Stability of spherically symmetric solutions in modified theories of gravity

In recent years, a number of alternative theories of gravity have been proposed as possible resolutions of certain cosmological problems or as toy models for possible but heretofore unobserved effects. However, the implications of such theories for the stability of structures such as stars have not been fully investigated. We use our “generalized variational principle,” described in a previous work [M. D. Seifert and R. M. Wald, Phys. Rev. D 75, 084029 (2007)], to analyze the stability of static spherically symmetric solutions to spherically symmetric perturbations in three such alternative theories: Carroll et al.’s $f(R)$ gravity, Jacobson and Mattingly’s “Einstein-æther theory,” and Bekenstein’s TeVeS theory. We find that in the presence of matter, $f(R)$ gravity is highly unstable; that the stability conditions for spherically symmetric curved vacuum Einstein-æther backgrounds are the same as those for linearized stability about flat spacetime, with one exceptional case; and that the “kinetic terms” of vacuum TeVeS theory are indefinite in a curved background, leading to an instability.

Figure 1

Representative plots of $Z(r)$ versus $r$ for the solutions described in 25. The solid and dashed lines correspond to $c_{123}=\pm 0.1$, respectively; in both cases, $c_2=0$ and $c_{14}=-0.1$. Note that the choice of parameters leading to the dashed plot would lead to instability in the asymptotic region.

Figure 2

Plot of $(\frac{2}{c_{14}}+1)Z_0(r)$, relevant for Einstein-æther theory with $c_{123}=0$. In this plot, we have chosen $c_{14}=-0.1$ and $c_2=0.1$.

Figure 3

Plot of $H(r)$ versus $r$ for TeVeS theory. In this plot, $k$ and $K$ are both chosen to be ${10}^{-2}$, and $m_s=z_g=1$.