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We study a thermodynamic potential during the classical gravitationalcollapse of a 4D (3+1) massless scalar field to a Schwarzschild black hole inisotropic coordinates. We track numerically the function $F(t)=-dI/dt=-L$,where $I$ is the total action of matter plus gravitation, $L$ the totalLagrangian and $t$ is the time measured by a stationary clock at infinity. Atlate stages in the collapse, this function can be identified with the freeenergy $F=E-TS$ of the black hole where $E$ is the ADM mass, $T$ the Hawkingtemperature and $S$ the entropy. From standard black hole thermodynamics, thefree energy of a 4D Schwarzschild black hole is equal to $E/2$. Our numericalsimulations show that at late stages of the collapse the function $-L$approaches a constant to within 5% of the value of $E/2$. We also presentnumerical results for the thermodynamics of 5D collapse where the free energyin this case is $E/3$. In both 4D and 5D, our numerical simulations show thatat late stages of the collapse, the metric fields are nonstationary in a thinregion just behind the event horizon (and are basically static everywhereelse). The entropy stems mostly from the nonstationary interior region wherethere is a significant dip (negative contribution) to the free energy.

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