Cauchy Horizon Research Articles

Energy emission from a high curvature region and its backreaction

A strong gravity naked singular region can give important clues towards understanding classical as well as spontaneous nature of General Relativity. We propose here a model for energy emission from a naked singular region in a self-similar dust spacetime by gluing two self-similar dust solutions at the Cauchy horizon. The energy is defined and evaluated as a surface energy of a null hypersurface, the null shell. Also included are scenarios of spontaneous creation or disappearance of a singularity, end of inflation, black hole formation and bubble nucleation. Our examples investigated here explicitly show that one can model unlimitedly luminous and energetic objects in the framework of General Relativity.

Charged anti-de Sitter BTZ black holes in Maxwell-f(T) gravity

Inspired by the Bañados, Teitelboim and Zanelli (BTZ) formalism, we discuss the Maxwell-[Formula: see text] gravity in [Formula: see text] dimensions. The main task is to derive exact solutions for a special form of [Formula: see text], with [Formula: see text] being the torsion scalar of Weitzenböck geometry. To this end, a triad field is applied to the equations of motion of charged [Formula: see text] and sets of circularly symmetric noncharged and charged solutions have been derived. We show that, in the charged case, the monopole-like and the [Formula: see text] terms are linked by a correlative constant despite the known results in teleparallel geometry and its extensions.[Formula: see text] Furthermore, it is possible to show that the event horizon is not identical with the Cauchy horizon due to such a constant. The singularities and the horizons of these black holes are examined: they are new and have no analogue in the literature due to the fact that their curvature singularities are soft. We calculate the energy content of these solutions by using the general vector form of the energy–momentum within the framework of [Formula: see text] gravity. Finally, some thermodynamical quantities, like entropy and Hawking temperature, are derived.

Electrovacuum solutions in nonlocal gravity

We consider the coupling of the electromagnetic field to a non-local gravity theory comprising of the Einstein-Hilbert action in addition to a non-local $R\, {\Box}^{-2} R$ term associated with a mass scale $m$. We demonstrate that in the case of the minimally coupled electromagnetic field, real corrections about the Reissner-Nordstr\"om background only exist between the inner Cauchy horizon and the event horizon of the black hole. This motivates us to consider the modified coupling of electromagnetism to this theory via the Kaluza ansatz. The Kaluza reduction introduces non-local terms involving the electromagnetic field to the pure gravitational non-local theory. An iterative approach is provided to perturbatively solve the equations of motion to arbitrary order in $m^2$ about any known solution of General Relativity. We derive the first-order corrections and demonstrate that the higher order corrections are real and perturbative about the external background of a Reissner-Nordstr\"om black hole. We also discuss how the Kaluza reduced action, through the inclusion of non-local electromagnetic fields, could also be relevant in quantum effects on curved backgrounds with horizons.

On the Occurrence of Mass Inflation for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant and an Exponential Price Law

In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-N\"{o}rdstrom black hole at infinity. We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in $L^2_{{\rm loc}}$, thus violating the Christodoulou-Chru\'sciel version of strong cosmic censorship.

Tidal Forces in Dyonic Reissner-Nördstrom Black Hole

This paper investigates the tidal as well as magnetic charge effects produced in dyonic Reissner-Nordström black hole. We evaluate Newtonian radial acceleration using radial geodesics for freely falling test particles. We establish system of equations governing radial and angular tidal forces using geodesic deviation equation and discuss their solutions for bodies falling freely towards this black hole. The radial tidal force turns out to be compressing outside the event horizon whereas the angular tidal force changes sign between event and Cauchy horizons unlike Schwarzschild black hole. The radial geodesic component starts decreasing in dyonic Reissner-Nordström black hole unlike Schwarzschild case. We conclude that magnetic charge strongly affects the radial as well as angular components of tidal force.

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.

Quasinormal Modes and Strong Cosmic Censorship.

The fate of Cauchy horizons, such as those found inside charged black holes, is intrinsically connected to the decay of small perturbations exterior to the event horizon. As such, the validity of the strong cosmic censorship (SCC) conjecture is tied to how effectively the exterior damps fluctuations. Here, we study massless scalar fields in the exterior of Reissner-Nordström-de Sitter black holes. Their decay rates are governed by quasinormal modes of the black hole. We identify three families of modes in these spacetimes: one directly linked to the photon sphere, well described by standard WKB-type tools; another family whose existence and time scale is closely related to the de Sitter horizon; finally, a third family which dominates for near-extremally charged black holes and which is also present in asymptotically flat spacetimes. The last two families of modes seem to have gone unnoticed in the literature. We give a detailed description of linear scalar perturbations of such black holes, and conjecture that SCC is violated in the near extremal regime.

Higher Dimensional Charged Black Hole Solutions in f(R) Gravitational Theories

We present, without any assumption, a class of electric and magnetic flat horizon D-dimension solutions for a specific class of f(R)=R+αR2, all of which behave asymptotically as Anti-de-Sitter spacetime. The most interesting property of these solutions is that the higher dimensions black holes, D>4, always have constant electric and magnetic charges in contrast to what is known in the literature. For D=4, we show that the magnetic field participates in the metric on equal foot as the electric field participates. Another interesting result is the fact that the Cauchy horizon is not identical with the event horizon. We use Komar formula to calculate the conserved quantities. We study the singularities and calculate the Hawking temperature and entropy and show that the first law of thermodynamics is always satisfied.

Spherically symmetric charged black holes in f(R) gravitational theories

In this study, we have derived electric and magnetic spherically symmetric black holes for the class $f(R)=R+\zeta R^{2}$ without assuming any restrictions on the Ricci scalar. These black holes asymptotically behave as the de Sitter spacetime under certain constrains. We have shown that the magnetic charge contributes in the metric spacetime similarly to the electric charge. The most interesting feature of some of these black holes is the fact that the Cauchy horizon is not identical to the event horizon. We have calculated the invariants of Ricci and Kretschmann scalars to investigate the nature of singularities of such black holes. Also, we have calculated the conserved quantities to match the constants of integration with the physical quantities. Finally, the thermodynamical quantities, like Hawking temperature, entropy, etc., have been evaluated and the validity of the first law of thermodynamics has been verified.

Cauchy horizon stability in a collapsing spherical dust cloud: II. Energy bounds for test fields and odd-parity gravitational perturbations

We analyze the stability of the Cauchy horizon associated with a globally naked, shell-focussing singularity arising from the complete gravitational collapse of a spherical dust cloud. In a previous work, we have studied the dynamics of spherical test scalar fields on such a background. In particular, we proved that such fields cannot develop any divergences which propagate along the Cauchy horizon. In the present work, we extend our analysis to the more general case of test fields without symmetries and to linearized gravitational perturbations with odd parity. To this purpose, we first consider test fields possessing a divergence-free stress-energy tensor satisfying the dominant energy condition, and we prove that a suitable energy norm is uniformly bounded in the domain of dependence of the initial slice. In particular, this result implies that free-falling observers co-moving with the dust particles measure a finite energy of the field, even as they cross the Cauchy horizon at points lying arbitrarily close to the central singularity. Next, for the case of Klein–Gordon fields, we derive point-wise bounds from our energy estimates which imply that the scalar field cannot diverge at the Cauchy horizon, except possibly at the central singular point. Finally, we analyze the behaviour of odd-parity, linear gravitational and dust perturbations of the collapsing spacetime. Similarly to the scalar field case, we prove that the relevant gauge-invariant combinations of the metric perturbations stay bounded away from the central singularity, implying that no divergences can propagate in the vacuum region. Our results are in accordance with previous numerical studies and analytic work in the self-similar case.

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.

Linear Waves in the Interior of Extremal Black Holes II

We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr–Newman black holes. These results are the analogues of results obtained in Gajic (Commun Math Phys 353(2), 717–770, 2017) for extremal Reissner–Nordström and stand in stark contrast to previously established results for the subextremal case, where the local energy was shown to generically blow up at the Cauchy horizon.

Analysis of linear waves near the Cauchy horizon of cosmological black holes

We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner–Nordström–de Sitter and Kerr–de Sitter spacetimes and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our results by modifying the spacetime beyond the Cauchy horizon in a suitable manner, which puts the wave equation into a framework in which a number of standard as well as more recent microlocal regularity and scattering theory results apply. In particular, the conormal regularity of waves at the Cauchy horizon—which yields the boundedness statement—is a consequence of radial point estimates, which are microlocal manifestations of the blue-shift and red-shift effects.

Some Remarks on the $$C^0$$ C 0 -(In)Extendibility of Spacetimes

The existence, established over the past number of years and supporting earlier work of Ori [14], of physically relevant black hole spacetimes that admit $C^0$ metric extensions beyond the future Cauchy horizon, while being $C^2$-inextendible, has focused attention on fundamental issues concerning the strong cosmic censorship conjecture. These issues were recently discussed in the work of Jan Sbierski [17], in which he established the (nonobvious) fact that the Schwarschild solution in global Kruskal-Szekeres coordinates is $C^0$-inextendible. In this paper we review aspects of Sbierski's methodology in a general context, and use similar techniques, along with some new observations, to consider the $C^0$-inextendibility of open FLRW cosmological models. We find that a certain special class of open FLRW spacetimes, which we have dubbed `Milne-like,' actually admit $C^0$ extensions through the big bang. For spacetimes that are not Milne-like, we prove some inextendibility results within the class of spherically symmetric spacetimes.

Holographic stress-energy tensor near the Cauchy horizon inside a rotating black hole

We investigate a stress-energy tensor for a CFT at strong coupling inside a small five-dimensional rotating Myers-Perry black hole with equal angular momenta by using the holographic method. As a gravitational dual, we perturbatively construct a black droplet solution by applying the "derivative expansion" method, generalizing the work of Haddad (arXiv:1207.2305), and analytically compute the holographic stress-energy tensor for our solution. We find that the stress-energy tensor is finite at both the future and past outer (event) horizons, and that the energy density is negative just outside the event horizons due to the Hawking effect. Furthermore, we apply the holographic method to the question of quantum instability of the Cauchy horizon since, by construction, our black droplet solution also admits a Cauchy horizon inside. We analytically show that the null-null component of the holographic stress-energy tensor negatively diverges at the Cauchy horizon, suggesting that a singularity appears there, in favor of strong cosmic censorship.

Entropy relations and bounds of horizons in modified gravity

We first give some entropy relations for black holes in modified gravity models, which are introduced as an elegant technique trick for handling various entropy bounds. For these entropy relations, some are mass-independent and universal, while others are not. Then we study the entropy bound of horizons in modified gravity, including Horava-Lifshitz gravity, massive gravity and Einstein-dilaton gravity. We focus on black holes with two or three physical horizons, containing the event horizon, Cauchy horizon and negative horizon which is physical for the observers in the negative radial coordinate region. In these modified gravity, entropy bounds are geometrical bounds which are related close to the cosmological radius for asymptotically (A)dS black holes, while they are Penrose-like inequalities for asymptotically flat black holes. Moreover, they depend on the constants characterizing the strength of modified terms in the actions. We also present the first law of thermodynamics and Smarr relations for horizons.

Black hole nonmodal linear stability under odd perturbations: The Reissner-Nordström case

Following a program on black hole nonmodal linear stability initiated in Phys.\ Rev.\ Lett.\ {\bf 112} (2014) 191101, we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordstr\"om (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 $\mathcal{F} =\delta (F^*_{\alpha \beta} F^{\alpha \beta})$ and $\mathcal{Q} =\delta (\tfrac{1}{48} C^*_{\alpha \beta \gamma \delta} C^{\alpha \beta \gamma \delta})$, where $C_{\alpha \beta \gamma \delta}$ is the Weyl tensor, $F_{\alpha \beta}$ the Maxwell field, a star denotes Hodge dual and $\delta$ means first order variation, and that the linearized Einstein-Maxwell equations are equivalent to a coupled system of wave equations for $\mathcal{F}$ and $\mathcal{Q}$. For nonnegative cosmological constant we prove that $\mathcal{F}$ and $\mathcal{Q}$ 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.

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.

Regular Reissner-Nordström black hole solutions from linear electrodynamics

In recent years, there have appeared in the literature a large number of static, spherically symmetric metrics, which are regular at the origin, asymptotically flat, and have both an event and a Cauchy horizon for a certain range of the parameters. They have been interpreted as regular black hole (BH) spacetimes, and their physical source attributed to electric or magnetic monopoles in a suitable chosen nonlinear electrodynamics. Here we show that these metrics can also be interpreted as exact solutions of the Einstein equations coupled to ordinary linear electromagnetism---i.e., as sources of the Reissner-Nordstr\om (RN) spacetime---provided the components of the effective energy-momentum tensor satisfy the dominant energy condition (DEC). We use some well-known regular BH metrics to construct nonsingular RN black holes, where the singularity at the RN center is replaced by a regular perfect fluid charged sphere (whose charge-to-mass ratio is not greater than 1) which is inside the RN inner horizon.

Tidal forces in Kiselev black hole

The aim of this paper is to examine the tidal forces occurring in a Kiselev black hole surrounded by radiation and dust fluids. It is noted that the radial and angular components of the tidal force change the sign between event and Cauchy horizons. We solve the geodesic deviation equation for radially free-falling bodies toward Kiselev black holes. We explain the geodesic deviation vector graphically and point out the location of the event and Cauchy horizons for specific values of the radiation and dust parameters.