Effects of a co-ordinate mapping on a particular glass of static, axially symmetric solutions to the Einstein equations

Summary

Theγ _A-metric in a canonical form is a solitonic solution to the Einstein field equations in vacuum and represents a semi-infinite Weyl source located on the lower half on the z-axis forγ ± 0;γ ± 1. The co-ordinate transformationr → (λr)ⁿ,t →λ ^n(1-λ)t withn = 1, 2,..., as λ → 0 is a mapping which maps the Weyl source with density γ/2 and length 2m into a semi-infinite line source with densityγ/2, located on the lower half of the z-axis. When terms of the orderλ ^n(3-λ) are kept, a nonvacuum solution is taken. It satisfies certain conditions of energy for different values of the angle θ and the parameter γ.