Symmetries of the Einstein Equations

TL;DR

This paper classifies all generalized symmetries of the vacuum Einstein equations in four dimensions, revealing that only simple scalings and diffeomorphisms are symmetries, which impacts the understanding of gravitational observables and integrability.

Contribution

The paper provides a complete classification of local generalized symmetries of the vacuum Einstein equations in four dimensions, showing no additional nontrivial symmetries exist.

Findings

Only constant metric scalings are symmetries.

Generalized spacetime diffeomorphisms are symmetries.

Large classes of gravitational observables are ruled out.

Abstract

Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are assumed to be local, \ie at a given spacetime point they are functions of the metric and an arbitrary but finite number of derivatives of the metric at the point. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions and find that the only generalized symmetry transformations consist of: (i) constant scalings of the metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ``observables'' for the gravitational field, and suggest that the vacuum Einstein equations are not integrable.