Recovering General Relativity from Massive Gravity

We obtain static, spherically symmetric, and asymptotically flat numerical solutions of massive gravity with a source. Those solutions show, for the first time explicitly, a recovery of the Schwarzschild solution of general relativity via the so-called Vainshtein mechanism.

Figure 1

Plot of the metric functions $-\nu$ and $\lambda$ vs $R/R_V$, in the full nonlinear system and the decoupling limit (DL), with a star of radius $R_{\odot }={10}^{-2}{R}_V$ and $m{R}_V={10}^{-2}$. For $R\ll R_V$, the numerical solution is close to the GR solution (where in particular $\nu \sim -\lambda \sim -R_S/R$ for $R>R_{\odot }$). For $R\gg R_V$, the solution enters a linear regime. Between $R_V$ and $m^{-1}$, where the DL is still a good approximation, one has $\nu \sim -2\lambda \sim -(4/3)(R_S/R)$. At distances larger than $m^{-1}$ the metric functions decay à la Yukawa as appearing more clearly in the inset. The latter shows the same solution but for larger values of $R/R_V$, and in the range of distance plotted there, the numerical solutions are indistinguishable from the analytic solutions of the linearized field equations (7, 8).

Figure 2

Plot of the ratio of $\nu$ to $-\lambda$ vs $R/R_V$, with a star of radius $R_{\odot }/R_V={10}^{-3}$ and $m{R}_V={10}^{-3}$. This shows the transition, at the Vainshtein radius, between a GR regime where $\nu \sim -\lambda$ to a regime where $\nu \sim -2\lambda$. At larger distance, the solution features the expected Yukawa cutoff.

Figure 3

Plot of $-\nu a^{-4}$ vs $R/R_V$ (the $a^{-4}$ factor is included for convenience such that, in the decoupling limit, all plotted theories would exactly coincide) for three different values of $a\equiv m{R}_V$ and source of radius $R_{\odot }={10}^{-3}{R}_V$. The solution with $a=0.005$ corresponds to a spherical source of size and density close to that of the Milky Way and a graviton Compton length of the order of the Hubble radius. In the range of distances plotted, this solution (with $a=0.005$) is well approximated by the DL solution. The inset shows a zoom on small distances. There, the behavior of our solutions agrees with the one of GR, and for $a=0.1$, a value which belongs to a parameter range investigated in Ref. 16, one can see that the solution departs from the DL, emphasizing the role of the nonlinearities of GR which become important.