Motion of small objects in curved spacetimes: An introduction to gravitational self-force

Abstract

In recent years, asymptotic approximation schemes have been developed to describe the motion of a small compact object through a vacuum background to any order in perturbation theory. The schemes are based on rigorous methods of matched asymptotic expansions, which account for the object's finite size, require no "regularization" of divergent quantities, and are valid for strong fields and relativistic speeds. Up to couplings of the object's multipole moments to the external background curvature, these schemes have established that at least through second order in perturbation theory, the object's motion satisfies a generalized equivalence principle: it moves on a geodesic of a certain smooth metric satisfying the vacuum Einstein equation. I describe the foundations of this result, particularly focusing on the fundamental notion of how a small object's motion is represented in perturbation theory. The three common representations of perturbed motion are (i) the "Gralla-Wald" description in terms of small deviations from a reference geodesic, (ii) the "self-consistent" description in terms of a worldline that obeys a self-accelerated equation of motion, and (iii) the "osculating geodesics" description, which utilizes both (i) and (ii). Because of the coordinate freedom in general relativity, any coordinate description of motion in perturbation theory is intimately related to the theory's gauge freedom. In this review I describe asymptotic solutions of the Einstein equations adapted to each of the three representations of motion, and I discuss the gauge freedom associated with each. I conclude with a discussion of how gauge freedom must be refined in the context of long-term dynamics.

100 pages, 3 figures, book contribution to Equations of Motion in Relativistic Gravity, edited by D. Puetzfeld et al., Fundamental Theories of Physics 179, Springer, 2015

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• Hamiltonian Formulation of the Conservative Self-Force Dynamics in the Kerr Geometry
• Faithful effective-one-body waveform of small-mass-ratio coalescing black hole binaries: the eccentric, nonspinning, case
• Comparing second-order gravitational self-force and effective one body waveforms from inspiralling, quasi-circular and nonspinning black hole binaries II: the large-mass-ratio case
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• Spin-multipole effects in binary black holes and the test-body limit
• Second-order perturbation theory: the problem of infinite mode coupling
• Towards a gravitational self force-informed effective-one-body waveform model for nonprecessing, eccentric, large-mass-ratio inspirals
• Second-order gravitational self-force in a highly regular gauge
• Spin-orbit precession for eccentric black hole binaries at first order in the mass ratio
• First Law of Compact Binary Mechanics with Gravitational-Wave Tails
• Gravitational waves from plunges into Gargantua
• Numerical-relativity validation of effective-one-body waveforms in the intermediate-mass-ratio regime
• Self-force correction to geodetic spin precession in Kerr spacetime
• Covariant Equations of Motion of Extended Bodies with Arbitrary Mass and Spin Multipoles
• Self-forces on static bodies in arbitrary dimensions
• Testing gravity with Extreme-Mass-Ratio Inspirals
• Spin and Quadrupole Couplings for High Spin Equatorial Intermediate Mass-ratio Coalescences
• Flux-balance laws for spinning bodies under the gravitational self-force
• Self-forces in arbitrary dimensions
• Foundations of the self-force problem in arbitrary dimensions
• Real modes and null memory contributions in effective-one-body models
• Second-order gravitational self-force in a highly regular gauge: Covariant and coordinate punctures
• Gravitational spin-orbit dynamics at the fifth-and-a-half post-Newtonian order
• Post-adiabatic self-force waveforms: slowly spinning primary and precessing secondary
• Optimizing the Placement of Numerical Relativity Simulations using a Mismatch Predicting Neural Network
• Toward Second-Order Self-Force for Eccentric Extreme-Mass Ratio Inspirals in Schwarzschild Spacetime
• Self-force on a static particle near a black hole
• Simple, efficient method of calculating the Detweiler-Whiting singular field to very high order
• Multipolar Particles in Helically Symmetric Spacetimes
• The first law of binary black hole scattering
• Particle-without-Particle: a practical pseudospectral collocation method for linear partial differential equations with distributional sources
• Covariant Equations of Motion Beyond the Spin-Dipole Particle Approximation
• Constants of motion in gravitational self-force theory
• Flux-balance laws in scalar self-force theory
• Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits
• Extreme mass-ratio inspirals as probes of scalar fields: eccentric equatorial orbits around Kerr black holes