The derivation of the MiSaTaQuWa equations of
motion presented in Section 5.3 was framed within the paradigm
introduced in Sections 5.1 and 5.2 to describe the motion of a
point scalar charge, and a point electric charge, respectively.
While this paradigm is well suited to fields that satisfy linear
wave equations, it is not the best conceptual starting point in the
nonlinear context of general relativity. The linearization of the
Einstein field equations with respect to the small parameter did allow us to use the same mathematical techniques
as in Sections 5.1 and 5.2, but the validity of the
perturbative method must be critically examined when the
gravitational potentials are allowed to be singular. So while
Equation (550
) does indeed give the
correct equations of motion when
is small, its
previous derivation leaves much to be desired. In this section I
provide another derivation that is entirely free of conceptual and
technical pitfalls. Here the point mass will be replaced by a
nonrotating black hole, and the perturbation’s singular behaviour
on the world line will be replaced by a well-behaved metric at the
event horizon. We will use the powerful technique of matched asymptotic expansions [35, 31, 58
, 19, 1, 20].
The problem presents itself with a clean
separation of length scales, and the method relies entirely on
this. On the one hand we have the length scale associated with the
small black hole, which is set by its mass . On the other hand we have the length scale
associated with the background spacetime in which the black hole
moves, which is set by the radius of curvature
; formally this is defined so that a typical
component of the background spacetime’s Riemann tensor is equal to
up to a numerical factor of order unity. We demand
that
. As before we assume that the
background spacetime contains no matter, so that its metric is a
solution to the Einstein field equations in vacuum.
For example, suppose that our small black hole of
mass is on an orbit of radius
around another black
hole of mass
. Then
and we take
to be much smaller than the
orbital separation. Notice that the time scale over which the
background geometry changes is of the order of the orbital period
, so that this does not constitute a
separate scale. Similarly, the inhomogeneity scale - the length
scale over which the Riemann tensor of the background spacetime
changes - is of order
and also does not constitute an independent scale. (In this
discussion we have considered
to be of order unity, so as
to represent a strong-field, fast-motion situation.)
Consider now a region of spacetime defined by
, where
is a constant that is much
larger than
; this region will be called the external zone (see Figure 10
). In the external
zone the gravitational field is dominated by the conditions in the
external universe, and the metric can be expressed as
The metric
returned by the procedure described in the preceding paragraph is a
functional of a world line
that represents the motion
of the small black hole in the background spacetime. Our goal is to
obtain a description of this world line, in the form of equations
of motion to be satisfied by the black hole; these equations will
be formulated in the background spacetime. It is important to
understand that fundamentally,
exists only as an
external-zone construct: It is only in the external zone that the
black hole can be thought of as moving on a world line; in the
internal zone the black hole is revealed as an extended object and
the notion of a world line describing its motion is no longer
meaningful.
Equations (555) and (556
) give two different
expressions for the metric of the same spacetime; the first is
valid in the internal zone
, while
the second is valid in the external zone
. The fact that
allows us to
define a buffer zone in which
is restricted to the interval
.
In the buffer zone
is simultaneously much larger than
and much smaller than
- a typical value
might be
- and Equations (555
, 556
) are simultaneously
valid. Since the two metrics are the same up to a diffeomorphism,
these expressions must agree. And since
is a
functional of a world line
while
contains no such information, matching the metrics necessarily determines the
motion of the small black hole in the
background spacetime. What we have here is a beautiful
implementation of the general observation that the motion of
self-gravitating bodies is determined by the Einstein field
equations.
It is not difficult to recognize that the metrics
of Equations (555, 556
) can be matched in the
buffer zone. When
in the internal zone, the metric of the
unperturbed black hole can be expanded as
, where
is the metric of flat
spacetime (in asymptotically inertial coordinates) and the symbol
means “and a term of the form…”. On the other hand,
dimensional analysis dictates that
be of the form
while
should be expressed as
. Altogether we obtain
Matching the metrics of Equations (555) and (556
) in the buffer zone
can be carried out in practice only after performing a
transformation from the external coordinates used to express
to the internal coordinates employed
for
. The details of this coordinate
transformation will be described in Section 5.4.4,
and the end result of matching - the MiSaTaQuWa equations of motion
- will be revealed in Section 5.4.5.
To flesh out the ideas contained in the
previous Section 5.4.1 we first calculate the
internal-zone metric and replace Equation (555) by a more concrete
expression. We recall that the internal zone is defined by
, where
is a suitable measure of distance from
the black hole.
We begin by expressing , the Schwarzschild metric of an isolated black hole
of mass
, in terms of retarded Eddington-Finkelstein
coordinates
, where
is retarded time,
the usual areal radius, and
are two angles spanning the two-spheres of constant
and
. The metric is given by
these are appropriate for a black hole immersed in a flat spacetime charted by retarded coordinates.
The corrections and
in Equation (555
) encode the
information that our black hole is not isolated but in fact
immersed in an external universe whose metric becomes
asymptotically. In the internal zone
the metric of the background spacetime can be expanded in powers of
and expressed in a form that can be directly
imported from Section 3.3. If we assume for the moment
that the “world line”
has no acceleration in the
background spacetime (a statement that will be justified shortly),
then the asymptotic values of
must
be given by Equations (210
, 211
, 212
, 213
):
where
and are the tidal gravitational fields that were first introduced in Section 3.3.8. Recall thatThe modified asymptotic values lead us to the following ansatz for the internal-zone metric:
The five unknown functionsWhy is the assumption of no acceleration justified? As I shall explain in the next paragraph (and you might also refer back to the discussion of Section 5.3.7), the reason is simply that it reflects a choice of coordinate system: Setting the acceleration to zero amounts to adopting a specific - and convenient - gauge condition. This gauge differs from the Lorenz gauge adopted in Section 5.3, and it will be our choice in this section only; in the following Section 5.4.3 we will return to the Lorenz gauge, and the acceleration will be seen to return to its standard MiSaTaQuWa expression.
Inspection of Equations (560) and (561
) reveals that the
angular dependence of the metric perturbation is generated entirely
by scalar, vectorial, and tensorial spherical harmonics of degree
. In particular,
contains no
and
modes, and this statement
reflects a choice of gauge condition. Zerilli has shown [63
] that a perturbation
of the Schwarzschild spacetime with
corresponds to a
shift in the mass parameter. As Thorne and Hartle have
shown [58], a black
hole interacting with its environment will undergo a change of
mass, but this effect is of order
and thus
beyond the level of accuracy of our calculations. There is
therefore no need to include
terms in
. Similarly, it was shown by Zerilli that odd-parity
perturbations of degree
correspond to a shift in the
black hole’s angular-momentum parameters. As Thorne and Hartle have
shown, a change of angular momentum is quadratic in the hole’s
angular momentum, and we can ignore this effect when dealing with a
nonrotating black hole. There is therefore no need to include
odd-parity,
terms in
. Finally, Zerilli
has shown that in a vacuum spacetime, even-parity perturbations of
degree
correspond to a change of coordinate
system - these modes are pure gauge. Since we have the freedom to
adopt any gauge condition, we can exclude even-parity,
terms from the perturbed metric. This leads us to
Equations (562
, 563
, 564
, 565
), which contain only
perturbation modes; the even-parity modes are
contained in those terms that involve
, while the
odd-parity modes are associated with
. The perturbed
metric contains also higher multipoles, but those come at a higher
order in
; for example, the terms of order
include
modes. We conclude that
Equations (562
, 563
, 564
, 565
) is a sufficiently
general ansatz for the perturbed metric in the internal zone.
There remains the task of finding the functions
,
,
,
, and
. For this it is sufficient to take, say,
and
as the only nonvanishing
components of the tidal fields
and
. And since the equations for even-parity and
odd-parity perturbations decouple, each case can be considered
separately. Including only even-parity perturbations,
Equations (562
)-(565
) become
This metric is then substituted into the vacuum
Einstein field equations, . Calculating the Einstein
tensor is simplified by linearizing with respect to
and discarding its derivatives with respect to
: Since the time scale over which
changes is of order
, the ratio between
temporal and spatial derivatives is of order
and therefore small in the internal zone; the
temporal derivatives can be consistently neglected. The field
equations produce ordinary differential equations to be satisfied
by the functions
,
, and
. Those are easily decoupled, and demanding that the
functions all approach unity as
and be
well-behaved at
yields the unique solutions
Following the same procedure, we arrive at
Substituting Equations (566It shall prove convenient to transform from the quasi-spherical coordinates
to a set of quasi-Cartesian coordinates
. The transformation rules are worked
out in Section 3.3.7
and further illustrated in Section 3.3.8. This gives
We next move on to the external zone and seek
to replace Equation (556) by a more concrete
expression; recall that the external zone is defined by
. As was pointed out in Section 5.4.1,
in the external zone the gravitational perturbation associated with
the presence of a black hole cannot be distinguished from the
perturbation produced by a point particle of the same mass; it can
therefore be obtained by solving Equation (493
) in a background
spacetime with metric
. The external-zone metric is decomposed as
We now place ourselves in the buffer zone (where
and where the matching will take place) and work
toward expressing
as
an expansion in powers of
. For this purpose we will
adopt the retarded coordinates
of
Section 3.3 and rely on the machinery
developed there.
We begin with , the metric of the
background spacetime. We have seen in Section 3.3.8 that if the world line
is a geodesic, if the vectors
are parallel transported on the world line, and if
the Ricci tensor vanishes on
, then the metric takes the
form given by Equations (207
, 208
, 209
). This form, however,
is too restrictive for our purposes: We must allow
to have an acceleration, and allow the basis vectors
to be transported in the most general way compatible with their
orthonormality property; this transport law is given by
Equation (138
),
To express the perturbation as an expansion in powers of
we first go back to Equation (498
) and rewrite it in the
form
At this stage we introduce the trace-reversed fields
and recognize that the metric perturbation obtained from Equations (577The first step of this computation is to
decompose in the tetrad
that is obtained by parallel transport of
on the null geodesic that links
to its corresponding retarded point
on the world line. (The vectors are parallel
transported in the background spacetime.) The projections are
The perturbation is now expressed as
and its components are obtained by involving
Equations (169) and (170
), which list the
components of the tetrad vectors in the retarded coordinates; this
is the second (and longest) step of the computation. Noting that
and
can both be set equal to zero in these
equations (because they would produce negligible terms of order
in
), and that
,
, and
can all be expressed in terms of the
tidal fields
,
,
,
, and
using Equations (204
, 205
, 206
), we arrive at
The external-zone metric is obtained by adding
as given by Equations (580
, 581
, 582
) to
as given by Equations (593
, 594
, 595
). The final result
is
Comparison of Equations (568, 569
, 570
) and
Equations (596
, 597
, 598
) reveals that the
internal-zone and external-zone metrics do no match in the buffer
zone. But as the metrics are expressed in two different coordinate
systems, this mismatch is hardly surprising. A meaningful
comparison of the two metrics must therefore come after a
transformation from the external coordinates
to the internal coordinates
. Our task in this section is to construct this
coordinate transformation. We shall proceed in three stages. The
first stage of the transformation,
, will be seen to remove unwanted terms of order
in
. The second stage,
, will remove all terms of order
in
. Finally, the third stage
will produce the desired
internal coordinates.
The first stage of the coordinate transformation is
and it affects the metric at ordersThe second stage of the coordinate transformation is
and it affects the metric at ordersThe third and final stage of the coordinate transformation is
where This transformation puts the metric in its final form Except for the terms involvingA precise match between and
is
produced when we impose the relations
The black hole’s acceleration vector can be constructed from the frame
components listed in Equation (616
). A straightforward
computation gives
Substituting Equations (616) and (617
) into
Equation (579
) gives the following
transport equation for the tetrad vectors: