The method of matched asymptotic expansions relies on the fact that the inner and outer expansion agree
term by term when re-expanded in the buffer region, where . To illustrate this idea of
matching, consider the forms of the two expansions in the buffer region. The inner expansion holds
constant (since
) while expanding for small
. But if
is replaced with its value
, the
inner expansion takes the form
, where each term has a
dependence on
that can be expanded in the limit
to arrive at the schematic forms
and
, where
signifies “plus terms
of the form” and the expanded quantities can be taken to be components in Fermi coordinates. Here we
have preemptively restricted the form of the expansions, since terms such as
must vanish because
they would have no corresponding terms in the outer expansion. Putting these two expansions together, we
arrive at
On the other hand, the outer expansion holds constant (since
is formally of the
order of the global external coordinates) while expanding for small
, leading to the form
. But very near the world line, each term in this expansion can be expanded
for small
, leading to
and
. (Again, we have restricted
this form because terms such as
cannot arise in the inner expansion.) Putting these two expansions
together, we arrive at
One can make use of this fact by first determining the inner and outer expansions as fully as possible,
then fixing any unknown functions in them by matching them term by term in the buffer region; this was
the route taken in, e.g., Refs. [130, 142
, 49
, 170]. However, such an approach is complicated by the
subtleties of matching in a diffeomorphism-invariant theory, where the inner and outer expansions are
generically in different coordinate systems. See Ref. [145
] for an analysis of the limitations of this approach
as it has typically been implemented. Alternatively, one can take the opposite approach, working in the
buffer region first, constraining the forms of the two expansions by making use of their matching, then
using the buffer-region information to construct a global solution; this was the route taken
in, e.g., Refs. [102, 83
, 144
]. In general, some mixture of these two approaches can be taken.
Our calculation follows Ref. [144
]. The only information we take from the inner expansion
is its general form, which is characterized by the multipole moments of the body. From this
information, we determine the external expansion, and thence the equation of motion of the world
line.
Over the course of our calculation, we will find that the external metric perturbation in the buffer region
is expressed as the sum of two solutions: one that formally diverges at and is entirely determined
from a combination of (i) the multipole moments of the internal background metric
, (ii) the
Riemann tensor of the external background
, and (iii) the acceleration of the world line
; and a
second solution that is formally regular at
and depends on the past history of the body and the
initial conditions of the field. At leading order, these two solutions are identified as the Detweiler–Whiting
singular and regular fields
and
, respectively, and the self-force is determined entirely by
. Along with the self-force, the acceleration of the world line includes the Papapetrou spin
force [138
]. This calculation leaves us with the self-force in terms of the the metric perturbation in
the neighbourhood of the body. In Section 23, we use the local information from the buffer
region to construct a global solution for the metric perturbation, completing the solution of the
problem.
Before proceeding, we define some notation. We use the multi-index notation .
Angular brackets denote the STF combination of the enclosed indices, and a tensor bearing a hat is an STF
tensor. To accommodate this, we now write the Fermi spatial coordinates as
, instead of
as they were written in previous sections. Finally, we define the one-forms
and
.
One should note that the coordinate transformation between Fermi coordinates and the
global coordinates is
-dependent, since Fermi coordinates are tethered to an
-dependent world line. If
one were using a regular expansion, then this coordinate transformation would devolve into a background
coordinate transformation to a Fermi coordinate system centered on a geodesic world line, combined with a
gauge transformation to account for the
-dependence. But in the self-consistent expansion, the
transformation is purely a background transformation, because the
-dependence in it is reducible to that
of the fixed world line.
Because the dependence on in the coordinate transformation cannot be reduced to a gauge
transformation, in Fermi coordinates the components
of the background metric become
-dependent. This dependence takes the explicit form of factors of the acceleration
and its
derivatives, for which we have assumed the expansion
. There is also
an implicit dependence on
in that the proper time
on the world line depends on
if written as a
function of the global coordinates; but this dependence can be ignored so long as we work consistently with
Fermi coordinates.
Of course, even in these -dependent coordinates,
remains the background metric of the outer
expansion, and
is an exact solution to the wave equation (21.7
). At first order we will, therefore,
obtain
exactly in Fermi coordinates, for arbitrary
. However, for some purposes an
approximate solution of the wave equation may suffice, in which case we may utilize the expansion of
. Substituting that expansion into
and
yields the buffer-region expansions
Now, we merely assume that in the buffer region there exists a smooth coordinate transformation
between the local coordinates and the Fermi coordinates
such that
,
,
and
. The buffer region corresponds to asymptotic infinity
(or
) in the internal
spacetime. So after re-expressing
as
, the internal background metric can be expanded as
We assume that the perturbation can be similarly expanded in powers of
at fixed
,
The expansion of may or may not hold the acceleration fixed. Regardless of this
choice, the general form of the expansion remains valid: incorporating the expansion of the
acceleration would merely shuffle terms from one coefficient to another. And since the internal metric
must equal the external metric
, the general form of the above expansions
of
and
completely determines the general form of the external perturbations:
To obtain a general solution to the Einstein equation, we write each as an expansion in terms of
irreducible symmetric trace-free pieces:
Now, since the wave equations (21.4) and (21.5
) are covariant, they must still hold in the new
coordinate system, despite the additional
-dependence. Thus, both equations could be solved for
arbitrary acceleration in the buffer region. However, due to the length of the calculations involved, we will
instead solve the equations
Unlike the wave equations, the gauge conditions (21.10) and (21.11
) already incorporate the expansion
of the acceleration. As such, they are unmodified by the replacement of the second-order wave
equation (21.5
) with its approximation (22.17
). So we can write
In what follows, the reader may safely assume that all calculations are lengthy unless noted otherwise.
In principle, solving the first-order Einstein equation in the buffer region is straightforward. One need
simply substitute the expansion of , given in Eq. (22.11
), into the linearized wave equation (22.16
)
and the gauge condition (22.18
). Equating powers of
in the resulting expansions then yields a
sequence of equations that can be solved for successively higher-order terms in
. Solving
these equations consists primarily of expressing each quantity in its irreducible STF form, using
the decompositions (B.3
) and (B.7
); since the terms in this STF decomposition are linearly
independent, we can solve each equation term by term. This calculation is aided by the fact that
, so that, for example, the wave operator
consists of a flat-space Laplacian
plus corrections of order
. Appendix B also lists many useful identities, particularly
,
, and the fact that
is an eigenvector of the flat-space Laplacian:
.
Before proceeding with the calculation, which consists mostly of tedious and lengthy algebra, we summarize
the results. The first-order perturbation consists of two pieces, which we will eventually identify with
the Detweiler–Whiting regular and singular fields. In the buffer-region expansion, the regular field consists
entirely of unknowns, which is to be expected since as a free radiation field, it must be provided by
boundary data. Only when we consider the global solution, in Section 23, will we express it in terms of a
tail integral. On the other hand, the singular field is locally determined, and it is characterized by
the body’s monopole moment
. More precisely, it is fully determined by the tidal fields
of the external background spacetime and the Arnowitt–Deser–Misner mass of the internal
background spacetime
. By itself the wave equation does not restrict the behaviour of
this monopole moment, but imposing the gauge condition produces the evolution equations
We now proceed to the details of the calculation. We begin with the most divergent term in the wave
equation: the order , flat-space Laplacian term
At the next order, , along with the acceleration of the world line and the time-derivative of the
mass, first appears in the Einstein equation. The order
term in the wave equation is
At the next order, , along with squares and derivatives of the acceleration, first appears in the
Einstein equation, and the tidal fields of the external background couple to
. The order
term in the wave equation becomes
Substituting this into the order terms in the gauge condition, we find
Thus, the order component of
is
To summarize the results of this section, we have , where
is given in Eq. (22.27
),
is given in Eq. (22.29
), and
is given in Eq. (22.38
).
In addition, we have determined that the ADM mass of the internal background spacetime
is time-independent, and that the acceleration of the body’s world line vanishes at leading
order.
Though the calculations are much lengthier, solving the second-order Einstein equation in the buffer region
is essentially no different from solving the first. We seek to solve the approximate wave equation (22.17),
along with the gauge condition (22.19
), for the second-order perturbation
; doing so will
also, more importantly, determine the acceleration
. In this calculation, the acceleration is set to
everywhere except in the left-hand side of the gauge condition,
, which is linear in
.
We first summarize the results. As at first order, the metric perturbation contains a regular, free radiation
field and a singular, bound field; but in addition to these pieces, it also contains terms sourced by the
first-order perturbation. Again, the regular field requires boundary data to be fully determined. And again,
the singular field is characterized by the multipole moments of the body: the mass dipole of the
internal background metric
, which measures the shift of the body’s centre of mass relative to the
world line; the spin dipole
of
, which measures the spin of the body about the world line;
and an effective correction
to the body’s mass. The wave equation by itself imposes no
restriction on these quantities, but by imposing the gauge condition we find the evolution equations
We now proceed to the details of the calculation. Substituting the expansion
and the results for To begin, the most divergent, order term in the wave equation reads
The most divergent, order terms in the gauge condition similarly involve only
; they read
The metric perturbation in this form depends on five free functions of time. However, from
calculations in flat spacetime, we know that order terms in the metric perturbation can
be written in terms of two free functions: a mass dipole and a spin dipole. We transform the
perturbation into this “canonical” form by performing a gauge transformation (cf. Ref. [45
]).
The transformation is generated by
, the effect of which is to
remove
and
from the metric. This transformation is a refinement of the
Lorenz gauge. (Effects at higher order in
and
will be automatically incorporated into the
higher-order perturbations.) The condition
then becomes
. The remaining two functions are related to the ADM momenta of the internal
spacetime:
At the next order, , because the acceleration is set to zero,
does not contribute to
, and
does not contribute to
. The wave equation hence reads
We next substitute and
into the order
terms in the gauge condition. The
-component becomes
Thus, the order term in
is given by
We next move to the order terms in the wave equation, and the order
terms in the
gauge condition, which read
Finally, we arrive at the order terms in the wave equation. At this order, the body’s tidal
moments become coupled to those of the external background. The equation reads
Foregoing the details, after some algebra we can read off the solution
where each one of the STF tensors is listed in Table 1. In solving Eq. (22.64), we also find that the logarithmic term in the expansion becomes uniquely
determined:
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We now move to the final equation in the buffer region: the order gauge condition. This condition
will determine the acceleration
. At this order,
first contributes to Eq. (22.19
):
We have now completed our calculation in the buffer region. In summary, the second-order perturbation in
the buffer region is given by , where
is given in Eq. (22.50
),
in Eq. (22.60
),
in Eq. (22.65
), and
in Eq. (22.68
). In addition, we have found evolution equations for an effective correction to
the body’s mass, given by Eq. (22.39
), and mass and spin dipoles, given by Eqs. (22.59
) and
(22.71
).
Eq. (22.71) is the principal result of our calculation. After simplification, it reads
Eq. (22.72) is a type of master equation of motion, describing the position of the body relative to a
world line of unspecified (though small) acceleration, in terms of the metric perturbation on the world line,
the tidal fields of the spacetime it lies in, and the spin of the body. It contains two types of accelerations:
and
. The first type is the second time derivative of the body’s mass dipole moment (or the
first derivative of its ADM linear momentum), as measured in a frame centered on the world line
. The
second type is the covariant acceleration of the world line through the external spacetime. In other words,
measures the acceleration of the body’s centre of mass relative to the centre of the
coordinate system, while
measures the acceleration of the coordinate system itself. As discussed
in Section 21, our aim is to identify the world line as that of the body, and we do so via the
condition that the mass dipole vanishes for all times, meaning that the body is centered on
the world line for all times. If we start with initial conditions
, then
the mass dipole remains zero for all times if and only if the world line satisfies the equation
In our self-consistent approach, we began with the aim of identifying by the condition that the
body must be centered about it for all time. However, we could have begun with a regular
expansion, in which the world line is taken to be the remnant
of the body in the outer
limit of
with only
fixed. In that case the acceleration of the world line would
necessarily be
-independent, so
would be the full acceleration of
. Hence, when
we found
, we would have identified the world line as a geodesic, and there would
be no corrections
for
. We would then have arrived at the equation of motion
Although perfectly valid, such an equation is of limited use. If the external background is curved, then
has meaning only if the body is “close” to the world line. Thus,
is a meaningful acceleration
only for a short time, since
will generically grow large as the body drifts away from the reference
world line. On that short timescale of validity, the deviation vector defined by
accurately points from
to a “corrected” world line
; that world line, the approximate equation of motion of
which is given in Eq. (22.73
), accurately tracks the motion of the body. After a short time,
when the mass dipole grows large and the regular expansion scheme begins to break down,
the deviation vector will no longer correctly point to the corrected world line. Errors will also
accumulate in the field itself, because it is being sourced by the geodesic, rather than corrected,
motion.
The self-consistent equation of motion appears to be more robust, and offers a much wider range of
validity. Furthermore, even beyond the above step, where we had the option to choose to set either or
to zero, the self-consistent expansion continues to contain within it the regular expansion. Starting
from the solution in the self-consistent expansion, one can recover the regular expansion, and its equation of
motion (22.74
), simply by assuming an expansion for the world line and following the usual steps of
deriving the geodesic deviation equation.
Regardless of which equation of motion we opt to use, we have now completed the derivation of the gravitational self-force, in the sense that, given the metric perturbation in the neighbourhood of the body, the self-force is uniquely determined by irreducible pieces of that perturbation. Explicitly, the terms that appear in the self-force are given by
This is all that is needed to incorporate the motion of the body into a dynamical system that can be numerically evolved; at each timestep, one simply needs to calculate the field near the world line and decompose it into irreducible pieces in order to determine the acceleration of the body. The remaining difficulty is to actually determine the field at each timestep. In the next section, we will use the formal integral representation of the solution to determine the metric perturbation at the location of the body in terms of a tail integral. However, before doing so, we emphasize some important features of the self-force and the field near the
body. First, note that the first-order external field splits into two distinct pieces. There is the singular
piece
, given by
Next, there is the Detweiler–Whiting regular field , given by
Now, the acceleration of the body is given by
which we can rewrite as where
We now turn to the question of how the world line transforms under a gauge transformation. We begin with
the equation of motion (22.41), presented again here:
Suppose that we had not chosen a world line for which the mass dipole vanishes, but instead had chosen
some “nearby” world line. Then Eq. (22.85) provides the relationship between the acceleration of that
world line, the mass dipole relative to it, and the first-order metric perturbations (we again neglect spin for
simplicity). The mass dipole is given by
, which has the covariant form
Now consider a gauge transformation generated by , where
is
bounded as
, and
diverges as
. More specifically, we assume the expansions
and
. (The dependence on
appears in the form of dependence on proper time
. Other dependences could appear, but it
would not affect the result.) This transformation preserves the presumed form of the outer
expansion, both in powers of
and in powers of
. The metric perturbations transform as
From these results, we find that the left- and right-hand sides of Eq. (22.88) transform in the same way:
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Living Rev. Relativity 14, (2011), 7
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