Field equations for spatially homogeneous space-times

It is shown that there is a natural tetrad for each Bianchi type cosmological model in which the metric and field equations take a simple and convenient form.     

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.     

Causality violation in spatially closed space-times

Some results are obtained which establish conditions under which causality violation cannot occur in space-times admitting a compact, but not necessarily acausal, spacelike hypersurface. For example, it is shown that causality violation can occur in a space-time admitting a compact spacelike slice and obeying fairly reasonable energy conditions only if it is “singular” in the sense that some “causal interval”J ⁺(p) ∩J ^? (q) has noncompact closure.     

Some spatially homogeneous string cosmological models

The Einstein field equations of massive strings are solved completely with and without a source free magnetic field for the Bianchi type I metric in a different basic form. Some physical properties of the models are studied.     

Some dynamical properties of Einstein space-times admitting a Gaussian foliation

We investigate the behavior of curves in the space of Riemannian metrics which corresponds to Einstein space-times admitting a Gaussian foliation. Different types of variational principles are formulated and some global dynamical properties of these space-times are obtained.Alexander-von-Humboldt-Fellow.     

The global structure of simple space-times

According to a standard definition of Penrose, a space-time admitting well-defined future and past null infinitiesI ⁺ andI ^– is asymptotically simple if it has no closed timelike curves, and all its endless null geodesics originate fromI ^– and terminate atI ⁺. The global structure of such space-times has previously been successfully investigated only in the presence of additional constraints. The present paper deals with the general case. It is shown thatI ⁺ is diffeomorphic to the complement of a point in some contractible open 3-manifold, the strongly causal regionI ₀ ⁺ ofI ⁺ is diffeomorphic to , and every compact connected spacelike 2-surface inI ⁺ is contained inI ₀ ⁺ and is a strong deformation retract of bothI ₀ ⁺ andI ⁺. Moreover the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincaré conjecture, are diffeomorphic to ³.     

Exact spatially homogeneous cosmologies

We consider perfect fluid spatially homogeneous cosmological models. Starting with a new exact solution of Blanchi type VIII, we study generalizations which lead to new classes of exact solutions. These new solutions are discussed and classified in several ways. In the original type VIII solution, the ratio of matter shear to expansion is constant, and we present a theorem which delimits those space-times for which this condition holds.     

Some examples of incomplete space-times

An example is given of a space-time which is timelike and spacelike complete but null incomplete. An example is also given of a space-time which is geodesically complete but contains an inextendible timelike curve of bounded acceleration and finite length. These two examples may be modified so that in each case they become globally hyperbolic and retain the stated properties. All of the examples are conformally equivalent to open subsets of the two-dimensional Minkowski space.     

Curvature and closed trapped surfaces in 4-dimensional space-times

The way a spacelike surfaceH sits in a 4-dimensional space-timeM may be measured by the average null curvature function _H and the shape function _H ofH. Relations between the shape function _H of a spaceiike surfaceH and curvature of a 4-dimensional space-time obeying the Einstein equation are investigated. Some relations between the shape functions of compact spacelike surfaces and infinite curvature are obtained and discussed. Assuming some curvature conditions, some results concerning the evolution of closed trapped surfaces from a restricted type of marginally trapped surfaces diffeomorphic toS ² are obtained.Based on Chapter 4 of the author's Ph.D. thesis.     

Nonscalar singularities in spatially homogeneous cosmologies

Singularities in vacuum spatially homogeneous cosmological models are investigated. It is shown that in general the curvature scalarR _* ^abcd R^*abcddiverge and that the only solutions which have curvature singularities at which this scalar does not diverge describe certain plane-wave space-times. It is argued that with matter present these nonscalar singularities are even less likely to occur. The exceptional case of Bianchi type VI_–1/9 is not considered.     

Intrinsic curvature of the 3-surfaces of homothety in spatially homothetic space-times

It is shown how the sign of the intrinsic scalar curvature of spatially homothetic space-times depends on their algebraic type, thus generalizing a corresponding result for the homogeneous case.     

Stability of certain spatially homogeneous cosmological models

We investigate the general behavior of spatially homogeneous cosmological models at large times. Using existing techniques from stability theory we discover that some known exact solutions are asymptotically stable despite possessing special properties. Some consequences of these results are then discussed.This essay received an honorable mention from the Gravity Research Foundation for the year 1984-Ed.     

Conformal properties of nonpeeling vacuum space-times

In a previous paper we investigated a class ofnonpeeling asymptotic vacuum solutions which were shown to admit finite expressions for the Winicour-Tamburino energy-momentum and angular momentum integrals. These solutions have the property that $$\psi _0 = O(r^{ - 3 - \in _0 } ), \in _0 \leqslant 2$$ and $$\psi _1 = O(r^{ - 3 - \in _1 } ), \in _1< \in _0 and \in _1< 1$$ withψ ₂,ψ ₃, andψ ₄ having the same asymptotic behavior as they do for peeling solutions. The above investigation was carried out in the physical space-time. In this paper we examine the conformal properties of these solutions, as well as the more general Couch-Torrence solutions, which include them as a subclass. For the Couch-Torrence solutions $$\begin{gathered} \psi _0 = O(r^{ - 2 - \in _0 } ) \hfill \\ \psi _1 = O(r^{ - 2 - \in _1 } ), \in _1< \in _0 {\text{ }}and \in _1 \leqslant 2 \hfill \\ \end{gathered} $$ and , $$\psi _2 = O(r^{ - 2 - \in _2 } ),{\text{ }} \in _2< \in _1 {\text{ }}and \in _2 \leqslant 1$$ withψ ₃ andψ ₄ behaving as they do for peeling solutions. It is our purpose to determine how much of the structure generally associated with peeling space-times is preserved by the nonpeeling solutions. We find that, in general, a three-dimensional null boundary (?⁺) can be defined and that the BMS group remains the asymptotic symmetry group. For the general Couch-Torrence solutions several physically and/or geometrically interesting quantities     

Some global properties of massless free fields

Elementary group-theoretical considerations show that global solutions to the massless free field equations are functions on the bundle of twistor dyads, rather than the bundle of conformal spin frames. Only in certain degenerate cases may they be thought of as ordinary spinor fields. This is the origin of the Grgin discontinuity.Supported in part by the University of Kansas Research Fund and by the Science Research Council     

Perfect fluid sources for spatially homogeneous spacetimes

Spatially homogeneous perfect fluid spacetimes are studied from a point of view which emphasizes the spatial geometry and the action of that subgroup of the spatial gauge group of the three-plus-one formulation of general relativity which is compatible with the spatial homogeneity. The specializations of the dynamics which correspond to the existence of additional spacetime symmetries are classified. An unconstrained set of gravitational and fluid variables is obtained by elimination of the gravitational constraints using an approach which obtains the gravitational evolution equations from a suitably modified Lagrangian/Hamiltonian formalism. A slightly different choice of variables is then described which allows one to take full advantage of the spatial gauge group and of the 1-parameter group of scale transformations of the unit of length.     

Two Bianchi type VIII spatially homogeneous cosmologies

We study two Bianchi type VIII analogues of Taub space and maximal analytic extensions of them. The first one has SL(2,R) as an isometry group, which acts transitively on spacelike hypersurfaces. The maximal extension has all of the pathological features of Taub-NUT space. The second one has the universal covering group of SL(2,R) as an isometry group. The maximal extension of the latter does not have these pathological properties and is geodesically complete.Work supported in part by NSF Grant MPS 74-16311 AO1     

A global conformal extension theorem for perfect fluid Bianchi space-times

A global extension theorem is established for isotropic singularities in polytropic perfect fluid Bianchi space-times. When an extension is possible, the limiting behaviour of the physical space-time near the singularity is analysed.