Discontinuities in spherically symmetric gravitational fields and shells of radiation

Abstract

Boundary conditions at a 3-space of discontinuity ∑ are considered from the point of view of Lichnerowicz. The validity of the O’Brien—Synge junction conditions is established for co-ordinates derivable from Lichnerowicz’s ‘admissible co-ordinates’ by a transformation which is uniformly differentiable across ∑. The co-ordinates r , θ , ϕ , t , used by Schwarzschild and most later authors when dealing with spherically symmetric fields, are shown to be of this type. In Schwarzschild’s co-ordinates, the components of the metric tensor can always be made continuous across Ʃ, and simple relations are derived connecting the jumps in their first derivatives. A spherical shell of radiation expanding in empty space is examined in the light of the above ideas, and difficulties encountered by Raychaudhuri in a previous treatment of this problem are cleared up. A particular model is then discussed in some detail.