Splitting theorems for spatially closed space-times

A Lorentzian splitting theorem is obtained for spatially closed spacetimes. The proof employs and extends some recent results of Bartnik and Gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space-times. A splitting theorem for spatially closedtime-periodic space-times, which generalizes a result first considered by Avez, is derived as a corollary.     

On the Locality of the No Hair Conjecture and the Measure of the Universe

We reanalize the recently proposed proof by Jensen and Stein-Schabes [1] of the No Hair Theorem for inhomogeneous spacetimes, putting a special emphasis on the asymptotic behaviour of the shear and curvature. We conclude that the theorem only holds locally and estimate the minimum size a region should be in order for it to inflate. We discuss in some detail the assumptions used in the theorem. In the last section we speculate about the possible measure of the set of spacetimes that would undergo inflation.     

On the Geometry and Mass of Static,Asymptotically AdS Spacetimes,and the Uniqueness of the AdS Soliton

We prove two theorems, announced in [6], for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with Ricci-flat conformal boundary, the convexity condition is associated with negative mass. The second theorem is a uniqueness theorem for the negative mass AdS soliton spacetime. This result lends support to the new positive mass conjecture due to Horowitz and Myers which states that the unique lowest mass solution which asymptotes to the AdS soliton is the soliton itself. This conjecture was motivated by a nonsupersymmetric version of the AdS/CFT correspondence. Our results add to the growing body of rigorous mathematical results inspired by the AdS/CFT correspondence conjecture. Our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ``ground state' spacetimes such as Minkowski spacetime, namely, the presence of a null line, or complete achronal null geodesic, and the totally geodesic null hypersurface that it determines. En route, we provide an analysis of the boundary data at conformal infinity for the Lorentzian signature static Einstein equations, in the spirit of the Fefferman-Graham analysis for the Riemannian signature case. This leads us to generalize to arbitrary dimension a mass definition for static asymptotically AdS spacetimes given by Chruciel and Simon. We prove equivalence of this mass definition with those of Ashtekar-Magnon and Hawking-Horowitz.     

Singularities and conjugate points in FLRW spacetimes

Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial curvature and scale factors that near the singularity have power law behavior or power law behavior times a logarithm. When the spatial curvature is negative, the theorem is applicable to a subset of these spacetimes.     

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.     

Birkhoff theorem and matter

Birkhoff’s theorem for spherically symmetric vacuum spacetimes is a key theorem in studying local systems in general relativity theory. However realistic local systems are only approximately spherically symmetric and only approximately vacuum. In a previous paper, we showed the theorem remains approximately true in an approximately spherically symmetric vacuum space time. In this paper we prove the other case: the theorem remains approximately true in a spherically symmetric, approximately vacuum space time.     

A Definition of Total Energy-Momenta and the Positive Mass Theorem on Asymptotically Hyperbolic 3-Manifolds. I

Two sets of asymptotically hyperbolic initial data are defined, which correspond to the spatial infinity in asymptotically AdS spacetimes and to the null infinity in asymptotically Minkowski spacetimes respectively. The positive mass theorem involving the total energy, the total linear momentum and the total angular momentum is established for these initial data sets.Research partially supported by National Natural Science Foundation of China under grant 10231050 and the innovation project of the Chinese Academy of Sciences.     

A family of strings with spin

Using an equivalence theorem, we discuss some stationary interiors for a string with spin density in a space with torsion. We show that there is a family of solutions characterized by the spin divergences and compare the solutions to string solutions in general relativistic spacetimes.     

Almost Birkhoff theorem in general relativity

We extend Birkhoff’s theorem for almost LRS-II vacuum spacetimes to show that the rigidity of spherical vacuum solutions of Einstein’s field equations continues even in the perturbed scenario.     

Generic cosmic-censorship violation in anti-de Sitter space

We consider (four-dimensional) gravity coupled to a scalar field with potential V(phi). The potential satisfies the positive energy theorem for solutions that asymptotically tend to a negative local minimum. We show that for a large class of such potentials, there is an open set of smooth initial data that evolve to naked singularities. Hence cosmic censorship does not hold for certain reasonable matter theories in asymptotically anti-de Sitter spacetimes. The asymptotically flat case is more subtle. We suspect that potentials with a local Minkowski minimum may similarly lead to violations of cosmic censorship in asymptotically flat spacetimes, but we do not have definite results.     

Homothetic motions and perfect fluid cosmologies

A theorem on homothetic (self-similar) motions in spacetimes with a perfect fluid is derived. The main result is that a perfect fluid cosmological model cannot have a non-trivial homothetic motion orthogonal to the fluid 4-velocity vector.     

Does general relativity allow an observer to view an eternity in a finite time?

I investigate whether there are general relativistic spacetimes that allow an observerµ to collect in a finite time all the data from the worldline of another observer, where the proper length of's worldline is infinite. The existence of these spacetimes has a bearing on certain problems in computation theory. A theorem shows that most standard spacetimes cannot accommodate this scenario. There are however spacetimes which can: anti-de Sitter spacetime is one example.     

Chronological Spacetimes without Lightlike Lines are Stably Causal

The statement of the title is proved. It implies that under physically reasonable conditions, spacetimes which are free from singularities are necessarily stably causal and hence admit a time function. Read as a singularity theorem it states that if there is some form of causality violation on spacetime then either it is the worst possible, namely violation of chronology, or there is a singularity. The analogous result: “Non-totally vicious spacetimes without lightlike rays are globally hyperbolic” is also proved, and its physical consequences are explored.     

On Maximal Surfaces in Certain Non-Flat 3-Dimensional Robertson-Walker Spacetimes

An upper bound for the integral, on a geodesic disc, of the squared length of the gradient of a distinguished function on any maximal surface in certain non-flat 3-dimensional Robertson-Walker spacetimes is obtained. As an application, a new proof of a known Calabi-Bernstein??s theorem is given.     

A Spin-Statistics Theorem for Quantum Fields¶on Curved Spacetime Manifolds¶in a Generally Covariant Framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial. Received: 1 March 2001 / Accepted: 28 May 2001     

Trajectories Joining Two Submanifolds under the Action of Gravitational and Electromagnetic Fields on Static Spacetimes

In this paper we present existence and multiplicity results for orthogonal trajectories joining two submanifolds under the action of gravitational and electromagnetic fields on static spacetimes. These trajectories are critical points of unbounded functionals and they can be found by using a variant of the saddle point theorem and the relative category theory.     

Global Foliations of Matter Spacetimes¶with Gowdy Symmetry

A global existence theorem, with respect to a geometrically defined time, is shown for Gowdy symmetric globally hyperbolic solutions of the Einstein–Vlasov system for arbitrary (in size) initial data. The spacetimes being studied contain both matter and gravitational waves. Received: 8 December 1998 / Accepted: 20 March 1999     

Existence of Axially Symmetric Static Solutions of the Einstein-Vlasov System

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.     

Non-existence of stationary,axisymmetric dust solutions of Einstein's equations on spatially compact manifolds

By using a theorem on the classifacation of SO(2)-actions on three-dimensional manifolds, it is shown that axisymmetric, stationary solutions of Einstein's equations do not exist on spacetimes with compact, space-like hypersurfaces, provided that the energy-momentum tensor is trace-free in the two-dimensional space of trajectories.     

Existence of Constant Mean Curvature Foliations in Spacetimes with Two-Dimensional Local Symmetry

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat. Received: 15 July 1996 / Accepted: 12 March 1997