Relativistic Mechanics in Gravitational Field within Oblate Spheroidal Coordinates Based upon Riemannian Geometry for Rotating Homogeneous Mass Distribution

Abstract:

The emergence of the geometrical theory of gravitation (general relativity) by Albert Einstein in his quest to unity special relativity and the Newtonian law of universal gravitation, has led to several Mathematical approaches, for the exact and analytical solution for all gravitational fields in nature. The first and the famous analytical solution was the Schwarzchild’s which can be constructed by finding a mapping where the metric tensor takes a simple form i.e. the vanishing of the non-diagonal elements. In this paper, we construct exact solution of the Einstein geometrical gravitational field equation using Riemannian metric tensor called the golden metric tensor that was first developed by Howusu, in the year 2009, for the rotating homogeneous mass distribution within oblate Spheroidal Coordinates. The equations of motion for test particles in the Oblate Spheroidal Geometry were derived using the coefficient of affine connection. Then the law of conservation of momentum and energy are equivalently formulated using the generalized Lagrangian as compared to the analytical solution of the Schwarzchild’s gravitational field. We also derived the planetary equation of motion in the equatorial plane of the Oblate Spheroidal body for this gravitational field.