Global Casimir Effect in the Schwarzschild Spacetime

In this paper, we study the vacuum quantum fluctuations of an uncharged massive scalar field in the Schwarzschild background and analyze its main physical effects at zero temperature. The procedure consists of calculating the energy eigenvalues starting from the exact solutions recently found for the dynamics of that field, considering the regime in which the particle is not absorbed by the black hole. From this result, we obtain the vacuum energy for the field, taking into account the respective degeneracies. Then we use the Abel-Plana formula valid for bosonic fields in order to regularize this infinite vacuum energy. Such a regularized quantity is the Casimir energy, which is computed numerically and presented graphically. The Casimir energy thus obtained does not take into account any boundaries artificially imposed on the system, just the nontrivial spacetime topology associated to the source and its singularity.