Perfect fluid spherically symmetric solutions in massive gravity

We study spherically symmetric solutions in massive gravity generated by matter sources with polytropic equation of state. We concentrate in the nonperturbative regime where the mass term nonlinearities are important, and present the main features of the solutions.

Figure 1

Values of $a_s$ (lower curves) and of $c_s$ (upper dashed) as function of $b_s$ for the vacuum solution with $ϵ=1/2$, 2, 8. The GR solutions correspond to the lines $a_s=b_s$ and $c_s=1$.

Figure 2

Central density dependence of parameters of the GR solutions: $r_{\mathrm{phys}}$, $r_s$, $r_H=2GM$ and $z_{\mathrm{phys}}=r_{\mathrm{phys}}/r_s$.

Figure 3

Profiles of two solutions with the same exterior behavior corresponding to $ϵ=8$, $b_s=0.870$, $c_s=1.075$: Left: The curves for $\overline{\rho }(0)=0.26$ (red) and $\overline{\rho }(0)=1.41$ (black). Added for comparison is the exterior $a(r)=b(r)$ curve of GR with the same mass and $r_s$ as the second solution: $GM/\ell =2.39$, $r_s/\ell =5.96$. Right: Fit of the exterior solutions (solid lines) by the asymptotic solutions (2.16) (dashed lines). Beyond $r/r_s=2$ the difference between the solutions cannot be resolved in the figure.

Figure 4

Characteristics of the solutions for $ϵ=8$. Left: Mass and radii of the solutions as functions of the density at the origin $\rho (0)$: $M$, $r_s$, $r_V$ and $r_{\mathrm{phys}}$. The dashed curves are the corresponding GR ones. Right: The same parameters plus $\overline{\rho }(0)$ vs $b_s$.

Figure 5

Characteristics of the solutions for $ϵ=2$. Left: Mass and radii of the solutions as functions of the density at the origin $\rho (0)$: $M$, $r_s$, $r_V$ and $r_{\mathrm{phys}}$. The dashed curves are the corresponding GR ones. Right: The same parameters plus $\overline{\rho }(0)$ vs $b_s$.

Figure 6

Characteristics of the solutions for $ϵ=2$, 8 together with the GR curves (dashed). Left: The mass parameters $M_I$ (lower blue lines) and $M$ (upper red lines) as a function of the central density. Right: The radial parameters $r_s$ (lower blue) and $r_V$ (upper red).