Thermodynamics of Сlassical Schwarzschild Black Holes

Abstract

This paper assesses the thermodynamic properties of classical Schwarzschild Black Holes. We review the thermodynamic cycle in the Geroch gedanken experiment in its relation to the classical limit of Hawking temperature, and conclude that a minimal distance requirement for pouring out radiation onto the event horizon in the gedanken experiment implies the assignment of temperature ${{T}_{{{\text{BH}}}}} \to 0$ asympotically rather than exact absolute zero $K$: a classical Schwarzschild horizon is an asymptotically zero Kelvin mechanical thermostat operated at asymptotically unit efficiency. We classify the gravitational heat pumping work done by horizon surface gravitation as perpetual motion of the third kind. The latter property does not, however, violate the Second Law, because it is impossible to extract gravitational energy from a classical Schwarzschild Black Hole in quasi-static equilibrium. The classical limit of the Bekenstein–Hawking (quantum) entropy formula is divergent as ${{S}_{{{\text{BH}}}}} \to \infty$, yet since $\delta {{S}_{{{\text{BH}}}}} \to 0$ the Nernst statement of the Third Law is not violated. Aided by the Bekenstein bound, it is shown that the Generalized Second Law persists to hold in its classical limit, when applied to the joint system of Schwarzschild Black Hole and infalling radiation gas in the Geroch gedanken experiment. The minimal horizon distance $\delta r$ is inferred from the Bekenstein bound, and ensures that the Geroch thermodynamic cycle be processed reversibly. The overall conclusion is that the laws of thermodynamics are not transcended by classical Schwarzschild Black Holes, despite their extremal thermodynamic properties.