
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 selfforce: a 2010 literature survey 

2.1 
Early work: DeWitt and
DeWitt; Smith and Will 

2.2 
Modesum method 

2.3 
Effectivesource
method 

2.4 
Quasilocal approach with “matched expansions” 

2.5 
Adiabatic
approximations 

2.6 
Physical consequences of the selfforce 
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 
4 
Coincidence limits 

4.1 
Computation of coincidence
limits 

4.2 
Derivation of Synge’s rule 
5 
Parallel propagator 

5.1 
Tetrad on


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

9 
Fermi normal coordinates 

9.1 
Fermi–Walker transport 

9.2 
Tetrad
and dual tetrad on 

9.3 
Fermi normal coordinates 

9.4 
Coordinate
displacements near 

9.5 
Metric near 

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 and the vector field


10.4 
Frame components of tensor fields on the world line 

10.5 
Coordinate
displacements near 

10.6 
Metric near 

10.7 
Transformation to angular
coordinates 

10.8 
Specialization to 
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 

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 

12.4 
Smooth part of 

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 
Lightcone 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 
Pointparticle limits and matched asymptotic
expansions 
21 
Selfconsistent 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 
Firstorder solution in the buffer region 

22.4 
Secondorder
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 
Secondorder expansions of the Ricci tensor 
B 
STF multipole
decompositions 

References 

Updates 

Figures 

Tables 