Living Reviews in Relativity

"The Motion of Point Particles in Curved Spacetime"
by
Eric Poisson and Adam Pound and Ian Vega  

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Abstract
1 Introduction and summary
1.1 Invitation
1.2 Radiation reaction in flat spacetime
1.3 Green’s functions in flat spacetime
1.4 Green’s functions in curved spacetime
1.5 World line and retarded coordinates
1.6 Retarded, singular, and regular electromagnetic fields of a point electric charge
1.7 Motion of an electric charge in curved spacetime
1.8 Motion of a scalar charge in curved spacetime
1.9 Motion of a point mass, or a small body, in a background spacetime
1.10 Case study: static electric charge in Schwarzschild spacetime
1.11 Organization of this review
2 Computing the self-force: a 2010 literature survey
2.1 Early work: DeWitt and DeWitt; Smith and Will
2.2 Mode-sum method
2.3 Effective-source method
2.4 Quasilocal approach with “matched expansions”
2.5 Adiabatic approximations
2.6 Physical consequences of the self-force
I General Theory of Bitensors
3 Synge’s world function
3.1 Definition
3.2 Differentiation of the world function
3.3 Evaluation of first derivatives
3.4 Congruence of geodesics emanating from ′ x
4 Coincidence limits
4.1 Computation of coincidence limits
4.2 Derivation of Synge’s rule
5 Parallel propagator
5.1 Tetrad on β
5.2 Definition and properties of the parallel propagator
5.3 Coincidence limits
6 Expansion of bitensors near coincidence
6.1 General method
6.2 Special cases
6.3 Expansion of tensors
7 van Vleck determinant
7.1 Definition and properties
7.2 Derivations
II Coordinate Systems
8 Riemann normal coordinates
8.1 Definition and coordinate transformation
8.2 Metric near x ′
9 Fermi normal coordinates
9.1 Fermi–Walker transport
9.2 Tetrad and dual tetrad on γ
9.3 Fermi normal coordinates
9.4 Coordinate displacements near γ
9.5 Metric near γ
9.6 Thorne–Hartle–Zhang coordinates
10 Retarded coordinates
10.1 Geometrical elements
10.2 Definition of the retarded coordinates
10.3 The scalar field r(x) and the vector field kα (x )
10.4 Frame components of tensor fields on the world line
10.5 Coordinate displacements near γ
10.6 Metric near γ
10.7 Transformation to angular coordinates
10.8 Specialization to aμ = 0 = R μν
11 Transformation between Fermi and retarded coordinates; advanced point
11.1 From retarded to Fermi coordinates
11.2 From Fermi to retarded coordinates
11.3 Transformation of the tetrads at x
11.4 Advanced point
III Green’s Functions
12 Scalar Green’s functions in flat spacetime
12.1 Green’s equation for a massive scalar field
12.2 Integration over the source
12.3 Singular part of g(σ)
12.4 Smooth part of g(σ)
12.5 Advanced distributional methods
12.6 Alternative computation of the Green’s functions
13 Distributions in curved spacetime
13.1 Invariant Dirac distribution
13.2 Light-cone distributions
14 Scalar Green’s functions in curved spacetime
14.1 Green’s equation for a massless scalar field in curved spacetime
14.2 Hadamard construction of the Green’s functions
14.3 Reciprocity
14.4 Kirchhoff representation
14.5 Singular and regular Green’s functions
14.6 Example: Cosmological Green’s functions
15 Electromagnetic Green’s functions
15.1 Equations of electromagnetism
15.2 Hadamard construction of the Green’s functions
15.3 Reciprocity and Kirchhoff representation
15.4 Relation with scalar Green’s functions
15.5 Singular and regular Green’s functions
16 Gravitational Green’s functions
16.1 Equations of linearized gravity
16.2 Hadamard construction of the Green’s functions
16.3 Reciprocity and Kirchhoff representation
16.4 Relation with electromagnetic and scalar Green’s functions
16.5 Singular and regular Green’s functions
IV Motion of Point Particles
17 Motion of a scalar charge
17.1 Dynamics of a point scalar charge
17.2 Retarded potential near the world line
17.3 Field of a scalar charge in retarded coordinates
17.4 Field of a scalar charge in Fermi normal coordinates
17.5 Singular and regular fields
17.6 Equations of motion
18 Motion of an electric charge
18.1 Dynamics of a point electric charge
18.2 Retarded potential near the world line
18.3 Electromagnetic field in retarded coordinates
18.4 Electromagnetic field in Fermi normal coordinates
18.5 Singular and regular fields
18.6 Equations of motion
19 Motion of a point mass
19.1 Dynamics of a point mass
19.2 Retarded potentials near the world line
19.3 Gravitational field in retarded coordinates
19.4 Gravitational field in Fermi normal coordinates
19.5 Singular and regular fields
19.6 Equations of motion
V Motion of a Small Body
20 Point-particle limits and matched asymptotic expansions
21 Self-consistent expansion
21.1 Introduction
21.2 Field equations in outer expansion
21.3 Field equations in inner expansion
22 General expansion in the buffer region
22.1 Metric expansions
22.2 The form of the expansion
22.3 First-order solution in the buffer region
22.4 Second-order solution in the buffer region
22.5 The equation of motion
22.6 The effect of a gauge transformation on the force
23 Global solution in the external spacetime
23.1 Integral representation
23.2 Metric perturbation in Fermi coordinates
23.3 Equation of motion
24 Concluding remarks
24.1 The motion of a point particle
24.2 The motion of a small body
24.3 Beyond first order
25 Acknowledgments
Appendices
A Second-order expansions of the Ricci tensor
B STF multipole decompositions
Open References References
Updates
Figures
Tables