Born-Infeld black holes in (A)dS spaces

We study some exact solutions in a $D(\ge 4)$-dimensional Einstein-Born-Infeld theory with a cosmological constant. These solutions are asymptotically de Sitter or anti-de Sitter, depending on the sign of the cosmological constant. Black hole horizon and cosmological horizon in these spacetimes can be a positive, zero or negative constant curvature hypersurface. We discuss the thermodynamics associated with black hole horizon and cosmological horizon. In particular we find that for the Born-Infeld black holes with Ricci flat or hyperbolic horizon in AdS space, they are always thermodynamically stable, and that for the case with a positive constant curvature, there is a critical value for the Born-Infeld parameter, above which the black hole is also always thermodynamically stable, and below which a unstable black hole phase appears. In addition, we show that although the Born-Infeld electrodynamics is nonlinear, both black hole horizon entropy and cosmological horizon entropy can be expressed in terms of the Cardy-Verlinde formula. We also find a factorized solution in the Einstein-Born-Infeld theory, which is a direct product of two constant curvature spaces: one is a two-dimensional de Sitter or anti-de Sitter space, the other is a ($D-2$)-dimensional positive, zero or negative constant curvature space.

Figure 1

The inverse Hawking temperature of a five-dimensional Born-Infeld black hole in AdS space for a fixed charged $q/l^2=0.5$ in the case of $k=0$.

Figure 2

The inverse Hawking temperature of a five-dimensional Born-Infeld black hole in AdS space for a fixed charged $q/l^2=0.5$ in the case of $k=-1$.

Figure 3

The inverse Hawking temperature of a five-dimensional Born-Infeld black hole in AdS space for a fixed charged $q/l^2=0.5$ in the case of $k=1$.

Figure 4

The inverse Hawking temperature of a five-dimensional Born-Infeld black hole in AdS space for a fixed charge $q/l^2=0.5$ for the case of $k=1$. Three curves from right to left correspond to the case $\beta l=0.40$, $0.22$ and $0.15$.