Black hole nonmodal linear stability: odd perturbations of Reissner-Nordström (original) (raw)
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Black hole nonmodal linear stability under odd perturbations: The Reissner-Nordström case
Physical Review D
Following a program on black hole nonmodal linear stability initiated in Phys. Rev. Lett. 112 (2014) 191101, we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordström (A)dS black hole. We show that all the gauge invariant information in the metric and Maxwell field perturbations is encoded in the spacetime scalars F = δ(F * αβ F αβ) and Q = δ(1 48 C * αβγδ C αβγδ), where C αβγδ is the Weyl tensor, F αβ the Maxwell field, a star denotes Hodge dual and δ means first order variation, and that the linearized Einstein-Maxwell equations are equivalent to a coupled system of wave equations for F and Q. For nonnegative cosmological constant we prove that F and Q are pointwise bounded on the outer static region. The fields are shown to diverge as the Cauchy horizon is approached from the inner dynamical region, providing evidence supporting strong cosmic censorship. In the asymptotically AdS case the dynamics depends on the boundary condition at the conformal timelike boundary and there are instabilities if Robin boundary conditions are chosen.
Black hole nonmodal linear stability: Even perturbations in the Reissner-Nordström case
Physical Review D
This paper is a companion of [Phys. Rev. D 95, 124041 (2017)] in which, following a program on black hole nonmodal linear stability initiated in Phys. Rev. Lett. 112 (2014) 191101, odd perturbations of the Einstein-Maxwell equations around a Reissner-Nordström (A)dS black hole were analyzed. Here we complete the proof of the nonmodal linear stability of this spacetime by analyzing the even sector of the linear perturbations. We show that all the gauge invariant information in the metric and Maxwell field even perturbations is encoded in two spacetime scalars: S, which is a gauge invariant combination of δ(C αβγǫ C αβγǫ) and δ(C αβγδ F αβ F γδ), and T , a gauge invariant combination of δ(∇µF αβ ∇ µ F αβ) and δ(∇µC αβγδ ∇ µ C αβγδ). Here C αβγδ is the Weyl tensor, F αβ the Maxwell field and δ means first order variation. We prove that S and T are are in one-one correspondence with gauge classes of even linear perturbations, and that the linearized Einstein-Maxwell equations imply that these scalar fields are pointwise bounded on the outer static region.
Black hole non-modal linear stability: the Schwarzschild (A)dS cases
Classical and Quantum Gravity, 2016
The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant Λ. Two gauge invariant combinations G± of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map [h αβ ] → (G− ([h αβ ]) , G+ ([h αβ ])) with domain the set of equivalent classes [h αβ ] under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of [h αβ ] in terms of (G−, G+) is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, G+ and G− are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there are choices of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar's duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the Λ = 0 case are explained in detail. CONTENTS
Physical Review D
We study the instability of a Reissner-Nordström-AdS (RNAdS) black hole under perturbations of a massive scalar field coupled to Einstein tensor. Calculating the potential of the scalar perturbations we find that as the strength of the coupling of the scalar to Einstein tensor is increasing, the potential develops a negative well outside the black hole horizon, indicating an instability of the background RNAdS. We then investigate the effect of this coupling on the quasinormal modes. We find that there exists a critical value of the coupling that triggers the instability of the RNAdS. We also find that as the charge of the RNAdS is increased towards its extremal value, the critical value of the derivative coupling is decreased.
Gravitational instability of the inner static region of a Reissner–Nordström black hole
Classical and Quantum Gravity, 2010
Reissner-Nordström black holes have two static regions: r > ro and 0 < r < ri, where ri and ro are the inner and outer horizon radii. The stability of the exterior static region has been established long time ago. In this work we prove that the interior static region is unstable under linear gravitational perturbations, by showing that field perturbations compactly supported within this region will generically excite a mode that grows exponentially in time. This result gives an alternative reason to mass inflation to consider the space time extension beyond the Cauchy horizon as physically irrelevant, and thus provides support to the strong cosmic censorship conjecture, which is also backed by recent evidence of a linear gravitational instability in the interior region of Kerr black holes found by the authors. The use of intertwiners to solve for the evolution of initial data plays a key role, and adapts without change to the case of super-extremal Reissner-Nordström black holes, allowing to complete the proof of the linear instability of this naked singularity. A particular intertwiner is found such that the intertwined Zerilli field has a geometrical meaning-it is the first order variation of a particular Riemann tensor invariant-. Using this, calculations can be carried out explicitely for every harmonic number.
Nonmodal Linear Stability of the Schwarzschild Black Hole
Physical Review Letters, 2014
A proof is given that the space L of solutions of the linearized vacuum Einstein equation around a Schwarzschild black hole is parametrized by two scalar fields, which are gauge invariant combinations of perturbed algebraic and differential invariants of the Weyl tensor and encode the information on the odd (−) and even (+) sectors L±. These fields measure the distortion of the geometry caused by a generic perturbation and are shown to be pointwise bounded on the outer region r ≥ 2M .
Instability of the Cauchy horizon of Reissner-Nordström black holes
Physical Review D, 1979
The stability of the inner Reissner-Nordstrom geometry is studied with test massless integer-spin. fields.. In contrast to previous mathematical treatments we present physical arguments for the processes involved and show that ray tracing and simple first-order scattering suffice to elucidate nsost of the results. Monochromatic waves which are of small amplitude aiid ingoing near the outer horizon develop infinite energy densities near the inriei' Cauchy horizon (as measured by a freely falling observer'); Previous work has shown that certain. derivatives of the field in a general (nonmonochromatic) disturbance must fall off exponentially near the inner (Cauchy) horizon (r. = r) if energy densities are to r~ain finite. Thus the solution is unstable to physically reasonable perturbations which arise outside the black hole because such perturbations, if localized near past nu11 infinity (g), cannot be localized near r+, the outer horizon. The mass-energy of an infalling disturbance would generate multipole moments on the biack hole. Price, 'Sibgatullin, and Alekseev have shown that such moments are radiated away as "tails" which travel outward and are rescattered inward yielding a wave field with a time dependence t I', p & 0. This decay in time:is sufficiently slow that the tails yield infinite energy densities on the Cauchy horizon. (The amplification of the low-frequency tails upon interacting with t4e time-dependent potential between the horizons is an important feature guaranteeing the infinite energy density.) The interior structure of the analytically extended solution is thus disrupted by finite external disturbances. Gursel et al. have further shown that even perturbations which are. localized as they cross the outer horizon produce singularities at the inner horizon. By a raytracing scheme we are able to show that this singUIlarity arises when the incoming radiation is first scattered for r & r+ {i. e. , just inside the outer horizon), whence the exponentially small scattered radiation is efficiently rescattered when the potential becomes strong. The exponentially small first scattering near the outer horizon is translated by the second. scattering into exponentially decaying waves near the inner horizon. Their exponential decay is, however, so slow that the resultant energy density is singular on the horizon.
Thermodynamic and tachyonic instability for asymptotically flat black holes
Cornell University - arXiv, 2022
It was confirmed that the negative modes of the Euclidean section for asymptotically flat black holes reveal the thermodynamic instability of these black holes in the grand canonical ensemble (GCE). These include Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman black holes. In this work, we develop the relation between thermodynamic instability in the GCE and tachyonic instability for asymptotically flat black holes, where the latter is the onset for obtaining black holes with scalar hair. This implies that the tachyonic instability of black holes when introducing scalar coupling to the Gauss-Bonnet term or Maxwell term reflects thermodynamic instability of these black holes in the GCE. We go on further to consider the Schwarzschild-AdS black hole.
Linear Stability of Black Holes and Naked Singularities
Universe, 2022
A review of the current status of the linear stability of black holes and naked singularities is given. The standard modal approach, that takes advantage of the background symmetries and analyze separately the harmonic components of linear perturbations, is briefly introduced and used to prove that the naked singularities in the Kerr–Newman family, as well as the inner black hole regions beyond Cauchy horizons, are unstable and therefore unphysical. The proofs require a treatment of the boundary condition at the timelike boundary, which is given in detail. The nonmodal linear stability concept is then introduced, and used to prove that the domain of outer communications of a Schwarzschild black hole with a non-negative cosmological constant satisfies this stronger stability condition, which rules out transient growths of perturbations, and also to show that the perturbed black hole settles into a slowly rotating Kerr black hole. The encoding of the perturbation fields in gauge invar...
Strong cosmic censorship conjecture for a charged AdS black hole
arXiv (Cornell University), 2022
The strong cosmic censorship conjecture states (SCCC) that one cannot extend spacetime beyond the Cauchy horizon with a square-integrable connection. This conjecture was postulated to save the deterministic nature of the most successful theory of gravitation, general relativity. In order to explore the validation/violation of the SCCC for the charged anti-de Sitter black hole spacetime, we compute the ratio of the imaginary part of the quasinormal mode frequencies and the surface gravity at the Cauchy horizon both analytically and numerically. The lowest value of which defines the key parameter β determining the fate of SCCC where β < 1/2 indicates validation and else violation. We show that β > 1/2 for a charged AdS black hole with the dissipative boundary conditions in the near extremal region. Thus the SCCC is violated for this spacetime.