Non-linear Stability of the Kerr–Newman–de Sitter Family of Charged Black Holes

Abstract

We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein–Maxwell equations: if one perturbs the initial data of a slowly rotating Kerr–Newman–de Sitter (KNdS) black hole, then in a neighborhood of the exterior region of the black hole, the metric and the electromagnetic field decay exponentially fast to their values for a possibly different member of the KNdS family. This is a continuation of recent work of the author with Vasy on the stability of the Kerr–de Sitter family for the Einstein vacuum equations. Our non-linear iteration scheme automatically finds the final black hole parameters as well as the gauge in which the global solution exists; we work in a generalized wave coordinate/Lorenz gauge, with gauge source functions lying in a suitable finite-dimensional space. We include a self-contained proof of the linear mode stability of Reissner–Nordström–de Sitter black holes, building on work by Kodama–Ishibashi. In the course of our non-linear stability argument, we also obtain the first proof of the linear (mode) stability of slowly rotating KNdS black holes using robust perturbative techniques.

Appendices

Appendix A: Review of b-Geometry and b-Analysis

We only give a very brief account of the aspects of b-geometry and b-analysis which are used in the present paper; for a more detailed overview, we refer the reader to [61, “Appendix A”] as well as to Melrose’s book [84] on the subject.

Fix a smooth connected $(n+1)$-dimensional manifold M with non-empty boundary $\partial M$. We denote by $\mathcal V_{\mathrm {b}}(M)\subset \mathcal V(M)$ the space of b-vector fields, smooth vector fields on M which are tangent to $\partial M$. Away from $\partial M$, these are simply ordinary smooth vector fields. Near the boundary, with $(\tau ,x^1,\ldots ,x^n)$ denoting adapted local coordinates near a point in $\partial M$, namely with $\partial M$ given by the vanishing of $\tau$, a b-vector field V takes the form

$$\begin{aligned} V = a \tau \partial _\tau + \sum _{j=1}^n b_j \partial _{x^j},\quad a,b_j\in {\mathcal C}^\infty (M). \end{aligned}$$

Correspondingly, b-vector fields are the space of sections of a natural vector bundle ${}^{{\mathrm {b}}}TM\rightarrow M$, called b-tangent bundle, which over the interior $M^\circ$ is naturally isomorphic to the standard tangent bundle, and which near the boundary in the above coordinates has the basis $\{\tau \partial _\tau ,\partial _{x^1},\ldots ,\partial _{x^n}\}$; in particular, $\tau \partial _\tau$ is non-vanishing at $\tau =0$ as a b-vector field. One can check that $\tau \partial _\tau$ is in fact well-defined, i.e. independent of the choice of adapted local coordinates. The space $\mathrm {Diff}_{\mathrm {b}}^*(M)$ of b-differential operators is the universal enveloping algebra of $\mathcal V_{\mathrm {b}}(M)$, thus elements of $\mathrm {Diff}_{\mathrm {b}}^m(M)$ are finite linear combinations (with ${\mathcal C}^\infty (M)$ coefficients) of products of up to m b-vector fields. If $E,F\rightarrow M$ are two smooth vector bundles, one can more generally define m-th order b-differential operators $\mathrm {Diff}_{\mathrm {b}}^m(M;E,F)$ mapping ${\mathcal C}^\infty (M;E)$ into ${\mathcal C}^\infty (M;F)$, e.g. using local trivializations of E and F.

The dual bundle ${}^{{\mathrm {b}}}T^*M$ of ${}^{{\mathrm {b}}}TM$, called the b-cotangent bundle, is correspondingly spanned by $\frac{d\tau }{\tau },dx^1,\ldots ,dx^n$; here $\frac{d\tau }{\tau }$ is smooth (and non-degenerate) as a b-1-form up to $\tau =0$. A smooth b-metric g on M is then a smooth section of the second symmetric tensor power $S^2\,{}^{{\mathrm {b}}}T^*M$; in local coordinates as above, this means that

$$\begin{aligned} g = g_{00}\,\frac{d\tau ^2}{\tau ^2} + 2 g_{0j}\,\frac{d\tau }{\tau }\otimes _s dx^j + g_{ij}\,dx^i\otimes _s dx^j,\quad g_{\mu \nu }\in {\mathcal C}^\infty (M). \end{aligned}$$

If M arises as the compactification of a manifold $M^\circ$ without boundary as in equation (3.17), then a smooth b-metric on M is asymptotically stationary on $M^\circ$ in the following sense: letting $t:=-\log \tau$, we have $\frac{d\tau }{\tau }=-dt$, and a smooth function $a\in {\mathcal C}^\infty (M)$, having a Taylor expansion in powers of $\tau$, has a Taylor expansion on $M^\circ$ in powers of $e^{-t}$; thus, $g=g_0+\widetilde{g}$ with

$$\begin{aligned} g_0=g_{00}(0,x)\,dt^2 - 2 g_{0j}(0,x)\,dt\otimes _s dx^j + g_{ij}(0,x)\,dy^i\otimes _s dy^j,\quad \widetilde{g}=\mathcal O(e^{-t}), \end{aligned}$$

approaches the stationary metric $g_0$ exponentially fast as $t\rightarrow \infty$. Conversely, if $M^\circ$ is equipped with a metric g approaching a stationary metric exponentially fast at some rate $\alpha >0$, then g extends to be a smooth b-metric on M plus an error term (in general non-smooth) of size $\tau ^\alpha$. In the case of interest in the present paper, this remainder term will be conormal, or more generally lie in a weighted b-Sobolev space which we discuss further below.

We further have the b-differential ${}^{{\mathrm {b}}}d$, acting between sections of the exterior powers $\Lambda ^k\,{}^{{\mathrm {b}}}T^*M$; they are defined by extension of the usual exterior differential d from $M^\circ$; thus, acting on functions, one has

$$\begin{aligned} {}^{{\mathrm {b}}}da = (\tau \partial _\tau a)\frac{d\tau }{\tau } + (\partial _{x^j}a)dx^j, \end{aligned}$$

and in general ${}^{{\mathrm {b}}}d\in \mathrm {Diff}_{\mathrm {b}}^1(M;\Lambda ^k\,{}^{{\mathrm {b}}}T^*M,\Lambda ^{k+1}\,{}^{{\mathrm {b}}}T^*M)$.

On M, we naturally have the b-density bundle ${}^{{\mathrm {b}}}\Omega ^1(M)$, with local trivialization induced by $|\frac{d\tau }{\tau }dx^1\ldots dx^n|$; fixing a nowhere vanishing b-density $\nu$ on M, this allows us to define the $L^2$ space $L^2_{\mathrm {b}}(M;\nu )\equiv L^2(M;\nu )$. We drop the density $\nu$ from the notation from now on. (For compact M, different choices of $\nu$ lead to equivalent norms.) For integer $k\ge 0$ and real $\alpha \in \mathbb {R}$, we then define the weighted b-Sobolev space

$$\begin{aligned} H_{{\mathrm {b}}}^{k,\alpha }(M)&= \{ u\in \tau ^\alpha L^2_{\mathrm {b}}(M) :V_1\ldots V_j u\in \tau ^\alpha L^2_{\mathrm {b}}(M), \\&\qquad \qquad 0\le j\le k,\ V_\ell \in \mathcal V_{\mathrm {b}}(M),\ 1\le \ell \le j\}. \end{aligned}$$

For compact M, $H_{{\mathrm {b}}}^{k,\alpha }(M)$ can be endowed with a Hilbert space structure by means of a finite collection of b-vector fields which span ${}^{{\mathrm {b}}}T_p M$ over every $p\in M$; the norms for any two such collections are equivalent. For non-integer $s\in \mathbb {R}$, the space $H_{{\mathrm {b}}}^{s,\alpha }(M)$ is defined using duality, that is $H_{{\mathrm {b}}}^{s,\alpha }(M)^*=H_{{\mathrm {b}}}^{-s,-\alpha }(M)$, and interpolation. We point out that the definition of the space $H_{{\mathrm {b}}}^{s,\alpha }(M)$ as a Hilbert space for M compact does not require the choice of a metric. Elements of the space $H_{{\mathrm {b}}}^{\infty ,\alpha }(M)=\bigcap _{s\in \mathbb {R}}H_{{\mathrm {b}}}^{s,\alpha }(M)$ are called conormal (with respect to $L^2_{\mathrm {b}}$); for M compact, this space carries a natural Fréchet space structure. Near a point on $\partial M$, using coordinates $(\tau ,x^1,\ldots ,x^n)$ as above, and letting $t=-\log \tau$, the space $H_{{\mathrm {b}}}^{s,\alpha }(M)$ is locally the same (as a Hilbert space, up to equivalence of norms) as the space $e^{-\alpha t}H^s(M^\circ )$, where the Sobolev space on $M^\circ$ is defined by testing with products of the vector fields $\partial _t,\partial _{x^1},\ldots ,\partial _{x^n}$.

Suppose next that $\Omega \subset M$ is a non-empty open subset of M. One can then define the space of supported distributions ${\dot{\mathscr {D}}}(\Omega )$ as the space of distributions $u\in \mathscr {D}(M)={\dot{\mathcal C}}^\infty (M;\Omega ^1 M)^*$ with ${\text {supp}}u\subset \Omega$. (The same definition applies for M without boundary.) We then define ${\dot{H}}_{{\mathrm {b}}}^{s,\alpha }(\Omega )=H_{{\mathrm {b}}}^{s,\alpha }(M)\cap {\dot{\mathscr {D}}}(\Omega )$; this thus consists of elements of $H_{{\mathrm {b}}}^{s,\alpha }(M)$ which are supported in ${\bar{\Omega }}$. The space of extendible distributions, ${\bar{\mathscr {D}}}(\Omega )$, is equal to the space of restrictions $u|_\Omega$ for $u\in \mathscr {D}(M)$; we likewise define ${\bar{H}}_{{\mathrm {b}}}^{s,\alpha }(\Omega )=H_{{\mathrm {b}}}^{s,\alpha }(M)|_{\Omega }$, with the natural (quotient) norm. Thus, elements of ${\bar{H}}_{{\mathrm {b}}}^{s,\alpha }(\Omega )$ automatically have extensions to $H_{{\mathrm {b}}}^{s,\alpha }(M)$ (with the same norm).

If $E\rightarrow M$ is a smooth vector bundle, weighted b-Sobolev spaces $H_{{\mathrm {b}}}^{s,\alpha }(M;E)$ are defined using local trivializations of E; for $\Omega \subset M$ as above, one can likewise define spaces ${\dot{H}}_{{\mathrm {b}}}^{s,\alpha }(M;E)$ and ${\bar{H}}_{{\mathrm {b}}}^{s,\alpha }(M;E)$ of supported and extendible sections of E over $\Omega$.

Appendix B: Explicit Expressions for the Mode Stability Analysis

In this appendix, we list the explicit formulas for a number of functions arising in Sect. 5; we recall that the quantities x, y, z, m were defined in (5.47), H in (5.46), $\tilde{c}$ in (5.55), $a_+$ in (5.56), and $\hat{c}$ in (5.66).

The expressions for the functions used in equation (5.49) are then:

$$\begin{aligned} V_\Phi&= \frac{\mu }{r^2 H^2}\bigl (9 x^3 - 9(2 y+6 z-m)x^2 + (72 z^2-8(4 m-3)z + 3 m^2)x \\&\qquad \qquad + 8(9 x z-12 z^2-m z)y - 32 z^3 + 24 m z(z+1) + m^2(m+2)\bigr ), \nonumber \\ F_\Phi&= -\frac{8 Q\mu }{r^3 H^2}\bigl (2(3 x-8 z)y+2 x z-3 x^2+6 x+m(m+4)\bigr ). \nonumber \end{aligned}$$

(B. 1)

The functions appearing in equation (5.50) are given as follows:

$$\begin{aligned} P_{X 0}&= \bigl (6(4 z+m)x-64 z^2-16 m z\bigr )y + 27 x^3 - 24(5 z-m)x^2 \\&\qquad + \bigl (152 z^2-2(35 m-12)z+3 m(3 m+2)\bigr )x - 64 z^3 + 48 m z^2 \nonumber \\&\qquad - 8 m(m-2) z + 2 m^2 (m+2), \nonumber \\ P_{X 1}&= 2(4 z+m)y + 9 x^2 - (16 z - 5 m + 6)x + 8 z^2 - 6 m z - 4 m, \nonumber \\ P_{X\mathcal A}&= -4(4 z+m)y - 18 x^2 + 4(8 z-m+6)x \nonumber \\&\qquad - 16 z^2 + 4(m-4)z + 2 m(m+6), \nonumber \\ P_{Y 0}&= 2\bigl (18 x^2-3(28 z-m)x+96 z^2-8 m z\bigr )y + 9 x^3 - 6(10 z-m)x^2 \nonumber \\&\qquad + \bigl (120 z^2-2(11 m-12)z + 3 m(m+2)\bigr )x \nonumber \\&\qquad - 64 z^3 + 16(m-4)z^2 - 8 m(m+2)z, \nonumber \\ P_{Y 1}&= 2(6 x-12 z+m)y+3 x^2-(12 z+m+6)x + 8 z^2 + 2(m+8)z, \nonumber \\ P_{Y\mathcal A}&= -4(6 x-12 z+m)y - 6 x^2 + 4(6 z-m)x \nonumber \\&\qquad - 16 z^2 + 4(m-4)z - 2 m(m+2), \nonumber \\ P_Z&= (-6 x+16 z)y+3 x^2+(-2 z+3 m)x-(4 m+8)z-2 m. \nonumber \end{aligned}$$

(B. 2)

The functions used in equation (5.65) are:

$$\begin{aligned} P_X&= \bigl (9 x-36 z+3(12 y-6-m)\bigr )x - 6(12 z-m)y + 6(4 z+m+8)z, \nonumber \\ P_Y&= -3(9 x-16 z+5 m-6)x-6(4 z+m)y - 6(4 z-3 m)z + 12 m, \nonumber \\ P_\mathcal A&= -8(\tilde{c}+m r)z. \end{aligned}$$

(B. 3)

The functions appearing in equation (5.67) take the following form:

$$\begin{aligned} P_{X+}&= -8 z H \mu - 3\bigl (9 x-8(5 z-m)\bigr )x^2 - 3 m(3 m+2+2 y)x \nonumber \\&\qquad + 2(35 m-12 y-12)x z+64 z^3 - 8 z^2(19 x-8 y+6 m) \nonumber \\&\qquad + 8 m(2 y+m-2)z - 2 m^2(m+2) \nonumber \\&\quad - \Bigl (4 z-\frac{\tilde{c}}{r}\Bigr )\bigl ((9 x-16 z+2 m-12)x+2(m+4 z)y \nonumber \\&\qquad + 2(4 z-m+4)z - m(m+6)\bigr ), \nonumber \\ P_{X-}&= 81(4 z-x)x^4 - 18 x^4(4 m+3\hat{c}) \nonumber \\&\qquad - 3 x^3\bigl (144 z^2-2 z(36\hat{c}+23 m)+m(16\hat{c}+3(3 m+2)+6 y)\bigr ) \nonumber \\&\qquad - 2 x^2\bigl (-96 z^3+8 z^2(18\hat{c}+m)-2 m z(35\hat{c}+5 m+24) \nonumber \\&\qquad + 3 m(-8 y z+(m+2)(m+3\hat{c})+2\hat{c}(y-2))\bigr ) \nonumber \\&\qquad -4 m^2(m+2)\hat{c}\,x-64 x z^3(m-2\hat{c})-16 m(6\hat{c}+4 y-m+4)x z^2 \nonumber \\&\qquad + 8 m x z\bigl (m(3\hat{c}+m+6)+(4\hat{c}-2 m)y\bigr ), \nonumber \\ Q_+&= 12 x^2+\bigl (3(y-9 z-3)+5 m+2\hat{c}\bigr )x + 16 z^2 \nonumber \\&\qquad - 2 z(3 m+\hat{c}-4) + 2(m+\hat{c})y-2(2 m+\hat{c}), \nonumber \\ Q_-&= -\frac{16 r a_+ \mu Q}{r^2}+(9 x-16 z+5 m-6)x+8 z^2+2(m+4 z)y-6 m z-4 m, \nonumber \\ P_{Y+}&= -9 x^3+6 x^2(3 y+z+m-\hat{c}+3) + 16 m z^2 + 4 x z(-12 y-6 m+\hat{c}) \nonumber \\&\qquad +x\bigl (12(2 m+\hat{c})y+m(7 m+12)+12\hat{c}\bigr ) - 16 z\bigl ((3 m+2\hat{c})y-m-1\bigr ) \nonumber \\&\qquad + 4 m^2 y+2(m+2)^2(-2 z+m+\hat{c}-2) + 16 m z + 8(m-\hat{c}+2), \nonumber \\ P_{Y-}&= 81 x^4+54 x^3(6 y+4 z+m+\hat{c}+1) + 9 x^2\bigl (16 z^2-2 z(24 x-m+8\hat{c}) \nonumber \\&\qquad -6 x+24(\hat{c}-4 z)y+m(6 y+3 m+4\hat{c}+6)\bigr ) + 6 x\bigl (16 z^2(6 y-3 m+\hat{c}) \nonumber \\&\qquad +2 z(24(m-2\hat{c})y+m(m-3\hat{c}-12)) + 3 m(2 y+m+2)\hat{c}\bigr ) \nonumber \\&\qquad + 8z\bigl (24 m z^2-6 z(4(3 m-2\hat{c})y+m(m-4)) \nonumber \\&\qquad +3 m^2(2 y+m-\hat{c}+2) - 12 m \hat{c}\bigr ). \end{aligned}$$

(B. 4)