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, 32, 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.)
Let be a meaningful measure of distance from the small black hole, and let us consider a region of spacetime defined by , where is a constant that is much smaller than . This inequality defines a narrow world tube that surrounds the small black hole, and we shall call this region the internal zone (see Figure 10). In the internal zone the gravitational field is dominated by the black hole, and the metric can be expressed as[51, 59, 63].
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):
The modified asymptotic values lead us to the following ansatz for the internal-zone metric:
Why 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  that a perturbation of the Schwarzschild spacetime with corresponds to a shift in the mass parameter. As Thorne and Hartle have shown , 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
It 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
The 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 atcoordinate components of the metric perturbation in the retarded coordinates , expressed in terms of frame components of the tail fields and . The perturbation is expanded in powers of and it also involves the tidal gravitational fields of the background spacetime.
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
The second stage of the coordinate transformation is
The third and final stage of the coordinate transformation is
A 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:
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