Euclidean approach to the entropy for a scalar field in Rindler-like space-times

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 $C_{\beta }\times {\mathcal{M}}^N$, $C_{\beta }$ being the two-dimensional cone, making use of the $\zeta$-function regularization. Because of the presence of conical singularities, it is shown that the relation between the $\zeta$ function and the heat kernel is nontrivial 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) nonexistence of the Mellin transform is bypassed by Cheeger's analytical continuation of the $\zeta$ function on the 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 such as the brick-wall model, the optical conformal transformation techniques, or the canonical path-integral method.