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

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 Stückelberg 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.