Thermodynamics and stability of higher dimensional rotating (Kerr-)AdS black holes

We study the thermodynamic and gravitational stability of Kerr anti-de Sitter black holes in five and higher dimensions. We show, in the case of equal rotation parameters, $a_i=a$, that the Kerr-AdS background metrics become stable, both thermodynamically and gravitationally, when the rotation parameters $a_i$ take values comparable to the AdS curvature radius. In turn, a Kerr-AdS black hole can be in thermal equilibrium with the thermal radiation around it only when the rotation parameters become not significantly smaller than the AdS curvature radius. We also find with equal rotation parameters that a Kerr-AdS black hole is thermodynamically favored against the existence of a thermal AdS space, while the opposite behavior is observed in the case of a single nonzero rotation parameter. The five-dimensional case is however different and also special in that there is no high temperature thermal AdS phase regardless of the choice of rotation parameters. We also verify that at fixed entropy, the temperature of a rotating black hole is always bounded above by that of a nonrotating black hole, in four and five dimensions, but not in six and more dimensions (especially, when the entropy approaches zero or the minimum of entropy does not correspond to the minimum of temperature). In this last context, the six-dimensional case is marginal.

Figure 1

($d=5$) The free energy (solid lines) and temperature (dashed lines) as a function of horizon position, with ${\alpha }_1={\alpha }_2\equiv \alpha$. From top to bottom (free energy) or bottom to top (temperature): $\alpha =1/3,0.25,1/8,0.05$. In all plots we have set $4G=1$.

Figure 2

($d=5$) The free energy (solid lines) and temperature (dashed lines) vs horizon position, with ${\alpha }_1\equiv \alpha$, ${\alpha }_2=0$. From top to bottom (free energy) or bottom to top (temperature): $\alpha =0.4,0.25,1/6,0.1$.

Figure 3

($d=5$) The free energy vs temperature with ${\alpha }_1={\alpha }_2\equiv \alpha$. From top to bottom: $\alpha =1/3,0.2,0.1,0.05$.

Figure 4

($d=5$) The free energy vs temperature with ${\alpha }_1\equiv \alpha$, ${\alpha }_2=0$. From top to bottom: $\alpha =0.4,0.25,1/6,0.1$.

Figure 5

($d=6$) The free energy and temperature vs horizon position, with ${\alpha }_1={\alpha }_2=\alpha$. From top to bottom: $\alpha =0.4,1/3,0.25,0.1$.

Figure 6

($d=6$) The free energy and temperature vs horizon position, with ${\alpha }_1\equiv \alpha$, ${\alpha }_2=0$. From top to bottom: $\alpha =0.5,0.4,0.25,0.04$.

Figure 7

($d=6$) The free energy vs temperature with ${\alpha }_1={\alpha }_2\equiv \alpha$. From top to bottom: $\alpha =1/3,0.25,0.2,0.12$.

Figure 8

($d=6$) The free energy vs temperature with ${\alpha }_1=\alpha$, ${\alpha }_2=0$. From left to right: $\alpha =0.5,0.35,0.2,0.05$.

Figure 9

The free energy vs temperature with a single nonvanishing rotation parameter $\alpha$. From left to right $d=4$ (with $\alpha =0.3$) and $d=6,8,10$ (with $\alpha =0.5$).

Figure 10

The free energy vs temperature with equal rotation parameters, ${\alpha }_i=\alpha =0.4$. From left to right $d=4,6,8,10$.

Figure 11

The free energy vs temperature with a single nonvanishing rotation parameter $\alpha$. From left to right: $d=5,7,9,11$ (with $\alpha =0.6$).

Figure 12

The free energy vs temperature with equal rotation parameters, ${\alpha }_i=\alpha =0.5$. From left to right $d=5,7,9,11$.

Figure 13

($d=5$) The energy and temperature differentials vs horizon position, with ${\alpha }_1={\alpha }_2\equiv \alpha$. As $R\to 0$, $dE\to +\infty$ and $dT\to -\infty$. From left to right: $\alpha =0.1,1/6,0.25,1/3$.

Figure 14

($d=5$) The energy and temperature differentials vs horizon position, with ${\alpha }_1=\alpha$, ${\alpha }_2=0$. As $R\to 0$, $dE\to 0$ and $dT>0$. From top to bottom (free energy) or bottom to top (temperature) $\alpha =0.5,1/3,0.25,1/6$.

Figure 15

($d=5$) The temperature differential vs temperature with ${\alpha }_1={\alpha }_2=\alpha$; from top to bottom $\alpha =0.08,0.12,0.17,0.25$.

Figure 16

($d=5$) The specific heat vs temperature with ${\alpha }_1={\alpha }_2=\alpha$; from top to bottom $\alpha =0.17,1/3,0.5,0.6$. When $\alpha$ is $\lesssim 1/6$, then there would appear a new branch with almost constant specific heat at low temperature.

Figure 17

($d=5$) The specific heat vs horizon position with ${\alpha }_1=\alpha$, ${\alpha }_2=0$. From top to bottom: $\alpha =0.7,0.5$ (solid lines); and bottom to top: 0.3, 0.26 (dashed lines). With $\alpha \gtrsim 0.245$, the specific heat curve has a single branch and it is a monotonically increasing function of Hawking temperature.

Figure 18

($d=6$) The energy differential (solid lines) and temperature differential (dashed lines) vs horizon position, with ${\alpha }_1={\alpha }_2\equiv \alpha$. From left to right ($dE$ in the $dE>0$ region) or top to bottom ($dT$): $\alpha =0.5,0.2,0.1,1/15$.

Figure 19

($d=6$) The temperature differential vs temperature with ${\alpha }_1={\alpha }_2\equiv \alpha$. From top to bottom $\alpha =0.12,0.16,0.21,0.25$.

Figure 20

($d=6$) The specific heat vs temperature with ${\alpha }_1={\alpha }_2=\alpha$. From top to bottom $\alpha =0.25,0.5,0.6,0.7$.

Figure 21

The temperature vs entropy, in various dimensions with equal rotation parameters, ${\alpha }_i\equiv \alpha$. From top to bottom: $d=7,6,5,4$ with all ${\alpha }_i=0$ (dashed lines); from left to right: $d=6,7,4,5$ (solid lines) each with $\alpha =0.4$.

Figure 22

The temperature vs entropy with a single rotation parameter $\alpha$. From top to bottom: $d=7,6,5,4$ each with $\alpha =0.4$ (solid lines), and ${\alpha }_i=0$ (dashed lines).

Figure 23

The temperature vs entropy, in the small entropy region. From top to bottom (in the region $S>0.1$): $d=10,9,8,7$ with ${\alpha }_1=\cdots {\alpha }_{N-1}=0.01$, ${\alpha }_N=0.9$ (solid lines, Kerr-AdS) and ${\alpha }_i=0$ (dashed lines, Schwarzschild-AdS).