Kerr-Schild and generalized metric motions

TL;DR

This paper explores new types of metric motions, especially Kerr-Schild motions, revealing their algebraic structures, conditions for existence, and potential as symmetries, with implications for solving Einstein's equations.

Contribution

It provides a detailed analysis of Kerr-Schild motions, introduces a scheme for other metric motions, and relates these to solution methods in Einstein's field equations.

Findings

Kerr-Schild motions often form finite-dimensional Lie algebras

They are generally isometrizable, linking to isometries of different spacetimes

New candidate motions exhibit richer algebraic structures

Abstract

In this work we study in detail new kinds of motions of the metric tensor. The work is divided into two main parts. In the first part we study the general existence of Kerr-Schild motions --a recently introduced metric motion. We show that generically, Kerr-Schild motions give rise to finite dimensional Lie algebras and are isometrizable, i.e., they are in a one-to-one correspondence with a subset of isometries of a (usually different) spacetime. This is similar to conformal motions. There are however some exceptions that yield infinite dimensional algebras in any dimension of the manifold. We also show that Kerr-Schild motions may be interpreted as some kind of metric symmetries in the sense of having associated some geometrical invariants. In the second part, we suggest a scheme able to cope with other new candidates of metric motions from a geometrical viewpoint. We solve a set of new candidates which may be interpreted as the seeds of further developments and relate them with known methods of finding new solutions to Einstein's field equations. The results are similar to those of Kerr-Schild motions, yet a richer algebraical structure appears. In conclusion, even though several points still remain open, the wealth of results shows that the proposed concept of generalized metric motions is meaningful and likely to have a spin-off in gravitational physics.We end by listing and analyzing some of those open points.