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.

arXiv comment21 pages + appendices; v2: revised introduction, added conclusion. Accepted in PRD

References in corpus (34)

• Observation of Gravitational Waves from a Binary Black Hole Merger
• Unified Schemes for Radio-Loud Active Galactic Nuclei
• Black holes, gravitational waves and fundamental physics: a roadmap
• The motion of point particles in curved spacetime
• Density of asteroids
• Dynamical Friction in a Gaseous Medium
• Self-force and radiation reaction in general relativity
• Stellar disk in the galactic center -- a remnant of a dense accretion disk?
• Candidate Electromagnetic Counterpart to the Binary Black Hole Merger Gravitational Wave Event S190521g
• Metal accretion onto white dwarfs caused by Poynting-Robertson drag on their debris disks
• 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
• The Angular Clustering of WISE-Selected AGN: Different Haloes for Obscured and Unobscured AGN
• Forced motion near black holes
• Hydrodynamic Drag on a Compact Star Orbiting a Supermassive Black Hole
• 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 Epicycles : another approach to geodesic deviations
• Orbital decay of satellites crossing an accretion disc
• Perturbation Theory Calculation of the Black Hole Elastic Scattering Cross Section
• Relativistic Bondi accretion for stiff equations of state
• Making the Relativistic Dynamics Equation Covariant: Explicit Solutions for Motion under a Constant Force
• Uniform Acceleration in General Relativity
• Experimental Realization of a Relativistic Harmonic Oscillator
• Accelerated observers and the notion of singular spacetime
• Lagrangian formulation of the general relativistic Poynting-Robertson effect
• Relativistic Gas Drag on Dust Grains and Implications
• Relativistic Dynamical Friction in the Weak Scattering Limit
• Friction forces in cosmological models
• Quantum Drag Forces on a Sphere Moving Through a Rarefied Gas
• Schwarzschild black hole embedded in a dust field: scattering of particles and drag force effects

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