A covariant formulation of relativistic mechanics

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.

21 pages + appendices; v2: revised introduction, added conclusion. Accepted in PRD

References in corpus (12)

• Candidate Electromagnetic Counterpart to the Binary Black Hole Merger Gravitational Wave Event S190521g
• The influence of the hydrodynamic drag from an accretion torus on extreme mass-ratio inspirals
• Environmental Effects for Gravitational-wave Astrophysics
• Eccentricity evolution of compact binaries and applications to gravitational-wave physics
• Dynamical friction of black holes in ultralight dark matter
• Forced motion near black holes
• The last migration trap of compact objects in AGN accretion disc
• Relativistic dynamical friction in a collisional fluid
• The general relativistic Poynting-Robertson effect
• Relativistic Bondi accretion for stiff equations of state
• Relativistic Gas Drag on Dust Grains and Implications
• Friction forces in cosmological models

Cited by in corpus (4)

• Relativistic drag forces on black holes from scalar dark matter clouds of all sizes
• Fully Relativistic Treatment of Extreme Mass-Ratio Inspirals in Collisionless Environments
• Relativistic dynamical friction in stellar systems
• Captured are circularized: A relativistic treatment of extreme mass ratio inspirals crossing accretion disks