Maisalatee, A.U.Azos, M.M.Ewa, I.I.2023-12-142023-12-142021-04-071. Chifu EN. Gravitational fields exterior to a homogeneous spherical masses. The Abraham Zelmanov Journal. 2012;5:31- 67. 2. Howusu SXK . Einstein’s Geometrical Field Equations, Jos University Press Ltd;2010:34. 3. Chifu EN, Howusu SXK. Solution of Einstein’s geometrical gravitational field equations exterior to astrophysically real or hypothetical time varying distributions of mass within regions of spherical geometry. Progress in Physics. 2009;3:45-48. 4. Lumbi LW, Ewa II, Tsaku N. Einstein’s equations of motion for test particles exterior to spherical distributions of mass whose tensor field varies with time, radial distance and polar angle.Archives of Applied Science Research Library. 2014;6(5):36-41.https://keffi.nsuk.edu.ng/handle/20.500.14448/6002In 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.enEinstein’s equation; radial distance; polar angle; schwarzchild’s metric; tensorArticle