New tools for determining the light travel time in static, spherically symmetric spacetimes beyond the order $G^2$

This paper is mainly devoted to the determination of the travel time of a photon as a function of the positions of the emitter and the receiver in a large class of static, spherically symmetric spacetimes. Such a function - often called time transfer function - is of crucial interest for testing metric theories of gravity in the solar system. Until very recently, this function was known only up to the second order in the Newtonian gravitational constant $G$ for a 3-parameter family of static, spherically symmetric metrics generalizing the Schwarzschild metric. We present here two procedures enabling to determine - at least in principle - the time transfer function at any order of approximation when the components of the metric are expressible in power series of the Schwarzschild radius of the central body divided by the radial coordinate. These procedures exclusively work for light rays which may be described as perturbations in power series in $G$ of a Minkowskian null geodesic passing through the positions of the emitter and the receiver. It is shown that the two methodologies lead to the same expression for the time transfer function up to the third order in $G$. The second procedure presents the advantage of exclusively needing elementary integrations which may be performed with any symbolic computer program whatever the order of approximation. The vector functions characterizing the direction of light propagation at the points of emission and reception are derived up to the third order in $G$. The relevance of the third order terms in the time transfer function is briefly discussed for some solar system experiments.