Subextremal Reissner Research Articles

Black Hole and Equipotential Photon Surface Uniqueness in Four-Dimensional Asymptotically Flat Electrostatic Electro-Vacuum Spacetimes

AbstractWe study four-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani–Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal, and super-extremal Reissner–Nordström spacetimes. We also obtain new results for connected photon spheres and equipotential photon surfaces in the extremal case. Finally, we provide, up to a restriction on the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) equipotential photon surfaces in the super-extremal case, not yet treated in the literature.

Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström black holes

We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner–Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis’ results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.

Universality of the quantum energy flux at the inner horizon of asymptotically de Sitter black holes

Recently, it was found that the energy flux of a free scalar quantum field on a Reissner–Nordström–de Sitter spacetime has a quadratic divergence towards the inner horizon of the black hole. Moreover, the leading divergence was found to be state independent as long as the spectral gap of the wave equation on the spacetime is sufficiently large. In this work, we show that the latter result can be extended to all subextremal Reissner–Nordström–de Sitter and subextremal Kerr–de Sitter spacetimes with a positive spectral gap.

On holonomy singularities in general relativity and the Cloc0,1-inextendibility of space-times

This paper investigates the structure of gravitational singularities at the level of the connection. We show in particular that for FLRW space-times with particle horizons a local holonomy, which is related to a gravitational energy, becomes unbounded near the big-bang singularity. This implies the Cloc0,1-inextendibility of such FLRW space-times. Again using an unbounded local holonomy, we also give a general theorem establishing the Cloc0,1-inextendibility of spherically symmetric weak null singularities which arise at the Cauchy horizon in the interior of black holes. Our theorem does not presuppose the mass-inflation scenario and in particular applies to the Reissner–Nordström-Vaidya space-times, as well as to space-times which arise from small and generic spherically symmetric perturbations of two-ended subextremal Reissner–Nordström initial data for the Einstein–Maxwell scalar field system. In previous work, Luk and Oh proved the C2-formulation of strong cosmic censorship for this latter class of space-times—and based on their work we improve this to a Cloc0,1-formulation of strong cosmic censorship.

The Linear Stability of Reissner–Nordström Spacetime: The Full Subextremal Range $$|Q|<M$$

We prove the linear stability of subextremal Reissner–Nordström spacetimes as solutions to the Einstein–Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations derived in our previous works (Giorgi in Ann Henri Poincar, 21: 24852580, 2020; Giorgi in Class Quantum Grav 36:205001, 2019), where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range \(|Q|<M\), completely in physical space. Such decay estimates, together with the estimates for the gauge-dependent quantities of the perturbations obtained in Giorgi (Ann PDE 6:8, 2020), settle the problem of linear stability to gravitational and electromagnetic perturbations of Reissner–Nordström solution in the full subextremal range \(|Q|< M\).

Higher order linear stability and instability of Reissner–Nordström’s Cauchy horizon

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner–Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in \(C^l\), \(l\geqslant 1\), up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in \(C^1\) and \(H^1\).

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

We consider solutions of the massless scalar wave equation $\Box_g\psi=0$, without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It follows from previous analyses in the Kerr exterior that for solutions $\psi$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon $\mathcal H^+$. Using the derived decay rate, we show that $\psi$ is in fact uniformly bounded, $|\psi|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which $\psi$ in fact extends continuously. In analogy to our previous paper, [30], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner--Nordstr\"om backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and application of Sobolev embedding. In contrast to the Reissner--Nordstr\"om case the commutation leads to additional error terms that have to be controlled.

Photon sphere uniqueness in higher-dimensional electrovacuum spacetimes

We show a uniqueness result for the n-dimensional spatial Reissner–Nordström manifold: a static, electrovacuum, asymptotically flat system which is asymptotically Reissner–Nordström is a subextremal Reissner–Nordström manifold with positive mass, provided that its inner boundary is a (possibly disconnected) photon sphere that fulfils a suitably defined quasilocal subextremality condition.Our result implies a number of earlier uniqueness results for the Schwarzschild and the Reissner–Nordström manifolds in the static, (electro-)vacuum, asymptotically flat context, both for photon sphere and black hole inner boundaries, in the tradition of Bunting–Masood-ul Alaam (1987 Gen. Relativ. Gravit. 19 147–54) and Ruback (1988 Class. Quantum Grav. 5 L155–9). The proof relies on the ideas from those works, combined with newer techniques developed by Cederbaum and Galloway (2017 Commun. Anal. Geom. 25 303–20) and Cederbaum (in progress). The author kindly made a draft available to us.

Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

We obtain the second-order late-time asymptotics for the radiation field of solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner–Nordström families of black hole spacetimes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections have been heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman–Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for spherically symmetric compactly supported initial data (and hence, with a vanishing Newman–Penrose constant) explicitly obtained here appear to be new.

A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordstrom black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.

Stability and Instability of the Sub-extremal Reissner\u2013Nordstr\xf6m Black Hole Interior for the Einstein\u2013Maxwell\u2013Klein\u2013Gordon Equations in Spherical Symmetry

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting:Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry.Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry.This instability of the black hole interior can also be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data.

Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime

Adapting and extending the techniques developed in recent work with Vasy for the study of the Cauchy horizon of cosmological spacetimes, we obtain boundedness, regularity and decay of linear scalar waves on subextremal Reissner–Nordström and (slowly rotating) Kerr spacetimes, without any symmetry assumptions; in particular, we provide simple microlocal and scattering theoretic proofs of analogous results by Franzen. We show polynomial decay of linear waves relative to a Sobolev space of order slightly above 1/2. This complements the generic H_{\mathrm {loc}}^1 blow-up result of Luk and Oh.

Bounded Energy Waves on the Black Hole Interior of Reissner–Nordström–de Sitter

Motivated by the Strong Cosmic Censorship Conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation $\Box_g\phi=0$ on fixed subextremal Reissner--Nordstr\"om--de Sitter backgrounds $({\mathcal M}, g)$, without imposing symmetry assumptions on $\phi$. We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in $C^0\cap H^1_{loc}$. Our results correspond to another manifestation of the potential breakdown of Strong Cosmic Censorship in the positive cosmological constant setting.

Linear Waves in the Interior of Extremal Black Holes I

We consider solutions to the linear wave equation in the interior region of extremal Reissner–Nordström black holes. We show that, under suitable assumptions on the initial data, the solutions can be extended continuously beyond the Cauchy horizon and, moreover, that their local energy is finite. This result is in contrast with previously established results for subextremal Reissner–Nordström black holes, where the local energy was shown to generically blow up at the Cauchy horizon.

Superradiance on the Reissner–Nordstrøm metric

In this article, we study the superradiance of charged scalar fields on the sub-extremal Reissner–Nordstrøm metric, a mechanism by which such fields can extract energy from a static spherically symmetric charged black hole. A geometrical way of measuring the amount of energy extracted is proposed. Then we investigate the question numerically. The toy model and the numerical methods used in our simulations are presented and the problem of long time measurement of the outgoing energy flux is discussed. We provide a numerical example of a field exhibiting a behaviour analogous to the Penrose process: an incoming wave packet which splits, as it approaches the black hole, into an incoming part with negative energy and an outgoing part with more energy than the initial incoming one. We also show another type of superradiant solution for which the energy extraction is more important. Hyperradiant behaviour is not observed, which is an indication that the Reissner–Nordstrøm metric is linearly stable in the sub-extremal case.

On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion

This paper is the first part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein–Maxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal Reissner–Nordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a ‘suitably regular’ Lorentzian manifold. In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of a maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals. In the second part of this series (Costa et al , arXiv:1406.7253) we study the stability of the radius function at the Cauchy horizon. In the third and final paper (Costa et al ,arXiv:1406.7261) we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.