World function and time transfer: general post-Minkowskian expansions

TL;DR

This paper develops a recursive perturbative approach to expand the world function in general relativity, enabling precise modeling of light deflection and time transfer in weak gravitational fields, with applications to non-stationary metrics.

Contribution

It introduces a method to compute the world function and time transfer functions to any order in G using line integrals along unperturbed geodesics, applicable to complex spacetime geometries.

Findings

Derived explicit formulas for world function expansion up to G^2.

Demonstrated the method with a static, spherically symmetric metric.

Showed how to determine light ray directions from time transfer functions.

Abstract

In suitably chosen domains of space-time, the world function may be a powerful tool for modelling the deflection of light and the time/frequency transfer. In this paper we work out a recursive procedure for expanding the world function into a perturbative series of ascending powers of the Newtonian gravitational constant G. We show rigorously that each perturbation term is given by a line integral taken along the unperturbed geodesic between two points. Once the world function is known, it becomes possible to determine the time transfer functions giving the propagation time of a photon between its emission and its reception. We establish that the direction of a light ray as measured in the 3-space relative to a given observer can be derived from these time transfer functions, even if the metric is not stationary. We show how to deduce these functions up to any given order in G from the perturbative expansion of the world function. To illustrate the method, we carry out the calculation of the world function and of the time transfer function outside a static, spherically symmetric body up to the order G^2, the metric containing three arbitrary parameters.