Stability and Critical Phenomena of Black Holes and Black Rings

TL;DR

This work analyzes the thermodynamic stability and critical behavior of five-dimensional rotating black holes and black rings using the Poincaré turning-point method and Ruppeiner thermodynamic geometry. It shows that the small black ring is locally unstable, while the large black ring is more stable, with a single stability change occurring at x_min where the BR branches meet. Near extremality (x=1) the system exhibits critical-like scaling with well-defined exponents, whereas the x_min point reflects a stability transition rather than a conventional critical point. Divergences in thermodynamic curvature at x_min and x=1 corroborate the presence of critical-like fluctuations, though the interpretation is nuanced by non-extensivity and potential non-axisymmetric dynamical instabilities.

Abstract

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 Poincare (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 RG theory in statistical mechanics. We find that, at extremality, the behaviour of the system is formally very similar to a second order phase transition.

Figures (7)

• Figure 1: Plot of ${\cal A}/2\pi^2\mu^{3/2}$ against $x=\sqrt{27\pi/32G}J/M^{3/2}$ for the Myers-Perry BH (dashed line) and the two black rings (see Emparan:2001wn). At $x_{\rm min} = \sqrt{27/32}\approx 0.919$ 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 Kaburaki:TSKerrBH.
• 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.