Properties of Bigravity Solutions in a Solvable Class

We consider the properties of solutions in the bigravity theory for general models, which are parametrized by two parameters ${\alpha }_3$ and ${\alpha }_4$. Assuming that two metric tensors $g_{\mu \nu }$ and $f_{\mu \nu }$ satisfy the condition $f_{\mu \nu }=C^2{g}_{\mu \nu }$ where $C$ is a constant, we investigate the conditions for the parameters so that the solutions with $C\mathrm{\ne }1$ could exist. We also discuss the magnitude and the sign of corresponding cosmological constants. For the black hole solution, we consider the black hole entropy to which the massive spin-$2$ field contributes. In order to obtain the black hole entropy, we take an approach which uses the Virasoro algebra and the central charge corresponding to the surface term in the action.