Thermodynamic instability of rotating black holes

We show that the quasi-Euclidean sections of various rotating black holes in different dimensions possess at least one nonconformal negative mode when thermodynamic instabilities are expected. The boundary conditions of the fixed induced metric correspond to the partition function of the grand-canonical ensemble. Indeed, in the asymptotically flat cases, we find that a negative mode persists even if the specific heat at constant angular momenta is positive, since the stability in this ensemble also requires the positivity of the isothermal moment of inertia. We focus, in particular, on Kerr black holes, on Myers-Perry black holes in five and six dimensions, and on the Emparan-Reall black ring solution. We go on further to consider the richer case of the asymptotically AdS Kerr black hole in four dimensions, where thermodynamic stability is expected for a large enough cosmological constant. The results are consistent with previous findings in the nonrotation limit and support the use of quasi-Euclidean instantons to construct gravitational partition functions.

Figure 1

For the Kerr instanton, $\mathcal{I}$ is negative, decreasing monotonically away from $a=0$ and evaluating to $-12$ $r_0^{-2}$ at extremality $|a|=r_0/2$.

Figure 2

For the five-dimensional Myers-Perry instanton, $\mathcal{I}$ is always negative. The dashed lines represent contours of constant $\mathcal{I}{r}_0^2$.

Figure 3

$\mathcal{I}$ is negative across the entire range.

Figure 4

For the six-dimensional Myers-Perry instanton, $\mathcal{I}$ is always negative. The dashed lines represent contours of constant $\mathcal{I}{r}_0^2$.

Figure 5

Phase diagram of the Kerr-AdS black hole.

Figure 6

Graphic representation of $C_J$ (solid line with axis on the left) and $ϵ$ (dashed line with axis on the right).

Figure 7

The darker region indicates the region in the moduli space for which $C_J$ is negative, and the lighter region indicates where $ϵ$ is not positive definite.

Figure 8

Graphic representation of $C_J$ (solid line with axis on the left) and $ϵ$ (dashed line with axis on the right).

Figure 9

The darker region indicates the region in the moduli space for which $C_J$ is negative. The region in which $ϵ$ is not positive definite includes the lighter region.

Figure 10

Phase diagram of the Kerr-AdS black hole. In region III, the linear response functions $C_J$ and $ϵ$ are positive and we find $\mathcal{I}>0$. In regions II and V, $ϵ<0$ and $C_J>0$, while, in regions I and IV, $C_J<0$ and $ϵ>0$. We find a negative mode, i.e. $\mathcal{I}<0$, in regions I, II and VI. The thin region VI, amplified in the top-right corner, represents a puzzling feature since we find a negative mode when thermodynamic stability was expected.

Figure 11

In the Kerr-AdS case, up to three black holes can exist with a given temperature $T$, for a certain range of a fixed angular momentum $J$. In this example, $J=1$ and $\ell =10$. In fact, a fourth solution exists with small values of the energy $E$, but is beyond the limit $|a|=\ell$.