On the extendibility of Weyl coordinates

Extract

Axially symmetric static fields produced by one extended body are considered. The metric outside the body is the Weyl metric which can be written in so-called Weyl coordinates in the form ds² = a²(dr² + dz²) + b²dϕ² – c²dt² (a, b, c functions of (r, z) and r > 0) with b² = r²c⁻². Criteria are found under which these Weyl coordinates can be extended through the whole of the body so that ds² keeps the form above (although in general b² ǂ r²c⁻² inside the body). In the spherically symmetric (⇒ axially symmetric) case it is shown that the Weyl coordinates cover the whole of spacetime only if one allows r < 0 in a certain part of the body.