Higher-dimensional charged $f(R)$ black holes

We construct a new class of higher-dimensional black hole solutions of $f(R)$ theory coupled to a nonlinear Maxwell field. In deriving these solutions the traceless property of the energy-momentum tensor of the matter filed plays a crucial role. In $n$-dimensional spacetime the energy-momentum tensor of conformally invariant Maxwell field is traceless provided we take $n=4p$, where $p$ is the power of conformally invariant Maxwell Lagrangian. These black hole solutions are similar to higher-dimensional Reissner-Nordstrom anti-de Sitter black holes but only exist for dimensions which are multiples of four. We calculate the conserved and thermodynamic quantities of these black holes and check the validity of the first law of black hole thermodynamics by computing a Smarr-type formula for the total mass of the solutions. Finally, we study the local stability of the solutions and find that there is indeed a phase transition for higher-dimensional $f(R)$ black holes with conformally invariant Maxwell source.

Figure 1

The function $N(r)$ versus $r$ for $m=2$, $n=4$, $q=1$, and $R_0=-12$. $f^{\prime }(R_0)=-0.5$ (bold line) and $f^{\prime }(R_0)=-2$ (dashed line).

Figure 2

The function $N(r)$ versus $r$ for $m=2$, $q=1$, $n=4$, and $R_0=-12$. $f^{\prime }(R_0)=-0.2$ (bold line), $f^{\prime }(R_0)=-0.54$ (continuous line), and $f^{\prime }(R_0)=-0.8$ (dashed line).

Figure 3

The function $({\partial }^{2}M/\partial S^2{)}_Q$ versus $q$ for $l=1$, $n=4$, $r_{+}=0.8$. $f^{\prime }(R_0)=-0.5$ (bold line), $f^{\prime }(R_0)=0$ (continuous line), and $f^{\prime }(R_0)=0.5$ (dashed line).

Figure 4

The function $({\partial }^{2}M/\partial S^2{)}_Q$ versus $r_{+}$ for $l=1$, $f^{\prime }(R_0)=1$ and $n=4$. $q=0.4$ (bold line), $q=1$ (continuous line), and $q=2$ (dashed line).

Figure 5

The function $({\partial }^{2}M/\partial S^2{)}_Q$ versus $q$ for $l=1$, $r_{+}=0.8$, and $f^{\prime }(R_0)=1$. $n=4$ (bold line), $n=8$ (continuous line), and $n=12$ (dashed line).

Figure 6

The function $({\partial }^{2}M/\partial S^2{)}_Q$ versus $q$ for $l=1$, $n=8$, and $r_{+}=0.8$. $f^{\prime }(R_0)=-0.5$ (bold line), $f^{\prime }(R_0)=0$ (continuous line), and $f^{\prime }(R_0)=0.5$ (dashed line).

Figure 7

The function $({\partial }^{2}M/\partial S^2{)}_Q$ versus $q$ for $l=1$, $n=12$, and $r_{+}=0.8$. $f^{\prime }(R_0)=-0.5$ (bold line), $f^{\prime }(R_0)=0$ (continuous line), and $f^{\prime }(R_0)=0.5$ (dashed line).