Conformal killing vectors of plane symmetric four dimensional lorentzian manifolds

Suhail Khan,Ashfaque H Bokhari,Gulzar Ali Khan,Tahir Hussain

doi:10.1140/epjc/s10052-015-3758-1

Abstract

In this paper, we investigate conformal Killing vectors (CKVs) admitted by some plane symmetric spacetimes. Ten conformal Killing’s equations and their general forms of CKVs are derived along with their conformal factor. The existence of conformal Killing symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. Considering the cases of time-like and inheriting CKVs, we obtain spacetimes admitting plane conformal symmetry. Integrability conditions are solved completely for some known non-conformally flat and conformally flat classes of plane symmetric spacetimes. A special vacuum plane symmetric spacetime is obtained, and it is shown that for such a metric CKVs are just the homothetic vectors (HVs). Among all the examples considered, there exists only one case with a six dimensional algebra of special CKVs admitting one proper CKV. In all other examples of non-conformally flat metrics, no proper CKV is found and CKVs are either HVs or Killing’s vectors (KVs). In each of the three cases of conformally flat metrics, a fifteen dimensional algebra of CKVs is obtained of which eight are proper CKVs.

Highlights

IntroductionSome physically plausible solutions of Einstein’s field equations have been obtained in [30,31,32,33] under the assumption that the spacetime admit conformal Killing vectors (CKVs). At the geometric level it is well understood that application of CKVs makes possible coordinate choice to simplify the metric
Considering a general form of a plane symmetric spacetimes metric, ten conformal Killing’s equations are solved and a general form of conformal Killing vectors (CKVs) along with their conformal factor are obtained subject to twelve integrability conditions
When subjected to one integrability condition, it is shown that purely timelike CKVs are admitted

Summary

Introduction

Some physically plausible solutions of Einstein’s field equations have been obtained in [30,31,32,33] under the assumption that the spacetime admit CKVs. At the geometric level it is well understood that application of CKVs makes possible coordinate choice to simplify the metric. The. Page 3 of 9 523 wide range applications of CKVs in astrophysics and cosmology (as discussed above) and at geometric, dynamics and kinematic levels motivated us to explore the CKVs of plane symmetric non static spacetimes. In order to solve the above integrability conditions completely, we impose certain restrictions either on the metric functions or the CKV components To this end, we restrict the components of the CKVs to admit a particular form and present results . It is worth noting that one CKV is the usual KV giving rotational symmetry z∂y − y∂z, while all other CKVs are arbitrary functions of t and x

Conformal killing vectors of some special plane symmetric spacetime metrics

Summary and discussion