The Sources of the a and B Degenerate Static Vacuum Fields

TL;DR

This paper investigates the physical origins of certain static vacuum solutions in Einstein's equations, using image sources in Weyl coordinates to interpret the A and B metrics, with varying success.

Contribution

It introduces a method to interpret A and B static vacuum solutions via Newtonian image sources in Weyl coordinates, clarifying their physical meaning.

Findings

The image source interpretation works for some solutions.

The method fails when Weyl transform lacks proper properties at infinity.

Provides insights into the physical interpretation of static vacuum metrics.

Abstract

We attempt to a physical interpretation of some known static vacuum solutions of Einstein's equations, namely, the A and B metrics of Ehlers and Kundt. All of them have axial symmetry, so they can be transformed to the Weyl form. In Weyl coordinates $\ln\sqrt{-g_{44}}$ obeys a Laplace equation, and from this a source, called The Newtonian image source can be identified. We use the image sources to interpret the metrics. The procedure is sucessful in some cases. In others it fails because the Weyl transform does not have reasonable properties at infinity.