Peter Szekeres

Colliding Plane Gravitational Waves

The equations governing the collision of two plane gravitational waves are derived. The general exact solution representing this situation when both waves are linearly polarized are found, and some special solutions of possible physical interest are discussed in detail.

The abstract boundary—a new approach to singularities of manifolds

A new scheme is proposed for dealing with the problem of singularities in General Relativity. The proposal is, however, much more general than this. It can be used to deal with manifolds of any dimension which are endowed with nothing more than an affine connection, and requires a family C of curves satisfying a bounded parameter property to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family C will in general lead to different singularity structures. Our key notion is the abstract boundary or a-boundary of a manifold, which is defined for any manifold M and is independent of both the affine connection and the chosen family C of curves. The a-boundary is made up of equivalence classes of boundary points of M in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold (M,g) with a specified family C of curves, the abstract boundary points can then be split up into four main categories—regular, points at infinity, unapproachable points and singularities. Precise definitions are also provided for the notions of a removable singularity and a directional singularity. The pseudo-Riemannian manifold will be said to be singularity-free if its abstract boundary contains no singularities. The scheme passes a number of tests required of any theory of singularities. For instance, it is shown that all compact manifolds are singularity-free, irrespective of the metric and chosen family C. All geodesically complete pseudo-Riemannian manifolds are also singularity-free if the family C simply consists of all affinely parametrised geodesics. Furthermore, if any closed region is excised from a singularity-free manifold then the resulting manifold is still singularity-free. Numerous examples are given throughout the text. Problematic cases posed by Geroch and Misner are discussed in the context of the a-boundary and are shown to be readily accommodated.

Uniqueness of the Newman–Janis Algorithm in Generating the Kerr–Newman Metric

After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be “derived” by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einsteins field equations now known as the Kerr–Newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman–Janis algorithm works, many physicist considering it to be an ad hoc procedure or “fluke” and not worthy of further investigation. Contrary to this belief this paper shows why the Newman–Janis algorithm is successful in obtaining the Kerr–Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman–Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein–Maxwell equations is the Kerr–Newman metric.After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be “derived” by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein’s field equations now known as the Kerr-newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman-Janis algorithm works, many physicist considering it to be an ad hoc procedure or “fluke” and not worthy of further investigation. Contrary to this belief this paper shows why the Newman-Janis algorithm is successful in obtaining the Kerr-Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman-Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein-Maxwell equations is the Kerr-Newman metric. PACS numbers: 02.30.Dk, 04.20.Cv, 04.20.Jb, 95.30.Sf

Theorems on Shear-Free Perfect Fluids with Their Newtonian Analogues

In this paper we provide fully covariant proofs of some theorems on shear-free perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear.

The Curzon singularity. I: Spatial sections

Some new properties of geodesies and other curves lying in the spatial sectionst = const. of the Curzon solution are derived. These are shown to allow one to build up a new coordinate system in which the singularity appears unambiguously as a ring. A new region of spacelike infinity is also revealed on the “other side” of this ring, which can be approached by spatial geodesies threading through the ring.

Timelike and null focusing singularities in spherical symmetry: A solution to the cosmological horizon problem and a challenge to the cosmic censorship hypothesis

Extending the study of spherically symmetric metrics satisfying the dominant energy condition and exhibiting singularities of power-law type initiated in SI93, we identify two classes of peculiar interest: focusing timelike singularity solutions with the stress-energy tensor of a radiative perfect fluid (equation of state:

The Curzon singularity. II: Global picture

p={1\over 3} \rho

What is a shell-crossing singularity?

) and a set of null singularity classes verifying identical properties. We consider two important applications of these results: to cosmology, as regards the possibility of solving the horizon problem with no need to resort to any inflationary scenario, and to the Strong Cosmic Censorship Hypothesis to which we propose a class of physically consistent counter-examples.

Some properties of higher spin rest-mass zero fields in general relativity

A new coordinate system is presented for the Curzon metric such that a class of incomplete geodesics are extendible both in the past and the future into regions of flat space. All other incomplete geodesics terminate at the (infinite curvature) singularity, which appears as a ring with finite radius and is naked. The Curzon solution emerges in this picture as a peculiar kind of sandwich wave. This and other physical interpretations such as a possible collapse scenario are discussed.

Post-Newtonian Cosmology

A detailed discussion of Newtonian and general relativistic spherically symmetric dust solutions leads to the following suggested criteria for a singularity to be classified as a shell-cross: (1) All Jacobi fields have finite limits (in an orthonormal parallel propagated frame) as they approach the singularity. (2) The boundary region forms an essential C 2 singularity which is C 1 regular, that is it can be transformed away by a C 1 coordinate transformation.