Exact solutions of the Einstein vacuum field equations in Weyl co-ordinates

Abstract

Inconsistencies inherent in the interpretation of the canonical Weyl co-ordinates in the quadratic differential form for static, axially symmetric distributions of matter in general relativity as the timet and the cylindrical co-ordinatesr,z,ϕ are demonstrated in a number of ways. The most general Weyl quadratic differential form in which the metric is a function of only the radial co-ordinate is analyzed. This solution depends upon a single parameterσ, and reasons are given for interpreting this solution as an infinite line of positive mass for 0 <σ < 1/2; a homogeneous gravitational field forσ = +1, and an infinite sheet of negative mass forσ = −1. All solutions are given which correspond to flat space, and these are used to demonstrate that if one has two Weyl potentials each corresponding to a physical system, the sum of the two potentials is not the potential corresponding to the superposition of the two physical systems.