Black holes in $f(R)$ theories

In the context of $f(R)$ theories of gravity, we address the problem of finding static and spherically symmetric black hole solutions. Several aspects of constant curvature solutions with and without electric charge are discussed. We also study the general case (without imposing constant curvature). Following a perturbative approach around the Einstein-Hilbert action, it is found that only solutions of the Schwarzschild-(anti) de Sitter type are present up to second order in perturbations. Explicit expressions for the effective cosmological constant are obtained in terms of the $f(R)$ function. Finally, we have considered the thermodynamics of black holes in anti-de Sitter space-time and found that this kind of solution can only exist provided the theory satisfies $R_0+f(R_0)<0$. Interestingly, this expression is related to the condition which guarantees the positivity of the effective Newton’s constant in this type of theories. In addition, it also ensures that the thermodynamical properties in $f(R)$ gravities are qualitatively similar to those of standard general relativity.

Figure 1

Thermodynamical regions in the $(\alpha ,\beta )$ plane for Model I in $D=4$. Region 1 (left), Region 2 (right).

Figure 2

Thermodynamical regions in the $(\alpha ,\beta )$ plane for Model I in $D=5$. Region 1 (left), Region 2 (right).

Figure 3

Thermodynamical regions in the $(\alpha ,\beta )$ plane for Model I in $D=10$. Region 1 (left), Region 2 (right).

Figure 4

Thermodynamical regions in the $(\alpha ,q)$ plane for Model II in $D=4$. Region 1 (left), Region 2 (right).

Figure 5

Thermodynamical regions in the $(\alpha ,q)$ plane for Model II in $D=5$. Region 1 (left), Region 2 (right).

Figure 6

Thermodynamical regions in the $(\alpha ,q)$ plane for Model II in $D=10$. Region 1 (left), Region 2 (right).

Figure 7

Thermodynamical regions in the $(\alpha ,q)$ plane for Model III in $D=4$ (left) and $D=5$ (right).

Figure 8

Thermodynamical regions in the $(\alpha ,q)$ plane for Model III in $D=10$.

Figure 9

Thermodynamical regions in the $(|\alpha |,|\beta |)$ plane for Model IV in $D=4$ (left) and $D=5$ (right).

Figure 10

Thermodynamical regions in the $(|\alpha |,|\beta |)$ plane for Model IV in $D=10$.