Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I

We consider the covariant Klein-Gordon equation (□g_g+m²) φ = 0 of mass m ≥ 0 on the exterior Schwarzschild spacetime of mass M. We introduce and study a set of outer and inner wave operators Ω₀^±, Ω₁^± (constructed in detail elsewhere) describing the asymptotic behavior of classical solutions-Ω₀^± for large distances and Ω₁^± near the Schwarzschild radius-as t→±∞. We re-interpret Ω₁^± on the Kruskal spacetime as solving a characteristic initial value problem for data on the future/past right horizon H^±. As a by-product, we prove (since we require it here) a stronger result than previously known concerning the stability of the Schwarzschild black hole against linearized (scalar) perturbations. Using Ω₀⁺, Ω₁⁺ we construct in and out and horizon fields for the corresponding quantum problem. We give a construction for the Hartle-Hawking state ω_H and prove that it coincides on the in and out fields with a state of exact thermal equilibrium in Minkowski space, while its two-point function on the (right) horizons H^± is given (independently of m) by ω_H[(δ_U ø ̂)(U₁,ξ₁)(δ_U ø ̂)(U₂,ξ₂)] = (-δ(ξ₁,ξ₂)/16φM²)(U₁-U₂-iε{lunate})^-2 (U₁, U₂ ε{lunate} (-∞, 0), ξ₁, ξ₂ ε{lunate} S²) on H^- and by a similar formula (with U→V, (-∞, 0) → (0, ∞), etc.) on H⁺. We also construct the Unruh state ω_U as a product state, equal to the Minkowski vacuum on the in field, and with two-point function on H^- equal to that given above for ω_H. We prove that the restriction of this ω_U to the out field coincides with a particular state of thermal radiation in Minkowski space. A special feature of our treatment is that we relate the horizon behavior of φ̂ with the light-cone behavior of a massless free scalar field φ̂₁ in a two-dimensional flat spacetime: δ² ø ̂₁ δT²- δ² ø ̂₁ δX² = 0. In particular, using our classical characteristic-intial-value-problem results, we explain why the above expression for the two-point function of ∂_Uφ̂ on H is identical (modulo the trivial role played by S² and up to a scale factor of 2M) with the well-known two-point function for ∂_Uφ̂₁ (U=T-X) on the null line T + X = 0.