Collineations of the Ricci tensor

Abstract

Ricci collineations for the Ricci tensor which is constructed from a general spherically symmetric and static metric are classified for all possibilities of Rab(r) (such that Rab≠0 for a=b). It turns out that the only collineations admitted by this tensor can be ten, six, or four and there does not appear any case in between.

Teleparallel Killing Vectors of the Einstein Universe

“…It is shown that there arise 7 TP KVs which are the same in numbers as found in the context of GR [2][3]. The comparison shows that ξ (1) is the same in both the theories while ξ (2) , ξ (3) and ξ (4) in TPG are multiple of the corresponding KVs in GR by F (r). For F (r) = 1, that is, for r = ± √ 2R − 1, the first four KVs reduce to the basic four KVs of the spherical symmetry and coincide with those in GR.…”

“…As a pioneer, Petrov [1] solved the Killing equations for four-dimensional spaces. Bokhari and Qadir [2][3] were able to achieve a complete classification of static spherically symmetric spacetimes. Later, Qadir and Ziad extended this work for a complete classification of all spherically symmetric spacetimes [4][5] by removing the condition of staticity.…”

Ricci Collineations of the Bianchi Types I and Iii, and Kantowski–sachs Spacetimes

“…A complete classification of RCs was obtained for spherically symmetric static spacetimes [10], which were compared with KVs admitted by the corresponding spacetimes. Amir et al [11] have investigated the relationship between the RCVs and the KVs for these spacetimes in detail.…”

Symmetries of Energy-Momentum Tensor: Some Basic Facts

“…The infinite dimensionality may also lead to problems related to the orbits of the resulting local diffeomorphism [7]. 4. If the energy-momentum tensor is of rank 4, it may be regarded as a metric on M. Then the family of C 2 MCs is, in fact, a Lie algebra of smooth vector fields on M of finite dimension ≤ 10 and = 9.…”

“…Further, these solutions are classified according to the Lie algebra structure generated by these symmetries. The well-known connection between Killing vectors (KVs) and constants of the motion [2,3] has encouraged the search for general relations between collineations and conservation laws [4]. Curvature and the Ricci tensors are the important quantities which play a significant role in understanding the geometric structure of spacetime.…”