In this section we consider the motion of a
point particle of mass subjected to its own
gravitational field. The particle moves on a world line
in a curved spacetime whose background metric
is assumed to be a vacuum solution to the Einstein field
equations. We shall suppose that
is small, so that the
perturbation
created by the particle can also be
considered to be small; it will obey a linear wave equation in the
background spacetime. This linearization of the field equations
will allow us to fit the problem of determining the motion of a
point mass within the framework developed in Sections 5.1 and 5.2, and we shall obtain the
equations of motion by following the same general line of
reasoning. We shall find that
is not a geodesic of the
background spacetime because
acts on the particle and
induces an acceleration of order
; the motion is
geodesic in the test-mass limit only.
Our discussion in this first section is largely formal: As in Sections 5.1.1 and 5.2.1 we insert the point particle in the background spacetime and ignore the fact that the field it produces is singular on the world line. To make sense of the formal equations of motion will be our goal in the following Sections 5.3.2, 5.3.3, 5.3.4, 5.3.5, 5.3.6, and 5.3.7. The problem of determining the motion of a small mass in a background spacetime will be reconsidered in Section 5.4 from a different and more satisfying premise: There the small body will be modeled as a black hole instead of as a point particle, and the singular behaviour of the perturbation will automatically be eliminated.
Let a point particle of mass move on a world line
in a curved
spacetime with metric
. This is the total metric of the perturbed spacetime, and it depends on
as well as all other relevant parameters. At a later
stage of the discussion the total metric will be broken down into a
“background” part
that is independent of
, and a “perturbation” part
that is proportional to
. The world line is
described by relations
in which
is an arbitrary parameter - this will later be
identified with proper time
in the background spacetime. In this and the
following sections we will use
symbols to
denote tensors that refer to the perturbed spacetime; tensors in
the background spacetime will be denoted, as usual, by italic
symbols.
The particle’s action functional is
whereOn a formal level the metric is obtained by solving the Einstein field equations,
and the world line is determined by solving the equations of
energy-momentum conservation, which follow from the field
equations. From Equations (81
, 260
, 484
) we obtain
and additional manipulations reduce this to
where is the covariant
acceleration and
is a scalar field on the world line.
Energy-momentum conservation therefore produces the geodesic
equation
At this stage we begin treating as a formal expansion parameter, and we write
We have already stated that the particle is the
only source of matter in the spacetime, and the metric must therefore be a solution to the vacuum field
equations:
. Equations (483
, 488
, (489
) then imply
, in which both sides of the equation
are of order
. To simplify the expression of the
first-order correction to the Einstein tensor we introduce the
trace-reversed gravitational potentials
The equations of motion for the point mass are
obtained by substituting the expansion of Equation (487) into
Equations (485
) and (486
). The perturbed
connection is easily computed to be
, and this leads to
having once more selected proper time
(as measured in the background spacetime) as the parameter on the
world line. On the other hand, Equation (486
) gives
where is the
particle’s acceleration vector. Since it is clear that the
acceleration will be of order
, the second term can be
discarded and we obtain
Keeping the error term implicit, we shall express this in the equivalent form
which emphasizes the fact that the acceleration is orthogonal to the four-velocity.It should be clear that Equation (494) is valid only in a
formal sense, because the potentials obtained from
Equations (493
) diverge on the world
line. The nonlinearity of the Einstein field equations makes this
problem even worse here than for the scalar and electromagnetic
cases, because the singular behaviour of the perturbation might
render meaningless a formal expansion of
in powers of
. Ignoring this issue for the
time being (we shall return to it in Section 5.4), we will proceed as in
Sections 5.1 and 5.2 and attempt, with a careful
analysis of the field’s singularity structure, to make sense of
these equations.
To conclude this section I should explain why it
is desirable to restrict our discussion to spacetimes that contain
no matter except for the point particle. Suppose, in contradiction
with this assumption, that the background spacetime contains a
distribution of matter around which the particle is moving. (The
corresponding vacuum situation has the particle moving around a
black hole. Notice that we are still assuming that the particle
moves in a region of spacetime in which there is no matter; the
issue is whether we can allow for a distribution of matter somewhere else.) Suppose also that the
matter distribution is described by a collection of matter fields
. Then the field equations satisfied by the matter
have the schematic form
, and the
metric is determined by the Einstein field equations
, in which
stands for
the matter’s stress-energy tensor. We now insert the point particle
in the spacetime, and recognize that this displaces the background
solution
to a new solution (
. The perturbations are determined by
the coupled set of equations
and
. After linearization these take the
form of
where ,
,
, and
are suitable differential operators
acting on the perturbations. This is a coupled set of partial differential
equations for the perturbations
and
. These equations are linear, but they are much more
difficult to deal with than the single equation for
that was obtained in the vacuum case. And although
it is still possible to solve the coupled set of equations via a
Green’s function technique, the degree of difficulty is such that
we will not attempt this here. We shall, therefore, continue to
restrict our attention to the case of a point particle moving in a
vacuum (globally Ricci-flat) background spacetime.
The retarded solution to Equation (493) is
, where
is the retarded Green’s function
introduced in Section 4.5. After substitution of the
stress-energy tensor of Equation (490
) we obtain
For a more concrete expression we must take to be in a neighbourhood of the world line. The
following manipulations follow closely those performed in
Section 5.1.2 for the case of a scalar
charge, and in Section 5.2.2 for the case of an electric
charge. Because these manipulations are by now familiar, it will be
sufficient here to present only the main steps. There are two
important simplifications that occur in the case of a massive
particle. First, for the purposes of computing
to first order in
, it is sufficient to
take the world line to be a geodesic
of the background spacetime: The deviations from geodesic motion
that we are in the process of calculating are themselves of order
and would affect
at order
only. We shall therefore be allowed to set
With the understanding that is close to the world line (refer back to
Figure 9
), we substitute the
Hadamard construction of Equation (352
) into
Equation (495
) and integrate over
the portion of
that is contained in
. The result is
In the following Sections 5.3.3,
5.3.4, 5.3.5,
5.3.6, and 5.3.7,
we shall refer to as the gravitational potentials at
produced by a particle of mass
moving on the world line
, and to
as the gravitational
field at
. To compute this is our next task.
Keeping in mind that and
are related by
, a straightforward computation reveals that the
covariant derivatives of the gravitational potentials are given
by
We wish to express in the
retarded coordinates of Section 3.3, as an expansion in powers of
. For this purpose we decompose the field in the
tetrad
that is obtained by parallel transport
of
on the null geodesic that links
to
; this construction is detailed in
Section 3.3. Note that throughout this
section we set
, where
is the rotation tensor defined by Equation (138
): The tetrad vectors
are taken to be parallel transported on
. We recall from Equation (141
) that the parallel
propagator can be expressed as
. The expansion relies on Equation (166
) for
and Equation (168
) for
, both specialized to the case of geodesic motio,
. We shall also need
Making these substitutions in Equation (484) and projecting
against various members of the tetrad gives
The translation of the results contained in
Equations (505, 506
, 507
, 508
, 509
, 510
) into the Fermi normal
coordinates of Section 3.2 proceeds as in Sections 5.1.4 and 5.2.4, but is simplified by the
fact that here the world line can be taken to be a geodesic. We may
thus set
in Equations (224
) and (225
) that relate the
tetrad
to
, as well
as in Equations (221
, 222
, 223
) that relate the Fermi
normal coordinates
to the retarded coordinates.
We recall that the Fermi normal coordinates refer to a point
on the world line that is linked to
by a spacelike geodesic that intersects
orthogonally.
The translated results are
where all frame components are now evaluated atIt is then a simple matter to average these
results over a two-surface of constant and
. Using the area element of Equation (404
) and definitions
analogous to those of Equation (405
), we obtain
The singular gravitational potentials
are solutions to the wave equation of Equation (493To evaluate the integral of Equation (525) we take
to be close to the world line (see Figure 9
), and we invoke
Equation (373
) as well as the
Hadamard construction of Equation (379
). This gives
Differentiation of Equation (526) yields
To derive an expansion for we follow the general method of
Section 3.4.4 and introduce the functions
. We have that
where overdots indicate differentiation with
respect to and
. The
leading term
was worked out in Equation (501
), and the derivatives
of
are given by
and
according to Equations (503) and (360
). Combining these
results together with Equation (229
) for
gives
which becomes
and which is identical to Equation (503We proceed similarly to obtain an expansion for
. Here we introduce the
functions
and express
as
. The leading term
was computed in
Equation (502
), and
follows from Equation (359). Combining these
results together with Equation (229
) for
gives
We obtain the frame components of the singular
gravitational field by substituting these expansions into
Equation (527) and projecting
against the tetrad
. After some algebra we
arrive at
The difference between the retarded field of
Equations (505, 506
, 507
, 508
, 509
, 510
) and the singular
field of Equations (532
, 533
, 534
, 535
, 536
, 537
) defines the radiative
gravitational field
. Its tetrad components
are
The retarded gravitational field of a point particle is singular on the world line,
and this behaviour makes it difficult to understand how the field
is supposed to act on the particle and influence its motion. The
field’s singularity structure was analyzed in Sections 5.3.3
and 5.3.4, and in Section 5.3.5
it was shown to originate from the singular field
; the radiative field
was then shown
to be smooth on the world line.
To make sense of the retarded field’s action on
the particle we can follow the discussions of Section 5.1.6 and 5.2.6 and postulate that the self
gravitational field of the point particle is either , as worked out in Equation (523
), or
, as worked out in Equation (544
). These regularized
fields are both given by
The actual gravitational perturbation is obtained by inverting Equation (491
), which leads to
. Substituting
Equation (546
) yields
Equation (550) was first derived by
Yasushi Mino, Misao Sasaki, and Takahiro Tanaka in 1997 [39
]. (An incomplete
treatment had been given previously by Morette-DeWitt and
Ging [42].) An
alternative derivation was then produced, also in 1997, by Theodore
C. Quinn and Robert M. Wald [49
]. These equations
are now known as the MiSaTaQuWa equations of motion. It should be
noted that Equation (550
) is formally
equivalent to the statement that the point particle moves on a
geodesic in a spacetime with metric
, where
is the radiative metric perturbation obtained by
trace-reversal of the potentials
;
this perturbed metric is smooth on the world line, and it is a
solution to the vacuum field equations. This elegant interpretation
of the MiSaTaQuWa equations was proposed in 2002 by Steven
Detweiler and Bernard F. Whiting [23].
Quinn and Wald [50] have shown that under some conditions,
the total work done by the gravitational self-force is equal to the
energy radiated (in gravitational waves) by the particle.
The equations of motion derived in the
preceding Section 5.3.6 refer to a specific choice of
gauge for the metric perturbation produced by a
point particle of mass
. We indeed recall that back
at Equation (492
) we imposed the Lorenz
gauge condition
on the gravitational
potentials
. By virtue of this condition we found that the
potentials satisfy the wave equation of Equation (493
) in a background
spacetime with metric
. The hyperbolic nature of
this equation allowed us to identify the retarded solution as the
physically relevant solution, and the equations of motion were
obtained by removing the singular part of the retarded field. It
seems clear that the Lorenz condition is a most appropriate choice
of gauge.
Once the equations of motion have been
formulated, however, the freedom of performing a gauge
transformation (either away from the Lorenz gauge, or within the
class of Lorenz gauges) should be explored. A gauge transformation
will affect the form of the equations of motion: These must depend
on the choice of coordinates, and there is no reason to expect
Equation (550) to be invariant under
a gauge transformation. Our purpose in this section is to work out
how the equations of motion change under such a transformation.
This issue was first examined by Barack and Ori [8].
We introduce a coordinate transformation of the form
whereand this change can be interpreted as a gauge transformation of the metric perturbation created by the moving particle:
This, in turn, produces a change in the particle’s acceleration, whereTo compute the gauge acceleration we substitute
Equation (552) into
Equation (494
), and we simplify the
result by invoking Ricci’s identity,
, and the fact that
. The final
expression is