The motion of point particles in curved spacetime

Abstract

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

162 pages. Major update of the 2004 version (arXiv:gr-qc/0306052) initially published in Living Reviews in Relativity. This version contains a thorough literature review (as of 2010) and a lot of new material. v3 reflects referee comments

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• Gravitational-wave energy and other fluxes in ghost-free bigravity
• First Law of Mechanics for Spinning Compact Binaries: Dipolar Order
• The first law of binary black hole scattering
• Multipolar Particles in Helically Symmetric Spacetimes
• Tidal dissipation in binary neutron star inspirals from hyperon bulk viscosity: Phase modeling and parameter estimation bias
• Self-force on a static particle near a black hole
• Extreme mass ratio inspirals around topological stars
• Toward Second-Order Self-Force for Eccentric Extreme-Mass Ratio Inspirals in Schwarzschild Spacetime
• Measuring the gravitational field in General Relativity: From deviation equations and the gravitational compass to relativistic clock gradiometry
• Electromagnetic self-force in curved spacetime: New insights from the Janis-Newman algorithm
• Self-force on a scalar charge in a circular orbit about a Reissner-Nordström black hole
• Post-adiabatic waveforms from extreme mass ratio inspirals in the presence of dark matter
• Heat Kernel Asymptotics, Path Integrals and Infinite-Dimensional Determinants
• Damping of long wavelength gravitational waves by the intergalactic medium
• Geometric properties of a 2-D space-time arising in 4-D black hole physics
• On the Equivalence Principle and Electrodynamics of Moving Bodies
• Aspects of electrostatics in BTZ geometries
• The non-existence of the self-accelerating dipole, and related questions
• Spherical inspirals of spinning bodies into Kerr black holes
• Gravitational Waves Emitted by a Uniformly Accelerated Mass: The Role of Zero-Rindler-Energy Modes in the Classical and Quantum Descriptions
• Einstein's equations and the pseudo-entropy of pseudo-Riemannian information manifolds
• Radiating particle in the vicinity of the weakly charged Schwarzschild black hole
• Self-force on an arbitrarily coupled static scalar particle in a wormhole space-time
• Properties of an affine transport equation and its holonomy
• Self-action in Gravity
• Zero-point gravitational field equations
• Normal coordinates based on curved tangent space
• Spacetime atoms and extrinsic curvature of equi-geodesic surfaces
• Horizon Instability of the Extremal BTZ Black Hole
• Finite-Size Effects On The Self-Force
• Dirac branes for Dirichlet branes: Supergravity actions
• Electromagnetic self-force in the five dimensional Myers-Perry space time
• Covariant Equations of Motion Beyond the Spin-Dipole Particle Approximation
• Self-force of a scalar charge in the space-time of extreme charged anti-dilatonic wormhole
• Fixing three dimensional geometries from entanglement entropies of CFT${}_2$
• Free-falling motion of an elastic rigid rod towards a Schwarzschild black hole
• Algebraic symmetries of the observables on the sky: Variable emitters and observers
• Multipolar invariants and the eccentricity enhancement function parametrization of gravitational radiation
• A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT
• Regularization of a scalar charged particle for generic orbits in Kerr spacetime
• Electromagnetic fields and charges in expanding universes
• Quantum states for a minimum-length spacetime
• Discontinuous collocation methods and gravitational self-force applications
• Relieving scale disparity in binary black hole simulations
• Generalized nonlocal gravity framework based on Poincaré gauge theory
• Introduction to gravitational wave astronomy
• Post-Newtonian expansion of fluxes from a scalar charge on an inclined-spherical orbit about a Kerr black hole
• Scalar Self-force for High Spin Black Holes
• Self-force and fluid resonances
• Propagators in AdS for higher-derivative and nonlocal gravity: Heat kernel approach
• Shadow Ringing of Black Holes from Photon Sphere Quasinormal Modes
• Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities
• A Line Source in Minkowski for the de Sitter Spacetime Scalar Green's Function: Massive Case
• Self-force and the Huygens principle
• Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation
• First-Order Perturbative Hamiltonian Equations of Motion for a Point Particle Orbiting a Schwarzschild Black Hole
• The need for a nonlocal expansion in general relativity
• The precession of particle spin in spherical symmetric spacetimes
• Cosmological memory effect in scalar-tensor theories
• Spacetime Symmetries and Kepler's Third Law
• Flux-balance laws in scalar self-force theory
• Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution
• On the radiation field of a linearly accelerated charged particle in Born-Infeld theory
• A study of the radiation-reaction on a point charge that moves along a constant applied electric field in an electromagnetic Bopp-Landé-Thomas-Podolsky vacuum
• Self-force and the Schwarzschild star
• Generalized Perturbed Kepler Problem: Gravitational Wave Imprints from Eccentric Compact Binaries
• A frequency-domain implementation of the particle-without-particle approach to EMRIs
• Continuous body dynamics and the Mathisson-Papapetrou-Dixon equations
• On the self-force in electrodynamics and implications for gravity
• Conformal compactification and affine-null metric formulation of the Einstein equations
• Scalar self-interaction in the spacetime of a cosmic dispiration
• The Repulsive Effect of Covariant Effective Quantum Gravity
• Using thermodynamics to learn gravitational wave physics
• Post-Newtonian Dynamics of Radiating Charges: Canonical Formulation and Binary Inspiral Laws
• General-relativistic and non-ideal radiative cooling in neutron star magnetospheres
• LIV effects on the quantum stochastic motion in an acoustic FRW-geometry
• Secular evolution of orbital parameters for general bound orbits in Kerr spacetime
• Master Functions and Equations for Perturbations of Vacuum Spherically-Symmetric Spacetimes
• Hadamard Tail from Initial Data on the Light Cone
• Explicit triangular decoupling of the separated vector wave equation on Schwarzschild into scalar Regge-Wheeler equations
• Estimating High-Order Time Derivatives of Kerr Orbital Functionals
• Eternal binaries
• Fixing the AdS${}_3$ metric from the pure state entanglement entropies of CFT${}_2$
• Super Heat Kernel and One-Loop Divergence of Super Yang-Mills Theory in Conformal Supergravity
• The Einstein-Infeld-Hoffmann legacy in mathematical relativity. Part I: The classical motion of charged point particles
• World quantum gravity: Quantum test objects and Synge's world function
• The initial value problem for gravitational waves in conformal gravity
• Lectures on the Bondi--Metzner--Sachs group and related topics in infrared physics
• Local operators in the Sine-Gordon model: ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ and the stress tensor
• Exact solution of the DeWitt-Brehme-Hobbs equation in copropagating electromagnetic and gravitational waves
• Timing Gravity with Pulsars in the Strong Field
• Geodesic deviation to all orders via a tangent bundle formalism
• Notes from the bulk: metric dependence of the edge states of Chern-Simons theory
• Geometrical Interpretation of Electromagnetism in a 5-Dimensional Manifold
• Causal Poisson bracket via deformation quantization
• Semiclassical Gravity Beyond General Relativity: Insights from Torsion
• Hyperboloidal method for frequency-domain self-force calculations
• Multipole decomposition of the gravitational field of a point mass at the black hole horizon
• The Weyl anomaly in interacting quantum field theory on curved spacetimes
• Solvable two-particle systems with time-asymmetric interactions in de Sitter space
• On the order reduction
• Redshift factor and the small mass-ratio limit in binary black hole simulations
• Early Warning From Eccentric Compact Binaries: Template Initialization And Sub-dominant Mode Effects
• Spin Hall effects and the localization of massless spinning particles
• Adiabatic evolution due to the conservative scalar self-force during orbital resonances
• Prospects for detecting asteroid-mass primordial black holes in extreme mass-ratio inspirals with continuous gravitational waves
• Encounter between an extended hyperelastic body and a Schwarzschild black hole with quadrupole-order effects
• There and back again: Outspiraling motion in non-Kerr compact objects
• Vacuum polarization in the Schwarzschild black hole with a global monopole