Compact Cauchy Horizons Research Articles

Symmetries of Vacuum Spacetimes with a Compact Cauchy Horizon of Constant Nonzero Surface Gravity

We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to Moncrief–Isenberg and Friedrich–Rácz–Wald, where the generators of the Cauchy horizon were closed or densely filled a 2-torus. Consequently, the maximal globally hyperbolic vacuum development of generic initial data cannot be extended across a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant. Our result supports, thereby, the validity of the strong cosmic censorship conjecture in the considered special case. The proof consists of two main steps. First, we show that the Killing equation can be solved up to infinite order at the Cauchy horizon. Second, by applying a recent result of the first author on wave equations with initial data on a compact Cauchy horizon, we show that this Killing vector field extends to the globally hyperbolic region.

Wave equations with initial data on compact Cauchy horizons

We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant. Examples include the Misner spacetime and the Taub-NUT spacetime. We prove an energy estimate close to the Cauchy horizon for wave equations acting on sections of vector bundles. Using this estimate we prove that if a linear wave equation can be solved up to any order at the Cauchy horizon, then there exists a unique solution on the globally hyperbolic region. As a consequence, we prove several existence and uniqueness results for linear and non-linear wave equations without assuming analyticity or symmetry of the spacetime and without assuming that the generators close. We overcome in particular the essential remaining difficulty in proving that vacuum spacetimes with a compact Cauchy horizon with constant non-zero surface gravity necessarily admits a Killing vector field. This work is therefore related to the strong cosmic censorship conjecture.

Extension of Killing vector fields beyond compact Cauchy horizons

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

A classification theorem for compact Cauchy horizons in vacuum spacetimes

We establish a complete classification theorem for the topology and for the orbital type of the null generators of compact non-degenerate Cauchy horizons of time orientable smooth vacuum \(3+1\)-spacetimes. We show that one and only one of the following must hold: (i) all generators are closed, (ii) only two generators are closed and any other densely fills a two-torus, (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, (ii’) a lens space, (iii’) a two-torus bundle over a circle, or, (iv’) a three-torus. All the four possibilities are known to arise in examples. In the last case, (iv), (iv’), we show in addition that the spacetime is indeed flat Kasner, thus settling a problem posed by Isenberg and Moncrief for ergodic horizons. The results of this article open the door for a full parameterization of the metrics of all vacuum spacetimes with a compact Cauchy horizon. The method of proof permits direct generalizations to higher dimensions.

On the existence of Killing fields in smooth spacetimes with a compact Cauchy horizon

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.

Symmetries of Cosmological Cauchy Horizons with Non-Closed Orbits

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 field alluded to above. In this article we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon generating null geodesic densely fills a 2-torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as `ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the \textit{conjectures} that every such spacetime with an \textit{ergodic} horizon is trivially constructable from the flat Kasner solution by making certain `irrational' toroidal compactifications and that degenerate compact Cauchy horizons do not exist in the analytic case.

Area Theorem and Smoothness of Compact Cauchy Horizons

We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every compactly generated Cauchy horizon is smooth and compact. We explore the consequences of this result for time machines, topology change, black holes and cosmic censorship. For instance, it is shown that compact Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable dominant energy condition wherever there is some source content.

The vacuum conservation theorem

A version of the vacuum conservation theorem is proved which does not assume the existence of a time function nor demands stronger properties than the dominant energy condition. However, it is shown that a stronger stable version plays a role in the study of compact Cauchy horizons.

Smoothness of Compact Horizons

We prove that compact Cauchy horizons in a smooth spacetime satisfying the null energy condition are smooth. As an application, we consider the problem of determining when a cobordism admits Lorentzian metrics with certain properties. In particular, we prove a result originally due to Tipler without the smoothness hypothesis necessary in the original proof.

Symmetries of higher dimensional black holes

We prove that if a stationary, real analytic, asymptotically flat vacuum black hole spacetime of dimension n ⩾ 4 contains a non-degenerate horizon with compact cross-sections that are transverse to the stationarity generating Killing vector field then, for each connected component of the black hole's horizon, there is a Killing field which is tangent to the generators of the horizon. For the case of rotating black holes, the stationarity generating Killing field is not tangent to the horizon generators and therefore the isometry group of the spacetime is at least two dimensional. Our proof relies on significant extensions of our earlier work on the symmetries of spacetimes containing a compact Cauchy horizon, allowing now for non-closed generators of the horizon.

Fuchsian methods and spacetime singularities

Fuchsian methods and their applications to the study of the structure of spacetime singularities are surveyed. The existence question for spacetimes with compact Cauchy horizons is discussed. After some basic facts concerning Fuchsian equations have been recalled, various ways in which these equations have been applied in general relativity are described. Possible future applications are indicated.

On the Differentiability of Compact Cauchy Horizons

We construct an example of a compact Cauchy horizon that is not a differentiable manifold. This answers in the negative the question of whether a compact Cauchy horizon that arises from a space-like hypersurface is necessarily smooth.

Regularity of Horizons and the Area Theorem

We prove that the area of sections of future event horizons in space-times satisfying the null energy condition is non-decreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic space-time and there exists a conformal completion with a $ {\cal H} $ -regular $ {\cal J} $ +; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. We prove smoothness or analyticity of the relevant part of the event horizon when equality in the area inequality is attained – this has applications to the theory of stationary black holes, as well as to the structure of compact Cauchy horizons. In the course of the proof we establish several new results concerning the differentiability properties of horizons.

On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon

A rigidity theorem that applies to smooth electrovacuum 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 paper by Friedrich et al (1999 Commun. Math. Phys. 204 691-707). Here we enlarge the framework of the corresponding investigations by allowing the presence of other types 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]-Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proved that there exists a Killing vector field in a one-sided neighbourhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover, the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity.

On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

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 horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case.

Compact Null Hypersurfaces and Collapsing Riemannian Manifolds

Restrictions are obtained on the topology of a compact divergence - free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three - dimensional Riemannian metrics which collapses with bounded curvature and applying known results on the topology of manifolds which collapse. The result is then applied to general relativity, where it implies a restriction on the topology of smooth compact Cauchy horizons in spacetimes with various types of reasonable matter content.

Blow-up of test fields near Cauchy horizons

The behaviour of test fields near a compact Cauchy horizon is investigated. It is shown that solutions of nonlinear wave equations on Taub spacetime with generic initial data cannot be continued smoothly to both extensions of the spacetime through the Cauchy horizon. This is proved using an energy method. Similar results are obtained for the spacetimes of Moncrief containing a compact Cauchy horizon and for more general matter models.

Classical and quantum instability of compact Cauchy horizons in two dimensions

The author studies a class of two-dimensional model spacetimes with compact Cauchy horizons; these models retain some of the key features of the four-dimensional problem. All of the model Cauchy horizons are classically unstable against scalar perturbations and the mechanism of instability is the infinite blue-shifting of radiation near the horizon. The quantum instability of the horizon is purely non-radiative (particle creation remains bounded), and has exactly the same mechanism as in the classical case (infinite blue-shifting of the Casimir vacuum energy). Some specific conjectures on the four-dimensional problem are given.

Strong cosmic censorship in polarised Gowdy spacetimes

The strong cosmic censorship conjecture says that 'most' spacetimes developed as solutions of Einstein's equations from prescribed initial data cannot be extended outside of their domains of dependence. Here, the authors discuss results which show that if they restrict attention to the polarised Gowdy spacetimes, strong cosmic censorship holds. More specifically, these results show that in the space of Cauchy data for polarised Gowdy spacetimes, there is an open dense subset for which the maximal globally hyperbolic development is inextendible. Among the Gowdy spacetimes which can be extended, the authors find a set of them which admit a countable infinity of inequivalent classes of extensions. They also find that a polarised Gowdy spacetime (T3 or S2*S1) may be extended as a solution of Einstein's equations (with the Gowdy isometries) across a compact Cauchy horizon only if it is analytic.

Symmetry preserving deformations of generalized Taub–NUT space-times to all orders in perturbation theory

Generalized Taub–NUT (Newman–Unti–Tamburino) space-times have compact Cauchy horizons and (generically) admit one (spacelike) Killing field in their globally hyperbolic regions. A large family of such vacuum space-times can be defined on circle bundles over K×R, where K is a compact two-manifold, with the circular fibers of the bundle being defined by the orbits of the Killing field. For the simplest case of product circle bundles the symmetry preserving vacuum perturbations of such backgrounds to arbitrarily high order in perturbation theory are considered. The analytic form of the general solution of the nth-order perturbation equations for all n is derived under the restriction that the perturbations considered preserve the (one Killing field) symmetry of the background. The evolution equations are treated first and then the constraint equations are imposed, recovering along the way the well-known result that linearization instabilities arise only if one attempts to perturb from one (Killing) symmetry class to another. Gauge transformations, decompositions, and the natural symplectic structure associated with the perturbation formalism are also discussed. The possibility of extending these results to the case of symmetry breaking perturbations and of using the results to derive the asymptotic behavior of solutions near their singular boundaries is briefly discussed.