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

It is well-known that gravitationally induced vacuum polarization often violates the point-wise 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. I work in the test-field limit, and focus on conformally coupled massless scalar fields in Schwarzschild spacetime, using the Hartle–Hawking vacuum. I invoke a mixture of analytical and numerical techniques, and critically compare the qualitative behaviour to be expected from the Page approximation with that adduced from the numerical calculations of Anderson, Hiscock, and Samuel. I show that the various point-wise energy conditions are violated in a series of onion-like 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 other exensions should be straightforward.