Vacuum polarization in Schwarzschild spacetime

Abstract

The polarization of the vacuum induced by gravitation is studied for massless fields in the region exterior to the horizon of a Schwarzschild black hole. The renormalized value of $〈{\ensuremath{\varphi}}^{2}(x)〉$ is calculated according to the "covariant point-separation scheme" for each of the Boulware, Hartle-Hawking, and Unruh "vacua." The form of the renormalized expectation value of the stress tensor near the horizon and at infinity is discussed for each of these three states. It is found that the Unruh vacuum best approximates the state that would obtain following the gravitational collapse of a massive body in the sense that the expectation values of physical observables are finite, in a freely falling frame, on the future horizon and that this state is empty near infinity apart from an outgoing flux of a blackbody radiation. The response of an Unruh box is examined further in the light of the results obtained for the stress tensor. Finally it is shown by explicit solution of the linearized Einstein equations that the area of the horizon decreases at the rate expected from the flux at infinity.