Translation invariant time-dependent solutions to massive gravity II

TL;DR

This paper investigates translation-invariant solutions in ghost-free massive gravity with a specific mass term, revealing potential singularities and pathological behavior in time evolution, highlighting the need for careful treatment of the 3 mass term.

Contribution

It explicitly analyzes the constraints and time evolution of solutions with a 3 mass term, showing possible singularities and differences from previously studied cases.

Findings

Time-dependent solutions can develop finite-time singularities.

The lapse function may change sign, causing singular evolution.

The 3 mass term can lead to pathological behavior in massive gravity.

Abstract

This paper is a sequel to arXiv:1310.6560 [hep-th] and is also devoted to translation-invariant solutions of ghost-free massive gravity in its moving frame formulation. Here we consider a mass term which is linear in the vielbein (corresponding to a $\beta_3$ term in the 4D metric formulation) in addition to the cosmological constant. We determine explicitly the constraints, and from the initial value formulation show that the time-dependent solutions can have singularities at a finite time. Although the constraints give, as in the $\beta_1$ case, the correct number of degrees of freedom for a massive spin two field, we show that the lapse function can change sign at a finite time causing a singular time evolution. This is very different to the $\beta_1$ case where time evolution is always well defined. We conclude that the $\beta_3$ mass term can be pathological and should be treated with care.