Vacuum energy and the latent heat of AdS-Kerr black holes

Brian P. Dolan

Phys. Rev. D 90, 084002 – Published 2 October 2014

Abstract

Phase transitions for rotating asymptotically anti–de Sitter black holes in four dimensions are described in the $P - T$ plane, in terms of the Hawking temperature and the pressure provided by the cosmological constant. The difference between constant angular momentum and constant angular velocity is highlighted; the former has a second order phase transition while the latter does not. If the angular momentum is fixed there a line of first order phase transitions terminating at a critical point with a second order phase transition and vanishing latent heat, while if the angular velocity is fixed there is a line of first order phase transitions terminating at a critical point with infinite latent heat. For constant angular velocity the analytic form of the phase boundary is determined, latent heats derived and the Clapeyron equation verified.

Received 18 August 2014

DOI:https://doi.org/10.1103/PhysRevD.90.084002

Authors & Affiliations

Brian P. Dolan^*

Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh, EH14 4AS, United Kingdom and Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom

^*B.P.Dolan@hw.ac.uk

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Issue

Vol. 90, Iss. 8 — 15 October 2014

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Images

Figure 1
Left: The black hole free energy, for static black holes, at fixed $P = 1$ . The heat capacity is negative on the upper branch, positive on the lower branch and diverges at the cusp. The Hawking-Page temperature is where $G = 0$ . Right: The coexistence curve of the Hawking-Page phase transition is the lower (red) line; the heat capacity diverges on the upper (green) line. The lower branch of the free energy tends to minus infinity on the $P = 0$ axis.
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Figure 2
Left: The dimensionless free energy for a rotating black hole as a function of the dimensionless variable $t$ . The upper branch represents small black holes (negative heat capacity), the lower branch large black holes (positive heat capacity). Only negative $Ξ$ black holes are stable against Hawking-Page decay. The vertical axis is the dimensionless combination $Ω Ξ$ and the figure is drawn for $p = 4$ (other values of $p > 3$ move the position of the cusp but the shape of the figure is the same). Right: Phase diagram in the $p - t$ plane. The constraint that the Einstein universe at infinity is not rotating faster than the speed of light imposes the condition $p \geq 3$ . In the region below this line $Ξ_{BH}$ is single valued and positive. In the white region (upper left labeled “AdS phase”) there are no black hole solutions (a black hole in this region would have negative entropy). The right-hand boundary of the white region is the locus of points on which the heat capacity diverges (black line), to the right of this there is a wedge shaped region (dark gray, green) in which $Ξ_{BH}$ has two branches and is positive on both branches. AdS is still the preferred phase in this region as $0 = Ξ_{AdH} < Ξ_{BH}$ . In the gray region, labeled “Black hole phase,” one branch of $Ξ_{BH}$ becomes negative and black holes are more stable than AdS. The Hawking-Page phase transition occurs on the upper-left boundary of the region marked “Black hole phase,” where the lower branch of $Ξ_{BH}$ equals zero.
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Figure 3
Left: Gibbs free energy, $G / \sqrt{J}$ , as a function of $\tilde{t}$ for a black hole at constant $J$ with $\tilde{p} = < {\tilde{p}}_{c}$ . Right: Coexistence curve at fixed $J$ , in the $\tilde{p} - \tilde{t}$ plane. The curve ends at the critical point ${\tilde{t}}_{c} = 0.0417$ , ${\tilde{p}}_{c} = 0.144$ .
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Figure 4
Curves of constant $\tilde{p}$ in the $\tilde{s} - \tilde{t}$ plane. The lowest curve (green) is $\tilde{p} = 0$ . The the peaked curve (red) whose maximum is tangent to the third isobaric curve is the spinodal curve. The temperature of the phase transition is calculated using the Maxwell equal area rule.
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