Conformal Scalar Propagation inside the Schwarzschild Black Hole

TL;DR

This paper analytically extends the conformal scalar propagator inside a Schwarzschild black hole, matching it across the event horizon, and demonstrates its role in particle production in the Hartle-Hawking vacuum.

Contribution

It provides a novel analytic expression for the scalar propagator inside the black hole that depends explicitly on the space-time geometry and matches exterior solutions.

Findings

Propagator is analytically extended into the black hole interior.

The interior and exterior propagators are matched at the event horizon.

The resulting propagator describes particle production by the black hole.

Abstract

The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle-Hawking vacuum state for small values of the Schwarzschild radial coordinate above $ r = 2M$ is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below $ r = 2M$ and is an analytic expression which manifestly features its dependence on the background space-time geometry. This feature as well as the absence of any assumptions and prerequisites in the derivation render this Hartle-Hawking scalar propagator in the interior of the Schwarzschild black-hole geometry distinct from previous results. The two propagators obtained in the interior and in the exterior region of the Schwarzschild black hole are matched across the event horizon. The result of that match is a massless conformal scalar propagator in the Hartle-Hawking vacuum state which is shown to describe particle production by the Schwarzschild black hole.