Approximate solutions in General Relativity via deformation of embeddings

TL;DR

This paper explores how deformations of embedded Einstein space-times can generate approximate solutions to Einstein's equations, including wave-like and Kerr-like metrics, by analyzing infinitesimal changes in embedding functions.

Contribution

It introduces a systematic method for deriving approximate Einstein solutions through deformations of embedded space-times, extending to higher orders and specific metrics like Schwarzschild and Kerr.

Findings

First-order deformations yield solutions satisfying Einstein equations approximately.

Second and third order deformations produce wave-like and axial symmetry solutions.

The approach provides a geometric framework for approximating complex space-time metrics.

Abstract

A systematic study of deformations of four-dimensional Einsteinian space-times embedded in a pseudo-Euclidean space $E^N$ of higher dimension is presented. Infinitesimal deformations, seen as vector fields in $E^N$, can be divided in two parts, tangent to the embedded hypersurface and orthogonal to it; only the second ones are relevant, the tangent ones being equivalent to coordinate transformations in the embedded manifold. The geometrical quantities can be then expressed in terms of embedding functions $z^A$ and their infinitesimal deformations $v^A z^A \to {\tilde{z}}^A = z^A + \epsilon v^A$. The deformations are called Einsteinian if they keep Einstein equations satisfied up to a given order in $\epsilon$. The system so obtained is then analyzed in particular in the case of the Schwarzschild metric taken as the starting point, and some solutions of the first-order deformation of Einstein's equations are found. We discuss also second and third order deformations leading to wave-like solutions and to the departure from spherical symmetry towards an axial one (the approximate Kerr solution)