Microcanonical model for a gaz of evaporating black holes and strings, scattering amplitudes and mass spectrum

Abstract

We study the system formed by a gaz of black holes and strings within a microcanonical formulation. We derive the microcanonical content of the system: entropy, equation of state, number of components N, temperature T and specific heat. The pressure and the specific heat are negative reflecting the gravitational unstability and a non-homogeneous configuration. The asymptotic behaviour of the temperature for large masses emerges as the Hawking temperature of the system (classical or semiclassical phase) in which the classical black hole behaviour dominates, while for small masses (quantum black hole or string behavior) the temperature becomes the string temperature which emerges as the critical temperature of the system. At low masses, a phase transition takes place showing the passage from the classical (black hole) to quantum (string) behaviour. Within a microcanonical field theory formulation, the propagator describing the string-particle-black hole system is derived and from it the interacting four point scattering amplitude of the system is obtained. For high masses it behaves asymptotically as the degeneracy of states of the system (ie duality or crossing symmetry). The microcanonical propagator and partition function are derived from a (Nambu-Goto) formulation of the N-extended objects and the mass spectrum of the black-hole-string system is obtained: for small masses (quantum behaviour) these yield the usual pure string scattering amplitude and string-particle spectrum M_n\approx \sqrt{n}; for growing mass it pass for all the intermediate states up to the pure black hole behaviour. The different black hole behaviours according to the different mass ranges: classical, semiclassical and quantum or string behaviours are present in the model.