General relativistic solutions in minimal theory of bigravity

Abstract:

We investigate dynamical properties of static and spherically symmetric systems in the self-accelerating branch of the Minimal Theory of Bigravity (MTBG).  In the former part, we study the gravitational collapse of pressure-less dust and find special solutions, where, in both the physical and fiducial sectors, the exterior and interior spacetime geometries are given by the Schwarzschild spacetimes and the Friedmann-Lemaitre-Robertson-Walker universes dominated by pressure-less dust, respectively, with specific time slicings.

In the case that the Lagrange multipliers are trivial and have no jump across the matter interfaces in both the physical and fiducial sectors, the junction conditions across them remain the same as those in general relativity (GR). For simplicity, we foliate the interior geometry by homogeneous and isotropic spacetimes. We find interesting classes of exact solutions that represent gravitational collapse in MTBG.

In the spatially-flat case, under a certain tuning of the initial condition, we find exact solutions of matter collapse in which the two sectors evolve independently. In the spatially-closed case, once the matter energy densities and the Schwarzschild radii are tuned between the two sectors, we find exact solutions that correspond to the Oppenheimer-Snyder model in GR. In the latter part, we study odd-parity perturbations of the Schwarzschild-de Sitter solutions written in the spatially-flat coordinates.

For the higher-multipole modes $\ell\geq2$, we find that in general the system reduces to that of four physical modes, where two of them are dynamical and the remaining two are shadowy, i.e., satisfying only elliptic equations. In the case that the ratio of the lapse functions between the physical and fiducial sectors are equal to a constant determined by the parameters of the theory, the two dynamical modes are decoupled from each other but sourced by one of the shadowy modes.

Otherwise, the two dynamical modes are coupled to each other and sourced by the two shadowy modes. At least for the cases of collapse described in this paper, we find that the ratio of the lapse functions is determined by the properties of the collapse itself. On giving appropriate boundary conditions to the shadowy modes as to not strongly back-react/influence the dynamics of the master variables, in the high frequency and short wavelength limits, we show that the two dynamical modes do not suffer from ghost or gradient instabilities.