Gravitational vacuum polarization around static spherical stars

The gravitational vacuum polarization of conformally coupled quantum fields in the spacetime of a uniform-density, static, spherical star is studied using the approximation developed by Page, Brown, and Ottewill. Approximate vacuum stress-energy tensors are calculated for conformal massless scalar, spinor, and vector fields; in the case of vector fields, both dimensionally regularized and {ζ}-function results are given. Explicit algebraic forms for the stress-energy tensors are given for the interior of the star and for the exterior Schwarzschild region. If the vacuum stress energy is to be conserved and have the correct trace anomaly at the surface of the star, it is necessary that there be distributional terms in the vacuum stress energy at the surface. The nature and magnitude of these terms are determined. The semiclassical Einstein equations are solved in the exterior region of the star to first order in {}, and the first quantum corrections to Kepler's third law are found.