We wish to represent the motion of the body through the external background spacetime ,
rather than through the exact spacetime
. In order to achieve this, we begin by surrounding the
body with a (hollow, three-dimensional) world tube
embedded in the buffer region. We define the tube
to be a surface of constant radius
in Fermi normal coordinates centered on a world line
, though the exact definition of the tube is immaterial. Since there exists a diffeomorphism
between
and
in the buffer region, this defines a tube
. Now, the problem is the
following: what equation of motion must
satisfy in order for
to be “centered” about the
body?
How shall we determine if the body lies at the centre of the tube’s interior? Since the tube is close to the
small body (relative to all external length scales), the metric on the tube is primarily determined by the
small body’s structure. Recall that the buffer region corresponds to an asymptotically large spatial
distance in the inner expansion. Hence, on the tube, we can construct a multipole expansion of
the body’s field, with the form (or
– we will assume
in the buffer
region). Although alternative definitions could be used, we define the tube to be centered about
the body if the mass dipole moment vanishes in this expansion. Note that this is the typical
approach in general relativity: Whereas in Newtonian mechanics one directly finds the equation of
motion for the centre of mass of a body, in general relativity one typically seeks a world line
about which the mass dipole of the body vanishes (or an equation of motion for the mass dipole
relative to a given nearby world line) [66, 152
, 83
, 144
]. This definition of the world line is
sufficiently general to apply to a black hole. If the body is material, one could instead imagine a
centre-of-mass world line that lies in the interior of the body in the exact spacetime. This world line
would then be the basis of our self-consistent expansion. We use our more general definition to
cover both cases. See Ref. [173
] and references therein for discussion of multipole expansions
in general relativity, see Refs. [173, 174] for discussions of mass-centered coordinates in the
buffer region, and see, e.g., Refs. [160, 65] for alternative definitions of centre of mass in general
relativity.
As in the point-particle case, in order to determine the equation of motion of the world
line, we consider a family of metrics, now denoted , parametrized by
, such
that when
is given by the correct equation of motion for a given value of
, we have
. The metric in the outer limit is, thus, taken to be the general expansion
In the remainder of this section, we present a sequence of perturbation equations that arise in this expansion scheme, along with a complementary sequence for the inner expansion.
In the outer expansion, we seek a solution in a vacuum region outside of
. We specify
to
be an open set consisting of the future domain of dependence of the spacelike initial-data surface
, excluding the interior of the world tube
. This implies that the future boundary of
is a null surface
. Refer to Figure 11
for an illustration. The boundary of the domain
is
. The spatial surface
is chosen to intersect
at the initial time
.
Historically, in derivations of the self-force, solutions to the perturbative field equations were taken to be
global in time, with tail integrals extended to negative infinity, as we wrote them in the preceding sections.
But as was first noted in Ref. [144], because the self-force drives long-term, cumulative changes, any
approximation truncated at a given order will be accurate to that order only for a finite time; and this
necessites working in a finite region such as
. This is also true in the case of point charges and masses.
For simplicity, we neglected this detail in the preceding sections, but for completeness, we account for it
here.
Within this region, we follow the methods presented in the case of a point mass. We begin by reformulating
the Einstein equation such that it can be solved for an arbitrary world line. To accomplish this, we assume
that the Lorenz gauge can be imposed on the whole of , everywhere in
, such that
.
Here
Just as in the case of a point mass, this choice of gauge reduces the vacuum Einstein equation
to a weakly nonlinear wave equation that can be expanded and solved at fixed
. However, we now seek a
solution only in the region
, where the energy-momentum tensor vanishes, so the resulting sequence of
wave equations reads
Again as in the case of a point particle, we can easily write down formal solutions to the wave equations,
for arbitrary . Using the same methods as were used to derive the Kirchoff representation in
Section 16.3, we find
One should note several important properties of these integral representations: First, must lie in the
interior of
; an alternative expression must be derived if
lies on the boundary [153]. Second, the
integral over the boundary is, in each case, a homogeneous solution to the wave equation, while the integral
over the volume is an inhomogeneous solution. Third, if the field at the boundary satisfies the Lorenz gauge
condition, then by virtue of the wave equation, it satisfies the gauge condition everywhere; hence, imposing
the gauge condition to some order in the buffer region ensures that it is imposed to the same order
everywhere.
While the integral representation is satisfied by any solution to the associated wave equation, it does not
provide a solution. That is, one cannot prescribe arbitrary boundary values on and then arrive at a
solution. The reason is that the tube is a timelike boundary, which means that field data on it can
propagate forward in time and interfere with the data at a later time. However, by applying the wave
operator
onto Eq. (21.8
), we see that the integral representation of
is guaranteed to satisfy the
wave equation at each point
. In other words, the problem arises not in satisfying the wave equation
in a pointwise sense, but in simultaneously satisfying the boundary conditions. But since the tube is chosen
to lie in the buffer region, these boundary conditions can be supplied by the buffer-region expansion. And as
we will discuss in Section 23, because of the asymptotic smallness of the tube, the pieces of the
buffer-region expansion diverging as
are sufficient boundary data to fully determine the global
solution.
Finally, just as in the point-particle case, in order to split the gauge condition into a set
of exactly solvable equations, we assume that the acceleration of possesses an expansion
The outer expansion is defined not only by holding fixed, but also by demanding that the mass dipole
of the body vanishes when calculated in coordinates centered on
. If we perform a gauge transformation
generated by a vector
, then the mass dipole will no longer vanish in those coordinates. Hence, a
new world line
must be constructed, such that the mass dipole vanishes when calculated in
coordinates centered on that new world line. In other words, in the outer expansion we have the
usual gauge freedom of regular perturbation theory, so long as the world line is appropriately
transformed as well:
. The transformation law for the world line was first
derived by Barack and Ori [17
]; it was displayed in Eq. (1.49
), and it will be worked out again in
Section 22.6.
Using this gauge freedom, we now justify, to some extent, the assumption that the Lorenz
gauge condition can be imposed on the entirety of . If we begin with the metric in an
arbitrary gauge, then the gauge vectors
,
, etc., induce the transformation
For the inner expansion, we assume the existence of some local polar coordinates , such
that the metric can be expanded for
while holding fixed
,
, and
; to relate the
inner and outer expansions, we assume
, but otherwise leave the inner expansion completely
general.
This leads to the ansatz
where Since we are interested in the inner expansion only insofar as it informs the outer expansion, we shall
not seek to explicitly solve the perturbative Einstein equation in the inner expansion. See Ref. [144] for the
forms of the equations and an example of an explicit solution in the case of a perturbed black
hole.
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Living Rev. Relativity 14, (2011), 7
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