Conformal Killing Vectors Of Plane Symmetric Four Dimensional Lorentzian Manifolds

TL;DR

This paper classifies conformal Killing vectors in plane symmetric four-dimensional Lorentzian manifolds, revealing how these symmetries restrict metric functions and identifying cases with maximal conformal symmetry.

Contribution

It derives the general conformal Killing equations for plane symmetric spacetimes and solves integrability conditions for various classes, including conformally flat and non-flat metrics.

Findings

Identifies conditions under which CKVs exist in plane symmetric spacetimes.

Finds that most non-conformally flat metrics admit only homothetic or Killing vectors.

Discovers a unique case with a six-dimensional algebra of special CKVs.

Abstract

In this paper, we investigate conformal Killing's 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's 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.