Conformal Symmetries of Some Space Times in f(R) Theories of Gravity

Abstract

In this thesis, a study of conformal symmetries of some space times in the f(R) theories of gravity have been presented. The study includes static spherically symmetric, static plane symmetric, static cylindrically symmetric, Bianchi type I, II, III, V, Kantowski Sachs, Spatially homogeneous rotating space-times, a class of pp waves and non-static plane symmetric spacetimes. Initially, we have found some solutions of Einstein field equations (EFEs) using different fluid matters in the f(R) theories of gravity and then we have found conformal vector fields (CVFs) of the obtained solutions by means of direct integration technique. In the static spherically symmetric space-times, six cases have been discussed out of which there exists only one case for which CVFs become homothetic vector fields (HVFs) while in the rest of the cases CVFs become Killing vector fields (KVFs). In the static plane symmetric space-times, again there exist six cases. Out of these six cases, the space-times in five cases become conformally flat therefore admit fifteen independent CVFs while in the sixth case CVFs become KVFs. In the static cylindrically symmetric space-times, the dimension of CVFs turns out to be 4, 5, 6 and 15. In the Bianchi type I space-times, there exist fourteen cases. Out of these fourteen cases, the space-times in nine cases admit 4 and 5 HVFs while in five cases, space-times admit 6 and 15 CVFs. In Bianchi type II space-times, there exist seven cases while studying each case we found that in four cases, space-times admit proper HVFs while in rest of the three cases, CVFs become KVFs. In the Bianchi type V space-times, again there exist seven cases. From these seven cases, the space-times in six cases admit three KVFs while in the seventh case, the space-times become conformally flat therefore admit fifteen independent CVFs. In the Kantowski Sachs and Bianchi type III space-times, there exist eight cases. Studying each case in detail, we found that in six cases, space-times admit four and six KVFs while in two cases, the space-times admit proper CVFs of dimension six. In spatially homogeneous rotating space-times, there exist six cases. Within these six cases, there exist three cases in which space-times admit three and four KVFs. In two cases, the space-times admit four HVFs while in the remaining case, the space-time admits fifteen independent CVFs. In the pp-wave space-times, there exist ten cases. Studying each case, we found that in eight cases, space-times admit proper HVFs while in two cases, space-times admit proper CVFs. In non-static plane symmetric space-times, there exist, seven vi cases. From these seven cases, there exist two cases in which CVFs become KVFs. In three cases, space-times admit five HVFs while in the remaining two cases, the space-times admit six and fifteen independent CVFs.