The Internal and External Metrics of a Rotating Ellipsoid Under Post-Newtonianian Approximation

Under the post-Newtonian approximation, the internal and external metrics of a rigidly rotating oblate spheroid filled by a uniform and incompressible perfect fluid are obtained. And the analytic solutions of post-Newtonian metric components are derived by using the series expansion in an ellipsoidal coordinate system. For this specific problem, there are only finite terms remaining in the series expansion, so the obtained results can be used to study particle motion under these metrics.