Classification of Generalized Symmetries for the Vacuum Einstein Equations

TL;DR

This paper classifies all generalized symmetries of the vacuum Einstein equations in four dimensions, revealing that they are essentially diffeomorphisms and scalings, with no additional conservation laws.

Contribution

It provides a complete classification of natural and first-order generalized symmetries for vacuum Einstein equations, reducing higher-order cases to the first order.

Findings

Symmetries are infinitesimal generalized diffeomorphisms and scalings.

No non-trivial conservation laws are associated with these symmetries.

Use of spinorial coordinates on jet space is a novel analytical approach.

Abstract

A generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's ``exact set of fields'' for the vacuum equations.