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.
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Notes
This is the setting for the non-linear stability of the static model of de Sitter space, both for the Einstein vacuum equations, see [61, “Appendix C”], and for the Einstein–Maxwell system.
In any case, this is not the right equation to consider for charged black holes, as it ignores the coupling of the electromagnetic field and the metric tensor.
This can be deduced from [58, Theorem 1] applied to the equation \((\delta _g d+d\delta _g)A=0\).
In fact, for \(\gamma _3>0\) small, one can check that \(\Theta =0\). However—and this is crucial for Einstein–Maxwell—even a finite-dimensional family \(\Theta \) of gauge modifications would cause no problems.
The point is that this expression does not involve first derivatives of g, which has the advantage of making the gauge-fixed Einstein–Maxwell system principally scalar without having to assign different regularities to g and A.
Here, \(N^*(\Gamma \cap \{\sigma =1\})\) is the quotient \(\mathcal I/\mathcal I^2\), where \(\mathcal I\) is the space of \({\mathcal C}^\infty \) functions on \(\{\tau _s=0,\ \sigma =1\}\) vanishing on \(\Gamma \).
The results in [61] are stated for slowly rotating KdS spacetimes, but extend immediately to slowly rotating KNdS spacetimes as well.
See “Appendix A” for the definition of Sobolev spaces \({\bar{H}}^s\) of extendible distributions.
In the notation of [72], we discuss the case \(n=2\), \(\kappa ^2=2\) (see [72, equation (2\(\cdot \)12)]), \(q=Q\), \(\lambda =\Lambda /3\) (see [72, equation (2\(\cdot \)14)]), so \(E_0=Q/r^2\) (see [72, equation (2\(\cdot \)9)]), and there are a number of sign changes due to the different sign convention adopted here.
The O(3)-invariance, as opposed to merely SO(3)-invariance, rules out operators related to the orientation of \(\mathbb {S}^2\) such as .
These three calculations provide us with a complete orthogonal basis of \(L^2(\mathbb {S}^2;S^2T^*\mathbb {S}^2)\) and thus with the spectrum of acting on symmetric 2-tensors.
In these expressions, the operators \(\widehat{\delta }{}^*\), \(\iota _\rho \equiv \iota _{\rho ^{\widehat{\sharp }}}\) and \(\iota _\mathbf {X}\equiv \iota _{\mathbf {X}^{\widehat{\sharp }}}\), with \(\widehat{\sharp }\) indicating that one uses \({\widehat{g}}\) to compute the musical isomorphisms, the metric \({\widehat{g}}\) is fixed, i.e. these operators are not subject to the modifications by \((\updelta \dot{g},\updelta \dot{A})\); thus, for instance, \(\updelta (\widehat{\delta }{}^*\mathbf {X})=\widehat{\delta }{}^*(\updelta \mathbf {X})\).
By (5.16), this is the aspherical part of the tensor \(2 G_g\mathscr {L}_1(\dot{g},\dot{A})\), which is the sum of the linearized Einstein tensor and the contribution from the linearized stress-energy-momentum tensor.
There is a consistency condition for such a system to be well-posed. Indeed, differentiating the constraint, we find the necessary pointwise condition \((\gamma '+T^t\gamma )(v_0)-h'=0\) for all \(v_0\in \mathbb {R}^3\) with \(\gamma (v_0)=h\), which is equivalent to \(\gamma '+T^t\gamma =\alpha \gamma \) for a scalar \(\alpha \) satisfying \(\alpha h=h'\).
Another motivation for choosing the coefficients of X and Y equal is the requirement that the master variables we will describe below be regular at the event and cosmological horizons; see the discussion around equation (5.61).
The necessary symbolic calculations are quite lengthy and were performed using mathematica.
In [72, equation (C\(\cdot \)9b)], the factor r on the right hand side is extraneous.
Without further calculations, this almost follows directly from (5.57): if \((X,Y,\mathcal A)\) are arbitrary, i.e. not necessarily solutions of the linearized Einstein–Maxwell system, then the equation (5.57) is equal to some linear combination (with coefficients meromorphic in \(\sigma \)) of the components of the linearized Einstein–Maxwell system and their derivatives; thus, if the coefficients of this linear combination are regular at \(\sigma =0\), then we can indeed conclude that for a stationary perturbation \((X,Y,\mathcal A)\), the master variables \(\Psi _\pm ^0\) solve (5.69) in their domain of definition.
At frequencies \(\sigma \ne 0\), this is again automatic, as \(\Phi \) and \(\mathcal A\) are linear combinations of \(\Psi _\pm \) by the definition of the latter, and then X and Y can be recovered from \(\Phi \), \(\mathcal A\) and their derivatives by means of (5.50).
A simple calculation verifies that these quantities indeed solve (5.88).
There is an inconsequential typo in the matrix multiplying \(\gamma _2''\) in [61]; the matrix given here is the correct one.
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Acknowledgements
I would like to thank András Vasy, Maciej Zworski, Jim Isenberg, Sergiu Klainerman, and Yakov Shlapentokh-Rothman for valuable discussions and for their interest and support. I am very grateful to Mihalis Dafermos, and to an anonymous referee, for valuable comments on both content and exposition. I would also like to thank the Miller Institute at the University of California, Berkeley, for support.
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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
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
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
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
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
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:
The functions appearing in equation (5.50) are given as follows:
The functions used in equation (5.65) are:
The functions appearing in equation (5.67) take the following form:
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Hintz, P. Non-linear Stability of the Kerr–Newman–de Sitter Family of Charged Black Holes. Ann. PDE 4, 11 (2018). https://doi.org/10.1007/s40818-018-0047-y
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DOI: https://doi.org/10.1007/s40818-018-0047-y