A geometric relativistic dynamics under any conservative force

TL;DR

This paper develops a geometric relativistic dynamics framework for any static, conservative force, generalizing Newtonian mechanics and reproducing key tests of General Relativity.

Contribution

It introduces a new relativistic dynamics based on a generalized spacetime geometry derived from force potentials, extending Newtonian mechanics to relativistic regimes.

Findings

Reproduces classical tests of General Relativity

Provides a relativistic dynamics for any static, conservative force

Identifies and corrects deficiencies in Newtonian metric

Abstract

Riemann's principle "force equals geometry" provided the basis for Einstein's General Relativity - the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton's First Law - the \emph{Generalized Principle of Inertia} stating that: \emph{An inanimate object moves inertially, that is, with constant velocity, in \emph{its own} spacetime whose geometry is determined by the forces affecting it}. Classical Newtonian dynamics is treated within this framework, using a properly defined \emph{Newtonian metric} with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new \emph{Relativistic Newtonian Dynamics} for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.