IX Black Hole Workshop

Guimarães 19-20 December 2016

The thermodynamics of an extremal rotating thin shell, i.e., an extremal rotating ring in a (2+1)-dimensional spacetime with a negative cosmological constant is investigated. The outer and inner regions with respect to the shell are taken to be the BTZ spacetime and the vacuum AdS spacetime. By applying the first law of thermodynamics to the extremal thin shell one shows that the entropy of the shell is an arbitrary well-behaved function of the gravitational area A+ alone, S=S(A+). This indicates that extremal shells, here extremal rotating shells in a (2+1)-dimensional spacetime with a negative cosmological constant, are special while compared to the corresponding nonextremal shells. When the thin shell approaches its own gravitational radius and turns into an extremal rotating BTZ black hole, it is found that the entropy of the spacetime remains such a function of A+, S=S(A+), both when the local temperature of the shell at the gravitational radius is zero and nonzero. By the same rationale, it is vindicated by this analysis, that extremal black holes, here extremal BTZ black holes, have different properties from the corresponding nonextremal black holes. It is argued that for extremal black holes, in particular for extremal BTZ black holes, one should set that S(A+) is between 0 and A+/4, i.e., the extremal black hole entropy has values in-between 0 and the maximum entropy, the Bekenstein-Hawking entropy. Thus, rather than having just two entropies for the extremal black holes, as previous results have debated, namely 0 and A+/4, the extremal BTZ black hole entropy may have a continuous range of entropies, limited by precisely those two entropies. Surely, the entropy that a particular extremal black hole picks must depend on past processes, notably on how it was formed. It is argued that this holds for other types of black holes in other dimensions. It is also found a remarkable relation between the third law of thermodynamics and the impossibility for a massive body to reach the velocity of light. In addition, in the procedure, it becomes clear that there are two distinct angular velocities for the shell, the mechanical and thermodynamic angular velocities. Comments on the relationship between these two velocities will be given. In passing, we clarify, for a spacetime with a thermal shell, the meaning of the Tolman temperature formula at a generic radius and at the shell.