Stability and critical phenomena of black holes and black rings

We revisit the general topic of thermodynamical stability and critical phenomena in black hole physics, analyzing in detail the phase diagram of the five dimensional rotating black hole and the black rings discovered by Emparan and Reall. First we address the issue of microcanonical stability of these spacetimes and its relation to thermodynamics by using the so-called Poincaré (or “turning point”) method, which we review in detail. We are able to prove that one of the black ring branches is always locally unstable, showing that there is a change of stability at the point where the two black ring branches meet. Next we study divergence of fluctuations, the geometry of the thermodynamic state space (Ruppeiner geometry) and compute the appropriate critical exponents and verify the scaling laws familiar from renormalization group theory in statistical mechanics. We find that, at extremality, the behavior of the system is formally very similar to a second order phase transition.

Figure 1

Plot of $\mathcal{A}/2{\pi }^2{\mu }^{3/2}$ against $x=\sqrt{27\pi /32G}J/M^{3/2}$ for the Myers-Perry BH (dashed line) and the two black rings (see 21). At $x_{\min }=\sqrt{27/32}\approx 0.919$ black ring formation becomes possible. At $x=2\sqrt{2}/3\approx 0.943$ the entropy of the large BR exceeds that of the black hole.

Figure 2

Generic plot of a conjugate variable ${\beta }_a$ against ${\mu }^a$ along an equilibrium sequence. The point $P$ is a turning point, where a change of stability can take place. No changes of stability occur at $Q$, even if the slope changes sign there.

Figure 3

Conjugacy diagrams for the four dimensional Kerr black hole [15]. (a)  $\beta (M)$ at fixed $J$. $M_{\mathrm{ext}}$ stands for the minimal mass at fixed $J$ (the Kerr bound). $M\to \infty$ is the Schwarzschild limit. $M_D$ is the “Davies’ point” $J_D^2=(2\sqrt{3}-3)M_D^4$ [10]. (b) $\omega (J)$ at fixed mass. $J_{\mathrm{ext}}$ stands for the Kerr bound. As in the plot of $\beta (M)$, one finds a vertical asymptote there.

Figure 4

$H_{MM}(x)$, $H_{JJ}(x)$ and $\det H(x)$ for the rotating BH. The curve changing sign at $x=1/2$ is that of the element $H_{MM}$. $H_{JJ}$ and $\det H$ are always negative.

Figure 5

Plots of $H_{MM}$, $H_{JJ}$ and $\det H$ as a function of $x$ for both black rings. The horizontal axis corresponds to $H_{MM}=H_{JJ}=\det H=0$.(a) $H_{MM}(x)$, $H_{JJ}(x)$ and $\det H(x)$ for the small black ring. All quantities diverge both at the lower bound $x_{\min }$ and at the upper one $x=1$. (b) $H_{MM}(x)$, $H_{JJ}(x)$ and $\det H(x)$ for the large black ring. All quantities diverge at the lower bound $x_{\min }$ and tend asymptotically to zero at $x\to \infty$.

Figure 6

The conjugacy diagrams $\beta (M)$ and $\omega (J)$ for all phases of the BH/BR system along their corresponding equilibrium series. The dashed lines represent the BH-curves. We see the existence of a turning point in both diagrams at the point where both BR branches meet. The SBR and BH curves have a vertical asymptote at extremality. ${\beta }_{\mathrm{BH}}(M)$ has a minimum (not shown in the plot) at the value of $M$ that corresponds to $x=1/2$. (a) $\beta (M)$ at fixed J. $M_{\mathrm{ext}}$ stands for the minimal allowed mass at fixed angular momentum in the BH and SBR cases (the “Kerr” or extremal bound $x=1$). $M_{\min }$ is the mass at $x_{\min }$.(b) $\omega (J)$ at fixed mass.

Figure 7

Thermodynamic curvature scalar for the BH (dashed line), SBR and LBR phases. The physically relevant regions are those near $x_{\min }$ and $x=1$. The curvature is divergent at those points.