Non-vacuum Spacetimes Research Articles

Embeddings in non-vacuum spacetimes

A scheme is discussed for embedding n-dimensional, Riemannian manifolds in an (n + 1)-dimensional Einstein space. Criteria for embedding a given manifold in a spacetime that represents a solution to Einstein's equations sourced by a massless scalar field are also discussed. The embedding procedures are illustrated with a number of examples.

On certain quasi-local spin-angular momentum expressions for small spheres

The Ludvigsen-Vickers and two recently suggested quasi-local spin-angular momentum expressions, based on holomorphic and anti-holomorphic spinor fields, are calculated for small spheres of radius r about a point o. It is shown that, apart from the sign in the case of anti-holomorphic spinors in non-vacuum, the leading terms of all these expressions coincide. In non-vacuum spacetimes this common leading term is of order r4, and it is the product of the contraction of the energy-momentum tensor and an average of the approximate boost-rotation Killing vector that vanishes at o and of the 3-volume of the ball of radius r. In vacuum spacetimes the leading term is of order r6, and the factor of proportionality is the contraction of the Bel-Robinson tensor and another average of the same approximate boost-rotation Killing vector.

Limits of the energy - momentum tensor in general relativity

A limiting diagram for the Segre classification of the energy - momentum tensor is obtained and discussed in connection with a Penrose specialization diagram for the Segre types. A generalization of the coordinate-free approach to limits of Paiva et al to include non-vacuum spacetimes is made. Geroch's work on limits of spacetimes is also extended. The same argument also justifies part of the procedure for classification of a given spacetime using Cartan scalars.

THE WAVE EQUATIONS FOR THE WEYL TENSOR/SPINOR AND DIMENSIONALLY DEPENDENT TENSOR IDENTITIES

By reconciling the wave equation for the Weyl tensor with the corresponding wave equation for the Weyl spinor, we establish a new tensor identity—involving the sum of terms each consisting of a product of the Weyl and Ricci tensors—valid in four (and only four) dimensions. This enables us to give, for the first time, the correct and simplest form of the wave equation for the Weyl tensor in four-dimensional nonvacuum spacetimes. The wave equation for the Weyl tensor in n(> 4) dimensional nonvacuum spaces is also presented for the first time; we show that there does not exist an analogous n-dimensional tensor identity matching the four-dimensional one, and so it follows that there does not exist an analogous simplification of the Weyl wave equation in the n-dimensional case. It is also shown how our new identity, and some other recently discovered identities, relate to a large class of dimensionally dependent identities found some time ago by Lovelock.

Supersymmetric spacetimes in (2+1) adS-supergravity models

We find a class of (2+1)-dimensional spacetimes admitting Killing spinors appropriate to (2,0) adS-supergravity. The vacuum spacetimes include anti-de Sitter (adS) space and charged extreme black holes, but there are many others, including spacetimes of arbitrarily large negative energy that have only conical singularities, and the spacetimes of fractionally charged point particles. The non-vacuum spacetimes are those of self-gravitating solitons obtained by coupling (2,0) adS supergravity to sigma-model matter. We show, subject to a condition on the matter currents (satisfied by the sigma model), and a conjecture concerning global obstructions to the existence of certain types of spinor fields, that the mass of each supersymmetric spacetime saturates a classical bound, in terms of the angular momentum and charge, on the total energy of arbitrary field configurations with the same boundary conditions, although these bounds may be violated quantum mechanically.

The Karlhede classification of type-D nonvacuum spacetimes

The Karlhede classification indicates that it may be necesssary to go to the seventh covariant derivative of the Riemann tensor in order to classify a general type-D nonvacuum spacetime. The authors investigate the field equations directly to show that the worse case cannot in fact occur and so reduce the derivative bound from 7 to 6 for this class of spacetimes.

Linear instability of nonvacuum spacetimes.

We study the problem of linear instability in nonvacuum spacetimes. For vacuum spacetimes linear instability occurs when the spacetime has Killing vectors. In the nonvacuum case, one must prescribe how the sources are to vary. For one natural choice, we show that the signal for instability is the existence of integral constraint vector fields. These vector fields lead, as in the vacuum case, to nonlinear constraints on the first-order perturbations to the metric and momentum. For other choices for variations of the sources, we show how to modify the definition of integral constraint vectors appropriately. Since Robertson-Walker spacetimes have integral constraint vectors our results may have cosmological applications.

The high-frequency limit in general relativity

A precise characterization of the high-frequency limit in general relativity (vacuum space-times) is presented. The averaging schemes used in earlier works are avoided by the use of weak limits, giving the characterization the advantage that the high-frequency limit can be rigorously analyzed. Using this characterization, a theorem to the effect that a certain ‘‘effective stress-energy’’ tensor acts as a source of curvature of the smooth background space-time is proved. It is shown in the special case corresponding to the presence of a single wave that this tensor has the same form as the stress-energy tensor of a null fluid. Also, the extension of this characterization of the high-frequency limit to other fields, such as electromagnetic fields, and to nonvacuum space-times is briefly discussed.

Quasi-local mass for small surfaces

The authors calculate Penrose's quasi-local mass for a small sphere by expanding in powers of the affine parameter along the generators of a light cone. They then check the result and extend it to small surfaces of more general shape by techniques involving more direct use of spinor calculus. In a non-vacuum spacetime, the mass inside a small sphere is obtained from a 4-momentum Pa which is the product of the volume enclosed by the surface and the 4-vector Tabtb, where ta is a unit timelike vector orthogonal to a spanning 3-surface. In the vacuum case, the mass vanishes at the fifth order in the diameter of the surface, both for small spheres and for small surfaces of more general shape.

Exact power law solutions of the Einstein equations

An exact power law metric is discussed which arises when one considers the reduced Einstein equations for certain scale invariant variables associated with a spatially homogeneous or spatially self-similar vacuum or nonvacuum space-time. The metric contains a number of new solutions as well as many known ones.

On the Petrov classification of gravitational fields

An intuitive directional symmetry for a null direction with respect to a given local Lorentz observer is suggested. It is shown that the existence of a null direction with such a symmetry with respect to the given Lorentz observer implies that the null direction is a Debever- Penrose direction. A weak converse is also given. In the vacuum case it is shown that the existence of such a null direction is equivalent to the space-time being algebraically special in the Petrov classification and having that null direction as a repeated principal null direction. The results are generalized to non-vacuum space-times. Finally it is suggested how a vacuum space-time might be classified according to the type of symmetries it possesses.