Black hole free energy during charged collapse: a numerical study

We perform a numerical investigation of the thermodynamics during the collapse of a charged (complex) scalar field to a Reissner-Nordstr\"om (RN) black hole in isotropic coordinates. Numerical work on gravitational collapse in isotropic coordinates has recently shown that the negative of the total Lagrangian approaches the Helmholtz free energy F= E-TS of a Schwarzschild black hole at late times of the collapse (where E is the black hole mass, T the temperature and S the entropy). The relevant thermodynamic potential for the RN black hole is the Gibbs free energy G=E-TS-$\Phi_H$ Q where Q is the charge and $\Phi_H$ the electrostatic potential at the outer horizon. In charged collapse, there is a large outgoing matter wave which prevents the exterior from settling quickly to a static state. However, the interior region is not affected significantly by the wave. We find numerically that the interior contribution to the Gibbs free energy is entirely gravitational and accumulates in a thin shell just inside the horizon. The entropy is gravitational in origin and one observes dynamically that it resides on the horizon. We also compare the numerical value of the interior Lagrangian to the expected analytical value of the interior Gibbs free energy for different initial states and we find that they agree to within 10-13%. The two values are approaching each other so that their difference decreases with more evolution time.