Trace anomalies and the Hawking effect

Abstract

The general spherically symmetric, static solution of ${\nabla }_{\nu }{T}_{\mu }^{\nu } = 0$ in the exterior Schwarzschild metric is expressed in terms of two integration constants and two arbitrary functions, one of which is the trace of $T_{\mu \nu }$. One constant is the magnitude of $T_{\mathrm{tr}}$ at infinity, and the other is determined if the physically normalized components of $T_{\mu \nu }$ 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, $48M^2{r}^{-6\mathrm{}}$; we fix the coefficient for the scalar field by indirect arguments to be ${\left(2880\mathrm{}{\pi }^2\right)}^{-1\mathrm{}}$. 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 $T_{\theta }^{\theta }-\frac{T_{\alpha }^{\alpha }}{4}$. This, plus additional physical and mathematical considerations, leads us to a fairly definite, physically convincing qualitative picture of $〈T_{\mu \nu }〉$. Groundwork is laid for explicit calculations of $〈T_{\mu \nu }〉$.