Light-cone coordinates based at a geodesic world line

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.

Figure 1

Advanced coordinates of a point $x$ relative to a world line $\gamma$. The advanced time $v$ selects a particular light cone, the unit vector ${\Omega }^a≔{\widehat{x}}^a/r$ selects a particular generator of this light cone, and the advanced distance $r$ selects a particular point on this generator.