World function and time transfer: general post-Minkowskian expansions

In suitably chosen domains of spacetime, 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 3space relative to an observer can be derived from these time transfer functions, even if the metric is not stationary. We show how to derive 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 beta, gamma, delta.