Spatial and null infinity via advanced and retarded conformal factors

Details Spatial and null infinity via advanced and retarded conformal factors Spatial and null infinity via advanced and retarded conformal factors

2003-07-08

arxiv.org/abs/gr-qc/0307028v2

Phys.Rev. D68 (2003) 104015

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

11 revtex4 pages, 1 eps figure, correcting typos etc, unabridged (abridged version to be published)

Scientific paper

10.1103/PhysRevD.68.104015

A new approach to space-time asymptotics is presented, refining Penrose's idea of conformal transformations with infinity represented by the conformal boundary of space-time. Generalizing examples such as flat and Schwarzschild space-times, it is proposed that the Penrose conformal factor be a product of advanced and retarded conformal factors, which asymptotically relate physical and conformal null (light-like) coordinates and vanish at future and past null infinity respectively, with both vanishing at spatial infinity. A correspondingly refined definition of asymptotic flatness at both spatial and null infinity is given, including that the conformal boundary is locally a light cone, with spatial infinity as the vertex. It is shown how to choose the conformal factors so that this asymptotic light cone is locally a metric light cone. The theory is implemented in the spin-coefficient (or null-tetrad) formalism by a simple joint transformation of the spin-metric and spin-basis (or metric and tetrad). The advanced and retarded conformal factors may be used as expansion parameters near the respective null infinity, together with a dependent expansion parameter for both spatial and null infinity, essentially inverse radius. Asymptotic regularity conditions on the spin-coefficients are proposed, based on the conformal boundary locally being a smoothly embedded metric light cone. These conditions ensure that the Bondi-Sachs energy-flux integrals of ingoing and outgoing gravitational radiation decay at spatial infinity such that the total radiated energy is finite, and that the Bondi-Sachs energy-momentum has a unique limit at spatial infinity, coinciding with the uniquely rendered ADM energy-momentum.

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