Self-accelerating Massive Gravity: How Zweibeins Walk through Determinant Singularities

TL;DR

This paper investigates the behavior of massive gravity near determinant singularities, showing that continuous zweibein solutions can be constructed across these singularities, revealing new solution branches and stability features.

Contribution

It introduces a method to join zweibein solutions across determinant singularities, challenging previous assumptions and exploring stability on the self-accelerating branch.

Findings

Continuous zweibein solutions can cross determinant singularities.

The self-accelerating branch prevents solutions from becoming more pathological.

New solution branches emerge beyond traditional gauge choices.

Abstract

The theory of massive gravity possesses ambiguities when the spacetime metric evolves far from the non-dynamical fiducial metric used to define it. We explicitly construct a spherically symmetric example case where the metric evolves to a coordinate-independent determinant singularity which does not exist in the initial conditions. Both the metric and the vierbein formulation of the theory are ill-defined at this point. In unitary gauge, the chart of the spacetime ends at this point and does not cover the full spacetime whereas the spherically symmetric vierbeins, or zweibeins, of the fiducial metric become non-invertible and do not describe a valid metric. Nonetheless it is possible to continuously join a zweibein solution on the other side of the singularity which picks one of the degenerate solutions of the metric square root. This continuous solution is not the choice conventionally made in the previous literature. We also show that the Stueckelberg equations of motion on the self-accelerating branch prevent solutions from evolving to a more pathological situation in which the spacetime vierbeins lack a crucial symmetry with the fiducial vierbeins and real square roots fail to exist.