# Geometrical Formulation of Relativistic Mechanics

## TL;DR

This paper develops a geometrical approach to relativistic mechanics by deriving covariant equations of motion, a modified Lorentz transformation, and relativistic effects directly from the metric, providing a unified framework.

## Contribution

It introduces a metric-based formulation of relativistic mechanics, including covariant equations, a deformed Euler-Lagrange equation, and a modified Lorentz transformation.

## Key findings

- Derived covariant equations of motion from the metric
- Formulated a modified local Lorentz transformation
- Compared non-relativistic limits with conventional formulations

## Abstract

The relativistic Lagrangian in presence of potentials was formulated directly from the metric, with the classical Lagrangian shown embedded within it. Using it we formulated covariant equations of motion, a deformed Euler-Lagrange equation, and relativistic Hamiltonian mechanics. We also formulate a modified local Lorentz transformation, such that the metric at a point is invariant only under the transformation defined at that point, and derive the formulae for time-dilation, length contraction, and gravitational redshift. Then we compare our formulation under non-relativistic approximations to the conventional ad-hoc formulation, and we briefly analyze the relativistic Lienard oscillator and the spacetime it implies.