Recovering General Relativity from massive gravity

Abstract

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.

Figures (3)

• 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 $m \times R_V = 10^{-2}$. 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 $\nu \sim - 2 \lambda\sim -4/3 \times R_S/R$. At distances larger than $m^{-1}$ the metric functions decay à la Yukawa as appearing more clearly in the insert. 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 Eqs. (\ref{['nuL']}-\ref{['lL']}).
• 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 $m \times R_V = 10^{-3}$. 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 $-\nu \times a^{-4}$ vs. $R/R_V$ (the $a^{-4}$ factor is included for convenience such that, in the decoupling limit (DL), all plotted theories would exactly coincide) for three different values of $a \equiv m \times R_V$ 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 insert shows a zoom on small distances. There, the behaviour 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. Damour:2002gp, one can see that the solution departs from the DL, emphasizing the role of the non-linearities of GR which become important.