2PN Light Propagation in the Scalar-Tensor Theory: an $N$-Point-Masses Case

Within the framework of the scalar-tensor theory (STT), its second post-Newtonian (2PN) approximation is obtained with Chandrasekhar's approach. By focusing on an $N$-point-masses system as the first step, we reduce the metric to its 2PN form for light propagation. Unlike previous works, at 2PN order, we abandon the hierarchized hypothesis and do not assume two parametrized post-Newtonian (PPN) parameters $\gamma$ and $\beta$ to be unity. We find that although there exist $\gamma$ and $\beta$ in the 2PN metric, only $\gamma$ appears in the 2PN equations of light. As a simple example for applications, a gauge-invariant angle between the directions of two incoming photons for a differential measurement is investigated after the light trajectory is solved in a static and spherically symmetric spacetime. It shows the deviation from the general relativity (GR) $\delta\theta_{\mathrm{STT}}$ does \emph{not} depend on $\beta$ even at 2PN level in this circumstance, which is consistent with previous results. A more complicated application is light deflection in a 2-point-masses system. We consider a case that the light propagation time is much less than the time scale of its orbital motion and thus treat it as a static system. The 2-body effect at 2PN level originating from relaxing the hierarchized hypothesis is calculated. Our analysis shows the 2PN 2-body effect in the Solar System is one order of magnitude less than future $\sim 1$ nas experiments, while this effect could be comparable with 1PN component of $\delta\theta_{\mathrm{STT}}$ in a binary system with two Sun-like stars and separation by $\sim 0.1$ AU if an experiment would be able to measure $\gamma-1$ down to $\sim 10^{-6}$.