One-loop quantum gravity in Schwarzschild space-time

The quantum theory of linearized perturbations of the gravitational field of a Schwarzschild black hole is presented. The fundamental operators are seen to be the perturbed Weyl scalars Ψ${\mathrm{̇}}_0$ and Ψ${\mathrm{̇}}_4$ associated with the Newman-Penrose description of the classical theory. Formulas are obtained for the expectation values of the modulus squared of these operators in the Boulware, Unruh, and Hartle-Hawking quantum states. The differences between the renormalized expectation values of both ‖Ψ${\mathrm{̇}}_0$ ${\mathrm{\Vert }}^2$ and ‖Ψ${\mathrm{̇}}_4$ ${\mathrm{\Vert }}^2$ in the three quantum states are evaluated numerically.