Light propagation in the gravitational field of one arbitrarily moving pointlike body in the 2PN approximation

An analytical solution for the light trajectory in the near zone of the gravitational field of one pointlike body in arbitrary slow motion in the post-post-Newtonian approximation is presented in harmonic gauge. Expressions for total light deflection and time delay are given. The presented solution is a further step toward high-precision astrometry aiming at nanoarcsecond level of accuracy.

Figure 1

A geometrical representation of the light trajectory $\mathit{x}(t)$ of Eq. (41) through the gravitational field of one pointlike massive body $A$ moving along an arbitrary worldline in slow motion $v_A\ll c$. At the same instant of coordinate time the body’s position is ${\mathit{x}}_A(t)$ (gray sphere). However, since gravitational action travels with the finite speed of light, the light ray at $\mathit{x}(t)$ is influenced by the gravitational field generated by the body at its retarded position ${\mathit{x}}_A(t^{\text{ret}})$ (black sphere). The spatial vector ${\mathit{r}}_A(t^{\text{ret}})$ is defined by Eq. (32) and points from the massive body $A$ at its retarded position toward the exact photon’s position at instant $t$.