Gravitational vacuum polarization IV: Energy conditions in the Unruh vacuum

Building on a series of earlier papers [gr-qc/9604007,9604008,9604009], I investigate the various point-wise and averaged energy conditions in the Unruh vacuum. I consider the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field, work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. I construct a semi-analytic model for the stress-energy tensor that globally reproduces all known numerical results to within 0.8%, and satisfies all known analytic features of the stress-energy tensor. I show that in the Unruh vacuum (1) all standard point-wise energy conditions are violated throughout the exterior region—all the way from spatial infinity down to the event horizon, and (2) the averaged null energy condition is violated on all outgoing radial null geodesics. In a pair of appendices I indicate general strategy for constructing semi-analytic models for the stress-energy tensor in the Hartle–Hawking and Boulware states, and show that the Page approximation is in a certain sense the minimal ansatz compatible with general properties of the stress-energy in the Hartle–Hawking state. PACS: 04.60.+v 04.70.Dy