On Kerr-Schild Symmetries and Conservation Laws in General Relativity

TL;DR

This paper investigates the geometric conditions for Kerr-Schild symmetries and conservation laws in general relativity using the Newman-Penrose formalism, providing new insights into spacetime symmetries and conserved quantities.

Contribution

It formulates geometric constraints for Kerr-Schild groups and explores conditions for conservation laws in generalized Kerr-Schild spacetimes, linking symmetries with conserved currents.

Findings

Derived conditions for Kerr-Schild symmetries using Newman-Penrose formalism.

Identified restrictions on conservation laws in generalized Kerr-Schild spacetimes.

Demonstrated the feasibility of conditions with simple model examples.

Abstract

In the present work, the spin-coefficient formalism of Newman and Penrose is used to formulate geometric constraints for the existence of Kerr-Schild groups, i.e. continuous groups of generalized Kerr-Schild transformations. In addition, by characterizing the geometric structure of the deformed Einstein tensor of the generalized Kerr-Schild class, restrictions are imposed on the existence of apparent conservation laws in generic spacetimes, which are defined via considering special Kerr-Schild currents whose associated Kerr-Schild vector fields coincide with timelike Killing vector fields of pairs of stationary background geometries. The feasibility of the derived conditions is demonstrated by considering concrete, suitably simple models of generalized Kerr-Schild spacetimes.