Cosmological Pair Production of Charged and Rotating Black Holes

TL;DR

This work extends cosmological black hole pair production to charged and rotating holes by using Kerr-Newmann-deSitter end-states derived from a generalized C-metric, and by constructing complex instantons that interpolate to the Lorentzian solutions. The authors employ a Brown-York quasilocal action within a canonical partition function to compute creation rates, finding that the rates are suppressed relative to de Sitter space and that the entropy of rotating black hole spacetimes equals the log of their microstate count, proportional to the sum of horizon areas. A key methodological advance is the inclusion of angular-momentum boundary terms and the acceptance of complex instantons to ensure consistent matching with Lorentzian spacetimes. The results generalize the non-rotating case, reinforce the interpretation of black hole entropy as counting microstates, and highlight subtle distinctions between thermal equilibrium and full thermodynamic equilibrium in quantum cosmological tunneling processes.

Abstract

We investigate the general process of black hole pair creation in a cosmological background, considering the creation of charged and rotating black holes. We motivate the use of Kerr-Newmann-deSitter solutions to investigate this process, showing how they arise from more general C-metric type solutions that describe a pair of general black holes accelerating away from each other in a cosmological background. All possible KNdS-type spacetimes are classified and we examine whether they may be considered to be in full thermodynamic equilibrium. Instantons that mediate the creation of these space-times are constructed and we see that they are necessarily complex due to regularity requirements. Thus we argue that instantons need not always be real Euclidean solutions to the Einstein equations. Finally, we calculate the actions of these instantons and find that the standard action functional must be modified to correctly take into account the effects of the rotation. The resultant probabilities for the creation of the space-times are found to be real and consistent with the interpretation that the entropy of a charged and rotating black hole is the logarithm of the number of its quantum states.

Figures (9)

• Figure 1: The global structure for the KNdS solutions - with periodic identifications to ensure that $t=\hbox{constant}$ hypersurfaces contain only two black holes. As indicated the figure is repeated vertically and periodically identified horizontally. $r=r_c$ is the cosmological horizon, $r=r_o$ is the outer black hole horizon, and $r=r_i$ is the inner black hole horizon. The wavy lines at $r=0$ represent the ring singularity found there for $a \neq 0$. If $a = 0$ then this singularity may not be avoided and the space-time cuts off at $r=0$. Otherwise the singularity may be bypassed and we may proceed to negative values of $r$. $r=r_-$ is the (negative) fourth root of ${\mathcal{Q}}$.
• Figure 2: The allowed range of the KNdS parameters. The range is bounded by the planes $M=0$, $a^2 = 0$, $E_0^2 + G_0^2 = 0$, the cold solutions (the darkest sheet) and the rotating Nariai solutions (the lighter gray sheet). Also shown as a meshed sheet are the lukewarm solutions.
• Figure 3: The Penrose diagram for a two hole cold KNdS space-time. Opposite sides of the rectangle are identified. $r=r_c$ is the cosmological horizon and $r=r_{o,i}$ is the double black hole horizon. If $a=0$, then the space-time cuts off at the singularity at $r=0$. Otherwise, we may pass through the ring singularity to the negative values of $r$, including $r_-$, the fourth root of ${\mathcal{Q}}$.
• Figure 4: The Penrose diagram for the Nariai limit space-time. $\rho = \pm 1$ are the two cosmological horizons.
• Figure 5: Construction of the two horizon instanton. The radial/time sector is shown. The heavily dashed lines indicate that the solution continues in that direction.