Conformal Symmetries of the Ricci Tensor for Certain Spacetimes

Abstract

In this thesis, we have investigated conformal symmetries of the Ricci tensor, also known as conformal Ricci collineations (CRCs), for certain physically important spacetimes including kantowski-Sachs spacetimes, static spacetimes with maximal symmetric transverse spaces, non-static spherically symmetric spacetimes and locally rotationally symmetric Bianchi type I and V spacetimes. For each of these spacetimes, the CRC equations are solved in degenerate as well as non-degenerate cases. When the Ricci tensor is degenerate, it is observed that for all the above mentioned spacetimes, the Lie algebra of CRCs is in nite-dimensional. For non-degenerate Ricci tensor, it is shown that the spacetimes under consideration always admit a nite-dimensional Lie algebra of CRCs. For Kantowski-Sachs and locally rotationally symmetric Bianchi type V metrics, we obtain 15-dimensional Lie algebras of CRCs, which is the maximum dimension of conformal algebra for a spacetime. In case of static spacetimes with maximal symmetric transverse spaces, the dimension of Lie algebra of CRCs turned out to be 6, 7 or 15. Similarly, it is observed that non-static spherically symmetric spacetimes may possess 5, 6 or 15 CRCs for non-degenerate Ricci tensor. Finally, the dimension of Lie algebra of CRCs for locally rotationally symmetric Bianchi type I spacetimes is shown to be 7- or 15-dimensional. For all the above mentioned spacetimes, the CRCs are found subject to some highly non-linear di erential constraints. In order to show that the classes of CRCs are non-empty, some examples of exact form of the corresponding metric satisfying these constraints are provided.