Conformal geodesics in general relativity

Abstract

Conformal geodesics, space-time 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 ) Λ͠g 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 time-like infinity. In the first case we define an ∞-parameter family of (physical) Gauss systems near conformal infinity, in the second case a ten-parameter 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 space-time. For Λη 00 < 0 (De-Sitter type solutions) all solutions are characterized in terms of the physical space-time by their data on past time-like infinity. For Λ = 0 the conserved quantities of Newman and Penrose are characterized as the first non-trivial 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.