On the temperature of the quantum black hole

Abstract

A nontrivial peculiarity of general relativity is that when the horizon region of black holes is rendered harmless, the exterior doubles, resulting in a causally disconnected parallel universe. This intricacy plays a central role in 't Hooft's unitarity arguments, emphasising an exact identification between the physical universe and its duplicate on the other side of the horizon. However, it leads to another tension in the form of a factor of two correction in Hawking's temperature. This discrepancy is concerning because the Rindler temperature is universal and complies with the Bekenstein-Hawking entropy. We demonstrate that the mismatch in the Boltzmann factor gets fixed if the state that forms the corresponding density matrix adopts a generalised thermofield double structure. That leaves room for some interesting discussion.

Figures (3)

• Figure 1: Shown are the shaded right Rindler wedge ($\mathsf{RRW}$), i.e. Rindler space, left Rindler wedge ($\mathsf{LRW}$), expanding Kasner universe ($\mathsf{EKU}$), and contracting Kasner universe ($\mathsf{CKU}$). The thick line corresponds to the Rindler horizon. Constant $\varrho$ (dashed) and $t_\text{R}$ (dotted) lines are also depicted. Note that the entire Rindler space is causally connected.
• Figure 2: Maximally extended Fronsdal-Kruskal-Szekeres diagram of the Schwarzschild black hole solution. The coordinates are well behaved everywhere outside the curvature singularity shown as a thick, fuzzy (red) curve. In particular, the past and future horizon regions are non-singular. Shown are all four causally disconnected regions $\mathsf{I}$ - $\mathsf{IV}$. The physical universe corresponds to the shaded exterior region $\mathsf{I}$. The darkened area in the interior is blocked due to the curvature singularity at $r=0$. The dashed lines represent the past and future horizons at $r=r_\text{S}$. Constant $t$ and $r$ lines are depicted as well.
• Figure 3: Penrose diagrams for the extended Schwarzschild (left) and Fronsdal-Kruskal-Szekeres coordinates (right). Killing horizons (dotted lines), i.e., null hypersurfaces to which the corresponding Killing vector fields are tangent, are traversed by null arrows. The hyperbolic curve (dashed lines) in region $\mathsf{I}$ represents a fictitious cut between $\mathsf{sh}$ and $\mathsf{out}$.