On dRGT massive gravity with degenerate reference metrics

TL;DR

This paper extends dRGT massive gravity to include degenerate reference metrics by generalizing the inverse tensor, deriving modified equations of motion, and proving a generalized Birkhoff theorem for certain symmetric solutions.

Contribution

It introduces a generalized inverse for symmetric tensors on Lorentz manifolds, enabling derivation of equations of motion with degenerate reference metrics in dRGT gravity.

Findings

Derived equations of motion for degenerate reference metrics.

Proved a generalized Birkhoff theorem for specific symmetric solutions.

Identified conditions under which solutions are Schwarzschild-type or Nariai-Bertotti-Robinson-type.

Abstract

In dRGT massive gravity, to get the equations of motion, the square root tensor is assumed to be invertible in the variation of the action. However, this condition can not be fulfilled when the reference metric is degenerate. This implies that the resulting equations of motion might be different from the case where the reference metric has full rank. In this paper, by generalizing the Moore-Penrose inverse to the symmetric tensor on Lorentz manifolds, we get the equations of motion of the theory with degenerate reference metric. It is found that the equations of motion are a little bit different from those in the non-degenerate cases. Based on the result of the equations of motion, for the $(2+n)$-dimensional solutions with the symmetry of $n$-dimensional maximally symmetric space, we prove a generalized Birkhoff theorem in the case where the degenerate reference metric has rank $n$, i.e., we show that the solutions must be Schwarzschild-type or Nariai-Bertotti-Robinson-type under the assumptions.