Phase transition, critical behavior, and critical exponents of Myers-Perry black holes

The critical behavior of Myers-Perry black holes with equal angular momenta in even dimensions are studied. We include the corrections beyond the semiclassical approximation on Hawking temperature in the grand canonical ensemble. Having done so, we find that the critical behavior and critical exponents of Myers-Perry black holes correspond to those of a Van der Waals liquid-gas where this analogy holds in any dimension. Also, using Ehrenfest’s equations, we calculate the order of the phase transition in the semiclassical approximation for the canonical ensemble and beyond the semiclassical approximation for the grand canonical ensemble near the critical point. Finally, the Ruppeiner curvature formula is used to investigate the thermodynamic geometry of Myers-Perry black holes.

Figure 1

Specific heat $C_J$ with respect to $S$ for $d=6$ [red (solid) line], $d=8$ [blue (dashed) line], $d=10$ [black (dashed-dotted) line], and $J=0.1$.

Figure 2

Semiclassical Hawking temperature, $T$, with respect to $S$ for $d=6$ [red (solid) line], $d=8$ [blue (dashed) line], $d=10$ [black (dashed-dotted) line] and $J=0.1$.

Figure 3

Semiclassical Hawking temperature, $T$, with respect to $S$ for $d=8$ [blue (dashed) line], $d=10$ [black (dashed-dotted) line], and $\Omega =10$.

Figure 4

Specific heat $C_{\Omega }$ with respect to $S$ for $d=8$ [blue (dashed) line], $d=10$ [black (dashed-dotted) line] and $\Omega =10$.

Figure 5

Corrected Hawking temperature $\tilde{T}$ with respect to $S$ for $\Omega <{\Omega }_c$, ${\beta }_1=0.2$, and $d=8$ [blue (dashed) line], $d=10$ [black (dashed-dotted) line].

Figure 6

Corrected specific heat ${\tilde{C}}_{\Omega }$ with respect to $S$ for $\Omega <{\Omega }_c$, ${\beta }_1=0.2$ and $d=8$ [blue (dashed) line], $d=10$ [black (dashed-dotted) line].

Figure 7

The angular velocity $\Omega$ with respect to the angular momentum $J$ at $\tilde{T}={\tilde{T}}_c$ [blue (dashed) line], $\tilde{T}>{\tilde{T}}_c$ [black (dashed-dotted) line], and $\tilde{T}<{\tilde{T}}_c$ [red (solid) line] for $d=6$ and ${\beta }_1=0.2$.

Figure 8

Corrected specific heat ${\tilde{C}}_{\Omega }$ with respect to $S$ for ${\beta }_1=0.2$, $\Omega ={\Omega }_c$ and $d=6$ [red (solid) line], $d=8$ [blue (dashed) line], and $d=10$ [black (dashed-dotted) line].

Figure 9

The corrected Gibbs free energy with respect to the corrected temperature for ${\beta }_1=0.2$ and $\Omega <{\Omega }_c$ [red (solid) line], $\Omega ={\Omega }_c$ [blue (dashed) line], and $\Omega >{\Omega }_c$ [black (dashed-dotted) line].

Figure 10

The corrected heat capacity with respect to the corrected temperature ${\beta }_1=0.2$ and $\Omega <{\Omega }_c$ [red (solid) line], $\Omega ={\Omega }_c$ [blue (dashed) line], and $\Omega >{\Omega }_c$ [black (dashed-dotted) line].

Figure 11

Ruppeiner curvature scalar ($R$) for $\Omega ={\Omega }_c$, ${\beta }_1=0.2$, and $d=8$.