Complete Einstein’s Equations of Motion for Test Particles Exterior to Spherical Massive Bodies using a Varying Potential

In this article, a generalized varying gravitational scalar potential was used to completely define the metric tensors and coefficients of affine connections for spherical massive bodies whose tensor field varies with time, radial distance and polar angle. The completely defined metric tensors and coefficients of affine connections were used to study Einstein’s equations of motion for test particles within this field. The results obtained to the limit of c0 reduced to the corresponding Schwarzchild equations and to the limit of c2 , it contained additional terms not found in Schwarzchild equations which can be used in the study of blackhole and gravitational wave in this field and other astrophysical phenomena.