Entropy and topology for manifolds with boundaries

In this work a deep relation between topology and thermodynamical features of manifolds with boundaries is shown. The expression for the Euler characteristic, through the GaussBonnet integral, and the one for the entropy of gravitational instantons are proposed in a form which makes the relation between them selfevident. A generalization of Bekenstein-Hawking formula, in which entropy and Euler characteristic are related in the form S = χA/8, is obtained. This formula reproduces the correct result for extreme black hole, where the Bekenstein-Hawking one fails (S = 0 but A 6= 0). In such a way it recovers a unified picture for the black hole entropy law. Moreover, it is proved that such a relation can be generalized to a wide class of manifolds with boundaries which are described by spherically symmetric metrics (e.g. Schwarzschild , Reissner-Nordström, static de Sitter). #1 E-mail:[email protected] #2 E-mail:[email protected] 1