On explicit thermodynamic functions and extremal limits of Myers–Perry black holes

Abstract

We study thermodynamic geometries of Myers–Perry (MP) black holes with arbitrary number of angular momenta. This geometric method allows us to visualize thermodynamic state spaces of the MP black holes as wedges embedded in a Minkowski-like parameter space. The opening angles of these wedges are uniquely determined by the number of spacetime dimensions d, and the number of angular momenta associated with the MP black holes, n. The geometric structure captures extremal limits of the MP black holes, and hence serves as a method for identifying the black hole’s extremal limit. We propose that classification of the MP black hole solutions should based on these uncovered structures. In order for the ultraspinning regime to exist, at least one of the angular momenta has to be set to zero. Finally, we conjecture that the membrane phase of ultraspinning MP black holes is reached at the minimum temperature in the case where 2n<d−3 based on the thermodynamic curvature obtained.

Appendices

Appendix A: The MP black hole solution

The MP solutions have to be treated separately depending on whether the number of dimensions is odd or even. The black holes have (d−1)/2 angular momenta if d is odd and (d−2)/2 if d is even. The multiple-spin Kerr black hole’s metric in Boyer–Lindquist coordinates for odd d is given by

$$\begin{aligned} ds^2 & = -d\bar{t}^2 + \sum_{i=1}^N \bigl(r^2 + a^2_i\bigr) \bigl(d \mu^2_i + \mu^2_i\, d\bar{ \phi}^2_i\bigr) \\ &\quad {} +\frac{m r^2}{\varPi F}\Biggl(d\bar{t} + \sum _{i=1}^N a_i \mu^2_i\, d\bar{\phi}_i\Biggr)^2 + \frac{\varPi F}{\varPi- m r^2}\,dr^2, \end{aligned}$$

(A.1)

with the constraint

$$\begin{aligned} & F = 1 - \sum_{i=1}^N \frac{a^2_i \mu^2_i}{r^2 + a^2_i}, \end{aligned}$$

(A.2)

$$\begin{aligned} & \sum_{i=1}^N \mu^2_i = 1, \end{aligned}$$

(A.3)

where m is mass parameter, μ _i are directional cosines, and a _i are parameters. The function Π is defined as follows:

$$ \varPi= \prod_{i=1}^{(d-1)/2} \bigl(r^2 + a^2_i\bigr). $$

(A.4)

The metric is slightly modified for even d. The event horizons in the Boyer–Lindquist coordinates occur where g ^rr=1/g _rr vanishes. They are the largest roots of

$$\begin{aligned} & \varPi- m r = 0 \quad \text{even }d \end{aligned}$$

(A.5)

$$\begin{aligned} & \varPi- m r^2 = 0 \quad \text{odd }d. \end{aligned}$$

(A.6)

The areas of the event horizon are given by

$$\begin{aligned} A & = \frac{\varOmega_{(d-2)}}{r_+} \prod_i \bigl(r^2_+ + a^2_i\bigr) \quad \text{odd }d, \end{aligned}$$

(A.7)

$$\begin{aligned} A& = \varOmega_{(d-2)} \prod_i \bigl(r^2_+ + a^2_i\bigr) \quad \text{even }d. \end{aligned}$$

(A.8)

In d=5 there can be only two angular momenta associated with the Kerr black hole, thus the area of the event horizon reads

$$ A = \frac{2\pi^2}{r_+} \bigl(r^2_+ + a_1^2 \bigr) \bigl(r^2_+ + a^2_2\bigr). $$

(A.9)

The Bekenstein–Hawking entropy is given by $S= \frac {k_{B} A}{4G}$, and we can choose $k_{B} = \frac{1}{\pi}$ and $G = \frac {\varOmega_{(d-2)}}{4\pi}$ so that the Bekenstein–Hawking entropy for the MP black holes is simplified as

$$\begin{aligned} & S = \frac{1}{r_+} \prod_i \bigl(r^2_+ + a^2_i\bigr) \quad \text{odd }d, \end{aligned}$$

(A.10)

$$\begin{aligned} & S = \prod_i \bigl(r^2_+ + a^2_i\bigr) \quad \text{even }d. \end{aligned}$$

(A.11)

Appendix B: Extremal limits in multi-coordinates with all nonzero spin J equal

If 2n≥d−3 there will be an extremal limit at

$$ \frac{S}{J} \bigg|_{\mathrm{extr}} = 2 \sqrt{\frac{2n-d+3}{d-3}} . $$

(B.1)

This limit exists for 2n>d−3, which can be expressed in (M,S) coordinates as

$$ \frac{S^{d-3}}{M^{d-2}} \bigg|_{\mathrm{extr}} = \frac{2^{2d-n-4} (2n-d+3)^n}{ (d-2)^{d-2}n^n} $$

(B.2)

or in (M,J) coordinates

$$ \frac{J^{d-3}}{M^{d-2}} \bigg|_{\mathrm{extr}} = \frac{2^{d-n-1} (2n-d+3)^{\frac {2n-d+3}{2}}(d-3)^{\frac{d-3}{2}}}{(d-2)^{d-2}n^n} . $$

(B.3)

For the case 2n=d−3 case (requiring odd d) we have S _ext =0 and

$$ \frac{J^{d-3}}{M^{d-2}} \bigg|_{\mathrm{extr}} = \frac{2^{d-1}}{(d-2)^{d-2}} . $$

(B.4)

For even d with all $n = \frac{d}{2} -1$ spins we can also express

$$ \frac{S}{ J} \bigg|_{\mathrm{extr}} = \frac{2}{\sqrt{d-3}}, $$

(B.5)

and

$$ \frac{J^{d-3}}{M^{d-2}} \bigg|_{\mathrm{extr}} = \frac{2^{d-1} (d-3)^{\frac {d-3}{2}}}{(d-2)^{\frac{3(d-2)}{2}}} . $$

(B.6)

For odd d with all $n = \frac{d-1}{2}$ spins we have

$$ \frac{S}{ J} \bigg|_{\mathrm{extr}} = \frac{2\sqrt{2}}{\sqrt{d-3}}, $$

(B.7)

and

$$ \frac{J^{d-3}}{M^{d-2}} \bigg|_{\mathrm{extr}} = \frac{2^d (d-3)^{\frac{d-3}{2}}}{ (d-2)^{d-2} (d-1)^{\frac{d-1}{2}}}. $$

(B.8)

The Schwarzschild limit is when J=0 and this sets a physical bound to be

$$ \frac{S^{d-3}}{M^{d-2}} \leq \frac{4^{d-2}}{(d-2)^{d-2}}. $$

(B.9)

In Table 3 we present extremal limits in various coordinates.

Table 3 Extremal limits of MP black holes in various coordinates for all nonzero equal spins

Appendix C

We extend our discussion on the opening angles of the Weinhold metrics here with two subcases:

The subcase 2n=d−2

This applies to even dimensions, 4 with 1 spin, 6 with 2 spins etc. Here the transformation from u to σ is

$$ \sigma= \frac{1}{2} \biggl( \text{arcsinh}\frac{2 u}{\sqrt{d-3}} +\sqrt{d-4} \arctan\frac{2\sqrt{d-4} u}{\sqrt{d-3+4u^2}} \biggr). $$

(C.1)

Here we have $u_{\mathrm{extr}} = \frac{\sqrt{d-3}}{2}$ which gives

$$ \sigma_{\mathrm{extr}} = \frac{1}{2} \biggl( \text{arcsinh}1 +\sqrt {d-4} \arctan \frac{\sqrt{d-4}}{\sqrt{2}} \biggr). $$

(C.2)

The subcase 2n=d−1

This applies to dimension 5 with 2 spin, dimension 7 with 3 spins etc. Here the transformation is

$$\begin{aligned} \sigma& = \frac{\sqrt{d-1}}{\sqrt{2}\sqrt{d-2}} \biggl( \text {arcsinh}\frac{2\sqrt{2} u}{\sqrt{d-3}} \\ &\quad {} + \frac{\sqrt{d-5}}{\sqrt{2}} \arctan\frac{2\sqrt{d-5} u}{\sqrt {d-3+8u^2}} \biggr). \end{aligned}$$

(C.3)

In this case $u_{\mathrm{extr}} = \frac{\sqrt{d-3}}{2\sqrt{2}}$ so we obtain

$$ \sigma_{\mathrm{extr}} = \frac{\sqrt{d-1}}{\sqrt{2}\sqrt{d-2}} \biggl( \text{arcsinh}1 + \frac{\sqrt{d-5}}{\sqrt{2}} \arctan\frac{\sqrt{d-5}}{2} \biggr). $$

(C.4)