Global Regular Null Hypersurfaces in a Perturbed Schwarzschild Black Hole Exterior

Abstract

The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole. A significant point of the result is that the ambient spacetime metric is not required to be differentiable in all directions.

Appendix A. Proof of Lemma 3.5: Gronwall’s inequality

We consider the solution of the following propagation equation

$$\begin{aligned} \partial _s {}^{s}{\underline{f}}+ {}^{s}X^i \partial _i {}^{s}{\underline{f}}= {}^{s}{{\mathrm {re}}}\end{aligned}$$

(A.1)

and obtain estimates for ${}^{s}{\underline{f}}$ by integrating the equation. We first derive the propagation equation for the rotational derivatives of ${}^{s}{\underline{f}}$. Let $R_i, i=1,2,3$ be the rotational vector fields on the unit round sphere $({\mathbb {S}}^2, {\mathop {g}\limits ^{\circ }})$. In the 3-dimensional Euclidean space,

$$\begin{aligned} R_i = \sum _{j,k}\epsilon _{ijk} x^j \partial _k. \end{aligned}$$

Differentiating equation (A.1), we get

$$\begin{aligned} \partial _s R_i {}^{s}{\underline{f}}+ {}^{s}X\left( R_i {}^{s}{\underline{f}}\right) + \left[ R_i, {}^{s}X\right] {}^{s}{\underline{f}}= R_i {}^{s}{{\mathrm {re}}}. \end{aligned}$$

For the higher order derivatives, we have

$$\begin{aligned} \partial _s \left( R_{i_1} \cdots R_{i_m} {}^{s}{\underline{f}}\right) + {}^{s}X\left( R_{i_1} \cdots R_{i_m} {}^{s}{\underline{f}}\right) = {}^{R_{i_1}, \cdots , R_{i_m},s}{\mathrm {re}}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} {}^{R_{i_1}, \cdots , R_{i_m},s}{\mathrm {re}}&= R_{i_1} \cdots R_{i_n} {}^{s}{{\mathrm {re}}},\\&\quad - \sum _{\begin{array}{c} \{l_1,\cdots , l_s\}\cup \{k_1,\cdots ,k_{n-s}\} \\ =\{1,\cdots , m\}, \\ l_1<\cdots< l_s, \\ k_1< \cdots < k_{n-s}, \\ s\le m-1 \end{array}} [ R_{i_{k_1}}, \cdots , R_{i_{k_{n-s}}} , {}^{s}X] \left( R_{i_{l_1}} \cdots R_{i_{l_s}} {}^{s}{\underline{f}}\right) \end{aligned}$$

and

$$\begin{aligned} {[} R_{i_{k_1}}, \cdots , R_{i_{k_{n-s}}} , {}^{s}X] = {\mathcal {L}}_{R_{i_{k_1}}}\cdots {\mathcal {L}}_{R_{i_{k_{n-s}}}}{}^{s}X. \end{aligned}$$

The proof of Lemma 3.5 relies on the following lemma on the diffeomorphisms generated by the vector field ${}^{s}X$.

Lemma 5.3

Define the one parameter family of diffeomorphisms $\left\{ \varphi _s \right\}$ generated by ${}^{s}X$, i.e.

$$\begin{aligned} \varphi _s: {\mathbb {S}}^2 \rightarrow {\mathbb {S}}^2, \quad \frac{\mathrm {d}}{\mathrm {d}s} \varphi _s(\vartheta ) = {}^{s}X(\varphi _s(\vartheta )), \quad \varphi _{s=0}=\mathrm {Id} \end{aligned}$$

We define the push forward metric $g_s$ of ${\mathop {g}\limits ^{\circ }}$ via $\varphi _s$

$$\begin{aligned} g_s=(\varphi _s)_* {\mathop {g}\limits ^{\circ }}, \quad \mathrm {dvol}_{g_s} = (\varphi _s)_* \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}}. \end{aligned}$$

Then there exists a family of functions $\left\{ \phi _s \right\}$ such that

$$\begin{aligned} \mathrm {dvol}_{g_s} = \phi _s \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}}, \end{aligned}$$

and $\phi _s$ satisfies the equation

$$\begin{aligned} \partial _s \phi _s + {}^{s}X^i \partial _i \phi _s = - \phi _s {\mathop {\mathrm {div}}\limits ^{\circ }}{}^{s}X, \quad \left( \partial _s + {}^{s}X^i \partial _i \right) \log \phi _s = - {\mathop {\mathrm {div}}\limits ^{\circ }}{}^{s}X. \end{aligned}$$

(A.3)

Assume that

$$\begin{aligned} \sup _{{\mathbb {S}}^2} \big \vert {\mathop {\mathrm {div}}\limits ^{\circ }}{}^{s}X\big \vert \le \frac{k r_0}{(r_0+t)^2}, \end{aligned}$$

then

$$\begin{aligned} \big \vert \log \phi _s \big \vert \le k . \end{aligned}$$

Proof

It is sufficient to prove equation (A.3). It follows from

$$\begin{aligned} {\mathcal {L}}_{{}^{s}X} \mathrm {dvol}_{g_s} = {\mathcal {L}}_{{}^{s}X} \left( \phi _s \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}} \right) =0. \end{aligned}$$

$\square$

Applying the above lemma, we prove the $L^p$ estimate for ${}^{s}{\underline{f}}$.

Lemma 5.4

Under the assumptions of Lemma 3.5, there exists a constant c depending on p such that

$$\begin{aligned} \big \Vert {}^{s}{\underline{f}}\big \Vert _{L^p} \le \exp (c k ) \left\{ \big \Vert {}^{s=0}{\underline{f}} \big \Vert _{L^p} + \int _0^s \big \Vert {}^{s=t}{\mathrm {re}} \big \Vert _{L^p} \mathrm {d}t \right\} . \end{aligned}$$

Proof

We use the one parameter family defined in Lemma 5.3. Denote the pullbacks of ${}^{s}{\underline{f}}$ and ${}^{s}{{\mathrm {re}}}$ via $\varphi _s$ by ${}^{s}{\underline{f}}_{*}, {}^{s}{{\mathrm {re}}}_{*}$,

$$\begin{aligned} {}^{s}{\underline{f}}_{*} \left( \vartheta \right) = \varphi _s^* ({}^{s}{\underline{f}}) (\vartheta ) = {}^{s}{\underline{f}}\circ \varphi _s (\vartheta ), \quad {}^{s}{{\mathrm {re}}}_{*} \left( \vartheta \right) = \varphi _s^* ({}^{s}{{\mathrm {re}}}) (\vartheta ) = {}^{s}{{\mathrm {re}}}\circ \varphi _s (\vartheta ). \end{aligned}$$

Then we have the propagation equation

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s} {}^{s}{\underline{f}}_{*}(\vartheta ) = {}^{s}{{\mathrm {re}}}_{*} (\vartheta ). \end{aligned}$$

Therefore for $\big \Vert {}^{s}{\underline{f}}_{*} \big \Vert _{L^p}$, we have

$$\begin{aligned} \big \vert \frac{\mathrm {d}}{\mathrm {d}s} \left( \big \Vert {}^{s}{\underline{f}}_{*} \big \Vert _{L^p} \right) ^p \big \vert =&\big \vert p \int _{{\mathbb {S}}^2} {}^{s}{{\mathrm {re}}}_{*} \cdot \left( {}^{s}{\underline{f}}_{*} \right) ^{p-1} \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}} \big \vert \le p \big \Vert {}^{s}{{\mathrm {re}}}_{*} \big \Vert _{L^p} \cdot \big \Vert {}^{s}{\underline{f}}_{*} \big \Vert _{L^p}^{p-1} \end{aligned}$$

Therefore we have

$$\begin{aligned} \big \Vert {}^{s}{\underline{f}}_{*} \big \Vert _{L^p} \le \big \Vert {}^{s=0}{\underline{f}} \big \Vert _{L^p} + \int _0^s \big \Vert {}^{s=t}{\mathrm {re}}_{*} \big \Vert _{L^p} \mathrm {d}t. \end{aligned}$$

Between $L^p$ norms $\big \Vert {}^{s}{\underline{f}}_{*} \big \Vert _{L^p}$ and $\big \Vert {}^{s}{\underline{f}}\big \Vert _{L^p}$, we have that

$$\begin{aligned} \int _{{\mathbb {S}}^2} \left( {}^{s}{\underline{f}}_{*} \right) ^p \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}}&= \int _{{\mathbb {S}}^2} \left( (\varphi _s)^{*} {}^{s}{\underline{f}}\right) ^p \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}} = \int _{{\mathbb {S}}^2} \left( {}^{s}{\underline{f}}\right) ^p \mathrm {dvol}_{\left( \varphi _s \right) _{*} {\mathop {g}\limits ^{\circ }}} \\&= \int _{{\mathbb {S}}^2} \left( {}^{s}{\underline{f}}\right) ^p \mathrm {dvol}_{g_s} = \int _{{\mathbb {S}}^2} \left( {}^{s}{\underline{f}}\right) ^p \phi _s \mathrm {dvol}_{{\mathop {g}\limits ^{\circ }}}. \end{aligned}$$

Then by Lemma 5.3, we have

$$\begin{aligned} \exp (-c k) \big \Vert {}^{s}{\underline{f}}\big \Vert _{L^p} \le \big \Vert {}^{s}{\underline{f}}_{*} \big \Vert _{L^p} \le \exp (c k) \big \Vert {}^{s}{\underline{f}}\big \Vert _{L^p} \end{aligned}$$

since we have the $L^{\infty }$ estimate of ${\mathop {\mathrm {div}}\limits ^{\circ }}{}^{s}X$ from $\big \Vert {}^{s}X\big \Vert ^{n+1,p}$ by the Sobolev embedding. Similarly, for $\big \Vert {}^{s}{{\mathrm {re}}}_{*} \big \Vert _{L^p}$ and $\big \Vert {}^{s}{{\mathrm {re}}}\big \Vert _{L^p}$, they are also comparable with the constant $\exp (c k)$. Hence

$$\begin{aligned} \big \Vert {}^{s}{\underline{f}}\big \Vert _{L^p} \le \exp (c k) \left\{ \big \Vert {}^{s=0}{\underline{f}} \big \Vert _{L^p} + \int _0^s \big \Vert {}^{s=t}{\mathrm {re}} \big \Vert _{L^p} \mathrm {d}t \right\} . \end{aligned}$$

$\square$

Therefore Lemma 3.5 follows from equation (A.2) and lemma 5.4.