Obtention of a Gravitational Force from a Relativistic Solution to Binet's Equation: The Schwarzschild's Case

TL;DR

This paper introduces a method to derive gravitational forces from relativistic solutions using Binet's equation, specifically applied to Schwarzschild spacetime, bridging classical and relativistic gravitational descriptions.

Contribution

It presents a general procedure to obtain gravitational forces from relativistic spacetime metrics, providing classical expressions that highlight relativistic effects in gravitational phenomena.

Findings

Derived the gravitational force for Schwarzschild spacetime.

Reproduced Newton's law in the appropriate limit.

Identified two orbital velocities for test particles.

Abstract

Making use of the classical Binet's equation a general procedure to obtain the gravitational force corresponding to an arbitrary 4-dimensional spacetime is presented. This method provides, for general relativistic scenarios, classics expressions that may help to visualize certain effects that Newton's theory can not explain. In particular, the force produced by a gravitational field which source is spherically symmetrical (Schwarzschild's spacetime) is obtained. Such expression uses a redefinition of the classical reduced mass, in the limit case it can be reduced to Newton's Universal Law of Gravitation and it produces two different orbital velocities for test particles that asimptotically coincide with the Newtonian one. PACS: 04.25.Nx, 95.10.Ce, 95.30.Sf. Keywords: Universal gravitational law, perihelionshift, Schwarzschild potential, reduced mass.