For each of the three cases (scalar, electromagnetic, and gravitational) I have presented two different derivations of the equations of motion. The first derivation is based on a spatial averaging of the retarded field, and the second is based on a decomposition of the retarded field into singular and radiative fields. In the gravitational case, a third derivation, based on matched asymptotic expansions, was also presented. These derivations will be reviewed below, but I want first to explain why I have omitted to present a fourth derivation, based on energy-momentum conservation, in spite of the fact that historically, it is one of the most important.
Conservation of energy-momentum was used by
Dirac [25] to derive the
equations of motion of a point electric charge in flat spacetime,
and the same method was adopted by DeWitt and Brehme [24
] in their
generalization of Dirac’s work to curved spacetimes. This method
was also one of the starting points of Mino, Sasaki, and
Tanaka [39
] in their
calculation of the gravitational self-force. I have not discussed
this method for two reasons. First, it is technically more
difficult to implement than the methods presented in this review
(considerably longer computations are involved). Second, it is
difficult to endow this method with an adequate level of rigour, to
the point that it is perhaps less convincing than the methods
presented in this review. While the level of rigour achieved in
flat spacetime is now quite satisfactory [56
], I do not believe
the same can be said of the generalization to curved spacetimes.
(But it should be possible to improve on this matter.)
The method is based on the conservation equation
, where the stress-energy tensor
includes a contribution from the particle and a
contribution from the field; the particle’s contribution is a Dirac
functional on the world line, and the field’s contribution diverges
as
near the world line. (I am using retarded
coordinates in this discussion.) While in flat spacetime the
differential statement of energy-momentum conservation can
immediately be turned into an integral statement, the same is not
true in a curved spacetime (unless the spacetime possesses at least
one Killing vector). To proceed it is necessary to rewrite the
conservation equation as
where is a parallel propagator
from
to an arbitrary point
on the world line.
Integrating this equation over the interior of a world-tube segment
that consists of a “wall” of constant
and two “caps” of
constant
, we obtain
where is a three-dimensional
surface element and
an invariant, four-dimensional volume
element.
There is no obstacle in evaluating the wall
integral, for which reduces to the field’s
stress-energy tensor; for a wall of radius
the integral scales as
. The integrations over the
caps, however, are problematic: While the particle’s contribution
to the stress-energy tensor is integrable, the integration over the
field’s contribution goes as
and diverges. To properly regularize this integral requires great
care, and the removal of all singular terms can be achieved by mass
renormalization [24]. This
issue arises also in flat spacetime [25], and while it is
plausible that the rigourous distributional methods presented
in [56] could be generalized to curved
spacetimes, this remains to be done. More troublesome, however, is
the interior integral, which does not appear in flat spacetime.
Because
scales as
, this integral goes as
and it also diverges, albeit
less strongly than the caps integration. While simply discarding
this integral produces the correct equations of motion, it would be
desirable to go through a careful regularization of the interior
integration, and provide a convincing reason to discard it
altogether. To the best of my knowledge, this has not been
done.
To identify the strengths and weaknesses of the averaging method it is convenient to adopt the Detweiler-Whiting decomposition of the retarded field into singular and radiative pieces. For concreteness I shall focus my attention on the electromagnetic case, and write
Recall that this decomposition is unambiguous, and that the retarded and singular fields share the same singularity structure near the world line. Recall also that the retarded and singular fields satisfy the same field equations (with a distributional current density on the right-hand side), but that the radiative field is sourcefree.
To formulate equations of motion for the point
charge we temporarily model it as a spherical hollow shell, and we
obtain the net force acting on this object by averaging over the shell’s surface. (The averaging is
performed in the shell’s rest frame, and the shell is spherical in
the sense that its proper distance from the world line is the same
in all directions.) The averaged field is next evaluated on the
world line, in the limit of a zero-radius shell. Because the
radiative field is smooth on the world line, this yields
where
and
The equations of motion are then postulated to be
, where
is the particle’s bare mass. With the preceding
results we arrive at
,
where
is the particle’s observed
(renormalized) inertial mass.
The averaging method is sound, but it is not
immune to criticism. A first source of criticism concerns the
specifics of the averaging procedure, in particular, the choice of
a spherical surface over any other conceivable shape. Another
source is a slight inconsistency of the method that gives rise to
the famous “4/3 problem” [52]: The mass shift
is related to the shell’s electrostatic energy
by
instead of the expected
. This problem is likely due [45]
to the fact that the field that is averaged over the surface of the
shell is sourced by a point particle and not by the shell itself.
It is plausible that a more careful treatment of the near-source
field will eliminate both sources of criticism: We can expect that
the field produced by an extended spherical object will give rise
to a mass shift that equals the object’s electrostatic energy, and
the object’s spherical shape would then fully justify a spherical
averaging. (Considering other shapes might also be possible, but
one would prefer to keep the object’s structure simple and avoid
introducing additional multipole moments.) Further work is required
to clean up these details.
The averaging method is at the core of the approach followed by Quinn and Wald [49], who also average the retarded field over a spherical surface surrounding the particle. Their approach, however, also incorporates a “comparison axiom” that allows them to avoid renormalizing the mass.
The Detweiler-Whiting decomposition of the retarded field becomes most powerful when it is combined with the Detweiler-Whiting axiom, which asserts that
This axiom, which is motivated by the symmetric
nature of the singular field, and also its causal structure, gives
rise to the equations of motion , in
agreement with the averaging method (but with an implicit, instead
of explicit, mass shift). In this picture, the particle simply
interacts with a free radiative field (whose origin can be traced
to the particle’s past), and the procedure of mass renormalization
is sidestepped. In the scalar and electromagnetic cases, the
picture of a particle interacting with a radiative field removes
any tension between the nongeodesic motion of the charge and the
principle of equivalence. In the gravitational case the
Detweiler-Whiting axiom produces the statement that the point mass
moves on a geodesic in a spacetime whose metric
is nonsingular and a solution to the vacuum field
equations. This is a conceptually powerful, and elegant,
formulation of the MiSaTaQuWa equations of motion.
It is well known that in general relativity the motion of gravitating bodies is determined, along with the spacetime metric, by the Einstein field equations; the equations of motion are not separately imposed. This observation provides a means of deriving the MiSaTaQuWa equations without having to rely on the fiction of a point mass. In the method of matched asymptotic expansions, the small body is taken to be a nonrotating black hole, and its metric perturbed by the tidal gravitational field of the external universe is matched to the metric of the external universe perturbed by the black hole. The equations of motion are then recovered by demanding that the metric be a valid solution to the vacuum field equations. This method, which was the second starting point of Mino, Sasaki, and Tanaka [39], gives what is by far the most compelling derivation of the MiSaTaQuWa equations. Indeed, the method is entirely free of conceptual and technical pitfalls - there are no singularities (except deep inside the black hole) and only retarded fields are employed.
The introduction of a point mass in a nonlinear theory of gravitation would appear at first sight to be severely misguided. The lesson learned here is that one can in fact get away with it. The derivation of the MiSaTaQuWa equations of motion based on the method of matched asymptotic expansions does indeed show that results obtained on the basis of a point-particle description can be reliable, in spite of all their questionable aspects. This is a remarkable observation, and one that carries a lot of convenience: It is much easier to implement the point-mass description than to perform the matching of two metrics in two coordinate systems.
The concrete evaluation of the scalar, electromagnetic, and gravitational self-forces is made challenging by the need to first obtain the relevant retarded Green’s function. Successes achieved in the past were reviewed in Section 1.10, and here I want to describe the challenges that lie ahead. I will focus on the specific task of computing the gravitational self-force acting on a point mass that moves in a background Kerr spacetime. This case is especially important because the motion of a small compact object around a massive (galactic) black hole is a promising source of low-frequency gravitational waves for the Laser Interferometer Space Antenna (LISA) [32]; to calculate these waves requires an accurate description of the motion, beyond the test-mass approximation which ignores the object’s radiation reaction.
The gravitational self-acceleration is given by the MiSaTaQuWa expression, which I write in the form
where is the radiative part of the
metric perturbation. Recall that this equation is equivalent to the
statement that the small body moves on a geodesic of a spacetime
with metric
. Here
is the Kerr metric, and we wish to calculate
for a body moving in the Kerr spacetime. This
calculation is challenging and it involves a large number of
steps.
The first sequence of steps is concerned with the
computation of the (retarded) metric perturbation produced by a point particle moving on a specified
geodesic of the Kerr spacetime. A method for doing this was
elaborated by Lousto and Whiting [34] and
Ori [44], building on the
pioneering work of Teukolsky [57],
Chrzanowski [18], and
Wald [61]. The procedure
consists of
It is well known that the Teukolsky equation
separates when or
is expressed as a multipole
expansion, summing over modes with (spheroidal-harmonic) indices
and
. In fact, the procedure outlined above
relies heavily on this mode decomposition, and the metric
perturbation returned at the end of the procedure is also expressed
as a sum over modes
. (For each
,
ranges from
to
, and summation of
over this range is
henceforth understood.) From these, mode contributions to the
self-acceleration can be computed:
is obtained
from our preceding expression for the self-acceleration by
substituting
in place of
. These mode
contributions do not diverge on the world line, but
is discontinuous at the radial position of the
particle. The sum over modes, on the other hand, does not converge,
because the “bare” acceleration (constructed from the retarded
field
) is formally infinite.
The next sequence of steps is concerned with the
regularization of each by removing the contribution
from
[6, 7, 9, 11, 38, 21]. The singular field can be constructed
locally in a neighbourhood of the particle, and then decomposed
into modes of multipole order
. This gives rise to modes
for the singular part of the self-acceleration;
these are also finite and discontinuous, and their sum over
also diverges. But the true modes
of the self-acceleration
are continuous at the radial position of the particle, and their
sum does converge to the particle’s acceleration. (It might be
noted that obtaining a mode decomposition of the singular field
involves providing an extension of
on a sphere of
constant radial coordinate, and then integrating over the angular
coordinates. The arbitrariness of the extension introduces
ambiguities in each
, but the ambiguity
disappears after summing over
.)
The self-acceleration is thus obtained by first
computing from the metric perturbation derived
from
or
, then computing the counterterms
by mode-decomposing the singular field, and finally
summing over all
. This procedure is lengthy and involved, and thus
far it has not been brought to completion, except for the special
case of a particle falling radially toward a nonrotating black
hole [5]. In this
regard it should be noted that the replacement of the central Kerr
black hole by a Schwarzschild black hole simplifies the task
considerably. In particular, because there exists a practical and
well-developed formalism to describe the metric perturbations of a
Schwarzschild spacetime [51, 59, 63], there is no
necessity to rely on the Teukolsky formalism and the complicated
reconstruction of the metric variables.
The procedure described above is lengthy and
involved, but it is also incomplete. The reason is that the metric
perturbations that can be recovered from
or
do not by themselves sum up to the
complete gravitational perturbation produced by the moving
particle. Missing are the perturbations derived from the other
Newman-Penrose quantities:
,
, and
. While
and
can always be set to zero by an appropriate choice
of null tetrad,
contains such important physical
information as the shifts in mass and angular-momentum parameters
produced by the particle [60]. Because the mode
decompositions of
and
start at
, we might colloquially say that what is missing from
the above procedure are the “
and
” modes of the metric perturbations. It is not
currently known how the procedure can be completed so as to
incorporate all modes of the metric
perturbations. Specializing to a Schwarzschild spacetime eliminates
this difficulty, and in this context the low multipole modes have
been studied for the special case of circular orbits [43, 22].
In view of these many difficulties (and I choose to stay silent on others, for example, the issue of relating metric perturbations in different gauges when the gauge transformation is singular on the world line), it is perhaps not too surprising that such a small number of concrete calculations have been presented to date. But progress in dealing with these difficulties has been steady, and the situation should change dramatically in the next few years.
The successful computation of the gravitational
self-force is not the end of the road. After the difficulties
reviewed in the preceding Section 5.5.5
have all been removed and the motion of the small body is finally
calculated to order , it will still be necessary to obtain
gauge-invariant information associated with the body’s corrected
motion. Because the MiSaTaQuWa equations of motion are not by
themselves gauge-invariant, this step will necessitate going beyond
the self-force.
To see how this might be done, imagine that the small body is a pulsar, and that it emits light pulses at regular proper-time intervals. The motion of the pulsar around the central black hole modulates the pulse frequencies as measured at infinity, and information about the body’s corrected motion is encoded in the times-of-arrival of the pulses. Because these can be measured directly by a distant observer, they clearly constitute gauge-invariant information. But the times-of-arrival are determined not only by the pulsar’s motion, but also by the propagation of radiation in the perturbed spacetime. This example shows that to obtain gauge-invariant information, one must properly combine the MiSaTaQuWa equations of motion with the metric perturbations.
In the context of the Laser Interferometer Space
Antenna, the relevant observable is the instrument’s response to a
gravitational wave, which is determined by gauge-invariant
waveforms, and
. To calculate these
is the ultimate goal of this research programme, and the challenges
that lie ahead go well beyond what I have described thus far. To
obtain the waveforms it will be necessary to solve the Einstein
field equations to second order in
perturbation theory.
To understand this, consider first the
formulation of the first-order problem. Schematically, one
introduces a perturbation that satisfies a wave
equation
in the background spacetime, where
is the stress-energy tensor of the moving body,
which is a functional of the world line
. In first-order
perturbation theory, the stress-energy tensor must be conserved in
the background spacetime, and
must describe a
geodesic. It follows that in first-order perturbation theory, the
waveforms constructed from the perturbation
contain no information about the body’s corrected
motion.
The first-order perturbation, however, can be
used to correct the motion, which is now described by the world
line . In a naive implementation of the
self-force, one would now re-solve the wave equation with a
corrected stress-energy tensor,
, and
the new waveforms constructed from
would then
incorporate information about the corrected motion. This
implementation is naive because this information would not be
gauge-invariant. In fact, to be consistent one would have to
include all second-order terms in the
wave equation, not just the ones that come from the corrected
motion. Schematically, the new wave equation would have the form of
, and this is much more difficult to solve than the
naive problem (if only because the source term is now much more
singular than the distributional singularity contained in the
stress-energy tensor). But provided one can find a way to make this
second-order problem well posed, and provided one can solve it (or
at least the relevant part of it), the waveforms constructed from
the second-order perturbation
will be gauge invariant. In
this way, information about the body’s corrected motion will have
properly been incorporated into the gravitational waveforms.
The story is far from being over.