M.Sc. PHYSICS DISSERTATION
BLACK-HOLE FREE ENERGY DURING CHARGED COLLAPSE
Hugues Beauchesne
Physics Department,
Bishop's University
Thursday, May 17, 2012
1:30 p.m.
Hamilton 300
Recent numerical studies of spherically symmetric gravitational collapse of uncharged matter in both 4D (3+1-dimensions) and 5D has provided numerical evidence that the negative of the Lagrangian approaches the Helmholtz free energy F=E−TS of a Schwarzschild black-hole during the latter stages of collapse (where E, T and S are the mass, temperature and entropy of the black-hole respectively). In this thesis, we investigate the thermodynamics during the collapse of a complex (charged) scalar field to a Reisnner-Nordström (RN) black-hole in isotropic coordinates. The relevant thermodynamic potential for the RN black-hole is the Gibbs free energy G=E−TS−ΦHQ where ΦH is the electrostatic potential at the horizon and Q the charge. In charged collapse, there is a large outgoing matter wave that prevents the exterior region from settling quickly to a static state. However, the interior region is hardly affected by this outgoing wave. We find that the interior Gibbs free energy and the entropy accumulate in a thin slice just inside the horizon. We also find that the charge accumulates in a thin shell just inside the horizon. In accord with a charged shell configuration, the electrostatic potential is found to be constant in the interior. We conducted runs for different initial states. For all initial states, the graph representing the negative of the interior Lagrangian and the graph representing the analytical value of the interior Gibbs free energy approach each other with time. At the latest time achieved (at highest resolution), the values are within 10−13% of each other. The discrepancy would decrease further given more evolution time which is limited by the presence of sharp changes in the gradients of the metric and matter functions at the horizon. We develop a procedure for extending the evolution time in the exterior region where such sharp changes are not present. We find that the metric in the exterior approaches close to that of the RN black-hole, providing a further consistency check on our code.
