Riemannian Laplacian Formulation in Oblate Spheroidal Coordinate System Using the Golden Metric Tensor

Abstract

Studies have shown that most planetary bodies, the sun and earth inclusive, are better described as oblate spheroids rather than spherical. In this paper, using the Golden Metric tensor, we derive the expression for the Riemannian Laplacian in Oblate Spheroidal coordinate systems by the method of tensor analysis. The derived expression for the Riemannian Laplacian in Oblate Spheroidal coordinate systems reduce to pure Newtonian Laplacian in the limit c0 and contains post-Newtonian correction terms of all orders c-2. The result obtained here opens the way for applications of the Riemannian Laplacian to all bodies in the Spheroidal coordinate system.