Conformal Geodesics in General Relativity

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) = 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 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 Λeta₀₀ < 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.