Black Hole Thermodynamics in Carathéodory’s Approach

In this letter we show that the approach of Carathéodory to thermodynamics by means of Pfaffian forms can be applied to black hole thermodynamics and strongly links black hole thermodynamics to the standard thermodynamic formalism. The Pfaffian form δQrev ≡ dM − Ω dJ − Φ dQ, which is assumed to be the infinitesimal heat exchanged reversibly, is shown to be integrable; moreover, we show that it is a quasi-homogeneous form. As a consequence, an integrating factor is readily calculated. It is then shown that both the entropy and the temperature of a Kerr-Newman black hole can be recovered. No a priori knowledge of the laws of black hole mechanics is required. The Hawking effect is necessary in order to give an actual thermodynamic meaning to our calculation and in order to identify a undetermined multiplicative constant in the expression of the absolute temperature and the absolute entropy of the black hole. Also the problem of extremal black holes is shortly discussed.