One-loop Quantum Gravity in Schwarzschild Spacetime

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 ${\dot{\mathrm{\Psi }}}_0$ and ${\dot{\mathrm{\Psi }}}_4$ associated with the Newman-Penrose description of the classical theory. Formulae are obtained for the expectation values of the modulus squared of these operators in the Boulware, Unruh and Hartle-Hawking quantum states. Differences between the renormalized expectation values of both $\mid {\dot{\mathrm{\Psi }}}_0{\mid }^2$ and $\mid {\dot{\mathrm{\Psi }}}_4{\mid }^2$ in the three quantum states are evaluated numerically.