In Section 3 I introduce a number of
coordinate systems that play an important role in later parts of
the review. As a warmup exercise I first construct (in
Section 3.1) Riemann normal coordinates in
a neighbourhood of a reference point . I then move on (in
Section 3.2) to Fermi normal
coordinates [36], which
are defined in a neighbourhood of a world line
. The retarded coordinates, which are also based at a
world line and which were briefly introduced in Section 1.5, are covered systematically in
Section 3.3. The relationship between Fermi
and retarded coordinates is worked out in Section 3.4, which also locates the
advanced point
associated with a field point
. The presentation in Section 3 borrows heavily from Synge’s
book [55]. In fact, I am much
indebted to Synge for initiating the construction of retarded
coordinates in a neighbourhood of a world line. I have implemented
his program quite differently (Synge was interested in a large
neighbourhood of the world line in a weakly curved spacetime, while
I am interested in a small neighbourhood in a strongly curved
spacetime), but the idea is originally his.
In Section 4 I review the theory of Green’s
functions for (scalar, vectorial, and tensorial) wave equations in
curved spacetime. I begin in Section 4.1 with a pedagogical introduction
to the retarded and advanced Green’s functions for a massive scalar
field in flat spacetime; in this simple context the all-important
Hadamard decomposition [28] of the Green’s
function into “light-cone” and “tail” parts can be displayed
explicitly. The invariant Dirac functional is defined in
Section 4.2 along with its restrictions on
the past and future null cones of a reference point . The retarded, advanced, singular, and radiative
Green’s functions for the scalar wave equation are introduced in
Section 4.3. In Sections 4.4 and 4.5 I cover the vectorial and
tensorial wave equations, respectively. The presentation in
Section 4 is based partly on the paper by
DeWitt and Brehme [24
], but it is inspired
mostly by Friedlander’s book [27]. The
reader should be warned that in one important aspect, my notation
differs from the notation of DeWitt and Brehme: While they denote
the tail part of the Green’s function by
, I have
taken the liberty of eliminating the silly minus sign and I call it
instead
. The reader should also note that all
my Green’s functions are normalized in the same way, with a factor
of
multiplying a four-dimensional Dirac functional of
the right-hand side of the wave equation. (The gravitational
Green’s function is sometimes normalized with a
on the right-hand side.)
In Section 5 I compute the retarded, singular,
and radiative fields associated with a point scalar charge
(Section 5.1), a point electric charge
(Section 5.2), and a point mass
(Section 5.3). I provide two different
derivations for each of the equations of motion. The first type of
derivation was outlined previously: I follow Detweiler and
Whiting [23] and postulate that
only the radiative field exerts a force on the particle. In the
second type of derivation I take guidance from Quinn and
Wald [49
] and postulate that
the net force exerted on a point particle is given by an average of
the retarded field over a surface of constant proper distance
orthogonal to the world line - this rest-frame average is easily
carried out in Fermi normal coordinates. The averaged field is
still infinite on the world line, but the divergence points in the
direction of the acceleration vector and it can thus be removed by
mass renormalization. Such calculations show that while the
singular field does not affect the motion of the particle, it
nonetheless contributes to its inertia. In Section 5.4 I present an alternative
derivation of the MiSaTaQuWa equations of motion based on the
method of matched asymptotic expansions [35
, 31
, 58
, 19
, 1
, 20
]; the derivation
applies to a small nonrotating black hole instead of a point mass.
The ideas behind this derivation were contained in the original
paper by Mino, Sasaki, and Tanaka [39
], but the
implementation given here, which involves the retarded coordinates
of Section 3.3 and displays explicitly the
transformation between external and internal coordinates, is
original work.
Concluding remarks are presented in Section 5.5. Throughout this review I use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [40].