20 Point-particle limits and matched asymptotic expansions

The expansion presented in the previous section is based on an exact point-particle source. But in the full, nonlinear theory, no distributional solution would exist for such a source [80]. Although the expansion nevertheless yields a well-behaved linear approximation, it is ill-behaved beyond that order, since the second- and higher-order Einstein tensors will contain products of delta functions, which have no meaning as distributions. It may be possible to overcome this limitation using more advanced methods such as Colombeau algebras [164], which allow for the multiplication of distributions, but little work has been done to that end. Instead, the common approach, and the one we shall pursue here, has been to abandon the fiction of a point particle in favor of considering an asymptotically small body. As we shall see, we can readily generalize the self-consistent expansion scheme to this case. Furthermore, we shall find that the results of the previous section are justified by this approach: at linear order, the metric perturbation due to an asymptotically small body is precisely that of a point particle moving on a world line with an acceleration given by the MiSaTaQuWa equation (plus higher-order corrections).

In order for the body to be considered “small”, its mass and size must be much smaller than all external lengthscales. We denote these external scales collectively as ℛ, which we may define to be the radius of curvature of the spacetime (were the small body removed from it) in the region in which we seek an approximation. Given this definition, a typical component of the spacetime’s Riemann tensor is equal to 2 1 ∕ℛ up to a numerical factor of order unity. Now, we consider a family of metrics gαβ(𝜀) containing a body whose mass scales as 𝜀 in the limit 𝜀 → 0; that is, 𝜀 ∼ m ∕ℛ. If each member of the family is to contain a body of the same type, then the size of the body must also approach zero with 𝜀. The precise scaling of size with 𝜀 is determined by the type of body, but it is not generally relevant. What is relevant is the “gravitational size” – the length scale determining the metric outside the body – and this size always scales linearly with the mass. If the body is compact, as is a neutron star or a black hole, then its gravitational size is also its actual linear size. In what remains, we assume that all lengths have been scaled by ℛ, such that we can write, for example, m ≪ 1. Our goal is to determine the metric perturbation and the equation of motion produced by the body in this limit.

Point-particle limits such as this have been used to derive equations of motion many times in the past, including in derivations of geodesic motion at leading order [100, 79, 64] and in constructing post-Newtonian limits [74Jump To The Next Citation Point]. Perhaps the most obvious means of approaching the problem is to first work nonperturbatively, with a body of arbitrary size, and then take the limit. Using this approach (though with some restrictions on the body’s size and compactness) and generalized definitions of momenta, Harte has calculated the self-force in the case of scalar [89Jump To The Next Citation Point] and electromagnetic [90Jump To The Next Citation Point] charge distributions in fixed backgrounds, following the earlier work of Dixon [57Jump To The Next Citation Point, 58Jump To The Next Citation Point, 59Jump To The Next Citation Point]. However, while this approach is conceptually compelling, at this stage it applies only to material bodies, not black holes, and has not yet been presented as part of a systematic expansion of the Einstein equation. Here, we focus instead on a more general method.

Alternatively, one could take the opposite approach, essentially taking the limit first and then trying to recover the higher-order effects, by treating the body as an effective point particle at leading order, with finite size effects introduced as higher-order effective fields, as done by Galley and Hu [76, 75Jump To The Next Citation Point]. However, while this approach is computationally efficient, allowing one to perform high-order calculations with (relative) ease, it requires methods such as dimensional regularization and mass renormalization in order to arrive at meaningful results. Because of these undesirable requirements, we will not consider it here.

In the approach we review, we make use of the method of matched asymptotic expansions [46Jump To The Next Citation Point, 47Jump To The Next Citation Point, 102Jump To The Next Citation Point, 103, 174Jump To The Next Citation Point, 130Jump To The Next Citation Point, 129, 2, 142Jump To The Next Citation Point, 49Jump To The Next Citation Point, 74Jump To The Next Citation Point, 170Jump To The Next Citation Point, 83Jump To The Next Citation Point, 82Jump To The Next Citation Point, 144Jump To The Next Citation Point, 145Jump To The Next Citation Point]. Broadly speaking, this method consists of constructing two different asymptotic expansions, each valid in a specific region, and combining them to form a global expansion. In the present context, the method begins with two types of point-particle limits: an outer limit, in which 𝜀 → 0 at fixed coordinate values (we will slightly modify this in a moment); and an inner limit, in which 𝜀 → 0 at fixed values of R&tidle; := R∕𝜀, where R is a measure of radial distance from the body. In the outer limit, the body shrinks toward zero size as all other distances remain roughly constant; in the inner limit, the small body keeps a constant size while all other distances blow up toward infinity. Thus, the inner limit serves to “zoom in” on a small region around the body. The outer limit can be expected to be valid in regions where R ∼ 1, while the inner limit can be expected to be valid in regions where R&tidle; ∼ 1 (or R ∼ 𝜀), though both of these regions can be extended into larger domains.

More precisely, consider an exact solution gαβ on a manifold ℳ 𝜀 with two coordinate systems: a local coordinate system X α = (T,R, ΘA ) that is centered (in some approximate sense) on the small body, and a global coordinate system x α. For example, in an extreme-mass-ratio inspiral, the local coordinates might be the Schwarzschild-type coordinates of the small body, and the global coordinates might be the Boyer–Lindquist coordinates of the supermassive Kerr black hole. In the outer limit, we expand g αβ for small 𝜀 while holding xα fixed. The leading-order solution in this case is the background metric gαβ on a manifold ℳE; this is the external spacetime, which contains no small body. It might, for example, be the spacetime of the supermassive black hole. In the inner limit, we expand g αβ for small 𝜀 while holding &tidle; A (T, R,Θ ) fixed. The leading-order solution in this case is the metric body gα β on a manifold ℳI; this is the spacetime of the small body if it were isolated (though it may include slow evolution due to its interaction with the external spacetime – this will be discussed below). Note that ℳ E and ℳ I generically differ: in an extreme-mass-ratio inspiral, for example, if the small body is a black hole, then ℳI will contain a spacelike singularity in the black hole’s interior, while ℳE will be smooth at the “position” where the small black hole would be. What we are interested in is that “position” – the world line in the smooth external spacetime ℳE that represents the motion of the small body. Note that this world line generically appears only in the external spacetime, rather than as a curve in the exact spacetime (g αβ,ℳ 𝜀); in fact, if the small body is a black hole, then obviously no such curve exists.

Determining this world line presents a fundamental problem. In the outer limit, the body vanishes at 𝜀 = 0, leaving only a remnant, 𝜀-independent curve in ℳE. (Outside any small body, the metric will contain terms such as m ∕R, such that in the limit m → 0, the limit exists everywhere except at R = 0, which leaves a removable discontinuity in the external spacetime; the removal of this discontinuity defines the remnant world line of the small body.) But the true motion of the body will generically be 𝜀-dependent. If we begin with the remnant world line and correct it with the effects of the self-force, for example, then the corrections must be small: they are small deviation vectors defined on the remnant world line. Put another way, if we expand g αβ in powers of 𝜀, then all functions in it must similarly be expanded, including any representation of the motion, and in particular, any representative world line. We would then have a representation of the form α α α z (t,𝜀) = z(0)(t) + 𝜀z(1)(t) + ..., where α z(1)(t) is a vector defined on the remnant curve described by zα (t) (0). The remnant curve would be a geodesic, and the small corrections would incorporate the self-force and finite-size effects [83Jump To The Next Citation Point] (see also [102Jump To The Next Citation Point]). However, because the body will generically drift away from any such geodesic, the small corrections will generically grow large with time, leading to the failure of the regular expansion. So we will modify this approach by performing a self-consistent expansion in the outer limit, following the same scheme as presented in the point-particle case. Refs. [144Jump To The Next Citation Point, 145Jump To The Next Citation Point, 143Jump To The Next Citation Point] contain far more detailed discussions of these points.

Regardless of whether the self-consistent expansion is used, the success of matched asymptotic expansions relies on the buffer region defined by 𝜀 ≪ R ≪ 1 (see Figure 10View Image). In this region, both the inner and outer expansions are valid. From the perspective of the outer expansion, this corresponds to an asymptotically small region around the world line: R ≪ 1. From the perspective of the inner expansion, it corresponds to asymptotic spatial infinity: 1∕R&tidle; = 𝜀∕R ≪ 1. Because both expansions are valid in this region, and because both are expansions of the same exact metric gαβ and hence must “match,” by working in the buffer region we can use information from the inner expansion to determine information about the outer expansion (or vice versa). We shall begin by solving the Einstein equation in the buffer region, using information from the inner expansion to determine the form of the external metric perturbation therein. In so doing, we shall determine the acceleration of the small body’s world line. Finally, using the field values in the buffer region, we shall construct a global solution for the metric perturbation.

In this calculation, the structure of the body is left unspecified. Our only condition is that part of the buffer region must lie outside the body, because we wish to solve the Einstein field equations in vacuum. This requires the body to be sufficiently compact. For example, our calculation would fail for a diffuse body such as our Sun; likewise, it would fail if a body became tidally disrupted. Although we will detail only the case of an uncharged body, the same techniques would apply to charged bodies; Gralla et al. [82Jump To The Next Citation Point] have recently performed a similar calculation for the electromagnetic self-force on an asymptotically small body in a flat background spacetime. Using very different methods, Futamase et al. [73] have calculated equations of motion for an asymptotically small charged black hole.

The structure of our discussion is as follows: In Section 21, we present the self-consistent expansion of the Einstein equation. Next, in Section 22, we solve the equations in the buffer region up to second order in the outer expansion. Last, in Section 23, we discuss the global solution in the outer expansion and show that it is that of a point particle at first order. Over the course of this calculation, we will take the opportunity to incorporate several details that we could have accounted for in the point-particle case but opted to neglect for simplicity: an explicit expansion of the acceleration vector that makes the self-consistent expansion properly systematic, and a finite time domain that accounts for the fact that large errors eventually accumulate if the approximation is truncated at any finite order. For more formal discussions of matched asymptotic expansions in general relativity, see Refs. [104, 145Jump To The Next Citation Point]; the latter reference, in particular, discusses the method as it pertains to the motion of small bodies. For background on the use of matched asymptotic expansions in applied mathematics, see Refs. [63Jump To The Next Citation Point, 96, 109, 111, 178]; the text by Eckhaus [63] provides the most rigorous treatment.

View Image

Figure 10: A small body, represented by the black disk, is immersed in a background spacetime. The internal zone is defined by R ≪ 1, while the external zone is defined by R ≫ 𝜀. Since 𝜀 ≪ 1, there exists a buffer region defined by 𝜀 ≪ R ≪ 1. In the buffer region 𝜀∕R and R are both small.

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