Lagrangian Solution of Schwarzschild-like Metric for an Elliptical Object
Abstract
The Lagrangian method was applied for a linearly transformed geodesic line element of a Schwarzschild-like solution instead of the tensor method. The solution shows that it is not only valid for spherical objects but also it is more comprehensive for elliptical celestial objects. Two types of kinetic and potential energy are the basis of the calculation. Hamiltonian and Lagrangian equality show that the problem has no potential energy. With this transformed geodesic line element, we obtained a new coefficient for the meridional advance of an experimental particle in Schwarzschild spacetime in terms of period, eccentricity, and mean distance. This new perigee equation is not only valid for the Schwarzschild metric (for a spherical object), but also more accurate for the Schwarzschild-like metric (for elliptical objects).