New method for determining the light travel time in static, spherically symmetric spacetimes. Calculation of the terms of order $G^3$

TL;DR

This paper introduces a new iterative method to calculate photon travel times in static, spherically symmetric spacetimes, providing explicit third-order terms crucial for high-precision gravitational tests.

Contribution

It develops a novel perturbative approach to compute light travel times up to third order in $G$, including explicit expressions for impact parameters and travel times.

Findings

Derived explicit third-order terms for light travel time.

Confirmed the existence of a third-order enhanced term in conjunction.

Showed the method's relevance for precise measurements of the PPN parameter γ.

Abstract

A new iterative method for calculating the travel time of a photon as a function of the spatial positions of the emitter and the receiver in the field of a static, spherically symmetric body is presented. The components of the metric are assumed to be expressible in power series in $m/r$, with $m$ being half the Schwarzschild radius of the central body and $r$ a radial coordinate. The procedure exclusively works for a light ray which may be described as a perturbation in powers of $G$ of a Minkowskian null geodesic, with $G$ being the Newtonian gravitational constant. It is shown that the expansion of the travel time of a photon along such a ray only involves elementary integrals whatever the order of approximation. An expansion of the impact parameter in power series of $G$ is also obtained. The method is applied to explicitly calculate the perturbation expansions of the light travel time and the impact parameter up to the third order. The full expressions yielding the terms of order $G^3$ are new. The expression of the travel time confirms the existence of a third-order enhanced term when the emitter and the receiver are in conjunction relative to the central body. This term is shown to be necessary for determining the post-Newtonian parameter $\gamma$ at a level of accuracy of $10^{-8}$ with light rays grazing the Sun.