Geometric Entropy and Curvature Coupling in Conical Spaces: ζ Function Approach

The local ζ−function approach is implemented to regularize the natural path integral definition of the geometric entropy in the large mass black hole Euclidean manifold. The case of a massless field coupled with the (off-shell) singular curvature is considered. It is proved that the geometric entropy is independent of the curvature coupling parameter ξ avoiding negative values obtained in other approaches. PACS number(s): 04.62.+v, 04.70.Dy Introduction This short paper is devoted to develop a possible definition of the geometric entropy, employing the path integral and the ζ-function regularization in the case of a scalar field propagating in a Euclidean manifold containing conical singularities, taking care to consider any possible choice of the coupling parameter ξ with the conical curvature. The geometric entropy in such manifolds has been considered in many papers due to its relation to the black hole entropy [1, 2, 3, 4, 5, 6, 7, 8, 9] and the interesting remark that ultraviolet divergences exactly coincides with the divergences of the gravitational constant in perturbation theory of quantum gravity [10, 11, 8] at the Hawking-Unruh temperature. Unfortunately, some severe difficulties in developing these topics have been pointed out as far as the non-minimal coupling is concerned [11, 12, 13, 9]. In that case, the geometric entropy seems to become negative whenever ξ > 1/6. However, the geometric entropy, which is related to a completely (Ricci-) flat manifold, seems to be dependent on ξ, the coupling parameter with the Euclidean manifold (vanishing!) curvature. e-mail: [email protected]