Spherically symmetric spacetimes in $f(R)$ gravity theories

We study both analytically and numerically the gravitational fields of stars in $f(R)$ gravity theories. We derive the generalized Tolman-Oppenheimer-Volkov equations for these theories and show that in metric $f(R)$ models the parametrized post-Newtonian parameter ${\gamma }_{\mathrm{PPN}}=1/2$ is a robust outcome for a large class of boundary conditions set at the center of the star. This result is also unchanged by the introduction of dark matter in the solar system. We find also a class of solutions with ${\gamma }_{\mathrm{PPN}}\approx 1$ in the metric $f(R)=R-{\mu }^4/R$ model, but these solutions turn out to be unstable and decay in time. On the other hand, the Palatini version of the theory is found to satisfy the solar system constraints. We also consider compact stars in the Palatini formalism, and show that these models are not inconsistent with polytropic equations of state. Finally, we comment on the equivalence between $f(R)$ gravity and scalar-tensor theories and show that many interesting Palatini $f(R)$ gravity models cannot be understood as a limiting case of a Jordan-Brans-Dicke theory with $\omega \to -3/2$.

Figure 1

Shown is the density profile of the star (solid line) and the appropriate pressure needed to support this configuration (dashed line) (in units $r_{\odot }^{-4}$). The inset figure shows the equation of state $p=p(\rho )$ corresponding to this solution.

Figure 2

Shown are the functions $A$ (red, lower curves) and $B$ (blue, upper curves) for the metric $f(R)=R-{\mu }^4/R$ model (solid lines) and for GR (dotted lines). Also shown is the function $d-d_{\mathrm{vac}}$ (dashed green line), where $d_{\mathrm{vac}}$ is the asymptotic value of $d$ in vacuum.

Figure 3

Shown is ${\gamma }_{\mathrm{PPN}}$ for a metric $f(R)$ gravity (solid line) and the corresponding solution in GR (dashed line).

Figure 4

Shown are the functions $d$ (solid line) and $A-A_0$ (dashed line), where $A_0$ is chosen such that for GR $A\sim 1/r$ at $r\to \infty$. The dotted line shows the Palatini value of $d$, which corresponds to $R\approx 8\pi G\rho$.

Figure 5

Shown are the metric functions $A$ (red, lower line) and $B$ (blue, upper line) for the Palatini $f(R)=R-{\mu }^4/R$ model, which completely overlap with the GR solution.

Figure 6

Shown is the logarithm of the ratio $K_{\mathrm{sing}}/K_{\mathrm{GR}}$ for a polytropic star with $\Gamma =5/3$ in a Palatini $f(R)=R-{\mu }^4/R$ model, as a function of $\Delta r\equiv r_{\mathrm{out}}-r$, where $K_{\mathrm{GR}}=-2GM/r^3$ is the coefficient of radial tidal acceleration in the Schwarzschild solution. The dashed curve shows the corresponding density in units ${10}^{14}{\rho }_{\mathrm{crit}}$, which goes to zero at the sharp edge of the star $\Delta r=0$. We have chosen $M=2M_{\odot }$ and $r_{\mathrm{out}}=10\mathrm{km}$.

Figure 7

Shown is the effective potential $V_{\mathrm{eff}}(\phi )$ (in units ${\mu }^2{M}_{\mathrm{Pl}}^2$) as a function of $\phi /M_{\mathrm{Pl}}\equiv \sqrt{3/2}\log (F/F_{\min })$ for the model $f(R)=R-{\mu }^4/R+(\alpha /2{\mu }^2)R^2$. The solid line corresponds to $\alpha =0.8$ with $\rho =5{\rho }_{\mathrm{crit}}$ and the dashed line to $\alpha =0.01$ with $\rho =0.1{\rho }_{\mathrm{crit}}$.