0^N,&c_0^{N+1}=c_0^N+4(N+1), \end{aligned}$$

(A.5)

and the additional relations

$$\begin{aligned} a_j^{N+1}&=a_j^N+a_{j-1}^N+c_{j-1}^N, \end{aligned}$$

(A.6)

$$\begin{aligned} b_j^{N+1}&=b_{j}^N+b_{j-1}^N, \end{aligned}$$

(A.7)

$$\begin{aligned} c_{j}^{N+1}&=c_j^N+2c_{j-1}^N. \end{aligned}$$

(A.8)

This proves equation (7.1).

To find the explicit expressions for $a_j^N, b_j^N$ and $c_j^N$, we need to solve the recurrence relations above. From (A.5), we read off that

$$\begin{aligned} a_0^{N}=N(N+1),&b_0^N=1,&c_0^N=1+2N(N+1) \end{aligned}$$

for all $N\ge 0$. From this and (A.7), we can then read off that $b_j^N=\left( {\begin{array}{c}N\\ j\end{array}}\right)$ for all $N \ge 0$ and $0\le j\le N$. Similarly, by writing ${\tilde{c}}_j^N=2^{-j}c_j^N$, we find from (A.8) that

$$\begin{aligned}{\tilde{c}}_j^N=\left( {\begin{array}{c}N\\ j\end{array}}\right) +4\left( {\begin{array}{c}N+1\\ j+2\end{array}}\right) .\end{aligned}$$

Finally, plugging in the expression for $c_j^N$ into (A.6), one finds

$$\begin{aligned}a_j^N=(2^j-1)\left( {\begin{array}{c}N\\ j\end{array}}\right) +(2^{j+2}-2)\left( {\begin{array}{c}N+1\\ j+2\end{array}}\right) .\end{aligned}$$

This completes the proof of Proposition 7.1. $\square$

Proof of Proposition 7.2

The proof follows along the same lines as the previous one, with the difference that one now obtains the linear system

$$\begin{aligned} {\underline{a}}_0^{N+1}={\underline{a}}_0^N+2(N+1),&{\underline{b}}_0^{N+1}={\underline{b}}_0^N,&{\underline{c}}_0^{N+1}={\underline{c}}_0^N+4(N+1), \end{aligned}$$

(A.9)

and the additional relations

$$\begin{aligned} {\underline{a}}_j^{N+1}={\underline{a}}_j^N-{\underline{a}}_{j-1}^N-{\underline{c}}_{j-1}^N,&{\underline{b}}_j^{N+1}={\underline{b}}_{j}^N-{\underline{b}}_{j-1}^N,&{\underline{c}}_{j}^{N+1}={\underline{c}}_j^N-2{\underline{c}}_{j-1}^N. \end{aligned}$$

(A.10)

One can relate this to the previous system (A.5)–(A.8) by writing $\underline{{{\tilde{b}}}}_j^N=(-1)^{j}{\underline{b}}_j^N$ etc. $\square$

1.3 A.3 Proof of Lemma 8.3

Proof

Equation (8.57) clearly holds for $n=0$. Assume that it holds for a fixed n. We shall show that it also holds for $n+1$. Letting $[\cdot ,\cdot ]$ denote the usual commutator, we have

$$\begin{aligned}&(r^2\partial _u)^{n+1} f=(r^2T-r^2\partial _v)(r^2T-r^2\partial _v)^nf\nonumber \\&\quad = \sum _{k=0}^n (-1)^{n-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \sum _{i=0}^{k-1}\alpha ^{(n,k)}_i(r+{\mathscr {O}}(1))^i(r^2 T)^{k+1-i}\right) [r^2\partial _v]^{n-k}f\nonumber \\&\qquad + \sum _{k=0}^n (-1)^{n+1-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \sum _{i=0}^{k-1}\alpha ^{(n,k)}_i(r+{\mathscr {O}}(1))^i(r^2 T)^{k-i}\right) [r^2\partial _v]^{n+1-k}f\nonumber \\&\qquad - \sum _{k=0}^n (-1)^{n+1-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \sum _{i=0}^{k-1}\alpha ^{(n,k)}_i\left[ r^2\partial _v,(r+{\mathscr {O}}(1))^i(r^2 T)^{k-i}\right] \right) [r^2\partial _v]^{n-k}f \end{aligned}$$

(A.11)

by commuting $r^2\partial _v$ past the other terms. We now write

$$\begin{aligned} \begin{aligned}&\left[ r^2\partial _v,(r+{\mathscr {O}}(1))^i(r^2 T)^{k-i}\right] f\\&\quad = \left[ r^2\partial _v,(r+{\mathscr {O}}(1))^i\right] (r^2 T)^{k-i}f+(r+{\mathscr {O}}(1))^i\left[ r^2\partial _v,(r^2 T)^{k-i}\right] f, \end{aligned} \end{aligned}$$

(A.12)

and compute

$$\begin{aligned} \left[ r^2\partial _v,(r+{\mathscr {O}}(1))^i\right] f=iD r^{i+1}f+{\mathscr {O}}(r^i)f, \end{aligned}$$

(A.13)

as well as

$$\begin{aligned} \begin{aligned} \left[ r^2\partial _v,(r^2 T)^{k-i}\right] f=(k-i)\left[ r^2\partial _v,r^2 T\right] (r^2T)^{k-i-1}f =(k-i)2Dr (r^2T)^{k-i} f, \end{aligned} \end{aligned}$$

(A.14)

where we used that $[V_1,V_2^n]=n[V_1,V_2]V_2^{n-1}$, which holds true if $[[V_1,V_2],V_2]=0$. Plugging the above identities back into (A.11), one then recovers (8.57) for $n+1$ in a standard way. This gives rise to recurrence relations for the $\alpha _i^{(n,k)}$, which can be solved explicitly. We leave this to the interested reader.

Alternatively, we could have used the non-commutative binomial theorem to write

$$\begin{aligned} (r^2T-r^2\partial _v)^n f=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( r^2T+[r^2\partial _v, \cdot ]\right) ^k(-r^2\partial _v)^{n-k} f, \end{aligned}$$

(A.15)

and computed the commutators directly. $\square$

1.4 A.4 Proof of Lemma 10.2

Proof

Let us inductively assume that

$$\begin{aligned} {\int _{r_0}^{r}\frac{1}{r_{(k)}^2}\dots \int _{r_0}^{r_{(2)}}\frac{1}{r_{(1)}^2}\mathop {}\!\mathrm {d}r_{(1)}\cdots \mathop {}\!\mathrm {d}r_{(k)}}=\frac{1}{k!}\left( \frac{1}{r_0}-\frac{1}{r}\right) ^k \end{aligned}$$

(A.16)

for any $r\ge r_0>0$. This holds true for $k=1$. Going from k to $k+1$, we have

$$\begin{aligned} {\int _{r_0}^{r}\frac{1}{r'^2}\int _{r_0}^{r'}\frac{1}{r_{(k)}^2}\dots \int _{r_0}^{r_{(2)}}\frac{1}{r_{(1)}^2}\mathop {}\!\mathrm {d}r_{(1)}\cdots \mathop {}\!\mathrm {d}r_{(k)}\mathop {}\!\mathrm {d}r'}=\int _{r_0}^r \frac{1}{r'^2}\frac{1}{k!}\left( \frac{1}{r_0}-\frac{1}{r'}\right) ^k\mathop {}\!\mathrm {d}r'. \end{aligned}$$

(A.17)

We write $x=1/r$ and $x_0=1/r_0$ in the integral above to obtain

$$\begin{aligned} \int _{r_0}^r \frac{1}{r'^2} \frac{1}{k!}\left( \frac{1}{r_0}-\frac{1}{r'}\right) ^k\mathop {}\!\mathrm {d}r'=(-1)^{k+1}\int _{x_0}^x \frac{1}{k!}(x'-x_0)^k\mathop {}\!\mathrm {d}x'=\frac{1}{(k+1)!}(x_0-x)^{k+1} \end{aligned}$$

(A.18)

for any $r\ge r_0>0$. This concludes the proof. $\square$

1.5 A.5 Proof of Lemma 10.3

Proof

For convenience, we recall the definition

$$\begin{aligned} \mathrm {sum}(L,p,j):=\sum _{i=0}^j (-1)^{i} \frac{(2L-j+i)!}{i!(L+p-j+i)!(j-i)!}. \end{aligned}$$

(A.19)

We begin with the crucial observation that if $p=L$ and $j\ge 1$, then

$$\begin{aligned} \mathrm {sum}(L,L,j)=\sum _{i=0}^j (-1)^{i} \frac{1}{i!(j-i)!} =\sum _{i=0}^j\frac{1}{j!}\left( {\begin{array}{c}j\\ i\end{array}}\right) (-1)^i\cdot 1^j=\frac{1}{j!}(1-1)^j=0. \end{aligned}$$

(A.20)

The idea is to interpret the above sum as a function of some variable x evaluated at $x=1$: Consider, for instance, the case where $p=L-1$. Then

$$\begin{aligned} \begin{aligned} \mathrm {sum}(L,L-1,j)&=\sum _{i=0}^j (-1)^i\frac{2L+i-j}{i!(j-i)!} =\left. \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}x}\right| _{x=1}\sum _{i=0}^j (-1)^i\frac{1}{i!(j-i)!}x^{2L+i-j}\\&=(-1)^j\left. \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}x}\right| _{x=1}\left( x^{2L-j}\sum _{i=0}^j (-1)^{j-i}x^i\frac{1}{i!(j-i)!}\right) \\&=(-1)^j\left. \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}x}\right| _{x=1}\left( x^{2L-j}\frac{(x-1)^j}{j!}\right) . \end{aligned} \end{aligned}$$

(A.21)

Similarly, if $p=L-k$ for some $k\ge 0$, then we obtain inductively that

$$\begin{aligned} \mathrm {sum}(L,L-k,j)=(-1)^j\left. \frac{\mathop {}\!\mathrm {d}^k}{\mathop {}\!\mathrm {d}x^k}\right| _{x=1}\left( x^{2L-j}\frac{(x-1)^j}{j!}\right) . \end{aligned}$$

(A.22)

We then compute the k-th derivative above using the Leibniz rule:

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}^k}{\mathop {}\!\mathrm {d}x^k}\left( x^{2L-j}\frac{(x-1)^j}{j!}\right) =\sum _{n=0}^k \left( {\begin{array}{c}k\\ n\end{array}}\right) \frac{(2L-j)!x^{2L-j-n}}{(2L-j-n)!}\cdot \frac{(x-1)^{j-(k-n)}}{(j-(k-n))!}, \end{aligned}$$

(A.23)

where we use the convention that $\frac{1}{(-n)!}=0$ for all $n\in {\mathbb {N}}^+$. In particular, upon evaluating the expression (A.23) at $x=1$, we get

$$\begin{aligned} \left. \frac{\mathop {}\!\mathrm {d}^k}{\mathop {}\!\mathrm {d}x^k}\right| _{x=1}\left( x^{2L-j}\frac{(x-1)^j}{j!}\right) =\sum _{n=0}^k \left( {\begin{array}{c}k\\ n\end{array}}\right) \frac{(2L-j)!}{(2L-j-n)!}\cdot \delta _{k-n,j}=\left( {\begin{array}{c}k\\ k-j\end{array}}\right) \frac{(2L-j)!}{(2L-k)!}, \end{aligned}$$

(A.24)

which proves the second formula (10.50) of the proposition.

On the other hand, if $p=L+1$, we integrate the corresponding sum (instead of differentiating):

$$\begin{aligned} \begin{aligned} \mathrm {sum}(L,L+1,j)&=\sum _{i=0}^j (-1)^i\frac{1}{i!(j-i)!(2L+1-j+i)}\\&=\int _0^1\sum _{i=0}^j (-1)^i\frac{1}{i!(j-i)!}x^{2L+i-j}\mathop {}\!\mathrm {d}x =\int _0^1 x^{2L-j}\frac{(1-x)^j}{j!}\mathop {}\!\mathrm {d}x, \end{aligned} \end{aligned}$$

(A.25)

which is manifestly positive. Equation (10.49) then follows inductively. $\square$