Gravity Theories with Background Fields and Spacetime Symmetry Breaking

TL;DR

This paper reviews effective gravitational theories with fixed background fields that break spacetime symmetry, discussing explicit and spontaneous breaking, their excitations, and the role of Noether identities in symmetry restoration.

Contribution

It provides a comprehensive overview of how background fields break spacetime symmetry in gravity theories, highlighting differences between explicit and spontaneous breaking and the use of Stuckelberg fields.

Findings

Spontaneous breaking leads to dynamical background fields with Nambu-Goldstone and Higgs excitations.

Explicit breaking results in nondynamical backgrounds and additional metric or vierbein excitations.

Noether identities are key to understanding symmetry breaking and restoration mechanisms.

Abstract

An overview is given of effective gravitational field theories with fixed background fields that break spacetime symmetry. The behavior of the background fields and the types of excitations that can occur depend on whether the symmetry breaking is explicit or spontaneous. For example, when the breaking is spontaneous, the background field is dynamical and massless Nambu--Goldstone and massive Higgs excitations can appear. However, if the breaking is explicit, the background is nondynamical, and in this case additional metric or vierbein excitations occur due to the loss of local symmetry, or these excitations can be replaced by dynamical scalar fields using a Stuckelberg approach. The interpretation of Noether identities that must hold in each case differs, depending on the type of symmetry breaking, and this affects the nature of the consistency conditions that must hold. The Noether identities also shed light on why the Stuckelberg approach works, and how it is able to restore the broken spacetime symmetry in a theory with explicit breaking.