Symmetric Two-Point Function for a Massive, Minimally Coupled Scalar Field in Two-Dimensional Schwarzschild de-Sitter Spacetime

The computation of the symmetric two-point function for a quantized, massive, minimally coupled scalar field in two-dimensional Schwarzschild de-Sitter spacetime will be discussed. The calculation is done in the static patch between the black hole and cosmological horizons. The field is in the generalized Unruh state, which is specified by modes that are positive frequency with respect to the Kruskal time coordinate relevant to the past black hole horizon, and by modes that are positive frequency with respect to the Kruskal time coordinate relevant to the past cosmological horizon. Since the mode equation is not separable in Kruskal coordinates, it is necessary to expand these solutions in terms of a complete set of solutions to the mode equation that can be attained via separation of variables. Due to the existence of a non-trivial effective potential, the radial part of the mode equation must be solved numerically.