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

Abstract

We consider the covariant Klein-Gordon equation (□ g g + m 2) 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 Ω 0 ± , Ω 1 ± (constructed in detail elsewhere) describing the asymptotic behavior of classical solutions— Ω 0 ± for large distances and Ω 1 ± near the Schwarzschild radius—as t→±∞. We re-interpret Ω 1 ± 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 Ω 0 + , Ω 1 + 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 ̂ )(U 1,ξ 1)(δ U ø ̂ )(U 2,ξ 2)] = (−δ(ξ 1,ξ 2)/16φM 2)(U 1−U 2−iϵ) −2 ( U 1, U 2 ϵ (−∞, 0), ξ 1, ξ 2 ϵ S 2) 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 φ̂ 1 in a two-dimensional flat spacetime: δ 2 ø ̂ 1 δT 2 − δ 2 ø ̂ 1 δX 2 = 0. In particular, using our classical characteristic-intial-value-problem results, we explain why the above expression for the two-point function of ∂ U φ̂ on H is identical (modulo the trivial role played by S 2 and up to a scale factor of 2 M) with the well-known two-point function for ∂ U φ̂ 1 ( U= T− X) on the null line T + X = 0.