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 R indler-Like space-time is investigated within the conical Euclidean ap proach in the manifold C(beta)xM(N), C-beta being the two-dimensional cone, making use of the zeta-function regularization. Because of the p resence of conical singularities, it is shown that the relation betwee n the zeta function and the heat kernel is nontrivial and, as first po inted out by Cheeger, requires a separation between small and large ei genvalues of the Laplace operator. As a consequence, in the massless c ase, the (naive) nonexistence of the Mellin transform is bypassed by C heeger'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 functi on and they arise in removing the smearing cutoff. With a simple choic e of the smearing function, horizon divergences in the thermodynamical quantities are recovered and these are similar to the divergences fou nd by means of off-shell methods such as the brick-wall model, the opt ical conformal transformation techniques, or the canonical path-integr al method.