The Hadamard renormalization prescription is used to derive a two-dimensional analog of the renormalized stress tensor for a minimally coupled scalar field in Schwarzschild–de Sitter space-time. In the two-dimensional analog the minimal coupling reduces to the conformal coupling and the stress tensor is found to be determined by the (nonlocal) contribution of the anomalous trace and some additional parameters in close relation to the work by Christensen and Fulling. To properly relate the stress tensor to the state of outwards signals coming from the direction of the black hole at late times we propose a cutoff hypothesis that excludes the contribution of the anomalous trace close to the black hole horizon. The corresponding cutoff scale is found to be related to the equilibrium temperature of the cosmological horizon in a leading order estimate. Finally, we establish a relation between the radiation temperature of the black hole horizon at large distance from the hole and the anomalous trace and determine the correction term to the Hawking temperature due to the presence of the cosmological horizon.

• Received 20 May 1997

DOI:https://doi.org/10.1103/PhysRevD.56.4633