The motion of a point electric charge in flat spacetime was the subject of active investigation since the early
work of Lorentz, Abrahams, Poincaré, and Dirac [56], until Gralla, Harte, and Wald produced a definitive
derivation of the equations motion [82
] with all the rigour that one should demand, without recourse to
postulates and renormalization procedures. (The field’s early history is well related in Ref. [154
].) In 1960
DeWitt and Brehme [54
] generalized Dirac’s result to curved spacetimes, and their calculation was
corrected by Hobbs [95
] several years later. In 1997 the motion of a point mass in a curved
background spacetime was investigated by Mino, Sasaki, and Tanaka [130
], who derived an
expression for the particle’s acceleration (which is not zero unless the particle is a test mass); the
same equations of motion were later obtained by Quinn and Wald [150
] using an axiomatic
approach. The case of a point scalar charge was finally considered by Quinn in 2000 [149
], and this
led to the realization that the mass of a scalar particle is not necessarily a constant of the
motion.
This article reviews the achievements described in the preceding paragraph; it is concerned with
the motion of a point scalar charge , a point electric charge
, and a point mass
in
a specified background spacetime with metric
. These particles carry with them fields
that behave as outgoing radiation in the wave zone. The radiation removes energy and angular
momentum from the particle, which then undergoes a radiation reaction – its world line cannot
be simply a geodesic of the background spacetime. The particle’s motion is affected by the
near-zone field which acts directly on the particle and produces a self-force. In curved spacetime the
self-force contains a radiation-reaction component that is directly associated with dissipative
effects, but it contains also a conservative component that is not associated with energy or
angular-momentum transport. The self-force is proportional to
in the case of a scalar charge,
proportional to
in the case of an electric charge, and proportional to
in the case of a point
mass.
In this review we derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely self-contained, and all relevant materials are developed ab initio. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (Part I), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Part II). One must next formulate a theory of Green’s functions in curved spacetimes (Part III), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Part IV). A dedicated reader, correctly skeptical that sense can be made of a point mass in general relativity, will also want to work through the last portion of the review (Part V), which provides a derivation of the equations of motion for a small, but physically extended, body; this reader will be reassured to find that the extended body follows the same motion as the point mass. The review is very long, but the satisfaction derived, we hope, will be commensurate.
In this introductory section we set the stage and present an impressionistic survey of what the review contains. This should help the reader get oriented and acquainted with some of the ideas and some of the notation. Enjoy!
Let us first consider the relatively simple and well-understood case of a point electric charge moving in
flat spacetime [154, 101
, 171]. The charge produces an electromagnetic vector potential
that satisfies
the wave equation
An immediate difficulty presents itself: the vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.
Difficult but not impossible. To find a way around this problem we note first that the situation
considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the
time-reversal invariance of Maxwell’s theory. A specific time direction was adopted when, among all
possible solutions to the wave equation, we chose , the retarded solution, as the physically relevant
solution. Choosing instead the advanced solution
would produce a time-reversed picture in which the
radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear
superposition
Our second key observation is that while the potential of Eq. (1.2) does not exert a force on the charged
particle, it is just as singular as the retarded potential in the vicinity of the world line. This follows from the
fact that
,
, and
all satisfy Eq. (1.1
), whose source term is infinite on the world line. So
while the wave-zone behaviours of these solutions are very different (with the retarded solution describing
outgoing waves, the advanced solution describing incoming waves, and the symmetric solution
describing standing waves), the three vector potentials share the same singular behaviour near the
world line – all three electromagnetic fields are dominated by the particle’s Coulomb field and
the different asymptotic conditions make no difference close to the particle. This observation
gives us an alternative interpretation for the subscript ‘S’: it stands for ‘singular’ as well as
‘symmetric’.
Because is just as singular as
, removing it from the retarded solution gives rise to a
potential that is well behaved in a neighbourhood of the world line. And because
is known not to
affect the motion of the charged particle, this new potential must be entirely responsible for the radiation
reaction. We therefore introduce the new potential
The self-action of the charge’s own field is now clarified: a singular potential can be removed from
the retarded potential and shown not to affect the motion of the particle. What remains is a well-behaved
potential
that must be solely responsible for the radiation reaction. From the regular potential we
form an electromagnetic field tensor
and we take the particle’s equations of motion
to be
To establish that the singular field exerts no force on the particle requires a careful analysis that is
presented in the bulk of the paper. What really happens is that, because the particle is a monopole source
for the electromagnetic field, the singular field is locally isotropic around the particle; it therefore exerts no
force, but contributes to the particle’s inertia and renormalizes its mass. In fact, one could do without a
decomposition of the field into singular and regular solutions, and instead construct the force by using the
retarded field and averaging it over a small sphere around the particle, as was done by Quinn and Wald
[150]. In the body of this review we will use both methods and emphasize the equivalence of
the results. We will, however, give some emphasis to the decomposition because it provides a
compelling physical interpretation of the self-force as an interaction with a free electromagnetic
field.
To see how Eq. (1.5) can eventually be generalized to curved spacetimes, we introduce a new layer of
mathematical formalism and show that the decomposition of the retarded potential into singular and
regular pieces can be performed at the level of the Green’s functions associated with Eq. (1.1
). The
retarded solution to the wave equation can be expressed as
From the retarded and advanced Green’s functions we can define a singular Green’s function by
and a regular two-point function by By virtue of Eq. (1.8 Equation (1.6) implies that the retarded potential at
is generated by a single event in
spacetime: the intersection of the world line and
’s past light cone (see Figure 1
). We shall call
this the retarded point associated with
and denote it
;
is the retarded time, the
value of the proper-time parameter at the retarded point. Similarly we find that the advanced
potential of Eq. (1.7
) is generated by the intersection of the world line and the future light cone
of the field point
. We shall call this the advanced point associated with
and denote
it
;
is the advanced time, the value of the proper-time parameter at the advanced
point.
In a curved spacetime with metric the wave equation for the vector potential becomes
The causal structure of the Green’s functions is richer in curved spacetime: While in flat spacetime the
retarded Green’s function has support only on the future light cone of , in curved spacetime its support
extends inside the light cone as well;
is therefore nonzero when
, which denotes
the chronological future of
. This property reflects the fact that in curved spacetime, electromagnetic
waves propagate not just at the speed of light, but at all speeds smaller than or equal to the
speed of light; the delay is caused by an interaction between the radiation and the spacetime
curvature. A direct implication of this property is that the retarded potential at
is now
generated by the point charge during its entire history prior to the retarded time
associated
with
: the potential depends on the particle’s state of motion for all times
(see
Figure 2
).
Similar statements can be made about the advanced Green’s function and the advanced solution to the
wave equation. While in flat spacetime the advanced Green’s function has support only on the past light
cone of , in curved spacetime its support extends inside the light cone, and
is nonzero
when
, which denotes the chronological past of
. This implies that the advanced potential
at
is generated by the point charge during its entire future history following the advanced time
associated with
: the potential depends on the particle’s state of motion for all times
.
The physically relevant solution to Eq. (1.13) is obviously the retarded potential
, and as in
flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike
nature of the source, and the presence of spacetime curvature does not change the fact that the potential
diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the
retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall
attempt to decompose the retarded solution into a singular part that exerts no force, and a regular part
that produces the entire self-force.
To decompose the retarded Green’s function into singular and regular parts is not a straightforward task
in curved spacetime. The flat-spacetime definition for the singular Green’s function, Eq. (1.9), cannot be
adopted without modification: While the combination half-retarded plus half-advanced Green’s functions
does have the property of being symmetric, and while the resulting vector potential would
be a solution to Eq. (1.13
), this candidate for the singular Green’s function would produce a
self-force with an unacceptable dependence on the particle’s future history. For suppose that
we made this choice. Then the regular two-point function would be given by the combination
half-retarded minus half-advanced Green’s functions, just as in flat spacetime. The resulting potential
would satisfy the homogeneous wave equation, and it would be regular on the world line, but it
would also depend on the particle’s entire history, both past (through the retarded Green’s
function) and future (through the advanced Green’s function). More precisely stated, we would
find that the regular potential at
depends on the particle’s state of motion at all times
outside the interval
; in the limit where
approaches the world line, this
interval shrinks to nothing, and we would find that the regular potential is generated by the
complete history of the particle. A self-force constructed from this potential would be highly
noncausal, and we are compelled to reject these definitions for the singular and regular Green’s
functions.
The proper definitions were identified by Detweiler and Whiting [53], who proposed the following
generalization to Eq. (1.9
):
The potential constructed from the singular Green’s function can now be seen to depend on the
particle’s state of motion at times
restricted to the interval
(see Figure 3
). Because this
potential satisfies Eq. (1.13
), it is just as singular as the retarded potential in the vicinity of the world line.
And because the singular Green’s function is symmetric in its arguments, the singular potential can be
shown to exert no force on the charged particle. (This requires a lengthy analysis that will be presented in
the bulk of the paper.)
The Detweiler–Whiting [53] definition for the regular two-point function is then
From the regular potential we form an electromagnetic field tensor and the
curved-spacetime generalization to Eq. (1.4
) is
To flesh out the ideas contained in the preceding subsection we add yet another layer of mathematical formalism and construct a convenient coordinate system to chart a neighbourhood of the particle’s world line. In the next subsection we will display explicit expressions for the retarded, singular, and regular fields of a point electric charge.
Let be the world line of a point particle in a curved spacetime. It is described by parametric
relations
in which
is proper time. Its tangent vector is
and its acceleration is
; we shall also encounter
.
On we erect an orthonormal basis that consists of the four-velocity
and three spatial vectors
labelled by a frame index
. These vectors satisfy the relations
,
, and
. We take the spatial vectors to be Fermi–Walker transported on the
world line:
, where
Consider a point in a neighbourhood of the world line
. We assume that
is sufficiently close
to the world line that a unique geodesic links
to any neighbouring point
on
. The two-point
function
, known as Synge’s world function [169
], is numerically equal to half the squared geodesic
distance between
and
; it is positive if
and
are spacelike related, negative if they are
timelike related, and
is zero if
and
are linked by a null geodesic. We denote its gradient
by
, and
gives a meaningful notion of a separation vector (pointing from
to
).
To construct a coordinate system in this neighbourhood we locate the unique point on
which is linked to
by a future-directed null geodesic (this geodesic is directed from
to
); we
shall refer to
as the retarded point associated with
, and
will be called the retarded time. To
tensors at
we assign indices
,
, …; this will distinguish them from tensors at a generic point
on the world line, to which we have assigned indices
,
, …. We have
and
is a null vector that can be interpreted as the separation between
and
.
The retarded coordinates of the point are
, where
are the frame components
of the separation vector. They come with a straightforward interpretation (see Figure 4
). The invariant
quantity
To tensors at we assign indices
,
, …. These tensors will be decomposed in a tetrad
that is constructed as follows: Given
we locate its associated retarded point
on the world line, as well as the null geodesic that links these two points; we then take the
tetrad
at
and parallel transport it to
along the null geodesic to obtain
.
The retarded solution to Eq. (1.13) is
From the vector potential we form the electromagnetic field tensor , which we decompose in the
tetrad
introduced at the end of Section 1.5. We then express the frame components of the
field tensor in retarded coordinates, in the form of an expansion in powers of
. This gives
The expansion of near the world line does indeed reveal many singular terms. We first
recognize terms that diverge when
; for example the Coulomb field
diverges as
when we
approach the world line. But there are also terms that, though they stay bounded in the limit, possess a
directional ambiguity at
; for example
contains a term proportional to
whose limit
depends on the direction of approach.
This singularity structure is perfectly reproduced by the singular field obtained from the potential
The difference between the retarded and singular fields defines the regular field . Its frame
components are
We have argued in Section 1.4 that the self-force acting on a point electric charge is produced by the
regular field, and that the charge’s equations of motion should take the form of ,
where
is an external force also acting on the particle. Substituting Eq. (1.32
) gives
Equation (1.33) is the result that was first derived by DeWitt and Brehme [54
] and later corrected by
Hobbs [95
]. (The original version of the equation did not include the Ricci-tensor term.) In flat spacetime
the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere
within the domain of integration), and Eq. (1.33
) reduces to Dirac’s result of Eq. (1.5
). In curved
spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an
external force: it is then given by the tail integral, which represents radiation emitted earlier and coming
back to the particle after interacting with the spacetime curvature. This delayed action implies that in
general, the self-force is nonlocal in time: it depends not only on the current state of motion of the particle,
but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that
the physical process that leads to Eq. (1.33
) is simply an interaction between the charge and a
free electromagnetic field
; it is this field that carries the information about the charge’s
past.
The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the
electromagnetic theory. The particle’s charge produces a scalar field
which satisfies a wave
equation
The scalar field of Eq. (1.35) diverges on the world line and its singular part
must be removed
before Eqs. (1.36
) and (1.37
) can be evaluated. This procedure produces the regular field
, and it is
this field (which satisfies the homogeneous wave equation) that gives rise to a self-force. The gradient of the
regular field takes the form of
Substitution of Eq. (1.38) into Eqs. (1.36
) and (1.37
) gives the equations of motion of a point scalar
charge. (At this stage we introduce an external force
and reduce the order of the differential
equation.) The acceleration is given by
In flat spacetime the Ricci-tensor term and the tail integral disappear and Eq. (1.40) takes
the form of Eq. (1.5
) with
replacing the factor of
. In this simple case
Eq. (1.41
) reduces to
and the mass is in fact a constant. This property remains true in
a conformally flat spacetime when the wave equation is conformally invariant (
):
in this case the Green’s function possesses only a light-cone part and the right-hand side of
Eq. (1.41
) vanishes. In generic situations the mass of a point scalar charge will vary with proper
time.
The case of a point mass moving in a specified background spacetime presents itself with a serious
conceptual challenge, as the fundamental equations of the theory are nonlinear and the very notion of a
“point mass” is somewhat misguided. Nevertheless, to the extent that the perturbation created by
the point mass can be considered to be “small”, the problem can be formulated in close analogy with what
was presented before.
We take the metric of the background spacetime to be a solution of the Einstein field equations
in vacuum. (We impose this condition globally.) We describe the gravitational perturbation
produced by a point particle of mass
in terms of trace-reversed potentials
defined by
Equations of motion for the point mass can be obtained by formally demanding that the motion be
geodesic in the perturbed spacetime with metric . After a mapping to the background
spacetime, the equations of motion take the form of
We now remove from the retarded perturbation and postulate that it is the regular field
that should act on the particle. (Note that
satisfies the same wave equation as the retarded
potentials, but that
is a free gravitational field that satisfies the homogeneous wave equation.) On the
world line we have
The equations of motion of Eq. (1.48) were first derived by Mino, Sasaki, and Tanaka [130
], and then
reproduced with a different analysis by Quinn and Wald [150
]. They are now known as the MiSaTaQuWa
equations of motion. As noted by these authors, the MiSaTaQuWa equation has the appearance of the
geodesic equation in a metric
. Detweiler and Whiting [53
] have contributed the more
compelling interpretation that the motion is actually geodesic in a spacetime with metric
. The
distinction is important: Unlike the first version of the metric, the Detweiler-Whiting metric is regular on
the world line and satisfies the Einstein field equations in vacuum; and because it is a solution
to the field equations, it can be viewed as a physical metric — specifically, the metric of the
background spacetime perturbed by a free field produced by the particle at an earlier stage of its
history.
While Eq. (1.48) does indeed give the correct equations of motion for a small mass
moving in a
background spacetime with metric
, the derivation outlined here leaves much to be desired – to what
extent should we trust an analysis based on the existence of a point mass? As a partial answer to this
question, Mino, Sasaki, and Tanaka [130
] produced an alternative derivation of their result, which involved
a small nonrotating black hole instead of a point mass. In this alternative derivation, the metric of the black
hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the
background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the
vacuum field equations determines the motion of the black hole: it must move according to Eq. (1.48
). This
alternative derivation (which was given a different implementation in Ref. [142
]) is entirely free of
singularities (except deep within the black hole), and it suggests that that the MiSaTaQuWa
equations can be trusted to describe the motion of any gravitating body in a curved background
spacetime (so long as the body’s internal structure can be ignored). This derivation, however,
was limited to the case of a non-rotating black hole, and it relied on a number of unjustified
and sometimes unstated assumptions [83
, 144
, 145
]. The conclusion was made firm by the
more rigorous analysis of Gralla and Wald [83
] (as extended by Pound [144
]), who showed
that the MiSaTaQuWa equations apply to any sufficiently compact body of arbitrary internal
structure.
It is important to understand that unlike Eqs. (1.33) and (1.40
), which are true tensorial equations,
Eq. (1.48
) reflects a specific choice of coordinate system and its form would not be preserved under a
coordinate transformation. In other words, the MiSaTaQuWa equations are not gauge invariant, and they
depend upon the Lorenz gauge condition
. Barack and Ori [17
] have shown that under a
coordinate transformation of the form
, where
are the coordinates of the background
spacetime and
is a smooth vector field of order
, the particle’s acceleration changes according to
, where
The gravitational self-force possesses a physical significance that is not shared by its scalar and electromagnetic analogues, because the motion of a small body in the strong gravitational field of a much larger body is a problem of direct relevance to gravitational-wave astronomy. Indeed, extreme-mass-ratio inspirals, involving solar-mass compact objects moving around massive black holes of the sort found in galactic cores, have been identified as promising sources of low-frequency gravitational waves for space-based interferometric detectors such as the proposed Laser Interferometer Space Antenna (LISA [115]). These systems involve highly eccentric, nonequatorial, and relativistic orbits around rapidly rotating black holes, and the waves produced by such orbital motions are rich in information concerning the strongest gravitational fields in the Universe. This information will be extractable from the LISA data stream, but the extraction depends on sophisticated data-analysis strategies that require a detailed and accurate modeling of the source. This modeling involves formulating the equations of motion for the small body in the field of the rotating black hole, as well as a consistent incorporation of the motion into a wave-generation formalism. In short, the extraction of this wealth of information relies on a successful evaluation of the gravitational self-force.
The finite-mass corrections to the orbital motion are important. For concreteness, let us assume that the
orbiting body is a black hole of mass and that the central black hole has a mass
. Let us also assume that the small black hole is in the deep field of the large hole, near the
innermost stable circular orbit, so that its orbital period
is of the order of minutes. The gravitational
waves produced by the orbital motion have frequencies
of the order of the mHz, which is well within
LISA’s frequency band. The radiative losses drive the orbital motion toward a final plunge into the large
black hole; this occurs over a radiation-reaction timescale
of the order of a year, during which
the system will go through a number of wave cycles of the order of
. The role of the
gravitational self-force is precisely to describe this orbital evolution toward the final plunge. While at any
given time the self-force provides fractional corrections of order
to the motion of the small
black hole, these build up over a number of orbital cycles of order
to produce a large
cumulative effect. As will be discussed in some detail in Section 2.6, the gravitational self-force is
important, because it drives large secular changes in the orbital motion of an extreme-mass-ratio
binary.
One of the first self-force calculations ever performed for a curved spacetime was presented by Smith and
Will [163]. They considered an electric charge
held in place at position
outside a Schwarzschild
black hole of mass
. Such a static particle must be maintained in position with an external force that
compensates for the black hole’s attraction. For a particle without electric charge this force is directed
outward, and its radial component in Schwarzschild coordinates is given by
, where
is the
particle’s mass,
is the usual metric factor, and a prime indicates differentiation with
respect to
, so that
. Smith and Will found that for a particle of charge
, the external
force is given instead by
. The second term is contributed by the
electromagnetic self-force, and implies that the external force is smaller for a charged particle. This
means that the electromagnetic self-force acting on the particle is directed outward and given by
The repulsive nature of the electromagnetic self-force acting on a static charge outside a black
hole is unexpected. In an attempt to gain some intuition about this result, it is useful to recall
that a black-hole horizon always acts as perfect conductor, because the electrostatic potential
is necessarily uniform across its surface. It is then tempting to imagine that the self-force
should result from a fictitious distribution of induced charge on the horizon, and that it could be
estimated on the basis of an elementary model involving a spherical conductor. Let us, therefore,
calculate the electric field produced by a point charge
situated outside a spherical conductor of
radius
. The charge is placed at a distance
from the centre of the conductor, which is
taken at first to be grounded. The electrostatic potential produced by the charge can easily
be obtained with the method of images. It is found that an image charge
is
situated at a distance
from the centre of the conductor, and the potential is given
by
, where
is the distance to the charge, while
is the distance to
the image charge. The first term can be identified with the singular potential
, and the
associated electric field exerts no force on the point charge. The second term is the regular
potential
, and the associated field is entirely responsible for the self-force. The regular
electric field is
, and the self-force is
. A simple computation returns
The computation of the self-force in the black-hole case is almost as straightforward. The exact solution
to Maxwell’s equations that describes a point charge situated
and
in the Schwarzschild
spacetime is given by
The identification of Copson’s potential with the singular potential is dictated by the fact that its
associated electric field
is isotropic around the charge
and therefore exerts no force. The
self-force comes entirely from the monopole potential, which describes a (fictitious) charge
situated at
. Because this charge is of the same sign as the original charge
, the self-force is
repulsive. More precisely stated, we find that the regular piece of the electric field is given by
We should remark that the identification of and
with the Detweiler–Whiting singular and
regular fields, respectively, is a matter of conjecture. Although
and
satisfy the essential
properties of the Detweiler–Whiting decomposition – being, respectively, a regular homogenous
solution and a singular solution sourced by the particle – one should accept the possibility
that they may not be the actual Detweiler–Whiting fields. It is a topic for future research to
investigate the precise relation between the Copson field and the Detweiler–Whiting singular
field.
It is instructive to compare the electromagnetic self-force produced by the presence of a grounded
conductor to the self-force produced by the presence of a black hole. In the case of a conductor, the total
induced charge on the conducting surface is , and it is this charge that is responsible for the
attractive self-force; the induced charge is supplied by the electrodes that keep the conductor grounded. In
the case of a black hole, there is no external apparatus that can supply such a charge, and the total
induced charge on the horizon necessarily vanishes. The origin of the self-force is therefore
very different in this case. As we have seen, the self-force is produced by a fictitious charge
situated at the centre of black hole; and because this charge is positive, the self-force is
repulsive.
After a detailed review of the literature in Section 2, the main body of the review begins in Part I
(Sections 3 to 7) with a description of the general theory of bitensors, the name designating tensorial
functions of two points in spacetime. We introduce Synge’s world function and its derivatives in
Section 3, the parallel propagator
in Section 5, and the van Vleck determinant
in
Section 7. An important portion of the theory (covered in Sections 4 and 6) is concerned with the
expansion of bitensors when
is very close to
; expansions such as those displayed in
Eqs. (1.23
) and (1.24
) are based on these techniques. The presentation in Part I borrows heavily
from Synge’s book [169
] and the article by DeWitt and Brehme [54
]. These two sources use
different conventions for the Riemann tensor, and we have adopted Synge’s conventions (which
agree with those of Misner, Thorne, and Wheeler [131
]). The reader is therefore warned that
formulae derived in Part I may look superficially different from those found in DeWitt and
Brehme.
In Part II (Sections 8 to 11) we introduce a number of coordinate systems that play an important role
in later parts of the review. As a warmup exercise we first construct (in Section 8) Riemann normal
coordinates in a neighbourhood of a reference point . We then move on (in Section 9) to Fermi
normal coordinates [122], 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 10. The relationship between Fermi and retarded coordinates
is worked out in Section 11, which also locates the advanced point
associated with a
field point
. The presentation in Part II borrows heavily from Synge’s book [169]. In fact,
we are much indebted to Synge for initiating the construction of retarded coordinates in a
neighbourhood of a world line. We have implemented his program quite differently (Synge was
interested in a large neighbourhood of the world line in a weakly curved spacetime, while we are
interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally
his.
In Part III (Sections 12 to 16) we review the theory of Green’s functions for (scalar, vectorial, and
tensorial) wave equations in curved spacetime. We begin in Section 12 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 [88] of the Green’s function into
“light-cone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in
Section 13 along with its restrictions on the past and future null cones of a reference point . The
retarded, advanced, singular, and regular Green’s functions for the scalar wave equation are
introduced in Section 14. In Sections 15 and 16 we cover the vectorial and tensorial wave
equations, respectively. The presentation in Part III is based partly on the paper by DeWitt
and Brehme [54
], but it is inspired mostly by Friedlander’s book [71]. The reader should be
warned that in one important aspect, our notation differs from the notation of DeWitt and
Brehme: While they denote the tail part of the Green’s function by
, we have taken the
liberty of eliminating the silly minus sign and call it instead
. The reader should also
note that all our 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 Part IV (Sections 17 to 19) we compute the retarded, singular, and regular fields associated with a
point scalar charge (Section 17), a point electric charge (Section 18), and a point mass (Section 19). We
provide two different derivations for each of the equations of motion. The first type of derivation was
outlined previously: We follow Detweiler and Whiting [53] and postulate that only the regular field
exerts a force on the particle. In the second type of derivation we take guidance from Quinn and
Wald [150
] 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 Part V (Sections 20 to 23), we show that at linear order in the body’s mass , an
extended body behaves just as a point mass, and except for the effects of the body’s spin, the
world line representing its mean motion is governed by the MiSaTaQuWa equation. At this
order, therefore, the picture of a point particle interacting with its own field, and the results
obtained from this picture, is justified. Our derivation utilizes the method of matched asymptotic
expansions, with an inner expansion accurate near the body and an outer expansion accurate
everywhere else. The equation of motion of the body’s world line, suitably defined, is calculated by
solving the Einstein equation in a buffer region around the body, where both expansions are
accurate.
Concluding remarks are presented in Section 24, and technical developments that are required in Part V are relegated to Appendices. Throughout this review we use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [131].
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Living Rev. Relativity 14, (2011), 7
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