G\"{o}del-symmetric backgrounds and explicit spacetime symmetry breaking

TL;DR

This paper explores explicit spacetime symmetry breaking in gravity, focusing on G"odel backgrounds within the Standard-Model Extension, demonstrating consistency with constraints and analyzing causality violations through background-dependent critical radii.

Contribution

It develops the isometry-based approach for explicit symmetry breaking in gravity, applying it to G"odel metrics and revealing new insights into causality and constraints.

Findings

Consistent dynamics with constraint equations

Background-dependent critical radius for causality

Emergence of causal and non-causal regions

Abstract

Explicit breaking of diffeomorphism symmetry with nondynamical background fields in gravitational theories can lead to inconsistencies between the equations of motion and the underlying pseudo-Riemannian geometry. These theories produce a constraint equation that follows from the modified Einstein's equations and the contracted Bianchi identity which has no simple solution and raises questions about the consistency of backgrounds structures in extensions to general relativity. In contrast, spontaneous symmetry breaking where fields acquire a vacuum expectation value are shown to avoid these problems. Nevertheless, there are some approaches that consistently implement explicit diffeomorphism breaking by: i) introducing geometrical restrictions which may dynamically restore diffeomorphism invariance, ii) using the St\"{u}ckelberg formalism where auxiliary scalar fields can carry degrees of freedom associated to spontaneously broken symmetries, and more recently by iii) exploiting the isometries of a given gravitational configuration in the presence of a background; which ultimately requires the background fields to be Lie-dragged along the Killing vectors. In this work, we further develop and apply the latter approach focusing on a concrete example involving the G\"{o}del metric and the minimal gravitational Standard-Model Extension. We show that the resulting dynamics is fully consistent with the constraint equation and produce new Noether's identities. Furthermore we investigate causality violations by analyzing the critical radius, which is found to depend explicitly on the background fields. This dependence allows for the emergence of both causal and non-causal regions in spacetime.