Conformal Geodesics in General Relativity
Abstract
Conformal geodesics, spacetime curves which are related to conformal structures in a similar way as geodesics are related to metric structures, are discussed. `Conformal normal coordinates', `conformal Gauss systems' and their associated `normal connections', `normal frames' and `normal metrics' are introduced and used to study: (i) asymptotically simple solutions of Ric(g) = Λ g near conformal infinity, (ii) asymptotically simple solutions of Ric(g) = 0 with a past null infinity, which can be represented as the future null cone of a point i^, past timelike infinity. In the first case we define an ∞parameter family of (physical) Gauss systems near conformal infinity, in the second case a tenparameter family of (physical) Gauss systems covering a neighbourhood of i^. The behaviour of physical geodesics can be analysed in a particularly simple way in these coordinate systems. Each of these systems allows an extremely simple transition from the conformal analysis to the physical description of spacetime. For Λeta_{00} < 0 (DeSitter type solutions) all solutions are characterized in terms of the physical spacetime by their data on past timelike infinity. For Λ = 0 the conserved quantities of Newman and Penrose are characterized as the first nontrivial coefficient, given by the value of the rescaled Weyl tensor at i^, in an expansion of the physical field in a Gauss system of the type considered before.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 November 1987
 DOI:
 10.1098/rspa.1987.0139
 Bibcode:
 1987RSPSA.414..171F