The recovery of General Relativity in massive gravity via the Vainshtein mechanism

TL;DR

The paper investigates static, spherically symmetric solutions in a nonlinear Pauli-Fierz massive gravity framework with a flat reference metric, focusing on the Vainshtein mechanism’s capacity to recover GR at small scales while permitting modifications at large distances. It develops analytic insight via perturbation theory and the decoupling limit, and complements these with robust numerical solutions of the full nonlinear equations using relaxation and shooting methods. The results demonstrate a non-singular GR-recovering solution with a Vainshtein crossover and Yukawa decay, while showing that Q-scaling in the DL does not extend to regular full solutions; the work also reveals non-perturbative hairs that render infinity behavior non-unique. A weak-field approximation is introduced to capture both the Vainshtein crossover and Yukawa decay, providing a practical bridge between analytic and numerical descriptions and guiding future explorations of stability and dense-object regimes.

Abstract

We study in detail static spherically symmetric solutions of non linear Pauli-Fierz theory. We obtain a numerical solution with a constant density source. This solution shows a recovery of the corresponding solution of General Relativity via the Vainshtein mechanism. This result has already been presented by us in a recent letter, and we give here more detailed information on it as well as on the procedure used to obtain it. We give new analytic insights upon this problem, in particular for what concerns the question of the number of solutions at infinity. We also present a weak field limit which allows to capture all the salient features of the numerical solution, including the Vainshtein crossover and the Yukawa decay.

Figures (9)

• 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{['linsolution']}). This plot was already presented as the Fig.1 of Ref. us.
• Figure 2: Plot of the difference between the numerical solution for $\lambda$, $\lambda_{NUM}$ and the one of GR, $\lambda_{GR}$. This is compared to the computation of the same quantity using the first term in the expansion in $m^2$ proposed by Vainshtein, $|\Delta \lambda_{V}|=\frac{\sqrt{2}}{3}\left(m R \right)^{2} \sqrt{\frac{R_{S}}{R}}.$ Both functions are plotted vs.$R/R_V$, for a source of radius $R_{\odot} =10^{-3} R_V$ and a choice of parameters such that $a\equiv m \times R_V = 10^{-2}$. It can be seen that the numerical solution agrees very well with the Vainshtein's expression given by Eq. (\ref{['Chapitre SSS lSR']}) for $R \ll R_V$. For $R \sim R_V$, the expansion in $m^2$ is, as expected, no longer a good approximation of the solution.
• 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) inside the source of radius $R_{\odot} = 10^{-3} R_V$, for three different values of $a \equiv m \times R_V$. Along with the numerical solution, the GR analytical solution for $\nu$ is plotted for each value of $a$. It can be seen that the numerical solutions perfectly agree with the GR behaviour, while slightly differing form the DL expression for large $a$. This illustrate the fact that inside the source, the nonlinearities coming from the GR terms of Eq. (\ref{['EOMFULL']}), and which are not taken into account in the DL, are important. For small $a$, both the GR and the numerical solutions agree with the DL solution, which encodes for most of the physics, in agreement with the fact that the DL corresponds to the limit $a\to 0$.
• Figure 4: Plot of the functions $P/\rho$ and $P_{GR}/\rho$vs.$R/R_V$, for a source of radius $R_{\odot} =10^{-2} R_V$, of pressure $P$, and of constant density $\rho$, and for a parameters choice such that $a\equiv m \times R_V = 10^{-2}$. The function $P/\rho$ if found numerically, while $P_{GR}/\rho$ is the pressure computed in GR and is given by the analytical expression (\ref{['pression gr']}). The two curves are indistinguishable from each other, illustrating the fact that the Vainshtein's conjecture is valid inside the star.
• Figure 5: Plot of $a^{-2}\mu$ vs. $R/R_V$ (the $a^{-2}$ factor is included for convenience such that, in the decoupling limit (DL), all plotted theories would exactly coincide) for a source of radius $R_{\odot} = 10^{-3} R_V$ for three different values of $a \equiv m \times R_V= 0.005,0.05,0.1$ as well as in the DL case (which corresponds to $a\rightarrow 0$). The three DL regimes of Eq. (\ref{['table behavior vainshtein solution']}) are clearly distinguishable: for $R<R_{\odot}$, $\mu\sim \text{const}$; for $R_{\odot}<R\ll R_{V}$, $\mu\sim\sqrt{(8R_{V})/(9R)}$; for $R_{V}\ll R\ll m^{-1}$, $\mu\sim2R_{V}^{3}/(3R^{3})$. For $R\gg m^{-1}$, we observe the Yukawa decay of the linear solution (\ref{['linsolution']}). At the resolution of the picture, the thee numerical solutions are hard to distinguish for distances $R$ smaller than $m^{-1}$. The next figure shows a zoom of this figure at small distances (namely inside the source) where the differences between the three solutions are easily seen