Gravitational vacuum polarization. III. Energy conditions in the (1+1)-dimensional Schwarzschild spacetime

Building on techniques developed in the two preceding papers, I investigate the various pointwise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field in the (1+1)-dimensional Schwarzschild spacetime. Because the stress-energy tensors are analytically known, I can get exact results for the Hartle-Hawking, Boulware, and Unruh vacua. This exactly solvable model serves as a useful check on my (3+1)-dimensional investigations wherein I had to resort to a mixture of analytic approximations and numerical techniques. The key results in 1+1 dimensions are that (1) NEC is satisfied outside the event horizon for the Hartle-Hawking vacuum and violated for the Boulware and Unruh vacua, and (2) DEC is violated everywhere in the spacetime (for any quantum state, not just the standard vacuum states).