Stability of maximal hypersurfaces in spacetimes: new general conditions and application to relevant spacetimes

We obtain new simple sufficient conditions to ensure the stability and strong stability of maximal hypersurfaces (without boundary) immersed in an arbitrary spacetime. Several applications to maximal hypersurfaces in a spatially open or closed spacetime endowed with an infinitesimal causal symmetry are also given.     

On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

The global characteristic Cauchy problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth “on each side” of the initial value hypersurface. A uniqueness result in Sobolev regularity \(H^{1/2+\varepsilon }_{\mathrm {loc}}\) is proved among solutions supported in the union of the causal past and future of the initial value hypersurface, and whose product with the indicator function of the causal future (resp. past) of the hypersurface is past compact (resp. future compact). An explicit representation formula for solutions is obtained, which prominently features an invariantly defined, densitised version of the null expansion of the hypersurface. Finally, applications to quantum field theory on curved spacetimes are briefly discussed.     

Lightlike hypersurfaces of Lorentzian manifolds with distinguished screen

We first study the properties of the lightlike mean curvature on a lightlike hypersurface in a Lorentzian manifold. Then, we show the existence of a large class of lightlike hypersurfaces admitting a distinguished screen and study some of their properties. In particular, we find integrability conditions for distinguished screen distributions and give applications in a spacetime which obeys the null energy condition.     

Boundary sources in the Doran-Lobo-Crawford spacetime

We take a null hypersurface (causal horizon) generated by a congruence of null geodesics as the boundary of the Doran-Lobo-Crawford spacetime to be the place where the Brown-York quasilocal energy is located. The components of the outer and inner stress tensors are computed and shown to depend on time and on the impact parameter b of the test-particle trajectory. The spacetime is a solution of Einstein’s equations with an anisotropic fluid as source. The surface energy density σ on the boundary is given by the same expression as that obtained previously for the energy stored on a Rindler horizon. For time intervals long compared to b (when the stretched horizon tends to the causal one), the components of the stress tensors become constant.      

Gravitational fields with groups of motions on two-dimensional transitivity hypersurfaces in a model with matter and a magnetic field

For gravitational fields with metrics which admit of groups of motions multiply — transitive on 2-dimensional space-like invariant varieties, the exact solutions of the Einstein gravitational equations are given for the case when the sources of the gravitational field are dust-like matter and a magnetic field. A magnetic field is orientated along a direction orthogonal to transitivity hypersurface. The solutions contain arbitrary functions. In the case of transitivity hypersurface of positive curvature and in the absence of a magnetic field, the solution is reduced to the Tolman spherically symmetric solution for dust-like matter. The conditions are studied under which the solutions with a magnetic field become asymptotically isotropic and approach the flat and the open Friedmann models. The case of transitivity hypersurfaces with signature (+ –) is also considered.     

Beltrami-de Sitter时空中标量和旋量粒子的量子理论

参照在Minkowski时空中,从粒子的相对论性经典理论过渡到量子理论,建立标量粒子和旋量粒子的相对论性波动方程的方案,在Beltrami-de Sitter时空中建立了de Sitter不变的标量粒子和旋量粒子的相对论性量子力学的基本方程,它们恰恰分别是Beltrami-de Sitter时空中的Klein-Gordon方程和Dirac方程。在Beltrami-anti de Sitter时空的同时类空超曲面簇上求解了这些方程,得到了分立的本征值和相应的本征函数。 关键词：     

Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes

A new technique is introduced in order to solve the following question:When is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice? (Generalized Robertson-Walker spacetimes extend classical Robertson-Walker ones to include the cases in which the fiber has not constant sectional curvature.) First, we determine when this hypersurface must be compact. Then, all these compact hypersurfaces in (necessarily spatially closed) spacetimes are shown to be totally umbilical and, except in very exceptional cases, slices. This leads to proof of a new Bernstein-type result. The power of the introduced tools is also shown by reproving and extending several known results.     

Quantization of Maxwell’s Equations on Curved Backgrounds and General Local Covariance

We develop a quantization scheme for Maxwell??s equations without source on an arbitrary oriented four-dimensional globally hyperbolic spacetime. The field strength tensor is the key dynamical object and it is not assumed a priori that it descends from a vector potential. It is shown that, in general, the associated field algebra can contain a non-trivial centre and, on account of this, such a theory cannot be described within the framework of general local covariance unless further restrictive assumptions on the topology of the spacetime are taken.     

Maximal hypersurfaces and foliations of constant mean curvature in general relativity

We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to spacetimes that are vacuum and non-flat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe; i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez' original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions; we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat, Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.     

A New Distributional Approach to Signature Change

Colombeau's generalized functions are used to adapt the distributional approach to singular hypersurfaces in general relativity with signature change. Equations governing the dynamics of a singular hypersurface are derived and a specific non-vanishing form for the energy-momentum tensor of the singular hypersurface is obtained. It is shown that matching in the case of de Sitter space in the Lorentzian sector is possible along the boundary with minimum radius but leads to the vanishing of the energy-momentum tensor of the singular hypersurface.     

Cohomogeneity One Einstein-Sasaki 5-Manifolds

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.     

Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on , and are equivalent to the usual constraint equations that satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.     

Worldtube conservation laws for the null-timelike evolution problem

I treat the worldtube constraints which arise in the null-timelike initial-boundary value problem for the Bondi-Sachs formulation of Einstein’s equations. Boundary data on a worldtube and initial data on an outgoing null hypersurface determine the exterior spacetime by integration along the outgoing null geodsics. The worldtube constraints are a set of conservation laws which impose conditions on the integration constants. I show how these constraints lead to a well-posed initial value problem governing the extrinsic curvature of the worldtube, whose components are related to the integration constants. Possible applications to gravitational waveform extraction and to the well-posedness of the null-timelike initial-boundary value problem are discussed.     

The Langlands program and string modular K3 surfaces

A number theoretic approach to string compactification is developed for Calabi–Yau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic two-form of a particular type of K3 surface can be expressed in terms of modular forms constructed from the worldsheet theory. The process of deriving string physics from spacetime geometry can be reversed, allowing the construction of K3 surface geometry from the string characters of the partition function. A general argument for K3 modularity is given by combining mirror symmetry with the proof of the Shimura–Taniyama conjecture.     

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.     

On the existence of perturbed robertson-walker universes

Solutions of the full nonlinear field equations of general relativity near the Robertson-Walker universes are examined, together with their relation to linearized perturbations. A method due to Choquet-Bruhat and Deser is used to prove existence theorems for solutions near Robertson-Walker constraint data of the constraint equations on a spacelike hypersurface. These theorems allow one to regard the matter fluctuations as independent quantities, ranging over certain function spaces. In the k = ?1 case the existence theory describes perturbations which may vary within uniform bounds throughout space. When k = +1 a modification of the method leads to a theorem which clarifies some unusual features of these constraint perturbations. The k = 0 existence theorem refers only to perturbations which die away at large distances. The connection between linearized constraint solutions and solutions of the full constraints is discussed. For k = ±1 backgrounds, solutions of the linearized constraints are analyzed using transverse-traceless decompositions of symmetric tensors. Finally the time-evolution of perturbed constraint data and the validity of linearized perturbation theory for Robertson-Walker universes are considered.     

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     

General relativistic axisymmetric rotating systems: Coordinates and equations

The Einstein equations for rotating axisymmetric configurations in asymptotically flat spacetimes are put in a form suitable for numerical calculations of dynamics. The discussion is motivated by the astrophysical problem of gravitational collapse with generation of gravitational radiation and possibly black hole formation. In the context of topologically spherical coordinates there are two spatial gauge conditions which greatly simplify the Einstein equations and are compatible with regularity at the origin. We focus on one, the radial gauge, which generalizes Schwarzschild coordinates and is asymptotically a transverse-traceless gauge for gravitational radiation. The shift vector equation and the Hamiltonian constraint are parabolic equations in the radial gauge, rather than the usual elliptic equations. Two hypersurface conditions are explored in detail, the maximal hypersurface condition and another “polar” hypersurface condition which fits naturally with the radial gauge.     

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.