perihelion shift in special relativity?

22 Mar 2023

Would there be a perihelion shift in a weakly perturbed metric with the line element

\[\mathrm{d}s^2 = \left(1+2\frac{\Phi}{c^2}\right)c^2\mathrm{d}t^2 - \left(1-2\frac{\Phi}{c^2}\right)\mathrm{d}x_i\mathrm{d}x^i\]

for a non-relativistically or relativistically orbiting planet? The gravitational potential is given by \(\Phi = -GM/r\).