Euclidean Approach to the Entropy for a Scalar Field in Rindler-like Space-Times

TL;DR

This paper investigates the off-shell entropy of a massless scalar field in Rindler-like space-times using a Euclidean approach, addressing the effects of conical singularities and divergences in thermodynamical quantities.

Contribution

It introduces a novel analysis of the zeta-function regularisation in conical manifolds, highlighting the role of eigenvalue separation and smeared traces in entropy calculations.

Findings

Conical singularities complicate the zeta-function and heat kernel relation.

Cheeger's analytical continuation bypasses Mellin transform issues.

Horizon divergences depend on the choice of smearing function.

Abstract

The off-shell entropy for a massless scalar field in a D-dimensional Rindler-like space-time is investigated within the conical Euclidean approach in the manifold $C_\be\times\M^N$, $C_\be$ being the 2-dimensional cone, making use of the zeta-function regularisation. Due to the presence of conical singularities, it is shown that the relation between the zeta-function and the heat kernel is non trivial and, as first pointed out by Cheeger, requires a separation between small and large eigenvalues of the Laplace operator. As a consequence, in the massless case, the (naive) non existence of the Mellin transform is by-passed by the Cheeger's analytical continuation of the zeta-function on manifold with conical singularities. Furthermore, the continuous spectrum leads to the introduction of smeared traces. In general, it is pointed out that the presence of the divergences may depend on the smearing function and they arise in removing the smearing cutoff. With a simple choice of the smearing function, horizon divergences in the thermodynamical quantities are recovered and these are similar to the divergences found by means of off-shell methods like the brick wall model, the optical conformal transformation techniques or the canonical path integral method.