# Reformulation of the symmetries of first-order general relativity

## TL;DR

This paper uncovers a new internal gauge symmetry in higher-dimensional general relativity, generalizing known symmetries and revealing deeper gauge-theoretic structures, including its relation to diffeomorphisms and matter coupling effects.

## Contribution

It introduces a novel internal gauge symmetry for n-dimensional Palatini action, extending three-dimensional translations, and explores its algebra, relation to diffeomorphisms, and dependence on matter fields.

## Key findings

- Discovered a new gauge symmetry generalizing 3D translations.
- Expressed diffeomorphisms as combinations of this symmetry and Lorentz transformations.
- Showed the symmetry's dependence on the Immirzi parameter and matter coupling.

## Abstract

We report a new internal gauge symmetry of the n-dimensional Palatini action with cosmological term (n>3) that is the generalization of three-dimensional local translations. This symmetry is obtained through the direct application of the converse of Noether's second theorem on the theory under consideration. We show that diffeomorphisms can be expressed as linear combinations of it and local Lorentz transformations with field-dependent parameters up to terms involving the variational derivatives of the action. As a result, the new internal symmetry together with local Lorentz transformations can be adopted as the fundamental gauge symmetries of general relativity. Although their gauge algebra is open in general, it allows us to recover, without resorting to the equations of motion, the very well-known Lie algebra satisfied by translations and Lorentz transformations in three dimensions. We also report the analog of the new gauge symmetry for the Holst action with cosmological term, finding that it explicitly depends on the Immirzi parameter. The same result concerning its relation to diffeomorphisms and the open character of the gauge algebra also hold in this case. Finally, we consider the non-minimal coupling of a scalar field to gravity in n dimensions and establish that the new gauge symmetry is affected by this matter field. Our results indicate that general relativity in dimension greater than three can be thought of as a gauge theory.