A Universal Tool for Determining the Time Delay and the Frequency Shift of Light: Synge's World function

Abstract

In almost all of the studies devoted to the time delay and the frequency shift of light, the calculations are based on the integration of the null geodesic equations. However, the above-mentioned effects can be calculated without integrating the geodesic equations if one is able to determine the bifunction $\Omega(x_A, x_B)$ giving half the squared geodesic distance between two points $x_A$ and $x_B$ (this bifunction may be called Synge's world function). In this lecture, $\Omega(x_A, x_B)$ is determined up to the order $1/c^3$ within the framework of the PPN formalism. The case of a stationary gravitational field generated by an isolated, slowly rotating axisymmetric body is studied in detail. The calculation of the time delay and the frequency shift is carried out up to the order $1/c^4$. Explicit formulae are obtained for the contributions of the mass, of the quadrupole moment and of the internal angular momentum when the only post-Newtonian parameters different from zero are $\beta$ and $\gamma$. It is shown that the frequency shift induced by the mass quadrupole moment of the Earth at the order $1/c^3$ will amount to $10^{-16}$ in spatial experiments like the ESA's Atomic Clock Ensemble in Space mission. Other contributions are briefly discussed.