Gravitational vacuum polarization. I. Energy conditions in the Hartle-Hawking vacuum

When a quantum field theory is constructed on a curved background spacetime, the gravitationally induced vacuum polarization typically induces a nonzero vacuum expectation value for the quantum stress-energy tensor. It is well known that this gravitational vacuum polarization often violates the pointwise energy conditions and sometimes violates the averaged energy conditions. In this paper I begin a systematic attack on the question of where and by how much the various energy conditions are violated. To keep the discussion manageable, I work in the test-field limit, and focus on conformally coupled massless scalar fields in Schwarzschild spacetime, using the Hartle-Hawking vacuum. The discussion invokes a mixture of analytical and numerical techniques, and critically compares the qualitative behavior to be expected from the Page approximation with that adduced from the numerical calculations of Anderson, Hiscock, and Samuel. I show that the various pointwise energy conditions are violated in a series of onionlike layers located between the unstable photon orbit and the event horizon, the sequence of violations being DEC, WEC, and (NEC+SEC). Furthermore, the ANEC is violated for some of the null geodesics trapped in this region. Having established the basic machinery in this paper, the Boulware vacuum will be treated in a companion paper, while studies of the Unruh vacuum should be straightforward, as should extensions to nonconformal couplings, massive scalars, and Reissner-Nordström geometries.