A covariant formulation of relativistic mechanics

TL;DR

This paper develops a covariant framework for relativistic mechanics, enabling consistent equations of motion for various forces, including conservative, dissipative, and variable-mass effects, with applications to astrophysical scenarios.

Contribution

It introduces a covariant formulation of relativistic forces, extending known models and applying them to complex astrophysical phenomena like accretion and black hole infall.

Findings

Covariant formulations for Hooke's law and constant force.

Extension of Epstein drag, Chandrasekhar's dynamical friction, and Poynting-Robertson drag.

Numerical trajectories for black hole infall with accretion disk.

Abstract

Accretion disks surrounding compact objects, and other environmental factors, deviate satellites from geodetic motion. Unfortunately, setting up the equations of motion for such relativistic trajectories is not as simple as in Newtonian mechanics. The principle of general (or Lorentz) covariance and the mass-shell constraint make it difficult to parametrize physically adequate 4-forces. Here, we propose a solution to this old problem. We apply our framework to several conservative and dissipative forces. In particular, we propose covariant formulations for Hooke's law and the constant force, and compute the drag due to gravitational and hard-sphere collisions in dust, gas and radiation media. We recover and covariantly extend known forces such as Epstein drag, Chandrasekhar's dynamical friction and Poynting-Robertson drag. Variable-mass effects are also considered, namely Hoyle-Lyttleton accretion and the variable-mass rocket. We conclude with two applications: 1. The free-falling spring. We find that Hooke's law corrects the deviation equation by an effective Anti-de Sitter tidal force; 2. Black hole infall with drag. We numerically compute some trajectories on a Schwarzschild background supporting a dust-like accretion disk.