Trace anomalies and the Hawking effect

Abstract

The general spherically symmetric, static solution of ${\ensuremath{\nabla}}_{\ensuremath{\nu}}{T}_{\ensuremath{\mu}}^{\ensuremath{\nu}} = 0$ in the exterior Schwarzschild metric is expressed in terms of two integration constants and two arbitrary functions, one of which is the trace of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$. One constant is the magnitude of ${T}_{\mathrm{tr}}$ at infinity, and the other is determined if the physically normalized components of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ are finite on the future horizon. The trace of the stress tensor of a conformally invariant quantum field theory may be nonzero (anomalous), but must be proportional (here) to the Weyl scalar, $48{M}^{2}{r}^{\ensuremath{-}6}$; we fix the coefficient for the scalar field by indirect arguments to be ${(2880{\ensuremath{\pi}}^{2})}^{\ensuremath{-}1}$. In the two-dimensional analog, the magnitude of the Hawking blackbody effect at infinity is directly proportional to the magnitude of the anomalous trace (a multiple of the curvature scalar); a knowledge of either number completely determines the stress tensor outside a body in the final state of collapse. In four dimensions, one obtains instead a relation constraining the remaining undetermined function, which we choose as ${T}_{\ensuremath{\theta}}^{\ensuremath{\theta}}\ensuremath{-}\frac{{T}_{\ensuremath{\alpha}}^{\ensuremath{\alpha}}}{4}$. This, plus additional physical and mathematical considerations, leads us to a fairly definite, physically convincing qualitative picture of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$. Groundwork is laid for explicit calculations of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$.