Light-cone coordinates based at a geodesic world line

TL;DR

This paper develops a coordinate system based on light cones centered on a geodesic in curved spacetime, expressing the metric as an expansion involving the Riemann tensor, with applications to cosmology and magnetic universes.

Contribution

It introduces a new light-cone coordinate system centered on a geodesic, expanding the metric in terms of Riemann tensor components for arbitrary curved spacetimes.

Findings

Coordinate system constructed and expressed as a Riemann tensor expansion.

Applied to spatially-flat cosmology with a comoving world line.

Applied to Melvin's magnetic universe with an axisymmetric observer.

Abstract

Continuing work initiated in an earlier publication [Phys. Rev. D 69, 084007 (2004)], we construct a system of light-cone coordinates based at a geodesic world line of an arbitrary curved spacetime. The construction involves (i) an advanced-time or a retarded-time coordinate that labels past or future light cones centered on the world line, (ii) a radial coordinate that is an affine parameter on the null generators of these light cones, and (iii) angular coordinates that are constant on each generator. The spacetime metric is calculated in the light-cone coordinates, and it is expressed as an expansion in powers of the radial coordinate in terms of the irreducible components of the Riemann tensor evaluated on the world line. The formalism is illustrated in two simple applications, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer placed on the axis of symmetry of Melvin's magnetic universe.