1 Introduction

When studying linear and nonlinear wave equations on black hole spacetimes, such as the Kerr spacetime and Kerr–de Sitter spacetime, quasinormal modes play a prominent role. Indeed, for linear equations, within certain limitations corresponding to trapped null-geodesics, solutions have an asymptotic expansion at timelike infinity in quasinormal modes. Such expansions, or the corresponding decay or lack thereof statements, have a long history which in the mathematics literature goes back to Sá Barreto and Zworski [SBZ97], Bony and Häfner [BH08], Dyatlov [Dya11, Dya12], Vasy [Vas13], Shlapentokh-Rothman [SR15], Hintz and Vasy [HV15] and Gajic and Warnick [GW20]. In the physics literature the importance of these has been clear even longer, going back to Regge and Wheeler [RW57], Vishveshwara [Vis70], Zerilli [Zer70], Whiting [Whi89], Kodama, Ishibashi and Seto [KIS00] and others. For nonlinear equations the non-decaying quasinormal modes become an obstacle to solvability; for equations with gauge freedom, such as Einstein’s equation, it is non-decaying modes that are not ‘pure gauge’ that play an analogous role [HV18].

Quasinormal modes are solutions of the homogeneous wave equation which are eigenfunctions of the covariant derivative along appropriate Killing vector fields. A key consideration for applications is that for similar covariant eigenfunctions as the forcing (right hand side of the wave equation), there should be a satisfactory Fredholm theory. In this case the covariant eigenvalues (resonances) form a discrete set, and the corresponding eigenspaces are finite dimensional. Since Fredholm theory is global, this necessitates working relative to Killing vector fields with suitable global behavior.

Recently Galkowski and Zworski [GZ21] showed that quasinormal modes for non-rotating black holes are real analytic at the horizon; indeed they obtained a substantially stronger microlocal result. In this paper we generalize their result to the case of rotating black holes whose importance is underlined by their ubiquity. Our proof relies crucially on the ability to locally transform the rotating black hole quasimode problem to the non-rotating one, and thus being able to apply the result of [GZ21]. This transformation is facilitated by locally considering analogues of quasinormal modes with respect to a different Killing vector field that is lightlike on the horizon; this change is very simple in the Kerr and Kerr–de Sitter case as we discuss below, but in fact works in general for non-degenerate Killing horizons under an additional condition, as is also described below. While these modes are with respect to a different Killing vector field, we can in fact relate these to the quasinormal modes with respect to the original globally well-behaved vector field to obtain the real analyticity result. Indeed, a key feature of the Kerr–de Sitter setting is the presence of two horizons, and the well-behaved Killing vector fields with respect to each of these horizons, while globally well-defined, are ill-behaved at the other horizon. Thus, it is of central importance for our approach to be able to work locally near a horizon to obtain the analyticity conclusions.

It is conceivable that the results of [GZ21] could be proven in the more general setting of [Vas13], which would imply analyticity in the case of rotating black holes. However, such a proof would be significantly more technically involved than the proof we give here. The proof of analytic hypoellipticity in [GZ21] relies on a microlocal normal form of Haber [Hab14], which in turn relies on the relevant Lagrangian (in our case the conormal bundle of the horizon) being radial with respect to the Hamilton vector field. For rotating black holes, there is more intricate internal dynamics and one could therefore not directly apply the results of Haber. Indeed, our reduction to [GZ21] can be considered as a way of ‘straightening’ the dynamics and thus bringing it to a model form.

Furthermore, it could perhaps be possible to prove analytic hypoellipticity of Keldysh-type operators more explicitly using ODE theory and separation of variables and thereby avoid referring to [GZ21]. This was the approach by Lebeau and Zworski in [LZ19] (see also the work by Zuily in [Zui17]), where an explicit class of Keldysh type operators where studied, and certain values for the subprincipal symbol had to be excluded. However, the main purpose of this paper is to reduce the hypoanalyticity of quasinormal modes in Kerr(-de Sitter) spacetimes to the irrotational, i.e. Keldysh-type, case. Applying the ODE approach directly in the rotational case seems cumbersome and it is not clear to us whether it would work.

In the rest of the introduction we describe the precise results in the rotating black hole setting, as well as the generalization to non-degenerate Killing horizons. Then in Sect. 2 we discuss geometric aspects of these Killing horizons. In Sect. 3 we then prove our general local result. In Sect. 4 we use these local results to obtain a global result for joint modes of two Killing vector fields on Kerr and Kerr–de Sitter spacetimes. Finally, in Sect. 5 we show how these results imply the real analyticity of the quasinormal modes on Kerr and Kerr–de Sitter spacetimes, with modes taken with respect to the standard Killing vector field.

1.1 Kerr and Kerr–de Sitter spacetime

Fix three parameters \(a \in {\mathbb {R}}\) and \(m, \Lambda \ge 0\), such that the polynomial

$$\begin{aligned} \mu (r):= \left( r^2 + a^2\right) \left( 1 -\frac{\Lambda r^2}{3}\right) - 2mr \end{aligned}$$
(1)

has four distinct real roots \(r_-< r_C< r_e < r_c\) if \(\Lambda > 0\) and two distinct real roots \(r_C < r_e\) if \(\Lambda = 0\). The latter condition is equivalent to \(\left|a\right| < m\).

Assuming \(\Lambda > 0\), the domain of outer communication in the sub-extermal Kerr–de Sitter spacetime is given in Boyer-Lindquist coordinates by the real analytic spacetime

$$\begin{aligned} {\mathbb {R}}_t \times (r_e, r_c)_r \times S^1_\phi \times (0, \pi )_\theta , \end{aligned}$$

with real analytic metric

$$\begin{aligned} \begin{aligned} g&= (r^2 + a^2 \cos ^2(\theta ))\left( \frac{\mathrm dr^2}{\mu (r)} + \frac{\mathrm d\theta ^2}{c(\theta )}\right) \\&\quad +\frac{c(\theta )\sin ^2(\theta )}{b^2\left( r^2 + a^2 \cos ^2(\theta )\right) }\left( a \mathrm dt - \left( r^2 + a^2\right) \mathrm d\phi \right) ^2 \\&\quad - \frac{\mu (r)}{b^2\left( r^2 + a^2 \cos ^2(\theta )\right) }\left( \mathrm dt - a \sin ^2(\theta )\mathrm d\phi \right) ^2, \end{aligned} \end{aligned}$$
(2)

where

$$\begin{aligned} b:= 1 + \frac{\Lambda a^2}{3}, \quad c(\theta ):= 1 + \frac{\Lambda a^2}{3} \cos ^2(\theta ). \end{aligned}$$

The domain of outer communication in the subextremal Kerr spacetime is defined analogously, by passing to the limit \(\Lambda = 0\). We set \(r_c = \infty \) if \(\Lambda = 0\). The Boyer–Lindquist coordinates could be thought of as spherical coordinates around the black hole, where the black hole is centered at \(r = 0\). Even though they are not defined at the north and south poles \(\theta = 0\) and \(\pi \), it is straightforward to check that the metric (2) extends real analytically to

$$\begin{aligned} M:= {\mathbb {R}}_t \times (r_e, r_c)_r \times S^2_{\phi , \theta }. \end{aligned}$$

Since the metric extends real analytically, so does linear wave equations with a principal symbol given by the metric in these coordinates. We refer to [HV18, Sec. 3], for a more thorough discussion of the geometry of Kerr–de Sitter spacetimes.

These coordinates are singular at the roots of \(\mu (r)\). In order to define quasinormal modes, we need to extend this metric real analytically over the future event horizon and the future cosmological horizon, corresponding to the roots \(r = r_e\) and \(r = r_c\), respectively. This can be done, for instance, by the following coordinate change:

$$\begin{aligned} t_*&:= t - \Phi (r), \\ \phi _*&:= \phi - \Psi (r), \end{aligned}$$

where \(\Phi \) and \(\Psi \) satisfy

$$\begin{aligned} \Phi '(r)&= b \frac{r^2 + a^2}{\mu (r)} f(r), \\ \Psi '(r)&= b \frac{a}{\mu (r)} f(r). \end{aligned}$$

In the case \(\Lambda > 0\), we let \(f: (r_e-\delta , r_c+\delta ) \rightarrow {\mathbb {R}}\), for a small \(\delta > 0\), be a real analytic function such that

$$\begin{aligned} f(r_e) = - 1 \end{aligned}$$

and

$$\begin{aligned} f(r_c) = 1. \end{aligned}$$

In the case \(\Lambda = 0\), there is no cosmological horizon, so we instead assume that

$$\begin{aligned} \lim _{r \rightarrow \infty }f(r) = 1. \end{aligned}$$

The metric (2) extends real analytically to the manifold

$$\begin{aligned} M_*:= {\mathbb {R}}_{t_*} \times (r_e-\delta , r_c+\delta )_r \times S^2_{\phi _*, \theta }, \end{aligned}$$

and is given by

$$\begin{aligned} \begin{aligned} g_*&= (r^2 + a^2 \cos ^2(\theta )) \frac{1 - f(r)^2}{\mu (r)}\mathrm dr^2\\&\qquad - \frac{2}{b} f(r)(\mathrm dt_* - a \sin ^2(\theta ) \mathrm d\phi _*)\mathrm dr \\&\qquad - \frac{\mu (r)}{b^2\left( r^2 + a^2 \cos ^2(\theta )\right) }\left( \mathrm dt_* - a \sin ^2(\theta ) \mathrm d\phi _* \right) ^2 \\&\qquad + \frac{c(\theta )\sin ^2(\theta )}{b^2\left( r^2 + a^2 \cos ^2(\theta )\right) }\left( a\mathrm dt_* - \left( r^2 + a^2\right) \mathrm d\phi _*\right) ^2 \\&\qquad + (r^2 + a^2 \cos ^2(\theta ))\frac{\mathrm d\theta ^2}{c(\theta )}. \end{aligned} \end{aligned}$$
(3)

We will throughout the paper assume that f is chosen as in [PV, Rmk. 1.1], so that the hypersurfaces

$$\begin{aligned} \{t_* = c \} \times (r_e - \delta , r_c + \delta )_r \times S^2_{\phi _*, \theta } \end{aligned}$$

are spacelike, for all \(c \in {\mathbb {R}}\), and that \(\delta > 0\) is small enough so that the hypersurfaces

$$\begin{aligned}&{\mathbb {R}}_{t_*} \times \{r = r_e - \delta \} \times S^2_{\phi _*, \theta }, \\&{\mathbb {R}}_{t_*} \times \{r = r_c + \delta \} \times S^2_{\phi _*, \theta } \end{aligned}$$

are spacelike. The two real analytic lightlike hypersurfaces

$$\begin{aligned} {\mathcal {H}}_e^+&:= {\mathbb {R}}_{t_*} \times \{r_e\} \times S^2_{\phi _*, \theta }, \\ {\mathcal {H}}_c^+&:= {\mathbb {R}}_{t_*} \times \{r_c\} \times S^2_{\phi _*, \theta } \end{aligned}$$

are called the future event horizon and future cosmological horizon, respectively. Note that the real analytic Killing vector fields \(\partial _t\) and \(\partial _\phi \), in Boyer-Lindquist coordinates, extend to real analytic Killing vector fields \(\partial _{t_*}\) and \(\partial _{\phi _*}\) on \((M_*, g_*)\).

We will consider wave equations on complex tensors. Fixing \(r, s \in {\mathbb {N}}_0\), let \({\mathcal {T}}_r^s{\mathcal {U}}\) denote the complex (r, s)-tensors on an open subset \({\mathcal {U}}\subset M_*\) and let \(\nabla \) denote the Levi-Civita connection acting on \({\mathcal {T}}_r^s{\mathcal {U}}\). We let \(C^\infty ({\mathcal {T}}_r^s{\mathcal {U}})\) and \(C^\omega ({\mathcal {T}}_r^s{\mathcal {U}})\) denote the smooth and real analytic complex tensor fields, respectively. Let P be a wave operator, i.e. is a linear differential operator with principal symbol given by the dual metric, i.e.

$$\begin{aligned} P = - g^{\alpha \beta } \nabla _\alpha \nabla _\beta + \text {lower order terms}. \end{aligned}$$

More precisely, there are complex tensor fields

$$\begin{aligned} A&: T^*{\mathcal {U}}\otimes {\mathcal {T}}_r^s{\mathcal {U}}\rightarrow {\mathcal {T}}_r^s{\mathcal {U}}, \\ B&: {\mathcal {T}}_r^s{\mathcal {U}}\rightarrow {\mathcal {T}}_r^s{\mathcal {U}}, \end{aligned}$$

such that

$$\begin{aligned} P = \nabla ^*\nabla + A \circ \nabla + B. \end{aligned}$$

We consider solutions to wave equations \(Pu = f\), where the coefficients A and B are invariant under the Killing vector fields \(\partial _{t_*}\) and \(\partial _{\phi _*}\). This is a natural assumption for geometric wave equations, where A and B are typically given by curvature expressions. Our first main result is the following:

Theorem 1.1

Let \((M_*, g_*)\) be the subextremal Kerr(-de Sitter) spacetime, extended real analytically over the future event horizon (and future cosmological horizon if \(\Lambda > 0\)). Assume that

  • A and B are real analytic in \(M_*\),

  • \({\mathcal {L}}_{\partial _{t_*}}A = {\mathcal {L}}_{\partial _{\phi _*}}A = 0\) and \({\mathcal {L}}_{\partial _{t_*}}B = {\mathcal {L}}_{\partial _{\phi _*}}B = 0\) in \(M_*\).

If \(u \in C^\infty ({\mathcal {T}}_r^sM_*)\) satisfies

  1. (i)

    \(Pu \in C^\omega ({\mathcal {T}}_r^sM_*)\),

  2. (ii)

    \({\mathcal {L}}_{\partial _{t_*}} u = - i \sigma u\) for some \(\sigma \in {\mathbb {C}}\),

  3. (iii)

    \({\mathcal {L}}_{\partial _{\phi _*}} u = - i k u\) for some \(k \in {\mathbb {Z}}\),

then \(u \in C^\omega ({\mathcal {T}}_r^sM_*)\).

Smooth tensor fields satisfying (ii) and (iii) in Theorem 1.1 and \(Pu = 0\) are called quasinormal modes. For functions, these conditions are equivalent to assuming that

$$\begin{aligned} u(t_*, r, \phi _*, \theta ) = e^{- i(\sigma t_* + k \phi _*)}v(r, \theta ), \end{aligned}$$
(4)

which is perhaps the more common way to express quasinormal modes.

Combining Theorem 1.1 with the Fredholm theory developed by the second author in [Vas13] and [Vas] (see also [VZ00, Vas21]) and by both authors in [PV], we deduce our second main result, where we consider quasinormal modes only with respect to the Killing vector symmetry

$$\begin{aligned} \partial _{t_*} + \frac{a}{r_0^2 + a^2} \partial _{\phi _*}, \end{aligned}$$
(5)

where \(r_0 \in (r_e, r_c)\) is the unique point such that \(\mu '(r_0) = 0\), as opposed to modes with respect to both \(\partial _{t_*}\) and \(\partial _{\phi _*}\) separately (as in Theorem 1.1). Concretely, this means that quasinormal modes are supposed to satisfy

$$\begin{aligned} {\mathcal {L}}_{\partial _{t_*} + \frac{a}{r_0^2 + a^2}\partial _{\phi _*}} u = - i \sigma u. \end{aligned}$$

For solutions to linear scalar wave equations on any subextremal Kerr–de Sitter spacetime, there is an asymptotic expansion in these quasinormal modes up to an exponentially decaying term [PV, Thm. 1.5]. This extends the result of [Vas13], by removing restrictions on the angular momentum.

For the Fredholm theory to go through in the case \(\Lambda = 0\), we will need the induced operator on the modes to be a scattering operator with self-adjoint (i.e. real) scattering principal symbol near spatial infinity in the sense of Melrose [Mel94]. Let us use the convention that if \(\Lambda = 0\), then \(r_0 = \infty \), giving the standard notion of quasinormal modes on the Kerr spacetime. This amounts to making appropriate decay assumptions on A and B:

Theorem 1.2

Let \((M_*, g_*)\) be the subextremal Kerr(-de Sitter) spacetime, extended real analytically over the future event horizon (and future cosmological horizon if \(\Lambda > 0\)). Assume that

  • A and B are real analytic in \(M_*\),

  • \({\mathcal {L}}_{\partial _{t_*}}A = {\mathcal {L}}_{\partial _{\phi _*}}A = 0\) and \({\mathcal {L}}_{\partial _{t_*}}B = {\mathcal {L}}_{\partial _{\phi _*}}B = 0\) in \(M_*\).

If \(u \in C^\infty ({\mathcal {T}}_r^sM_*)\) satisfies

  1. (i)

    \(Pu = 0\),

  2. (ii)

    \({\mathcal {L}}_{\partial _{t_*} + \frac{a}{r_0^2 + a^2}\partial {_{{\phi }_*}}} u = - i \sigma u\) for some \(\sigma \in {\mathbb {C}}\),

  3. (iii)

    in case \(\Lambda = 0\) we also assume that \(\textrm{Im} \,\sigma \ge 0\) and

    • if \(\textrm{Im} \,\sigma > 0\), then assume that \(A, B \in {\mathcal {O}}_\infty \left( r^{-\epsilon }\right) \) and \(u|_{t_* = 0} \in {\mathcal {S}}'\),

    • if \(\sigma \in {\mathbb {R}}\backslash \{0\}\), then assume that \(P - P^* \in {\mathcal {O}}_\infty \left( r^{-1-\epsilon }\right) \) and \({A, B \in {\mathcal {O}}_\infty (r^{-\epsilon })}\) and that \(u|_{t_* = 0} \in r^{\frac{1}{2} - \epsilon }L^2\),

    • if \(\sigma = 0\), then assume that \(A \in {\mathcal {O}}_\infty \left( r^{-1 - \epsilon }\right) \) and \(B = {\mathcal {O}}_\infty \left( r^{-2 - \epsilon }\right) \) and \(u|_{t_* = 0} \in {\mathcal {S}}'\),

    for some \(\epsilon > 0\),

then \(u \in C^\omega ({\mathcal {T}}_r^sM_*)\).

Here we used the notation \({\mathcal {S}}'\) for tempered distributions and the notation \({F \in {\mathcal {O}}_\infty (r^\alpha )}\) for a complex tensor field F if and only if for all \(k \in {\mathbb {N}}_0\), there is a constant \(C_k > 0\), such that

$$\begin{aligned} \left|\nabla ^k F\right| \le C_k r^{\alpha -k}, \end{aligned}$$

where \(\left|\cdot \right|\) is the positive definite norm on complex tensors induced from the Euclidean metric \(\mathrm dt^2 + \mathrm dr^2 + r^2g_{S^2}\). The notation \(Q \in {\mathcal {O}}_\infty (r^\alpha )\) for a differential operator Q means that the coefficients of Q are in \({\mathcal {O}}_\infty (r^\alpha )\).

Remark 1.3

In the case when \(\Lambda = 0\), one could weaken the assumptions on u, A and B at spatial infinity in various ways and still get a Fredholm problem following the arguments of [Vas]. Indeed, the natural condition on \(u|_{t_* = 0}\) in the case \(\sigma \in {\mathbb {R}}\backslash \{0\}\) is formulated microlocally in terms of variable order Sobolev spaces, c.f. [Vas, Prop. 5.28]. Moreover, the threshold growth \(r^{\frac{1}{2}}\) could be adjusted depending on A and B, to allow for more general coefficients, see [Vas, Sec. 5.4.8]. We restrict for simplicity to this setting.

The restriction \(\textrm{Im} \,\sigma \ge 0\) for Kerr spacetimes is due to the lack of a directly applicable Fredholm theory for the Fourier conjugated (in \(-t_*\)) operators in this case, though alternatives are still available for studying these resonances. For functions, the condition (ii) in Theorem 1.2 is equivalent to assuming that

$$\begin{aligned} u(t_*, r, \phi _*, \theta ) = e^{-i\sigma t_*}w(r, \phi _*, \theta ), \end{aligned}$$

which should be compared with equation (4) above.

In the special case when \(a = 0\), the Kerr(-de Sitter) spacetime simplifies to the Schwarzschild(-de Sitter) spacetime. In this case, Theorem 1.1 and Theorem 1.2 can be immediately deduced from the framework developed by Galkowski–Zworski in [GZ21] as follows: Wave equations for modes with respect to (5) reduce in the coordinate system \((t_*, r, \phi _*, \theta )\) to a Keldysh type operator, exactly of the type studied in [GZ21]. Galkowski–Zworski prove in [GZ21, Thm. 1] (generalizing [Zui17, Thm. 1.3]) the analytic hypoellipticity of such operators, thus proving the real analyticity of quasinormal modes when \(a = 0\). In fact, if \(a = 0\), the argument goes through without assuming that the coefficients A and B are invariant under \(\partial _{\phi _*}\). Due to the rotation in the Kerr(-de Sitter) spacetime when \(a \ne 0\), this argument does not go through immediately. The key to be able to apply the analytic hypoellipticity theory by Galkowski–Zworski to the case \(a \ne 0\) is the main new idea of this paper and is described in the next subsection.

1.2 Non-degenerate Killing horizons

By checking the formula (3) for the extended metric \(g_*\), one observes that the Killing vector field

$$\begin{aligned} \partial _{t_*} + \frac{a}{r_0^2 + a^2} \partial _{\phi _*}, \end{aligned}$$
(6)

where \(r_0 \in (r_e, r_c)\) is the unique point such that \(\mu '(r_0) = 0\), is lightlike at the horizons if and only if \(a = 0\). This turns out to be exactly why the modes with respect to (6) satisfy the useful Keldysh type equation if and only if \(a = 0\). In the Kerr(-de Sitter) spacetime, the Killing vector fields

$$\begin{aligned}&\partial _{t_*} + \frac{a}{r_e^2 + a^2} \partial _{\phi _*}, \end{aligned}$$
(7)
$$\begin{aligned} \text {and }&\partial _{t_*} + \frac{a}{r_c^2 + a^2} \partial _{\phi _*} \text { (if } \Lambda > 0), \end{aligned}$$
(8)

are lightlike at the horizons \({\mathcal {H}}_e^+\) and \({\mathcal {H}}_c^+\) (if \(\Lambda > 0\)), respectively. We show that the mode solutions with respect to these Killing vector fields satisfy equations which are almost of Keldysh type. More precisely, the bicharacteristics associated to the mode equation will have the radial point structure assumed by Galkowski–Zworski in their analytic hypoellipticity result [GZ21, Thm. 2]. Now, if u satisfies the assumption of Theorem 1.1, then

$$\begin{aligned} {\mathcal {L}}_{\partial _{t_*} + \frac{a}{r_e^2 + a^2} \partial _{\phi _*}} u = - i \left( \sigma + \frac{a}{r_e^2 + a^2}k\right) u, \end{aligned}$$
(9)

and similarly with \(r_e\) replaced by \(r_c\). This shows that u is a mode solution with respect to both Killing vector fields (7) and (8). The analytic hypoellipticity result by Galkowski–Zworski thus shows that u is real analytic near the horizons \({\mathcal {H}}^+_e\) and \({\mathcal {H}}^+_c\) (if \(\Lambda > 0\)). This is the main step in the proof of Theorem 1.1, the rest follows by standard propagation of real analyticity for wave equations and analytic hypoellipticity of elliptic equations (c.f. [Mar02, Chapter 4]).

In fact, this method is not specific to the Kerr(-de Sitter) spacetime, but turns out to work for any Killing horizon in any real analytic spacetime, assuming the surface gravity of the Killing horizon is nowhere vanishing. Assume therefore that (M, g) is a real analytic spacetime, i.e. a time-oriented Lorentzian manifold, of dimension \(n + 1 \ge 2\), with sign convention \((-, +, \ldots , +)\) and with a real analytic lightlike hypersurface \({\mathcal {H}}\subset M\). We assume in particular that the metric g is real analytic.

Definition 1.4

A real analytic Killing vector field W on M, such that \(W|_{\mathcal {H}}\) is lightlike and tangent to \({\mathcal {H}}\), is called a horizon Killing vector field with respect to \({\mathcal {H}}\).

For each Killing horizon \({\mathcal {H}}\) and horizon Killing vector field W, it is straightforward to check that

$$\begin{aligned} \nabla _W W|_{\mathcal {H}}= \kappa W|_{\mathcal {H}}, \end{aligned}$$
(10)

for a real analytic function \(\kappa : {\mathcal {H}}\rightarrow {\mathbb {R}}\) such that \(W|_{\mathcal {H}}\kappa = 0\).

Definition 1.5

Given a Killing horizon \({\mathcal {H}}\) and a horizon Killing vector field W, the surface gravity is the real analytic function \(\kappa \) defined in (10).

The key assumption to prove real analyticity of quasinormal modes is that the surface gravity \(\kappa \) is nowhere vanishing. All horizons in Kerr(-de Sitter) spacetimes have surface gravity proportional to \(\mu '\) at the horizons, where \(\mu \) was defined in (1) (c.f. Step 1 in the proof of Theorem 1.1). This is the reason our result only applies to subextremal Kerr(-de Sitter) spacetimes, since subextremality makes sure that \(\mu '\) does not vanish at the roots of \(\mu \), i.e. at the horizons.

Remark 1.6

We note in Lemma A.1 that if

$$\begin{aligned} \textrm{Ric}(X, W)|_{\mathcal {H}}= 0 \end{aligned}$$
(11)

for all \(X \in T{\mathcal {H}}\) and \({\mathcal {H}}\) is connected, then the surface gravity \(\kappa \) is constant. In practice, the condition (11) is often satisfied. Indeed, it is for example satisfied if the spacetime satisfies the Einstein equation with a cosmological constant of any sign or if the spacetime satisfies the dominant energy condition (c.f. [Pet21a, Rmk. 1.16]). In case \(\kappa \) is constant, we get a dichotomy of non-degenerate Killing horizons, where \(\kappa \ne 0\), and degenerate Killing horizons, where \(\kappa = 0\).

As in the previous subsection, we fix \(r, s \in {\mathbb {N}}_0\) and consider linear wave operators on complex (r, s)-tensors \({\mathcal {T}}_r^sM\) and write

$$\begin{aligned} P = \nabla ^*\nabla + A \circ \nabla + B \end{aligned}$$

with complex tensor fields

$$\begin{aligned} A&: T^*M \otimes {\mathcal {T}}_r^sM \rightarrow {\mathcal {T}}_r^sM, \\ B&: {\mathcal {T}}_r^sM \rightarrow {\mathcal {T}}_r^sM. \end{aligned}$$

Our third main result in this paper is the following theorem:

Theorem 1.7

Assume that

  • (M, g) is a real analytic spacetime,

  • \({\mathcal {H}}\subset M\) is a real analytic lightlike hypersurface,

  • W is a real analytic horizon Killing vector field,

  • the surface gravity \(\kappa \) is nowhere vanishing,

  • A and B are real analytic and \({\mathcal {L}}_W A = 0\) and \({\mathcal {L}}_W B = 0\) on M.

If \(u \in C^\infty (T_r^sM)\) satisfies

  1. (i)

    \(Pu \in C^\omega ({\mathcal {T}}_r^sM)\),

  2. (ii)

    \({\mathcal {L}}_W u = - i \sigma u\) for some \(\sigma \in {\mathbb {C}}\),

then there is an open subset \({\mathcal {U}}\supset {\mathcal {H}}\), such that \(u \in C^\omega (T_r^s{\mathcal {U}})\).

Note that all assumptions in Theorem 1.7 are local. As explained above, we will apply Theorem 1.7 with

$$\begin{aligned} W = \partial _{t_*} + \frac{a}{r_e^2 + a^2} \partial _{\phi _*}, \end{aligned}$$

and with

$$\begin{aligned} W = \partial _{t_*} + \frac{a}{r_c^2 + a^2} \partial _{\phi _*}, \end{aligned}$$

if \(\Lambda > 0\), which will prove the main step in Theorem 1.1 and Theorem 1.2, namely the real analyticity near the horizons.

Our methods require the existence of a horizon Killing vector field. This allows to reduce the wave equation for the modes to the useful (almost) Keldysh form. Surprisingly, a horizon Killing vector field is quite often guaranteed to exist in vacuum spacetimes with horizons. Proving the existence of a horizon Killing vector field has been the central tool in various black hole uniqueness results for the subextremal Kerr spacetime. This line of argument was pioneered by Hawking, who showed that stationary real analytic vacuum black holes with a non-degenerate event horizon necessarily admit a horizon Killing vector field [Haw72, HE73]. This result was later generalized to higher dimensional analytic vacuum black holes by Hollands–Ishibashi–Wald [HIW07] and Moncrief-Isenberg [MI08].

There is an analogous problem for compact (also called cosmological) Cauchy horizons in vacuum spacetimes. A conjecture by Moncrief and Isenberg [MI83] states that any compact Cauchy horizon in a vacuum spacetime admits a horizon Killing vector field. The existence of a horizon Killing vector field in that setting would prove that vacuum spacetimes with compact Cauchy horizons are non-generic, which would support the Strong Cosmic Censorship Conjecture in cosmology. During the last decades, Moncrief and Isenberg have made important progress on their conjecture, assuming that the spacetime metric is real analytic [MI83, IM85, MI20].

Remarkably, the existence of a horizon Killing vector field does often not even rely on the real analyticity of the spacetime metric. Alexakis, Ionescu and Klainerman proved in [AIK10a] (see also [IK13]) an analogue of Hawking’s theorem, showing the existence of a horizon Killing vector field in a neighbourhood of any bifurcate horizon in smooth vacuum spacetimes, as opposed to real analytic. This result has been central in their approach to prove uniqueness of subextremal Kerr black holes [AIK10b, AIK14] in the smooth setting. For compact Cauchy horizons in smooth vacuum spacetimes, as opposed to real analytic, a horizon Killing vector field has been shown to exist by Petersen in [Pet21b], assuming that the surface gravity is a non-zero constant (extending [FRW99, Pet21a, PR]). The assumption on constant surface gravity has recently been shown to be equivalent to a weak non-degeneracy assumption for compact Cauchy horizons in vacuum spacetimes, see [BR21] and [GM22].

Though the above mentioned results mainly concern vacuum spacetimes without cosmological constant, one expects them to extend to the case of positive cosmological constant and electro-vacuum spacetimes as well (c.f. [Rac00]). In conclusion, studying wave equations close to non-degenerate horizons (bifurcate or constant non-zero surface gravity), one might in quite wide generality be able to pass to modes with respect to the horizon Killing vector field and analyze the (almost) Keldysh type equation they are known to satisfy by the arguments in this paper.

2 Suitable Coordinates Near Non-degenerate Killing Horizons

The first step towards proving Theorem 1.7 is to define appropriate coordinates near the lightlike hypersurface \({\mathcal {H}}\):

Proposition 2.1

Assume the same as in Theorem 1.7. Then, for any \(p \in {\mathcal {H}}\), there is a real analytic coordinate system \((x_0, \ldots , x_n)\), defined on an open neighborhood \({\mathcal {U}}\ni p\), such that

  • \(\partial _{x_0} = W|_{\mathcal {U}}\),

  • \(x_1\) is a defining function for \({\mathcal {U}}\cap {\mathcal {H}}\) (i.e. \({\mathcal {U}}\cap {\mathcal {H}}= x_1^{-1}(0)\) and \(\mathrm dx_1|_{{\mathcal {U}}\cap {\mathcal {H}}} \ne 0\)),

  • the metric g expressed in these coordinates satisfies

    $$\begin{aligned} g|_{x_1 = 0} = \begin{pmatrix} 0 &{} 1 &{} 0 &{} \ldots &{} 0 \\ 1 &{} 0 &{} 0 &{} \ldots &{} 0 \\ 0 &{} 0 &{} g_{22}|_{x_1 = 0} &{} \ldots &{} g_{2 n}|_{x_1 = 0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} g_{n2}|_{x_1 = 0} &{} \ldots &{} g_{nn}|_{x_1 = 0} \\ \end{pmatrix}, \end{aligned}$$
    (12)

    where

    $$\begin{aligned} \begin{pmatrix} g_{22}|_{x_1 = 0} &{} \ldots &{} g_{2 n}|_{x_1 = 0} \\ \vdots &{} \ddots &{} \vdots \\ g_{n2}|_{x_1 = 0} &{} \ldots &{} g_{nn}|_{x_1 = 0} \\ \end{pmatrix}, \end{aligned}$$
    (13)

    is positive definite and

    $$\begin{aligned} \partial _1 g_{00}|_{x_1 = 0} = - 2\kappa , \end{aligned}$$

    where \(\kappa \) is the (nowhere vanishing) surface gravity.

Remark 2.2

These coordinates are essentially the Gaussian null coordinates introduced by Moncrief-Isenberg in [MI83], with the extra condition that \(\partial _0\) is the horizon Killing vector field restricted to an open neighborhood. (This is precisely what is obtained a posteriori after the construction of the horizon Killing vector field in [MI83].)

Example 2.3

The simplest example of a spacetime satisfying all our assumptions is \(M = {\mathbb {R}}^{n+1}\), equipped with the real analytic Misner metric

$$\begin{aligned} g = 2\mathrm dx_1\mathrm dx_0 + x_1 \mathrm dx_0^2 + \sum _{j = 2}^n (\mathrm dx^j)^2, \end{aligned}$$

where \({\mathcal {H}}= \{x_1 = 0\}\), \(W = \partial _0\) and surface gravity \(\kappa = - \frac{1}{2}\).

Example 2.4

In fact, even in the subextremal Kerr(-de Sitter) spacetime, one can easily choose coordinates which almost satisfy the conditions in Proposition 2.1, with one (insignificant) difference. To define these, it will be convenient to introduce an intermediate coordinate system, which will only be defined near one of the horizons. Let us start with the event horizon. In terms of Boyer-Lindquist coordinates, define

$$\begin{aligned} {\tilde{t}}_*&:= t - {\tilde{\Phi }}(r), \\ {\tilde{\phi }}_*&:= \phi - {\tilde{\Psi }}(r), \end{aligned}$$

where \({\tilde{\Phi }}\) and \({\tilde{\Psi }}\) satisfy

$$\begin{aligned} {\tilde{\Phi }}'(r)&= - b \frac{r^2 + a^2}{\mu (r)}, \\ {\tilde{\Psi }}'(r)&= - b \frac{a}{\mu (r)}, \end{aligned}$$

near \(r = r_e\). This commonly used analytic coordinate system \((\tilde{t}_*, r, \tilde{\phi }_*, \theta )\) is defined near the future event horizon. Choose now the coordinates

$$\begin{aligned} x_0&= {\tilde{t}}_*, \\ x_1&= r - r_e, \\ x_2&= {\tilde{\phi }}_* - \frac{a}{r_e^2 + a^2}{\tilde{t}}_*, \\ x_3&= \theta , \end{aligned}$$

from which we get

$$\begin{aligned} \partial _{x_0} = \partial _{{\tilde{t}}_*} + \frac{a}{r_e^2 + a^2}\partial _{{\tilde{\phi }}_*}. \end{aligned}$$

Defining

$$\begin{aligned} \psi (x_3):= b \frac{r_e^2 + a^2}{r_e^2 + a^2 \cos ^2(x_3)}, \end{aligned}$$

one easily computes that the metric \(g_*\) at the future event horizon is given by

$$\begin{aligned} \psi g_*|_{x_1 = 0} = \begin{pmatrix} 0 &{}{} 1 &{}{} 0 &{}{} 0 \\ 1 &{}{} 0 &{}{} 0 &{}{} 0 \\ 0 &{}{} 0 &{}{} {g_*}_{22}|_{x_1 = 0} &{}{} 0 \\ 0 &{}{} 0 &{}{} 0 &{}{} {g_*}_{33}|_{x_1 = 0} \\ \end{pmatrix}, \end{aligned}$$
(14)

in these coordinates, where \({g_*}_{22}|_{x_1 = 0}, {g_*}_{33}|_{x_1 = 0} > 0\). Moreover, we have

$$\begin{aligned} \partial _1 (\psi {g_*}_{00})|_{x_1 = 0} = - 2 \kappa _e, \end{aligned}$$

where the surface gravity \(\kappa _e\) is given by

$$\begin{aligned} \kappa _e = \frac{\mu '(r_e)}{2b\left( r_e^2 + a^2\right) } > 0, \end{aligned}$$

c.f. the computation in Step 1 in the proof of Theorem 1.1. These coordinates coincide with the coordinates in Proposition 2.1, up to the multiplication by the positive conformal factor \(\psi \). Since conformal changes of the geometry only reparametrize the lightlike geodesics, \(\psi \) is irrelevant for the analysis. However, it is of course natural to construct the coordinates in Proposition 2.1 without a conformal factor. This would here correspond to changing \(x_1\) to \({\tilde{x}}_1\) by solving the geodesic equation

$$\begin{aligned} \nabla _{\partial _{{\tilde{x}}_1}}\partial _{{\tilde{x}}_1} = 0, \quad \partial _{{\tilde{x}}_1}|_{x_1 = 0} = \psi \partial _{x_1}|_{x_1 = 0}, \end{aligned}$$

and changing the remaining coordinates \(x_j\) to \({\tilde{x}}_j\) by demanding that

$$\begin{aligned}{}[\partial _{{\tilde{x}}_1}, \partial _{{\tilde{x}}_j}] = 0, \quad \partial _{{\tilde{x}}_j}|_{x_1 = 0} = \partial _{x_j}|_{x_1 = 0}. \end{aligned}$$

In this new coordinate system, we get precisely the conditions in Proposition 2.1. One analogously constructs similar coordinates near the future cosmological horizon.

Proof of Proposition 2.1

Let first \((x_0, x_2, \ldots , x_n)\) be real analytic coordinates in an open neighborhood \({\mathcal {V}}\subseteq {\mathcal {H}}\) of p, such that

$$\begin{aligned} \partial _0 = W|_{\mathcal {V}}. \end{aligned}$$

Now let L be the unique lightlike real analytic vector field (transversal to \({\mathcal {H}}\)) along \({\mathcal {V}}\) such that

$$\begin{aligned} g(L, L)|_{\mathcal {V}}= g(L, \partial _j)|_{\mathcal {V}}= 0, \quad g(L, \partial _0)|_{\mathcal {V}}= 1 \end{aligned}$$
(15)

for \(j = 2, \ldots , n\). Define now the real analytic coordinate \(x_1\) in an open neighborhood \({\mathcal {U}}\subset M\) of p, such that \({\mathcal {V}}= {\mathcal {U}}\cap {\mathcal {H}}\), by solving the geodesic equation in direction of L, i.e. we solve

$$\begin{aligned} \nabla _{\partial _1}\partial _1&= 0, \\ \partial _1|_{\mathcal {V}}&= L \end{aligned}$$

and set \(x_1 = 0\) at \({\mathcal {H}}\). It follows that \(x_1\) is a defining function for \({\mathcal {U}}\cap {\mathcal {H}}\). We also extend the other coordinates to \({\mathcal {U}}\) by demanding that

$$\begin{aligned}{}[\partial _1, \partial _0] = [\partial _1, \partial _j] = 0 \end{aligned}$$

in \({\mathcal {U}}\), for \(j = 2, \ldots , n\). The inverse function theorem for real analytic functions implies that this forms a coordinate system.

We now show that \(\partial _0 = W|_{\mathcal {U}}\). Recall first that \(\partial _0|_{x_1 = 0} = W|_{x_1 = 0}\). By uniqueness of ODE and since \([\partial _0, \partial _1] = 0\), it suffices to show that \([\partial _1, W|_{\mathcal {U}}] = 0\). This is equivalent to W leaving the integral curves of \(\partial _1\) invariant. Since W is a Killing vector field and the integral curves of \(\partial _1\) are geodesics, it thus suffices to prove that the initial velocity \(\partial _1|_{x_1 = 0}\) of the geodesics are invariant under W, i.e. that

$$\begin{aligned}{}[W, \partial _1]|_{x_1 = 0} = \left( \nabla _W\partial _1 - \nabla _{\partial _1}W \right) |_{x_1 = 0} = 0. \end{aligned}$$

Since \(W|_{x_1 = 0} = \partial _0|_{x_1 = 0}\), it follows that \(\nabla _W\partial _1|_{x_1 = 0} = \nabla _{\partial _0}\partial _1|_{x_1 = 0}\) and therefore, since \([\partial _0, \partial _1]|_{x_1 = 0} = 0\), it suffices to prove that \(\nabla _{\partial _1}W|_{x_1 = 0} = \nabla _{\partial _1}\partial _0|_{x_1 = 0}\). Using that W is a Killing vector field, we observe that

$$\begin{aligned} g(\nabla _{\partial _1} W, \partial _1)&= \frac{1}{2} {\mathcal {L}}_W g(\partial _1, \partial _1) \\&= 0. \end{aligned}$$

We also have

$$\begin{aligned} g(\nabla _{\partial _1} \partial _0, \partial _1)|_{x_1 = 0}&= g([\partial _1, \partial _0], \partial _1)|_{x_1 = 0} - g(\nabla _{\partial _0} \partial _1, \partial _1)|_{x_1 = 0} \\&= - \frac{1}{2} \partial _0 g(\partial _1, \partial _1)|_{x_1 = 0} \\&= 0, \end{aligned}$$

hence

$$\begin{aligned} g(\nabla _{\partial _1} W, \partial _1)|_{x_1 = 0} = 0 = g(\nabla _{\partial _1} \partial _0, \partial _1)|_{x_1 = 0}. \end{aligned}$$

Moreover, for all \(j = 0, 2, \ldots , n\), we have

$$\begin{aligned} g(\nabla _{\partial _1}W, \partial _j)|_{x_1 = 0}&= {\mathcal {L}}_Wg(\partial _1, \partial _j)|_{x_1 = 0} - g(\nabla _{\partial _j} W, \partial _1)|_{x_1 = 0} \\&= - g(\nabla _{\partial _j} \partial _0, \partial _1)|_{x_1 = 0} \\&= - g(\nabla _{\partial _0} \partial _j, \partial _1)|_{x_1 = 0} \\&= - \partial _0 g(\partial _j, \partial _1)|_{x_1 = 0} + g(\partial _j, \nabla _{\partial _0}\partial _1)|_{x_1 = 0}\\&= g(\nabla _{\partial _1} \partial _0, \partial _j)|_{x_1 = 0}, \end{aligned}$$

where we have used that \(g(\partial _j, \partial _1)|_{x_1 = 0}\) is constant by (15). This shows that

$$\begin{aligned} \nabla _{\partial _1} W|_{x_1 = 0} = \nabla _{\partial _1} \partial _0 |_{x_1 = 0}. \end{aligned}$$

Taken together, this shows our claim that \(\partial _0 = W|_{\mathcal {U}}\).

It is now clear that the metric has the form (12) at \(x_1 = 0\) and that the part (13) is positive definite. Using (10), we compute

$$\begin{aligned} \partial _1 g_{00}|_{x_1 = 0}&= 2 g(\nabla _{\partial _1}\partial _0, \partial _0)|_{x_1 = 0} \\&= 2\partial _0 g(\partial _1, \partial _0)|_{x_1 = 0} - 2 g(\partial _1, \nabla _{\partial _0}\partial _0)|_{x_1 = 0} \\&= - 2 g(\partial _1, \nabla _WW)|_{x_1 = 0} \\&= - 2 \kappa g(\partial _1, W)|_{x_1 = 0} \\&= - 2 \kappa . \end{aligned}$$

This finishes the proof. \(\square \)

3 Real Analyticity Near General Horizons

The goal of this section is to prove Theorem 1.7. In order to explain the idea, let us start by discussing the following example:

Example 3.1

The d’Alembert operator in Example 2.3 is given by

$$\begin{aligned} \Box = \partial _1 \left( x_1 \partial _1 - 2\partial _0\right) - \sum _{j=2}^n \partial _j^2. \end{aligned}$$

The condition (ii) in Theorem 1.7 is that

$$\begin{aligned} u(x_0, \ldots , x_n) = e^{- i \sigma x_0} v(x_1, \ldots , x_n). \end{aligned}$$

Such a mode solution to \(\Box u = 0\) must satisfy the reduced equation

$$\begin{aligned} \partial _1(x_1 \partial _1 v) - \sum _{j=2}^n \partial _j^2 v + 2 i\sigma \partial _1 v= 0. \end{aligned}$$

This is a Keldysh type equation on the quotient space

$$\begin{aligned} {\mathbb {R}}^{n+1}/{\sim } = {\mathbb {R}}^n, \end{aligned}$$

and [GZ21, Thm. 1] implies that v and hence u is real analytic.

The proof of Theorem 1.7 is a generalization of the argument in Example 3.1:

Proof of Theorem 1.7

Shrinking \({\mathcal {U}}\) if necessary, we can write the coordinates from Proposition 2.1 as

$$\begin{aligned} (x_0, \ldots , x_n): {\mathcal {U}}\rightarrow (-\epsilon , \epsilon )_{x_0} \times (-\delta , \delta )_{x_1} \times K_{x_2, \ldots , x_n} \subset {\mathbb {R}}^{n+1}, \end{aligned}$$

where \(K \subset {\mathbb {R}}^{n-1}\) is an open relatively compact subset and \(\epsilon , \delta > 0\) are sufficiently small. Since

$$\begin{aligned} \partial _0 = W|_{\mathcal {U}}\end{aligned}$$

is a Killing vector field, we would like to eventually reduce P in the \(x_0\)-direction. For this, we first set

$$\begin{aligned} {\mathcal {V}}:= {\mathcal {U}}/{\sim }, \end{aligned}$$

where \(p \sim q\) if and only if

$$\begin{aligned} (x_1(p), \ldots , x_n(p)) = (x_1(q), \ldots , x_n(q)), \end{aligned}$$

i.e. only \(x_0(p)\) and \(x_0(q)\) may differ. The induced coordinates on the quotient space are

$$\begin{aligned} (x_1, \ldots , x_n): {\mathcal {V}}\rightarrow (-\delta , \delta )_{x_1} \times K_{x_2, \ldots , x_n}, \end{aligned}$$

i.e. we have “dropped” the \(x_0\)-coordinate.

The complex (r, s)-tensors on \({\mathcal {U}}\) are complex linear combinations of basis elements of the form

$$\begin{aligned} e_\mathrm I:= \partial _{i_0} \otimes \ldots \otimes \partial _{i_r} \otimes \mathrm dx_{j_0} \otimes \ldots \otimes \mathrm dx_{j_s}, \end{aligned}$$

where \(\mathrm I:= (i_1, \ldots , i_r, j_1, \ldots , j_s)\), and we of course have

$$\begin{aligned} {\mathcal {L}}_{\partial _0} e_\mathrm I= 0. \end{aligned}$$

Let us define \(f:= Pu\) and write

$$\begin{aligned} u = \sum _{\mathrm I}u_\mathrm Ie_\mathrm I, \quad f = \sum _{\mathrm I}f_\mathrm Ie_\mathrm I. \end{aligned}$$

Since \(\partial _0 = W|_{\mathcal {U}}\) is a Killing vector field, we note that

$$\begin{aligned}{}[{\mathcal {L}}_{\partial _0}, \nabla ] = 0, \end{aligned}$$

and by the assumption in Theorem 1.7, we know that

$$\begin{aligned} {\mathcal {L}}_{\partial _0}A = {\mathcal {L}}_W A = 0, \quad {\mathcal {L}}_{\partial _0}B = {\mathcal {L}}_W B = 0. \end{aligned}$$

It thus follows that the wave equation \(Pu = f\), restricted to the subset \({\mathcal {U}}\), can be written as a linear system of scalar wave equations

$$\begin{aligned} \sum _{\alpha ,\beta = 0}^n -g^{\alpha \beta } \partial _\alpha \partial _\beta u_\mathrm I+ \sum _{\gamma = 0}^n\sum _{J}{\mathcal {A}}_{\mathrm I, \gamma }^\mathrm J\partial _\gamma u_\mathrm J+ \sum _\mathrm J{\mathcal {B}}_\mathrm I^\mathrm Ju_\mathrm J= f_\mathrm I, \end{aligned}$$
(16)

for each \(\mathrm I:= (i_1, \ldots , i_r, j_1, \ldots , j_s)\), where the coefficients

$$\begin{aligned} g^{\alpha \beta }, \quad {\mathcal {A}}_{\mathrm I, \gamma }^\mathrm J, \quad {\mathcal {B}}_\mathrm I^\mathrm J\end{aligned}$$

are independent of \(x_0\). By the mode condition (ii), we note that

$$\begin{aligned} \partial _0 u_\mathrm I= - i \sigma u_\mathrm I, \quad \partial _0 f_\mathrm I= - i \sigma f_\mathrm I, \end{aligned}$$

which implies that

$$\begin{aligned} u_\mathrm I= e^{- i \sigma x_0} u_\mathrm I|_{x_0 = 0}, \quad f_\mathrm I= e^{- i \sigma x_0} f_\mathrm I|_{x_0 = 0}. \end{aligned}$$

Inserting this into (16) gives a new system of equations

$$\begin{aligned} \sum _{i,j = 1}^n - g^{ij} \partial _i \partial _j u_\mathrm I|_{x_0 = 0} + \sum _{k = 1}^n\sum _J {\mathcal {C}}_{\mathrm I, k}^\mathrm J\partial _k u_\mathrm J|_{x_0 = 0} + \sum _\mathrm J{\mathcal {D}}_\mathrm I^\mathrm Ju_\mathrm J|_{x_0 = 0} = f_\mathrm I|_{x_0 = 0}, \end{aligned}$$

where the new coefficients \({\mathcal {C}}_{\mathrm I, k}^\mathrm J\) and \(\mathcal D_\mathrm I^\mathrm J\) are independent of \(x_0\). Note also that the sums now exclude derivatives in \(x_0\). We have thus shown that the equation \(Pu = f\) is equivalent to a system of equations

$$\begin{aligned} {\hat{P}} u|_{x_0 = 0} = f|_{x_0 = 0} \end{aligned}$$

on the quotient space

$$\begin{aligned} {\mathcal {V}}= {\mathcal {U}}/{\sim }, \end{aligned}$$

where the principal symbol of \({\hat{P}}\) is

$$\begin{aligned} \mathrm p(x_1, \ldots , x_n, \xi _1, \ldots , \xi _n) \textrm{Id}, \end{aligned}$$
(17)

where \(\textrm{Id}\) is the identity matrix and

$$\begin{aligned} \mathrm p(x_1, \ldots , x_n, \xi _1, \ldots , \xi _n):= \sum _{i,j = 1}^n g(x_1, \ldots , x_n)^{ij}\xi _i \xi _j \end{aligned}$$

for any \((x_1, \ldots , x_n, \xi _1, \ldots , \xi _n) \in T^* {\mathcal {V}}\).

This is where the information about the metric g in Proposition 2.1 becomes useful. We claim that first that

$$\begin{aligned} \{\mathrm p= 0\} \cap \{x_1=0\} = N^*\{x_1=0\}, \end{aligned}$$
(18)

where \( N^*\{x_1=0\}\) denotes the conormal bundle of the horizon \(\{x_1=0\}\). In order to compute the components \(g^{ij}\), for \(i,j = 1, \ldots , n\), we first need to invert the full matrix of metric components. By Proposition 2.1, we conclude that

$$\begin{aligned} g^{\alpha \beta }|_{{\mathcal {U}}\cap \{x_1 = 0\}} = \begin{pmatrix} 0 &{} 1 &{} 0 &{} \ldots &{} 0 \\ 1 &{} 0 &{} 0 &{} \ldots &{} 0 \\ 0 &{} 0 &{} g^{22}|_{x_1 = 0} &{} \ldots &{} g^{2 n}|_{x_1 = 0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} g^{n2}|_{x_1 = 0} &{} \ldots &{} g^{nn}|_{x_1 = 0} \\ \end{pmatrix} \end{aligned}$$
(19)

for \(\alpha , \beta = 0, \ldots , n\). The components appearing in (17) are given, at \(x_1 = 0\), by

$$\begin{aligned} g^{ij}|_{x_1 = 0} = \begin{pmatrix} 0 &{} 0 &{} \ldots &{} 0 \\ 0 &{} g^{22}|_{x_1 = 0} &{} \ldots &{} g^{2 n}|_{x_1 = 0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} g^{n2}|_{x_1 = 0} &{} \ldots &{} g^{nn}|_{x_1 = 0} \\ \end{pmatrix}. \end{aligned}$$

Since the matrix

$$\begin{aligned} \begin{pmatrix} g^{22}|_{x_1 = 0} &{} \ldots &{} g^{2 n}|_{x_1 = 0} \\ \vdots &{} \ddots &{} \vdots \\ g^{n2}|_{x_1 = 0} &{} \ldots &{} g^{nn}|_{x_1 = 0} \\ \end{pmatrix} \end{aligned}$$

is positive definite by Proposition 2.1, we have proven (18).

By standard microlocal analytic hypoellipticity at elliptic points in \(T^*{\mathcal {V}}\), see for example [Mar02, Thm. 4.2.2 & Exe. 4.6.4], we hence conclude that \(u_\mathrm I\) is microlocally real analytic everywhere at \(x_1 = 0\) except potentially at the conormal bundle \(N^* \{x_1 = 0\}\), i.e. the analytic wave front set at \(x_1 = 0\) is contained in the conormal bundle. We will show the real analyticity at the conormal bundle by applying [GZ21, Thm. 2], which requires a computation of the Hamiltonian vector field \(H_\mathrm p\) at \(N^* \{x_1 = 0\}\). For this, we first compute \(\partial _1 \mathrm p|_{N^* \{x_1 = 0\}}\). At an arbitrary point

$$\begin{aligned} q:= (0, x_2, \ldots , x_n, \xi _1, 0, \ldots , 0) \in N^* \{x_1 = 0\}, \end{aligned}$$

using (19) and Proposition 2.1, we compute

$$\begin{aligned} \partial _1\mathrm p|_q&= \partial _1g^{11}|_q (\xi _1)^2 \\&= - \sum _{\alpha , \beta = 0}^n g^{\alpha 1}(\partial _1g_{\alpha \beta })g^{\beta 1}|_q (\xi _1)^2 \\&= - \partial _1g_{00}|_q (\xi _1)^2 \\&= 2 \kappa (\xi _1)^2. \end{aligned}$$

We may now compute the Hamiltonian vector field as

$$\begin{aligned} H_\mathrm p|_q&= \sum _{j = 1}^n (\partial _{\xi _j}\mathrm p) \partial _j|_q - (\partial _j\mathrm p) \partial _{\xi _j}|_q \\&= - (\partial _1 \mathrm p) \partial _{\xi _1}|_q \\&= - 2 \kappa (\xi _1)^2 \partial _{\xi _1}|_q, \end{aligned}$$

where we recall that \(\kappa \) is nowhere vanishing. In particular

$$\begin{aligned} \mathrm d\mathrm p|_{N^* \{x_1 = 0\} \backslash \{0\}} \ne 0 \end{aligned}$$

and

$$\begin{aligned} H_\mathrm p|_{N^* \{x_1 = 0\} \backslash \{0\}} \parallel \xi \cdot \partial _\xi , \end{aligned}$$

which means that the assumptions in [GZ21, Thm. 2] are satisfied. Note here that [GZ21, Thm. 2] is only proven for scalar valued wave equations, but the argument goes through line by line for systems of equations with a scalar principal symbol, as in our case. Indeed, [GZ21, Thm. 2] relies on Haber’s normal form in [Hab14] only for the principal symbol. Thus, having scalar principal symbol suffices. Hence [GZ21, Thm. 2] implies that \(u_\mathrm I|_{t_* = 0}\) is microlocally real analytic also at the conormal bundle. It follows that \(u_\mathrm I|_{t_* = 0}\) is real analytic in an open subset containing \(\{x_1 = 0\}\). Consequently, \(u_\mathrm I\) and therefore u is real analytic in an open neighborhood containing p, which completes the proof. \(\square \)

4 Joint Quasinormal Modes

We continue by proving the next main result of this paper:

Proof of Theorem 1.1

Let us for simplicity restrict in this proof to the case of complex functions, as opposed to complex tensor fields of higher rank. This will make the proof more transparent and avoid the technical details involved with working with system of equations. All such technicalities are already present in the proof of Theorem 1.7 above. We thus consider functions of the form

$$\begin{aligned} u(t_*, r, \phi _*, \theta ) = e^{-i(\sigma t_* + k\phi _*)}v(r, \theta ), \end{aligned}$$

which are smooth on

$$\begin{aligned} {\mathcal {U}}= {\mathbb {R}}_{t_*} \times (r_e - \delta , r_c + \delta )_r \times S^2_{\phi _*, \theta } \subset M_*. \end{aligned}$$

We aim to prove that u is real analytic on \({\mathcal {U}}\).

Step 1: Real analyticity near the horizons. We would like to apply Theorem 1.7 with \({\mathcal {H}}= {\mathcal {H}}_{e/c}^+\) and therefore need to check that all assumptions of Theorem 1.7 are satisfied. Firstly, the Kerr(-de Sitter) spacetime is a vacuum spacetime with a non-negative cosmological constant, so the dominant energy condition is clearly satisfied. Moreover, the horizons \({\mathcal {H}}_{e/c}^+\) are real analytic lightlike hypersurfaces. Secondly, the Killing vector fields

$$\begin{aligned} W_{e/c}:= \partial _{t_*} + \frac{a}{r_{e/c}^2 + a^2} \partial _{\phi _*} \end{aligned}$$

are clearly lightlike at \({\mathcal {H}}_{e/c}^+\), respectively. Since A and B are invariant under \(\partial _{t_*}\) and \(\partial _{\phi _*}\), they are also invariant under \(W_{e/c}\). Further, the surface gravity \(\kappa _{e/c}\) of the horizons is computed using the extended metric in (3) as follows:

$$\begin{aligned} \partial _r g_* \left( W_{e/c}, W_{e/c}\right) |_{r_{e/c}}&= 2 g_*\left( \nabla _{\partial _r} W_{e/c}, W_{e/c}\right) |_{r_{e/c}} \\&= 2 g_*\left( \nabla _{W_{e/c}}\partial _r, W_{e/c}\right) |_{r_{e/c}} \\&= - 2 g_*\left( \partial _r, \nabla _{W_{e/c}} W_{e/c}\right) |_{r_{e/c}} \\&= - 2 \kappa _{e/c} \ g_*\left( \partial _r, W_{e/c}\right) |_{r_{e/c}} \\&= \mp \frac{2\kappa _{e/c}}{b} \frac{r_{e/c}^2 + a^2 \cos ^2(\theta )}{r_{e/c}^2 + a^2}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \partial _r g_* \left( W_{e/c}, W_{e/c}\right) |_{r_{e/c}}&= - \partial _r \left( \frac{\mu (r)}{b^2\left( r^2 + a^2 \cos ^2(\theta )\right) } \left( \frac{r^2 + a^2 \cos ^2(\theta )}{r_{e/c}^2 + a^2} \right) ^2 \right) |_{r_{e/c}} \\&\qquad + \partial _r \left( \frac{c(\theta )\sin ^2(\theta )}{b^2\left( r^2 + a^2 \cos ^2(\theta )\right) }\left( a - a \frac{r^2 + a^2}{r_{e/c}^2 + a^2} \right) ^2 \right) |_{r_{e/c}} \\&= - \frac{\mu '(r_{e/c})}{b^2} \frac{r_{e/c}^2 + a^2 \cos ^2(\theta )}{\left( r_{e/c}^2 + a^2 \right) ^2}. \end{aligned}$$

The surface gravity of the horizons \({\mathcal {H}}_{e/c}^+\) is thus given by

$$\begin{aligned} \kappa _{e/c} = \pm \frac{\mu '(r_{e/c})}{2b (r_{e/c}^2 + a^2)}. \end{aligned}$$

Since this is non-zero, we conclude that \({\mathcal {H}}_{e/c}^+\) are non-degenerate Killing horizons with respect to \(W_{e/c}\). Finally, we compute that

$$\begin{aligned} {\mathcal {L}}_{W_{e/c}} u&= {\mathcal {L}}_{\partial _{t_*}}u + \frac{a}{r_{e/c}^2 + a^2} {\mathcal {L}}_{\partial _{\phi _*}}u \\&= - i \sigma u - i \frac{a}{r_{e/c}^2 + a^2} k u \\&= - i \left( \sigma + \frac{a}{r_{e/c}^2 + a^2} k \right) u, \end{aligned}$$

which shows that u is a mode with respect to \(W_{e/c}\). We may therefore apply Theorem 1.7 and conclude that u is real analytic in open neighborhoods of the horizons, which are invariant under \(W_e\) and \(W_c\), respectively.

Step 2: Real analyticity in the domain of outer communication. We now prove real analyticity of u in the open subset

$$\begin{aligned} {\mathcal {W}}:= {\mathbb {R}}_{t_*} \times \left( r_e, r_c\right) _r \times S^2_{\phi _*, \theta }. \end{aligned}$$

(Recall that \(r_c = \infty \) in the Kerr spacetime). The Boyer-Lindquist coordinates \((t, r, \phi , \theta )\) are defined on this set and are convenient to work with. Since

$$\begin{aligned} \partial _t = \partial _{t_*}|_{\mathcal {W}}, \quad \partial _\phi = \partial _{\phi _*}|_{\mathcal {W}}, \end{aligned}$$

the conditions (ii) and (iii) in Theorem 1.1 imply that

$$\begin{aligned} u(t, r, \phi , \theta ) = e^{- i(\sigma t + k \phi )} w(r, \theta ), \end{aligned}$$

so we can equally well consider the modes with respect to the Boyer-Lindquist coordinates. The dual metric G of g in Boyer-Lindquist coordinates is

$$\begin{aligned} \begin{aligned} (r^2 + a^2 \cos ^2(\theta ))G&= \mu (r)\partial _r^2 + c(\theta )\partial _\theta ^2 + \frac{b^2}{c(\theta )\sin ^2(\theta )}\left( a \sin ^2(\theta )\partial _t + \partial _\phi \right) ^2 \\&\qquad - \frac{b^2}{\mu (r)} \left( (r^2 + a^2)\partial _t + a \partial _\phi \right) ^2. \end{aligned} \end{aligned}$$
(20)

We begin by proving real analyticity of w in the open subset

$$\begin{aligned} (r_e, r_c)_r \times (0, \pi )_\theta \end{aligned}$$

i.e. we leave out the north and the south pole of \(S^2_{\phi , \theta }\) for the moment. Since we have assumed that the coefficients of P are independent of t and \(\phi \), the function w satisfies an induced equation on \((r_e, r_c)_r \times (0, \pi )_\theta \), with principal part given by

$$\begin{aligned} \frac{1}{r^2 + a^2 \cos ^2(\theta )} \left( \mu (r)\partial _r^2 + c(\theta )\partial _\theta ^2 \right) . \end{aligned}$$
(21)

Since \(\mu (r), c(\theta ) > 0\) in this set, the induced equation for w is elliptic with real analytic coefficients. Standard analytic hypoellipticity, see for example [Mar02, Thm. 4.2.2 & Exe. 4.6.4] therefore implies that w is real analytic in \((r_e, r_c)_r \times (0, \pi )_\theta \) and hence u is real analytic in

$$\begin{aligned} {\mathbb {R}}_t \times \left( r_e, r_c\right) _r \times S^1_\phi \times (0, \pi )_\theta . \end{aligned}$$

We now turn to show that u is also real analytic at the north and south poles of \(S^2_{\phi , \theta }\), i.e. at the limits \(\theta = 0\) and \(\theta = \pi \), still with \(r \in (r_e, r_c)\). Note that the expression (21) does not extend smoothly to those points. We now write

$$\begin{aligned} u(t, r, \phi , \theta ) = e^{- i\sigma t} z(r, \phi , \theta ), \end{aligned}$$

i.e. the idea is to show real analyticity of

$$\begin{aligned} z(r, \phi , \theta ):= e^{- i k\phi }w(r, \theta ), \end{aligned}$$

which is smooth in

$$\begin{aligned} (r_e, r_c)_r \times S^2_{\phi , \theta }. \end{aligned}$$

Since the coefficients of P are independent of t, we get an induced equation for z, with real analytic coefficients and principal part

$$\begin{aligned} \mu (r)\partial _r^2 + c(\theta )\partial _\theta ^2 + \frac{b^2}{c(\theta )\sin ^2(\theta )} \partial _\phi ^2 - \frac{b^2}{\mu (r)} a^2 \partial _\phi ^2. \end{aligned}$$
(22)

We claim that this operator is elliptic at \(\theta = 0\) and \(\theta = \pi \). For this, we note that

$$\begin{aligned} c(\theta )\partial _\theta ^2 + \frac{b^2}{c(\theta )\sin ^2(\theta )} \partial _\phi ^2&= \left( c(\theta ) - \frac{b^2}{c(\theta )}\right) \partial _\theta ^2 + \frac{b^2}{c(\theta )} \left( \frac{1}{\sin ^2(\theta )} \partial _\phi ^2 + \partial _\theta ^2 \right) \\&= \frac{1}{c(\theta )} \left( c(\theta )^2 - b^2\right) \partial _\theta ^2 + \frac{b^2}{c(\theta )}G_{S^2} \\&= \frac{1}{c(\theta )} \left( \left( b - \frac{\Lambda a^2}{3}\sin ^2(\theta ) \right) ^2 - b^2\right) \partial _\theta ^2 + \frac{b^2}{c(\theta )}G_{S^2} \\&= h(\theta ) \sin ^2(\theta )\partial _\theta ^2 + \frac{b^2}{c(\theta )}G_{S^2}, \end{aligned}$$

for some function h, which extends real analytically to \(S^2\) and where \(G_{S^2}\) is the dual metric to the standard metric on \(S^2\). Since both \(\sin ^2(\theta )\partial _\theta ^2\) and \(G_{S^2}\) extend real analytically to \(S^2\), we can evaluate this expression at \(\theta = 0\) or \(\theta = \pi \) and conclude that (22) is simply

$$\begin{aligned} \mu (r)\partial _r^2 + b G_{S^2} \end{aligned}$$

at the north and the south pole of \(S^2\). Since \(\mu (r) > 0\) for \(r \in (r_e, r_c)\) and \(b > 0\), we conclude that (22) is indeed elliptic there as well. Again, standard real analytic hypoellipticity, as in for example [Mar02, Thm. 4.2.2 & Exe. 4.6.4], implies that z, and therefore u, is real analytic also at the north and the south pole if \(r \in (r_e, r_c)\). To sum up, we now know that u is real analytic in the domain of outer communication and slightly beyond the horizons, i.e. in a region of the form

$$\begin{aligned} {\mathbb {R}}_{t_*} \times (r_e - \epsilon , r_c + \epsilon )_r \times S^2_{\phi _*, \theta }. \end{aligned}$$

Step 3: The region beyond the horizons. It remains to prove real analyticity in the regions beyond the horizons (only the event horizon if \(\Lambda = 0\)). Consider first the region

$$\begin{aligned} {\mathbb {R}}_{t_*} \times \left( r_e - \delta , r_e \right) _r \times S^2_{\phi _*, \theta }, \end{aligned}$$

beyond the event horizon. We may use Boyer–Lindquist coordinates also here, since this region does not intersect any horizon, where the coordinates would not be defined. Let us again consider

$$\begin{aligned} z(r, \phi , \theta ):= e^{- ik\phi }w(r, \theta ), \end{aligned}$$

which by assumption is smooth in

$$\begin{aligned} \left( r_e - \delta , r_e\right) _r \times S^2_{\phi , \theta }. \end{aligned}$$

In this set, we have \(\mu (r) < 0\). Again, the coefficients of P are independent of t and the principal part of the induced equation for z can be read off from (22) to be

$$\begin{aligned} - \left|\mu (r)\right|\partial _r^2 + c(\theta )\partial _\theta ^2 + \left( \frac{b^2}{c(\theta )\sin ^2(\theta )} + \frac{b^2}{\left|\mu (r)\right|} a^2\right) \partial _\phi ^2. \end{aligned}$$

Since the operator

$$\begin{aligned} c(\theta )\partial _\theta ^2 + \left( \frac{b^2}{c(\theta )\sin ^2(\theta )} + \frac{b^2}{\left|\mu (r)\right|} a^2\right) \partial _\phi ^2. \end{aligned}$$

is elliptic on

$$\begin{aligned} \{ r \} \times S^2_{\phi , \theta }, \end{aligned}$$

for all \(r \in \left( r_e - \delta , r_e\right) \), we conclude that all inextendible bicharacteristics of the induced operator pass through all hypersurfaces

$$\begin{aligned} \{ r \} \times S^2_{\phi , \theta } \end{aligned}$$

precisely once. Moreover, the induced equation for z is a linear wave operator with real analytic coefficients on the

$$\begin{aligned} \left( r_e - \delta , r_e\right) _r \times S^2_{\phi _*, \theta }. \end{aligned}$$

and we know by Step 1 that z is real analytic in an open subset

$$\begin{aligned} (r_e - \epsilon , r_e) \times S^2_{\phi _*, \theta } \end{aligned}$$

for some \(\epsilon > 0\). Propagation of analytic singularities, see for example [Mar02, Thm. 4.3.7 & Exe. 4.6.4], therefore implies that z is real analytic in

$$\begin{aligned} \left( r_e - \delta , r_e\right) _r \times S^2_{\phi , \theta }, \end{aligned}$$

and hence u is real analytic in

$$\begin{aligned} {\mathbb {R}}_t \times \left( r_e - \delta , r_e\right) _r \times S^2_{\phi , \theta }. \end{aligned}$$

One similarly treats the subset

$$\begin{aligned} {\mathbb {R}}_t \times \left( r_c, r_c + \delta \right) _r \times S^2_{\phi , \theta }, \end{aligned}$$

in case \(\Lambda > 0\). This finishes the proof. \(\square \)

5 Standard Quasinormal Modes

We finish by proving our last main result:

Proof of Theorem 1.2

As in the proof of Theorem 1.1, let us for simplicity restrict to the case of complex functions, as opposed to complex tensors of higher rank. This will again make the proof more transparent and avoid technical details that are completely analogous to the corresponding part of the proof of Theorem 1.7. It is convenient to change coordinate system to one that is better suited for the quasinormal mode condition (ii) in Theorem 1.2. We introduce the new coordinate system \((\tau _*, r, \psi _*, \theta )\), where

$$\begin{aligned} \begin{pmatrix} \tau _* \\ \psi _* \end{pmatrix}:= \begin{pmatrix} t_* \\ \phi _* - \frac{a}{r_0^2 + a^2} t_* \end{pmatrix}, \end{aligned}$$
(23)

with again \(r_0 \in (r_e, r_c)\) is uniquely defined by

$$\begin{aligned} \mu '(r_0) = 0, \end{aligned}$$

and \(r_0 = \infty \) if \(\Lambda = 0\). Note that

$$\begin{aligned} \partial _{\tau _*} = \partial _{t_*} + \frac{a}{r_0^2 + a^2} \partial _{\phi _*}, \quad \partial _{\psi _*} = \partial _{\phi _*} \end{aligned}$$

are both Killing vector fields, since a and \(r_0\) are constant. It follows that u is a quasinormal mode if and only if

$$\begin{aligned} u(\tau _*, r, \psi _*, \theta ) = e^{-i \sigma \tau _*}z(r, \psi _*, \theta ). \end{aligned}$$

where z is smooth in

$$\begin{aligned} (r_e - \delta , r_c + \delta )_r \times S^2_{\psi _*, \theta }, \end{aligned}$$

where \(r_c = \infty \) if \(\Lambda = 0\). Since the coefficients of P are independent of \(t_*\) and \(\phi _*\), and therefore of \(\tau _*\), it follows that z satisfies a \(\tau _*\)-reduced equation

$$\begin{aligned} P_\sigma z = 0, \end{aligned}$$
(24)

in

$$\begin{aligned} (r_e - \delta , r_c + \delta )_r \times S^2_{\psi _*, \theta }, \end{aligned}$$

where

$$\begin{aligned} P_\sigma u:= e^{i \sigma \tau _*} P\left( e^{- i \sigma \tau _*} u \right) . \end{aligned}$$

We now further decompose into angular modes

$$\begin{aligned} z(r, \psi _*, \theta ) = \sum _{k \in {\mathbb {Z}}} e^{- ik\psi _*}v_k(r, \theta ), \end{aligned}$$
(25)

where we claim that each summand

$$\begin{aligned} e^{- ik\psi _*} v_k(r, \theta ) = \frac{1}{2\pi } e^{- ik\psi _*} \int _0^{2\pi } e^{iks}z(r, s, \theta ) ds \end{aligned}$$

is smooth on \((r_e - \delta , r_c + \delta )_r \times S^2_{\psi _*, \theta }\). Indeed, let u be the unique solution at a point

$$\begin{aligned} (s, r, p) \in {\mathbb {R}}_s \times (r_e - \delta , r_c + \delta )_r \times S^2_{\psi _*, \theta } \end{aligned}$$

to

$$\begin{aligned} \left( \partial _s + ik \right) u(s, r, p)&= \frac{1}{2\pi } z\left( r, \exp _p \left( s\partial _{\psi _*} \right) \right) , \\ u(0, r, p)&= 0. \end{aligned}$$

where \(\exp (s\partial _{\psi _*})\) denotes the flow along \(\partial _{\psi _*}\) at time s, starting at p. Then u is smooth and since

$$\begin{aligned} e^{- ik\psi _*} v_k(r, \theta ) = u(2\pi , r, p), \end{aligned}$$

where \(p = (\psi _*, \theta )\), it is smooth as claimed. Since the coefficients of P are independent of \(t_*\) and \(\phi _*\), and therefore of \(\psi _*\), and \(P_\sigma \) acts diagonally on the \(\psi _*\)-Fourier modes, it follows that

$$\begin{aligned} P_\sigma \left( e^{- ik\psi _*} v_k \right) = 0, \end{aligned}$$

in

$$\begin{aligned} (r_e - \delta , r_c + \delta )_r \times S^2_{\psi _*, \theta }, \end{aligned}$$

for each \(k \in {\mathbb {Z}}\).

Now, if \(\Lambda > 0\), then [PV, Thm. 2.1] implies that the operator \(P_\sigma \) is a Fredholm operator between appropriate function spaces containing \(e^{- ik\psi _*} v_k\). Since the kernel is finite dimensional, it follows that only finitely many such terms can be non-zero. We conclude that

$$\begin{aligned} z(r, \psi _*, \theta ) = \sum _{j =1}^N e^{- ik_j\psi _*}v_{k_j}(r, \theta ) \end{aligned}$$
(26)

and therefore

$$\begin{aligned} u(\tau _*, r, \psi _*, \theta )&= \sum _{j =1}^N e^{- i \left( \sigma \tau _* + k_j \psi _* \right) }v_{k_j}(r, \theta ) \\&= \sum _{j =1}^N e^{- i \left( \left( \sigma - \frac{a}{r_0^2 + a^2}k_j \right) t_* + k_j \phi _* \right) }v_{k_j}(r, \theta ). \end{aligned}$$

Each term satisfies the assumption of Theorem 1.1 and are therefore analytic. Hence the finite sum is also real analytic, concluding the proof when \(\Lambda > 0\).

In order to similarly proceed in the case \(\Lambda = 0\), we need to instead use the Fredholm theory developed in [Vas] (remember that the above coordinate change is trivial when \(\Lambda = 0\)) to deduce that in fact

$$\begin{aligned} z(r, \phi _*, \theta ) = \sum _{j =1}^N e^{- ik_j\phi _*}v_{k_j}(r, \theta ) \end{aligned}$$
(27)

is a finite sum. In this case we have \(r_c = \infty \) and the cosmological horizon is replaced by an asymptotically Euclidean end. For the analysis near the event horizon, the methods based on [Vas13] described above can be applied without changes. However, the analysis near the asymptotically Euclidean end cannot be based on [Vas13], we instead need to use a slight generalization of [Vas, Prop. 5.28]. Let us therefore briefly recall how the Fredholm problem was set up in [Vas, Prop. 5.28]. We begin by bordifying the space

$$\begin{aligned} (r_e - \delta , \infty )_r \times S^2_{\phi _*, \theta }, \end{aligned}$$

at \(r = \infty \) by introducing \(x:= \frac{1}{r}\), i.e. we radially compactify spacelike infinity. We thus write

$$\begin{aligned} {\mathcal {V}}:= \left[ 0, \frac{1}{r_e - \delta }\right) _x \times S^2_{\phi _*, \theta } \subset \overline{{\mathbb {R}}^3}, \end{aligned}$$

where \(\overline{{\mathbb {R}}^3}\) is the radially compactified \({\mathbb {R}}^3\). On these spaces, we define

$$\begin{aligned} {\mathcal {Y}}_{sc}^{s, l}:= \left\{ u|_{{\mathcal {V}}} \mid u \in H^{s, l}(\overline{{\mathbb {R}}^3}) \right\} , \end{aligned}$$

where s, l are variable order differential and decay orders (as \(x \rightarrow 0\)), which we will choose below. We refer to [Vas, Sec. 5.3.9] for the definition of variable order weighted Sobolev spaces \(H^{s, l}(\overline{{\mathbb {R}}^3})\). Note that near the spacelike hypersurface

$$\begin{aligned} \left\{ x = \frac{1}{r_e - \delta }\right\} , \end{aligned}$$

\({\mathcal {Y}}_{sc}^{s, l}\) is similar to \({\mathcal {Y}}^s\) introduced above. Analogous to above, define

$$\begin{aligned} {\mathcal {X}}_{sc}^{s, l}:= \left\{ u \in {\mathcal {Y}}_{sc}^{s,l} \mid {\hat{P}} u \in {\mathcal {Y}}_{sc}^{s-1,l+1} \right\} \end{aligned}$$

and consider

$$\begin{aligned} {\hat{P}}: {\mathcal {X}}_{sc}^{s, l} \rightarrow {\mathcal {Y}}_{sc}^{s-1,l+1}. \end{aligned}$$
(28)

The characteristic set of \({\hat{P}}\) has two components, one close to the event horizon and a scattering characteristic set at \(x = 0\), in particular, the characteristic set at fiber infinity near \(x = 0\) is empty. By the decay assumptions on A and B, the scattering principal symbol of \({\hat{P}}\) at \(x = 0\) is given by

$$\begin{aligned} \mathrm p|_{x = 0}(\xi ) = \left|\xi \right|^2 \textrm{Id}- \sigma ^2, \end{aligned}$$

for the fixed \(\sigma \) with \(\textrm{Im} \,\sigma \ge 0\), and any

$$\begin{aligned} \xi \in {}^{sc}T^*_{\{x = 0\}}{\mathcal {V}}. \end{aligned}$$

If \(\textrm{Im} \,\sigma > 0\) it follows that \(\sigma ^2 \notin [0, \infty )\), which implies that \({\hat{P}}\) is elliptic as set up in (28) and consequently a Fredholm operator for any order s, l. However, in case \(\sigma \in {\mathbb {R}}\backslash \{0\}\), there is a scattering characteristic set at \(x = 0\), given by all \(\xi \in {}^{sc}T^*_{\{x = 0\}}{\mathcal {V}}\) with \(\left|\xi \right| = \left|\sigma \right|\). As shown in [Vas, p. 311–314], the sets

$$\begin{aligned} L_\pm = \left\{ (y, \xi ) \in {}^{sc}T^*_{\{x = 0\}}{\mathcal {V}}\mid y = c \xi , \left|\xi \right|^2 = \sigma ^2, \pm c > 0\right\} , \end{aligned}$$

act as a source and a sink, respectively, for the Hamiltonian flow. It is also shown that (28) is a Fredholm operator (c.f. [Vas, Prop. 5.28]), if l is chosen such that either

$$\begin{aligned} l|_{L_+} < - \frac{1}{2} \text { and } l|_{L_-} > -\frac{1}{2} \end{aligned}$$

or the other way around (with \(L_+\) and \(L_-\) swapped). The decay assumption on \(u|_{t_* = 0}\) in Theorem 1.2 ensures that \(z|_{t_* = 0} \in {\mathcal {X}}^{s,l}_{sc}\), and therefore each summand in (25), is in \(\ker ({\hat{P}})\) as set up in (28). We have thus proven (26), for the case when \(\Lambda = 0\) and \(\sigma \ne 0\) (and \(\textrm{Im} \,\sigma \ge 0\)).

The case which remains is when \(\Lambda = \sigma = 0\). The structure of the operator \({\hat{P}}\) now changes drastically near \(x = 0\) and is more naturally thought of as a b-operator in the sense of Melrose [Mel93], see also [GH08, GH09]. We follow [Vas, Sec. 5.6] for the Fredholm theory. Concretely, we note that the fast decay assumptions on A and B ensure that

$$\begin{aligned} x^{-\frac{n-2}{2}}x^{-2} {\hat{P}} x^{\frac{n-2}{2}} \end{aligned}$$

is a b-operator with normal operator

$$\begin{aligned} - (x\partial _x)^2 + \Delta _h + \frac{(n-2)^2}{4} \end{aligned}$$

at \(x = 0\). Choose a smooth function \(f: [0, \infty ) \rightarrow [0, \infty )\), such that \(f(x) = x\) for \(x \le \epsilon \) and \(f(x) = 1\) for \(x \ge 2\epsilon \) and define

$$\begin{aligned} L:= f(x)^{-\frac{n-2}{2}}f(x)^{-2} {\hat{P}} f(x)^{\frac{n-2}{2}}, \end{aligned}$$

where \(\epsilon > 0\) small enough so that the component of the characteristic set of \({\hat{P}}\) away from \(x = 0\) is unaffected by this conjugation. We now define the spaces

$$\begin{aligned} {\mathcal {Y}}_b^{s, l}:= \{u|_{{\mathcal {V}}} \mid u \in H^{s, l}_b(\overline{{\mathbb {R}}^3})\}, \end{aligned}$$

where \(H^{s, l}_b(\overline{{\mathbb {R}}^3})\) is defined in [Vas, p. 353] and

$$\begin{aligned} {\mathcal {X}}_b^{s, l}:= \left\{ u \in {\mathcal {Y}}_b^{s,l} \mid L u \in {\mathcal {Y}}_b^{s-1,l} \right\} . \end{aligned}$$

By combining the discussion on [Vas, p. 361] (c.f. also [Vas, Thm. 5.11]) with the theory near the event horizon described above, we know that

$$\begin{aligned} L: {\mathcal {X}}_b^{s, l} \rightarrow {\mathcal {Y}}_b^{s-1, l} \end{aligned}$$

is a Fredholm for all \(s, l \in {\mathbb {R}}\), such that

$$\begin{aligned} l^2 - \frac{(n-2)^2}{4} \end{aligned}$$

is not an \(L^2\) eigenvalue of \(\Delta \) on the 2-sphere. Since the set of \(L^2\)-eigenvalues is discrete, we can choose l arbitrarily large and still have a Fredholm operator. It follows that the kernel of L is finite dimensional. Now, the kernel of \({\hat{P}}\) and the kernel of L are related just by a multiplication with \(f(x)^{\frac{n-2}{2}}\) and we have thus proven the \(\ker ({\hat{P}})\) is finite dimensional and consequently (26). This finishes the proof. \(\square \)