On quantization of free fields in stationary space-times

In Section 1 we analyse the structure of the infinite-dimensional Hamiltonian system described by the Klein-Gordon equation (free real scalar field) in stationary space-times with closed space sections; we give an existence and uniqueness theorem for the Lichnerowicz distribution kernelG ₁ together with its proper Fourier expansion, and we construct the Hilbert spaces of frequency-part solutions defined by means ofG ₁.In Section 2 an analysis, a theorem and a construction similar to the above are formulated for thefree real field spin 1, massm>0, in one kind of static space-times.In this letter, only results are given. For detailed proofs and further results, see reference [9], [10] and [11].     

Null cut loci of spacelike surfaces

The null cut locus of a spacelike submanifold of codimension 2 in a space-time is defined. In globally hyperbolic space-times, it is shown that the future (past) null cut locusc _n ⁺ (H) [c _n ^- (H)] of a compact, acausal, spacelike submanifoldH of codimension 2 is a closed subset of the space-time, and each pointx c _n ⁺ (H) is either a focal point ofH along some future-directed null geodesic meetingH orthogonally or there exist at least two null geodesics fromH tox, realizing the distance betweenH andx or both. Also, it can be shown that the assumptions of the Penrose's singularity theorem for open globally hyperbolic space-times may be weakened to the space-times which are conformal to an open subset of an open globally hyperbolic space-time.This study is based on Chapter 3 of the author's Ph.D. thesis.     

On the algebraic type of space-times possessing a nonsingular Killing horizon

The characteristic initial-value problem is discussed for the Einstein and Einstein-Maxwell equations when initial data are specified on two branches of a bifurcate Killing horizon. In the latter case it is assumed that the principal null directions of the Maxwell tensor coincide with those of a Killing bivector. A theorem specifying necessary and sufficient conditions for such space-times to be Petrov typeD is derived and consequences of this with respect to Israel's theorem and Robinson's theorem are discussed.Part of this work was initiated while the author attended the summer school on global analysis held at the International Center for Theoretical Physics, Trieste, Italy (1972).     

Some global properties of closed spatially homogeneous space-times

We consider some properties of the space-times which contain a spatially homogeneous domain of dependenceD(V), whereF is acompact achronal spatial hypersurface of homogeneity. For example, it is shown that ifV has a nonempty future Cauchy horizon then the timelike geodesies orthogonal toV are future incomplete and there is strong causality failure onH ⁺(V). Also, conditions for the global hyperbolicity of such space-times are obtained.     

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.     

Killing-Yano tensors in general relativity

It is shown that space-times admitting more than one independent Killing-Yano tensor belong to a small collection of highly idealised space-times. A new characterization of Robertson-Walker space-times arises as a corollary of the main theorem.     

Initial Data Engineering

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.Partially supported by a Polish Research Committee grant 2 P03B 073 24Partially supported by the NSF under Grants PHY-0099373 and PHY-0354659Partially supported by the NSF under Grant DMS-0305048 and the UW Royalty Research Fund     

On the algebraic structure of second order symmetric tensors in 5-dimensional space-times

A new approach to the algebraic classification of second order symmetric tensors in 5-dimensional space-times is presented. The possible Segre types for a symmetric two-tensor are found. A set of canonical forms for each Segre type is obtained. A theorem which collects together some basic results on the algebraic structure of the Ricci tensor in 5-dimensional space-times is also stated.     

Causality violations and singularities

We announce and justify two theorems (proofs will appear in Refs. 1 and 2): i) A generalization of the singularity theorem of Hawking and Penrose [3] to space-times with chronology violations. Although it is impossible to remove the chronology condition completely the announced theorem is in a well defined sense optimal: the chronology condition is replaced by a strictly weaker condition that cannot be removed because of counter examples, ii) If the chronology violating setV has compact closure and the strong energy and generic conditions hold, thenV is generated by incomplete null geodesics. It follows that if the region of causality violation does not extend to infinity thenV contains singularities.This essay received an honourable mention from the Gravity Research Foundation, 1989     

Quantum Oscillations Can Prevent the Big Bang Singularity in an Einstein-Dirac Cosmology

We consider a spatially homogeneous and isotropic system of Dirac particles coupled to classical gravity. The dust and radiation dominated closed Friedmann-Robertson-Walker space-times are recovered as limiting cases. We find a mechanism where quantum oscillations of the Dirac wave functions can prevent the formation of the big bang or big crunch singularity. Thus before the big crunch, the collapse of the universe is stopped by quantum effects and reversed to an expansion, so that the universe opens up entering a new era of classical behavior. Numerical examples of such space-times are given, and the dependence on various parameters is discussed. Generically, one has a collapse after a finite number of cycles. By fine-tuning the parameters we construct an example of a space-time which satisfies the dominant energy condition and is time-periodic, thus running through an infinite number of contraction and expansion cycles.     

Collision-free gases in Bianchi space-times

We study collision-free gases in Bianchi space-times. Spatially homogeneous distribution functions are found for all Bianchi types by supposing that the distribution functionf(x, p) is a function of the Killing vector constants of the motion only. Bianchi types I, VIII and IX only, lead to physical distributions. In types VIII and IX the average behaviour of the gas is that of a nonrotating viscous fluid. In an attempt to obtain physical spatially homogeneous distribution functions for all Bianchi types, we write the Liouville equation in a spatially homogeneous orthonormal tetrad. Furthermore, the general inhomogeneous solution of Liouville's equation in Bianchi type I is obtained, depending on constants of the motion that generalise the conserved quantities generated by Lorentz boosts in flat space-time.     

Metric and Ricci tensors for a certain class of space-times ofD type

The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

A few properties of a certain class of degenerate space-times

The properties are studied of a class of space-times determined by assuming the shape of the metric formds ² including disposable coordinate functions. It has been found that this class includes degenerate space-times with geodetic, null, shear-free congruences with nonvanishing expansion. The theorem has been proved that this class of solutions of the Einstein equations can easily be expanded to solutions of Einstein-Maxwell equations with a fairly general electromagnetic field. For a selected subclass relations are given between the functions determining the metric form, and two new explicit solutions with arbitrary functions of the Einstein-Maxwell equations with a cosmological constant are found.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Cosmic censorship and conformai transformations

A new definition of a nakedly singular space-time is proposed. Conformai transformations of general, vacuum space-times are considered for conformai factors which are proper mappings into (0, ). A space-time generated in this manner which is null convergent on the future Cauchy development of a partial Cauchy surface is shown to be not nakedly singular relative to that surface in the sense of the chosen definition. If the conformal factor is bounded from above then the untransformed, vacuum space-time is similarly not nakedly singular. A censorship theorem for null convergent, conformally flat space-times is obtained as a corollary to the principal result.     

On event horizons in static space-times

A proof of the (vacuum) Israel theorem on event horizons in static space-times is given employing the Newman-Penrose formalism. The theorem is extended to include the case of a static, massive, complex, scalar field.Supported in part by the National Research Council of Canada.     

A class of singular space-times

We present a singularity theorem for a certain class of space-times. The theorem contains an ‘energy’ condition stronger than Hawking's, but does not require any condition about Cauchy surfaces, normals or time orientability.     

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.     

The holonomy group in general relativity and the determination ofg ij fromT i j

We prove a relativistic version of deRham's theorem and use it to find the holonomy group of a large class of space-times. We also show that the concept of energy content needed above completely determinesg _ij under suitable assumptions. Thus it brings us closer to a theorem that will express Mach's principle in general relativity.     

The instability of the Kerr-like Cauchy horizons

A theorem is proved which shows that under a certain condition called the strong null convergence condition the topological structure of the Cauchy horizons of the type occurring in the Reissner-Nordström and Kerr space-times will have to change. This result together with an earlier argument due to Tipler shows that such horizons are unstable.