On the behavior of the bounds of the holographic theory for massive and massless particle systems (original) (raw)
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Aspects of holography or dimensional reduction in gravitational physics are discussed with reference to black hole thermodynamics. Degrees of freedom living on Isolated Horizons (as a model for macroscopic, generic, eternal black hole horizons) are argued to be topological in nature and counted, using their relation to two dimensional conformal field theories. This leads to the microcanonical entropy of these black holes having the Bekenstein-Hawking form together with finite, unambigious quantum spacetime corrections. Another aspect of holography ensues for radiant black holes treated as a standard canonical ensemble with Isolated Horizons as the mean (equilibrium) configuration. This is shown to yield a universal criterion for thermal stability of generic radiant black holes, as a lower bound on the mass of the equilibrium isolated horizon in terms of its microcanonical entropy. Saturation of the bound occurs at a phase boundary separating thermally stable and unstable phases with symptoms of a first order phase transition.
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We propose dual thermodynamics corresponding to black hole mechanics with the identifications E ′ → A/4, S ′ → M , and T ′ → T −1 in Planck units. Here A, M and T are the horizon area, mass and Hawking temperature of a black hole and E ′ , S ′ and T ′ are the energy, entropy and temperature of a corresponding dual quantum system. We show that, for a Schwarzschild black hole, the dual variables formally satisfy all three laws of thermodynamics, including the Planck-Nernst form of the third law requiring that the entropy tend to zero at low temperature. This is in contrast with traditional black hole thermodynamics, where the entropy is singular. Once the third law is satisfied, it is straightforward to construct simple (dual) quantum systems representing black hole mechanics. As an example, we construct toy models from one dimensional (Fermi or Bose) quantum gases with N ≃ M in a Planck scale box. In addition to recovering black hole mechanics, we obtain quantum corrections to the entropy, including the logarithmic correction obtained by previous papers. The energy-entropy duality transforms a strongly interacting gravitational system (black hole) into a weakly interacting quantum system (quantum gas) and thus provides a natural framework for the quantum statistics underlying the holographic conjecture.
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