Entropy and Action of Dilaton Black Holes

TL;DR

This paper calculates the entropy and action of dilaton black holes, confirming the universal relation S = A/4, and explains its origin through extrinsic curvature contributions, applicable to various black hole types.

Contribution

It provides a general explanation for the entropy formula S = A/4 for static, spherically symmetric black holes, based on extrinsic curvature contributions, extending previous case-by-case results.

Findings

Entropy equals one quarter of the horizon area for studied black holes.

Extrinsic curvature at the horizon accounts for the entropy, independent of charge or scalar fields.

The on-shell Lagrangian for extremal black holes vanishes, consistent with previous limits.

Abstract

We present a detailed calculation of the entropy and action of $U(1)~2$ dilaton black holes, and show that both quantities coincide with one quarter of the area of the event horizon. Our methods of calculation make it possible to find an explanation of the rule $S = A/4$ for all static, spherically symmetric black holes studied so far. We show that the only contribution to the entropy comes from the extrinsic curvature term at the horizon, which gives $S = A/4$ independently of the charge(s) of the black hole, presence of scalar fields, etc. Previously, this result did not have a general explanation, but was established on a case-by-case basis. The on-shell Lagrangian for maximally supersymmetric extreme dilaton black holes is also calculated and shown to vanish, in agreement with the result obtained by taking the limit of the expression obtained for black holes with regular horizon.The physical meaning of the entropy is discussed in relation to the issue of splitting of extreme black holes.