The post-Newtonian metric of a self-gravitating and collapsing thin spherical shell

We calculate the metric for a self-gravitating and collapsing infinitely-thin spherical shell under the theory of post-Newtonian approximation, and successfully recover the shell's energy-momentum tensor from the achieved metric. The post-Newtonian dynamic equations of the test particles inside and outside the shell are achieved respectively. It is demonstrated that the radial motion of the spherically-symmetric system can contribute to the second gravitational potential and the gravitomagnetic field, inducing the time-dependent gravitational effects.