A Simple Stationary Line Element for the Schwarzschild Geometry, and Some Applications

TL;DR

This paper introduces a simple, stationary line element for Schwarzschild geometry that smoothly extends through the horizon, enabling straightforward derivations of Hawking radiation and Penrose diagrams, with potential for further improvements.

Contribution

It presents a novel, simple stationary form of the Schwarzschild line element that facilitates analysis of black hole properties and radiation.

Findings

Line element is smooth through the horizon

Derivation of Hawking radiance is simplified

Complete Penrose diagram constructed with time-reversal symmetry

Abstract

Guided by a Hamiltonian treatment of spherically symmetric geometry, we find a remarkably simple -- stationary, but not static -- form for the line element of Schwarzschild (and Reissner-Nordstrom) geometry. The line element continues smoothly through the horizon; by exploiting this feature we are able to give a very simple and physically transparent derivation of the Hawking radiance. We construct the complete Penrose diagram by enforcing time-reversal symmetry. Finally we outline how an improved treatment of the radiance, including effects of self-gravitation, can be obtained.