Estelita Vaz

Ricci and matter collineations in space-time

A discussion of Ricci and matter collineations (mainly the former) is presented. A mathematical description of their dimensionality, differentiability, extendibility etc. is given. Examples of Ricci collineations are constructed particularly in decomposable space-times.

Matter collineations: The inverse. "symmetry inheritance" problem

Matter collineations, as a symmetry property of the energy‐momentum tensor Tab, are studied from the point of view of the Lie algebra of vector fields generating them. Most attention is given to space–times with a degenerate energy‐momentum tensor. Some examples of matter collineations are found for dust fluids (including Szekeres’s space–times), and null fluid space–times.

General spherically symmetric elastic stars in relativity

The relativistic theory of elasticity is reviewed within the spherically symmetric context with a view towards the modeling of star interiors possessing elastic properties such as the ones expected in neutron stars. Emphasis is placed on generality in the main sections of the paper, and the results are then applied to specific examples. Along the way, a few general results for spacetimes admitting isometries are reviewed, and their consequences are fully exploited in the case of spherical symmetry relating them next to the the case in which the material content of the spacetime is some elastic material. Specific examples are provided satisfying the dominant energy condition and admitting a constitutive equation, including a static two-layer star ‘toy model’ consisting of an elastic core surrounded by a perfect fluid corresponding to the interior Schwarzschild solution matched to the vacuum Schwarzschild solution. This paper extends and generalizes the pioneering work by Magli and Kijowski (Gen Relat Gravit 24:139, 1992), Magli (Gen Relat Gravit 25:1277, 1993; 25:441, 1993), and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers (Karlovini and Samuelsson in Class Quantum Grav 20:363, 2003; 21:1559, 21:4531, 2004).

Cylindrically symmetric static solutions of the Einstein field equations for elastic matter

The Einstein field equations are derived for a static cylindrically symmetric spacetime with elastic matter. The equations can be reduced to a system of two nonlinear ordinary differential equations and we present analytical and numerical solutions satisfying the dominant energy conditions. Furthermore, we show that the solutions can be matched at a finite radius to suitable Λ-vacuum exteriors given by the Linet-Tian spacetime.

Double warped space-times

An invariant characterization of double warped space–times is given in terms of Newman–Penrose formalism and a classification scheme is proposed. A detailed study of the conformal algebra of these space–times is also carried out and some remarks are made on certain classes of exact solutions.

Analysing the elasticity difference tensor of general relativity

The elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is here analysed. Principal directions associated with this tensor are compared with eigendirections of the material metric. Examples concerning spherically symmetric and axially symmetric space-times are then presented.

Symmetries and the Karlhede classification of type N vacuum solutions

The influence of symmetries on the invariant classification of a general type N vacuum spacetime is studied. It is shown that the existence of two independent symmetries (Killing vector fields/homothetic vector fields) reduces the upper bound on the Karlhede algorithm for such solutions from five to three (derivatives), as long as the vector fields obey a geometrical condition obtained here. The only known twisting type N vacuum metric, the Hauser metric, is such an example.

Killing pairs and the empty space field equations

The empty space field equations are investigated for each of the canonical forms obtained previously for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal. It is found that the rational first integral of the geodesic equation, corresponding to the Killing pair, is always necessarily the ration of two linear first integrals.

Some canonical forms for the metric of spacetimes admitting a rational first integral of the geodesic equation

Canonical forms are obtained for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal.

Curvature collineations for type-N Robinson-Trautman space-times

The metric of type-N Robinson-Trautman space-times is generated by a real functionP satisfying certain field equations. Canonical forms forP are obtained under the assumption that at least one curvature collineation exists. In order to give an example of the improper subgroup structure of a group of curvature collineations all the curvature collineations are determined for the space-times corresponding to one of the two canonical forms.