Abstract
Many discussions in the literature of spacetimes with more than one Killing horizon note that some horizons have positive and some have negative surface gravities, but assign to all a positive temperature. However, the first law of thermodynamics then takes a non-standard form. We show that if one regards the Christodoulou and Ruffini formula for the total energy or enthalpy as defining the Gibbs surface, then the rules of Gibbsian thermodynamics imply that negative temperatures arise inevitably on inner horizons, as does the conventional form of the first law. We provide many new examples of this phenomenon, including black holes in STU supergravity. We also give a discussion of left and right temperatures and entropies, and show that both the left and right temperatures are non-negative. The left-hand sector contributes exactly half the total energy of the system, and the right-hand sector contributes the other half. Both the sectors satisfy conventional first laws and Smarr formulae. For spacetimes with a positive cosmological constant, the cosmological horizon is naturally assigned a negative Gibbsian temperature. We also explore entropy-product formulae and a novel entropy-inversion formula, and we use them to test whether the entropy is a super-additive function of the extensive variables. We find that super-additivity is typically satisfied, but we find a counterexample for dyonic Kaluza-Klein black holes.
Highlights
Since the early days of black hole thermodynamics there have been suggestions that the thermodynamic of the inner, Cauchy, horizons of charged and or rotating black holes should be taken more seriously than it has been [1,2,3,4,5,6,7,8,9,10]
We shall begin by recalling the most physically convincing argument that Schwarzschild black holes have a temperature, and entropy. This was given by Hawking [46,47], who coupled a collapsing black-hole metric in an asymptotically-flat spacetime to a quantum field, and showed that if the quantum field was initially in its vacuum state, at late times it would emit particles with a thermal spectrum and temperature given by (1.3)
One may dispense with that region, and work with the exact vacuum Schwarzschild solution, obtaining the same result, by choosing an appropriate boundary condition for the quantum field on the past horizon
Summary
Since the early days of black hole thermodynamics there have been suggestions that the thermodynamic of the inner, Cauchy, horizons of charged and or rotating black holes should be taken more seriously than it has been [1,2,3,4,5,6,7,8,9,10]. With the development of String Theory approaches these suggestions have become more insistent [11,12,13,14,15,16,17] This interest increased considerably with the observation that the product of the areas and entropies of the inner and outer horizon takes in many examples a universal form which should be quantised at the quantum level [18,19,20,21,22]. A rather general feature of many asymptoticaly flat black holes with two horizons is that the product of the areas of the two horizons is independent of the mass, and given in terms of conserved charges and angular momenta, which may plausibly be quantised at the quantum level.
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