Numerical thermodynamic studies of classical gravitational collapse in 3+1 and 4+1 dimensions

Recent numerical work on the gravitational collapse of a five-dimemsional (5D) [4+1] Yang-Mills instanton has provided numerical evidence that the free energy F=E-TS=E/3 of a 5D Schwarzschild black hole of mass E can be obtained classically via the Lagrangian. Although there is no Hawking radiation, these numerical results suggest that the quantity TS has a classical meaning. We investigate this association for the physically relevant case of 3+1 dimensional collapse. We track numerically the negative of the total Lagrangian -L during the gravitational collapse of a massless scalar field to a Schwarzschild black hole in isotropic coordinates. We show that -L approaches the free energy F=E-TS=E/2 of a four-dimensional Schwarzschild black hole to within 5%. We also show that the matter contribution to the free energy tends towards zero so that the entropy at late stages of the collapse is gravitational in origin. The entropy S makes a negative contribution to the free energy and this feature is observed in our numerical simulation. There is a pronounced dip (negative contribution) in a thin slice just inside the event horizon precisely where the metric field is nonstationary. This is in accord with recent work suggesting black hole entropy is connected with the nonstationary phase space hidden behind the event horizon. We also obtain thermodynamic results for the 5D collapse of a massless scalar field which confirms that previous 5D results are universal and independent of the type of matter undergoing the collapse.