Curvature collineations in general relativity. I

Abstract

Although curvature collineations (curvature preserving transformations) have been studied within the context of general relativity for 20 years, there has been little attempt to study them systematically and there does not appear to have been a detailed mathematical investigation of their properties. This is the first of two papers that are intended as a contribution to this deficiency. This paper presents a discussion of the more general mathematical aspects of curvature collineations and suggests a program for studying them. The implementation of this program and the analysis of specific examples will be carried out in the second paper.

Symmetries of Energy-Momentum Tensor: Some Basic Facts

“…Also, this may lead to problems related to the orbits of the resulting local diffeomorphism [7,12]. In the light of such problems and the usefulness of the covariant definition for MC [10,11] it can be concluded that the covariant definition of MC should be preferred.…”

“…An affine and a conformal vector field ξ on M is uniquely determined by specifying it and its first covariant derivative and specifying it and its first and second covariant derivatives respectively at some m ∈ M. However, the value of ξ and its covariant derivatives of all orders at some m ∈ M may not be enough to determine uniquely a MC ξ on M. Thus two MCs that agree on a non-empty open subset of M may not agree on M. These features are also found in RCs and CCS. This leads to a problem of the extendibility of local MCs to the whole of M which is more complicated than that for affine and conformal vector fields [7].…”

“…If MCs are C ∞ then one recovers the Lie algebra structure but loses the non-smooth. The infinite dimensionality may also lead to problems related to the orbits of the resulting local diffeomorphism [7]. 4.…”

“…Curvature and the Ricci tensors are the important quantities which play a significant role in understanding the geometric structure of spacetime. A pioneer study of curvature collineations (CCs) and Ricci collineations (RCs) has been carried out by Katzin, et al [5] and a further classification of CCs and RCs has been obtained by different authors [6,7]. The theoretical basis for the study of affine including Killing and homothetic vector fields is well understood and many examples are given.…”

Teleparallel Killing Vectors of the Einstein Universe

“…A detail investigation of the spacetimes and their geometrical symmetries like Killing vectors (KVs), curvature collineations and RCs was made by different authors [9][10][11][12][13]. Using Lie algebra approach, Carot et al [14] discussed the physical properties of the spacetimes called matter collineations (MCs).…”