Gravity from the extension of spatial diffeomorphisms

TL;DR

This paper investigates extending spatial diffeomorphisms in classical field theories, finding that such extensions lead to either Einstein's gravity or ultralocal gravity, and do not support simple modifications for renormalizability.

Contribution

It demonstrates that extending spatial diffeomorphisms results in a constraint algebra equivalent to Einstein's or ultralocal gravity, ruling out certain simple modifications.

Findings

Extended symmetry leads to Einstein's or ultralocal gravity

No simple renormalizable modifications within this class

Constraint algebra remains consistent with known gravity theories

Abstract

The possibility of the extension of spatial diffeomorphisms to a larger family of symmetries in a class of classical field theories is studied. The generator of the additional local symmetry contains a quadratic kinetic term and a potential term which can be a general (not necessarily local) functional of the metric. From the perspective of the foundation of Einstein's gravity our results are positive: The extended constraint algebra is either that of Einstein's gravity, or ultralocal gravity. If our goal is a simple modification of Einstein's gravity that for example makes it perturbatively renormalizable, as has recently been suggested, then our results show that there is no such theory within this class.