The extendibility of spacetime and the existence of weak solutions to the Einstein field equations beyond Cauchy horizons, is a crucial ingredient to examine the limits of General Relativity. Strong Cosmic Censorship serves as a firewall for gravitation by demanding inextendibility of spacetime beyond the Cauchy horizon. For asymptotically flat spacetimes, the predominance of the blueshift instability and the subsequent formation of a mass-inflation singularity at the Cauchy horizon have, so far, substantiated the conjecture. Accelerating black holes, described by the $C-$metric, are exact solutions of the field equations without a cosmological constant, which possess an acceleration horizon with similar causal properties to the cosmological horizon of de Sitter spacetime. Here, by considering linear scalar field perturbations, we provide numerical evidence for the stability of the Cauchy horizon of charged accelerating black holes. In particular, we show that the stability of Cauchy horizons in accelerating charged black holes is connected to quasinormal modes, we discuss the regularity requirement for which weak solutions to the field equations exist at the Cauchy horizon and show that Strong Cosmic Censorship may be violated near extremality.
Quasi-local energies for constantly accelerating observers in Bañados, Teitelboim, and Zanelli (BTZ), Schwarzschild and Schwarzschild-de Sitter spacetimes are derived. The energies are expressed in terms of acceleration, cosmological constant, and area, quantities measurable by the observers. Based on results from quantum fields in curved spacetime for the redshifted Hawking temperature, entropy and thermodynamic-like laws are briefly explored in the three spacetimes.
An exact solution of $(2+1)$-dimensional Einstein gravity with cosmological constant is studied. The corresponding spacetime is interpreted as an accelerating BTZ spacetime. The proper acceleration, horizon structure, temperature and entropy are presented in detail. The metric being studied is very similar to the one studied by Astorino in arXiv:1101.2616, but the range of parameters is different which results in significant changes in the causal structures.
In the multihorizon black hole spacetimes, it is possible that there are degenerate Cauchy horizons with vanishing surface gravities. We investigate the stability of the degenerate Cauchy horizon in black hole spacetimes. Despite the asymptotic behavior of spacetimes (flat, anti-de Sitter, or de Sitter), we find that the Cauchy horizon is stable against the classical perturbations, but unstable quantum mechanically.
Self-accelerating solutions in massive gravity provide explicit, calculable examples that exhibit the general interplay between superluminality, the well-posedness of the Cauchy problem, and strong coupling. For three particular classes of vacuum solutions, one of which is new to this work, we construct the conformal diagram for the characteristic surfaces on which isotropic stress-energy perturbations propagate. With one exception, all solutions necessarily possess spacelike characteristics, indicating perturbative superluminality. Foliating the spacetime with these surfaces gives a pathologi…
Smooth spacetimes with a compact Cauchy horizon ruled by closed null geodesics are considered. The compact Cauchy horizon is assumed to be non-degenerate. Then, supporting the validity of Penrose's strong cosmic censor hypothesis, the existence of a smooth Killing vector field in a neighbourhood of the horizon on the Cauchy development side is shown.
We discuss the definition of conserved quantities in asymptotically locally de Sitter spacetimes. One may define an analogue of holographic charges at future and past infinity and at other Cauchy surfaces $I_t$ as integrals over the intersection of timelike surfaces $C$ and the Cauchy surface $I_t$. In general, the charges $Q^t$ defined on the Cauchy surface $I_t$ depend on $C$, but if gravitational flux is absent the charges are independent of $C$. On the other hand, if there is a net gravitational flux entering or leaving the spacetime region bounded by $C_1, C_2$ and two Cauchy surfaces the…
We present the first proof-of-principle Cauchy evolutions of asymptotically global AdS spacetimes with no imposed symmetries, employing a numerical scheme based on the generalized harmonic form of the Einstein equations. In this scheme, the main difficulty in removing all symmetry assumptions can be phrased in terms of finding a set of generalized harmonic source functions that are consistent with AdS boundary conditions. In four spacetime dimensions, we detail an explicit set of source functions that achieves evolution in full generality. A similar prescription should also lead to stable evol…
In this note we present a result establishing the existence of a compact CMC Cauchy surface from a curvature condition related to the strong energy condition.
We analyze the response of a detector with a uniform acceleration $α$ in $κ-$Minkowski spacetime using the first order perturbation theory. The monopole detector is coupled to a massless complex scalar field in such a way that it is sensitive to the Lorentz violation due to the noncommutativity of spacetime present in the $κ-$deformation. The response function deviates from the thermal distribution of Unruh temperature at the order of $1/κ$ and vanishes exponentially as the proper time of the detector exceeds a certain critical time, a logarithmic function of $κ$. This suggests that the Unruh …
There are several examples known of two dimensional spacetimes which are linearly stable when perturbed by test scalar classical fields, but which are unstable when perturbed by test scalar quantum fields. We elucidate the mechanism behind such instabilities by considering minimally coupled, massless, scalar, test quantum fields on general two dimensional spacetimes with Cauchy horizons which are classically stable. We identify a geometric feature of such spacetimes which is a necessary condition for obtaining a quantum mechanical divergence of the renormalized expected stress tensor on the Ca…
It is shown that in spite of a generally accepted concept, there exist nondiffeomorphic solutions to the Cauchy problem in nonempty spacetime, which implies the necessity for canonical complementary conditions. It is nonlocal quantum jumps that provide a canonical global structure of spacetime manifold and, by the same token, the canonical complementary conditions.
We study general S2xS1 Gowdy models with a regular past Cauchy horizon and prove that a second (future) Cauchy horizon exists, provided that a particular conserved quantity $J$ is not zero. We derive an explicit expression for the metric form on the future Cauchy horizon in terms of the initial data on the past horizon and conclude the universal relation $A\p A\f=(8πJ)^2$ where $A\p$ and $A\f$ are the areas of past and future Cauchy horizon respectively.
A rigidity theorem that applies to smooth electrovac spacetimes which represent either (A) an asymptotically flat stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics was given in a recent work \cite{frw}. Here we enlarge the framework of the corresponding investigations by allowing the presence of other type of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian] Hig…
We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the hor…
In this paper, we review results on the existence (and nonexistence) of constant mean curvature spacelike hypersurfaces in the cosmological setting, and discuss the connection to the spacetime splittng problem. It is a pleasure to dedicate this paper to Robert Bartnik, who has made fundamental contributions to this area.
We first review the accelerating, rotating and charged Plebanski-Demianski (PD) black hole, which includes the Kerr-Newman rotating black hole and the Taub-NUT spacetime. The main feature of this black hole is that it has 4 horizons like event horizon, Cauchy horizon and two accelerating horizons. In the non-extremal case, the surface area, entropy, surface gravity, temperature, angular velocity, Komar energy and irreducible mass on the event horizon and Cauchy horizon are presented for PD black hole. The entropy product, temperature product, Komar energy product and irreducible mass product a…
The motion of test particles along circular orbits in the vacuum $C$ metric is studied in the Frenet-Serret formalism. Special orbits and corresponding intrinsically defined geometrically relevant properties are selectively studied.
Chruściel, Isenberg, and Pollack constructed a class of vacuum cosmological spacetimes that do not admit Cauchy surfaces with constant mean curvature. We prove that, for sufficiently large values of the gluing parameter, these examples are both future and past null geodesically incomplete. The authors are honored to dedicate this paper to Robert Bartnik on the occasion of his 60th birthday.
In classical General Relativity, the values of fields on spacetime are uniquely determined by their values at an initial time within the domain of dependence of this initial data surface. However, it may occur that the spacetime under consideration extends beyond this domain of dependence, and fields, therefore, are not entirely determined by their initial data. This occurs, for example, in the well-known (maximally) extended Reissner-Nordström or Reissner-Nordström-deSitter (RNdS) spacetimes. The boundary of the region determined by the initial data is called the "Cauchy horizon." It is locat…
Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Cauchy surface, hence extending Chernov-Nemirovski's observation to arbitrary dimensions.
We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, (M,g_{ab}), with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain `base points' of the Cauchy horizon, which are defined as `past terminal accumulation points' of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's `Chronology Protection Conjecture', according to which the laws of physics prevent one from manufacturing a `time machine'. Specifically, we prove: Theorem 1:…
We derive a thermodynamic first law for the electrically charged C-metric with vanishing cosmological constant. This spacetime describes a pair of identical accelerating black holes each pulled by a cosmic string. Treating the "boost time" of this spacetime as the canonical time, we find a thermodynamic first law in which every term has an unambiguous physical meaning. We then show how this first law can be derived using Noetherian methods in the covariant phase space formalism. We argue that the area of the acceleration horizon contributes to the entropy and that the appropriate notion of ene…
We have investigated the motion of timelike particles along geodesic in the background of accelerating and rotating black hole spacetime. We confirmed that the chaos exists in the geodesic motion of the particles by Poincaré sections, the power spectrum, the fast Lyapunov exponent indicator and the bifurcation diagram. Moreover, we probe the effects of the acceleration and rotation parameters on the chaotic behavior of a timelike geodesic particle in the black hole spacetime. Our results show that the acceleration brings richer physics for the geodesic motion of particles.
We prove that the surface gravity of a compact non-degenerate Cauchy horizon in a smooth vacuum spacetime, can be normalized to a non-zero constant. This result, combined with a recent result by Oliver Petersen and István Rácz, end up proving the Isenberg-Moncrief conjecture on the existence of Killing fields, in the smooth differentiability class. The well known corollary of this, in accordance with the strong cosmic censorship conjecture, is that the presence of compact Cauchy horizons is a non-generic phenomenon. Though we work in 3+1, the result is valid line by line in any n+1-dimensions.
Isotropic cosmology built in the Riemann-Cartan spacetime is investigated. Properties of homogeneous isotropic cosmological models filled with usual gravitating matter and scalar fields are studied in the beginning of cosmological expansion near the limiting energy density. It is shown that cosmological models are regular not only with respect to the Hubble parameter and the energy density but also with respect to the torsion and curvature tensors.
Misner spacetime is among the simplest solutions of Einstein's equation that exhibits a Cauchy horizon with a smooth extension beyond it. Besides violating strong cosmic censorship, this extension contains closed timelike curves. We analyze the stability of the Cauchy horizon, and prove that neighboring spacetimes in one parameter families of solutions through Misner's in pure gravity, gravity coupled to a scalar field, or Einstein-Maxwell theory, end at the Cauchy horizon developing a curvature singularity.
The last seven years has produced a growing body of evidence which concludes that the Cauchy horizon in black hole-de Sitter spacetimes is classically stable when the surface gravity at the cosmological event horizon is greater than that at the Cauchy horizon. That stability persists for a finite, but non-zero, region of the black hole's parameter space, $(M,Q,J,Λ)$, suggests that black holes immersed in de Sitter space are counter-examples to the strong cosmic censorship hypothesis. In this review we chronicle that body of evidence and describe the first steps of a program of numerical work a…
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 $H^1_{\mathrm{loc}}$ blow-up result of Luk and Oh.
The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes whose orientable double cover is globally hyperbolic. These spacetimes have generalized Cauchy surfaces on which smooth initial data sets yield unique solutions. A more difficult problem is to characterize the class of spacetimes with closed timelike curves that admit a well-posed initial value problem. Examples of spacetimes with closed timelike curves are g…
The Cauchy horizon forming colliding wave solution due to Chandrasekhar and Xanthopoulos (CX) has been generalized by inclusion of the NUT (Newman - Unti - Tamburino) parameter. This is done by transforming the part of the inner horizon region of a Kerr-Newman-NUT black hole into the space of colliding waves. By taking appropriate combination of Killing vectors and analytically extending beyond the Cauchy horizon the time-like hyperbolic sigularities are resolved as well. This provides another example of its kind among the type - D metrics with special emphasis on the role of the NUT parameter…
It is shown that cosmological equations for homogeneous isotropic models deduced in the framework of the Poincaré gauge theory of gravity by certain restrictions on indefinite parameters of gravitational Lagrangian take at asymptotics the same form as cosmological equations of general relativity theory for $ΛCDM$-model. Terms related to dark matter and dark energy in cosmological equations of standard theory for $ΛCDM$-model are connected in considered theory with the change of gravitational interaction provoked by spacetime torsion.
The strong cosmic censorship conjecture proposes that starting from generic initial data on some Cauchy surface, the solutions of the Einstein equation should not be extendable across the boundary of the domain of dependence of that surface. For the case of the Reissner-Nordström-de Sitter spacetime this means that any perturbation should blow up sufficiently badly when approaching this boundary, called the Cauchy horizon. However, recent results indicate that for highly charged black holes classical scalar perturbations allow for a violation of strong cosmic censorship. In a recent paper arXi…
Homogeneous isotropic cosmological models built in the framework of the Poincaré gauge theory of gravity based on general expression of gravitational Lagrangian with indefinite parameters are analyzed. Special points of cosmological solutions for flat cosmological models at asymptotics and conditions of their stability in dependence of indefinite parameters are found. Procedure of numerical integration of the system of gravitational equations at asymptotics is considered. Numerical solution for accelerating Universe without dark energy is obtained.
In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In the second section, I work gradually towards a construction of the limit space. I prove the limit space is unique up to isometry. I als show that, in general, the limit space has quite complicated causal behaviour. This work prepares the final paper in which I shall study in more detail properties of the limit space and the moduli space of (compact) globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this paper a suitable definition of dimension of a Lorentz space in …
We first review spacelike stretched warped AdS$_3$ and we describe its black hole quotients by using accelerating and Poincaré coordinates. We then describe the maximal analytic extension of the black holes and present their causal diagrams. Finally, we calculate spacetime limits of the black hole phase space $(T_R,T_L)$. This is done by requiring that the identification vector $\partial_θ$ has a finite non-zero limit. The limits we obtain are the self-dual solution in accelerating or Poincaré coordinates, depending respectively on whether the limiting spacetimes are non-extremal or extremal, …
In recent work, it was shown that velocity-dependent forces between parallel fundamental strings moving apart in a D-dimensional spacetime implied an expanding universe in D-1-dimensional spacetime. Here we expand on this work to obtain exact solutions for various string/brane cosmological toy models.
We establish a complete classification theorem for the topology and for the null generators of compact non-degenerate Cauchy horizons of time orientable smooth vacuum $3+1$-spacetimes. We show that, either: (i) all generators are closed, or (ii) only two generators are closed and any other densely fills a two-torus, or (iii) every generator densely fills a two-torus, or (iv) every generator densely fills the horizon. We then show that, respectively to (i)-(iv), the horizon's manifold is either: (i') a Seifert manifold, or (ii') a lens space, or (iii') a two-torus bundle over a circle, or, (iv'…
We show the index 0 Fredholm property of the spectral family of the azimuthal harmonics of the charged Klein-Gordon operator on general cosmological slowly accelerating and rotating charged black hole type spacetimes, including the De Sitter-Kerr-Newman family, using a parametrix construction for abstract totally characteristic operators. We then present a numerical scheme to compute the meromorphic poles of the inverse of the spectral family and provide an explicit estimate of the numerical error.
A simple proof is given that every globally hyperbolic spacetime admits a smooth Cauchy steep time function. This result is useful in order to show that globally hyperbolic spacetimes can be isometrically embedded in Minkowski spacetimes and that they spit as a product. The proof is based on a recent result on the differentiability of Geroch's volume functions.
We define the local acceleration and jerk of a relativistic brane in an ambient spacetime, and construct from them a dimensionless parameter lambda that must be small for an interpretation of brane acceleration as local Unruh temperature. As examples, we discuss (i) open rotating branes, for which lambda>1 (ii) closed spherical branes expanding in Minkowski spacetime, for which lambda=0 when the worldvolume is either an Einstein static universe or de Sitter space, in which case the brane temperature equals the Gibbons-Hawking temperature, (iii) closed spherical branes in anti-de Sitter spaceti…
We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago we proved that, if the null geodesic generators of such a horizon were all \textit{closed} curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of infinite dimensional, analytic families of such `generalized Taub-NUT' spacetimes and show that, generically, they admitted \textit{only} the single (horizon-generating) Killing fie…
