Conformal Scalar Propagation on the Schwarzschild Black-Hole Geometry

TL;DR

This paper derives an analytic approximation for the scalar propagator near a Schwarzschild black hole's horizon in the Hartle-Hawking state, providing new insights into quantum field behavior in curved spacetime.

Contribution

It presents a novel analytic approximation to the scalar propagator in Schwarzschild geometry, valid near the horizon, improving understanding of quantum effects in black hole backgrounds.

Findings

Derived an analytic expression for the scalar propagator near the horizon.

Obtained a renormalized <φ^2(x)> expression consistent with known results.

Analyzed the influence of causal structure on scalar propagation.

Abstract

The vacuum activity generated by the curvature of the Schwarzschild black-hole geometry close to the event horizon is studied for the case of a massless, conformal scalar field. The associated approximation to the unknown, exact propagator in the Hartle-Hawking vacuum state for small values of the radial coordinate above $ r = 2M$ results in an analytic expression which manifestly features its dependence on the background space-time geometry. This approximation to the Hartle-Hawking scalar propagator on the Schwarzschild black-hole geometry is, for that matter, distinct from all other. It is shown that the stated approximation is valid for physical distances which range from the event horizon to values which are orders of magnitude above the scale within which quantum and backreaction effects are comparatively pronounced. An expression is obtained for the renormalised $ <\phi^2(x)>$ in the Hartle-Hawking vacuum state which reproduces the established results on the event horizon and in that segment of the exterior geometry within which the approximation is valid. In contrast to previous results the stated expression has the superior feature of being entirely analytic. The effect of the manifold's causal structure to scalar propagation is also studied.