21 Self-consistent expansion

21.1 Introduction

We wish to represent the motion of the body through the external background spacetime (g ,ℳ ) αβ E, rather than through the exact spacetime (gα β,ℳ 𝜀). In order to achieve this, we begin by surrounding the body with a (hollow, three-dimensional) world tube Γ embedded in the buffer region. We define the tube to be a surface of constant radius s = ℛ (𝜀) in Fermi normal coordinates centered on a world line γ ⊂ ℳE, though the exact definition of the tube is immaterial. Since there exists a diffeomorphism between ℳ E and ℳ I in the buffer region, this defines a tube Γ ⊂ ℳ I I. Now, the problem is the following: what equation of motion must γ satisfy in order for Γ I 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 ∑ R −n (or ∑ s−n – we will assume s ∼ R 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, 152Jump To The Next Citation Point, 83Jump To The Next Citation Point, 144Jump To The Next Citation Point]. 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. [173Jump To The Next Citation Point] 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 gE (x,𝜀;γ), parametrized by γ, such that when γ is given by the correct equation of motion for a given value of 𝜀, we have gE (x,𝜀;γ(𝜀)) = g(x). The metric in the outer limit is, thus, taken to be the general expansion

gαβ(x,𝜀) = gEαβ(x, 𝜀;γ) = gαβ(x) + hαβ(x,𝜀;γ ), (21.1 )
∑∞ h (x,𝜀;γ) = 𝜀nh(n)(x;γ). (21.2 ) αβ αβ n=1
In the point-particle case, solving Einstein’s equations determined the equation of motion of the particle’s world line; in this case, it will determine the world line γ for which the inner expansion is mass-centered. In this self-consistent expansion, the perturbations produced by the body are constructed about a fixed world line determined by the particular value of 𝜀 at which one seeks an approximation.

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.

21.2 Field equations in outer expansion

In the outer expansion, we seek a solution in a vacuum region Ω outside of Γ. We specify Ω ⊂ ℳE to be an open set consisting of the future domain of dependence of the spacelike initial-data surface Σ, excluding the interior of the world tube Γ. This implies that the future boundary of Ω is a null surface 𝒥. Refer to Figure 11View Image for an illustration. The boundary of the domain is ∂Ω := Γ ∪ 𝒥 ∪ Σ. The spatial surface Σ is chosen to intersect Γ at the initial time t = 0.

View Image

Figure 11: The spacetime region Ω is bounded by the union of the spacelike surface Σ, the timelike tube Γ, and the null surface 𝒥.

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. [144Jump To The Next Citation Point], 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 Ω. 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.

Field equations

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 h α β, everywhere in Ω, such that L [h] = 0 μ. Here

( ) α βν 1- αβ ν Lμ[h] := g μg − 2g gμ ∇ νhαβ (21.3 )
is the Lorenz-gauge operator that was first introduced in Sections 16.1 and 19.1; the condition L μ[h] = 0 is the same statement as ∇ νγμν = 0, where γ μν := hμν − 12gμνg αβhαβ is the “trace-reversed” metric perturbation. We discuss the validity of this assumption below.

Just as in the case of a point mass, this choice of gauge reduces the vacuum Einstein equation R = 0 μν to a weakly nonlinear wave equation that can be expanded and solved at fixed γ. However, we now seek a solution only in the region Ω, where the energy-momentum tensor vanishes, so the resulting sequence of wave equations reads

(1) Eμν[h ] = 0, (21.4 ) E [h(2)] = 2δ2R [h (1)], (21.5 ) μν μν ...
( ) E μν[h ] := gα gβ ∇γ∇ γ + 2R α β hαβ (21.6 ) μ ν μ ν
is the tensorial wave operator that was first introduced in Sections 16.1 and 19.1, and the second-order Ricci tensor δ2Rμν, which is quadratic in its argument, is shown explicitly in Eq (A.1View Equation). More generally, we can write the nth-order equation as
[ ] [ ] E μν h(n) = S(μnν)h (1),...,h(n−1),γ , (21.7 )
where the source term (n) Sμν consists of nonlinear terms in the expansion of the Ricci tensor.

Again as in the case of a point particle, we can easily write down formal solutions to the wave equations, for arbitrary γ. Using the same methods as were used to derive the Kirchoff representation in Section 16.3, we find

(n) 1 ∮ ( ′ ′ (n) (n) ′′) ′ 1 ∫ ′ ′ (n) hαβ = --- G+αβγ δ∇ μ′hγ′δ′ − hγ′δ′∇ μ′G+αβγ δ dS μ + --- G+αβγ δS γ′δ′dV ′. (21.8 ) 4 π∂Ω 4π Ω
Because 𝒥 is a future null surface, the integral over it vanishes. Hence, this formal solution requires only initial data on Σ and boundary data on Γ. Since Γ lies in the buffer region, the boundary data on it is determined by information from the inner expansion.

One should note several important properties of these integral representations: First, x must lie in the interior of Ω; an alternative expression must be derived if x 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 Γ 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 E αβ onto Eq. (21.8View Equation), we see that the integral representation of (n) hαβ is guaranteed to satisfy the wave equation at each point x ∈ Ω. 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 − n s 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 γ possesses an expansion

aμ(t,𝜀) = aμ (t) + 𝜀a μ (t;γ) + .... (21.9 ) (0) (1)
This leads to the set of equations
L(0)[h (1)] = 0, (21.10 ) μ [ ] [ ] L(μ1)h (1) = − L(μ0) h(2) , (21.11 ) . ..
or, more generally, for n > 0,
[ ] ∑n [ ] L (nμ) h(1) = − L(μn−m) h(m+1) . (21.12 ) m=1
In these expressions, L(0)[f ] μ is the Lorenz-gauge operator acting on the tensor field f αβ evaluated with μ aμ = a(0), (1) L μ [f] consists of the terms in Lμ[f] that are linear in μ a (1), and (n) Lμ [f ] contains the terms linear in a(n)μ, the combinations a(n−1)μa(1)ν, and so on. Imposing these gauge conditions on the solutions to the wave equations will determine the acceleration of the world line. Although we introduce an expansion of the acceleration vector in order to obtain a systematic sequence of equations that can be solved exactly, such an expansion also trivially eliminates the need for order-reduction of the resulting equations of motion, since it automatically leads to equations for a(1)μ in terms of a(0)μ, a(2)μ in terms of a(0)μ and a (1)μ, and so on.

Gauge transformations and the Lorenz condition

The outer expansion is defined not only by holding α x fixed, but also by demanding that the mass dipole of the body vanishes when calculated in coordinates centered on γ. If we perform a gauge transformation generated by a vector ξ(1)α(x;γ ), then the mass dipole will no longer vanish in those coordinates. Hence, a new world line γ ′ 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: (hαβ,γ) → (h′αβ,γ′). The transformation law for the world line was first derived by Barack and Ori [17Jump To The Next Citation Point]; it was displayed in Eq. (1.49View Equation), 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 hαβ. If we begin with the metric in an arbitrary gauge, then the gauge vectors 𝜀ξα(1)[γ], 𝜀2ξα(2)[γ], etc., induce the transformation

hαβ → h′ = hαβ + Δh αβ αβ = hαβ + 𝜀£ ξ(1)gαβ + 12𝜀2(£ξ(2) + £2ξ(1))gαβ + 𝜀2£ ξ(1)h (1α)β + O(𝜀3). (21.13 )
If ′ h αβ is to satisfy the gauge condition ′ Lμ[h ], then ξ must satisfy L μ[Δh ] = − L μ[h ]. After a trivial calculation, this equation becomes
∑ 𝜀n- α α[ (1)] 2 α[ (2)] 2 α [1 2 (1)] 3 n!□ ξ(n) = − 𝜀L h − 𝜀 L h − 𝜀 L 2£ ξ(1)g + £ ξ(1)h + O (𝜀 ). (21.14 ) n>0
Solving this equation for arbitrary γ, we equate coefficients of powers of 𝜀, leading to a sequence of wave equations of the form
□ ξα(n) = W (αn), (21.15 )
where α W (n) is a functional of α α ξ(1),...,ξ(n−1) and (1) (n) hαβ,...,h αβ. We seek a solution in the region Ω described in the preceding section. The formal solution reads
∫ ∮ α 1-- α α′ ′ 1-- ( α γ′ γ′ α ) μ′ ξ(n) = − 4π G +α′W (n)dV + 4π G+ γ′∇ μ′ξ(n) − ξ(n)∇ μ′G+ γ′ dS . (21.16 ) Ω ∂Ω
From this we see that the Lorenz gauge condition can be adopted to any desired order of accuracy, given the existence of self-consistent data on a tube Γ of asymptotically small radius. We leave the question of the data’s existence to future work. This argument was first presented in Ref. [145Jump To The Next Citation Point].

21.3 Field equations in inner expansion

For the inner expansion, we assume the existence of some local polar coordinates X α = (T, R,ΘA ), such that the metric can be expanded for 𝜀 → 0 while holding fixed &tidle;R := R ∕𝜀, ΘA, and T; to relate the inner and outer expansions, we assume R ∼ s, but otherwise leave the inner expansion completely general.

This leads to the ansatz

gαβ(T,R&tidle;, ΘA, 𝜀) = gbαoβdy(T,R&tidle;, ΘA ) + H αβ(T, &tidle;R, ΘA, 𝜀), (21.17 )
where H αβ at fixed values of &tidle; A (T,R, Θ ) is a perturbation beginning at order 𝜀. This equation represents an asymptotic expansion along flow lines of constant R ∕𝜀 as 𝜀 → 0. It is tensorial in the usual sense of perturbation theory: the decomposition into gbαoβdy and H αβ is valid in any coordinates that can be decomposed into 𝜀-independent functions of the scaled coordinates plus O (𝜀) functions of them. As written, with body gα β depending only on the scaled coordinates and independent of 𝜀, the indices in Eq. (21.17View Equation) can be taken to refer to the unscaled coordinates (T,R, ΘA ). However, writing the components in the scaled coordinates will not alter the form of the expansion, but only introduce an overall rescaling of spatial components due to the spatial forms transforming as, e.g., &tidle; dR → 𝜀d R. For example, if the body is a small Schwarzschild black hole of ADM mass 𝜀m (T ), then in scaled Schwarzschild coordinates &tidle; A (T,R, Θ ), body &tidle; A g αβ (T, R,Θ ) is given by
ds2 = − (1 − 2m (T )∕&tidle;R )dT 2 + (1 − 2m (T)∕R&tidle;)−1𝜀2dR&tidle;2 + 𝜀2R&tidle;2 (dΘ2 + sin ΘdΦ2 ). (21.18 )
As we would expect from the fact that the inner limit follows the body down as it shrinks, all points are mapped to the curve R = 0 at 𝜀 = 0, such that the metric in the scaled coordinates naturally becomes one-dimensional at 𝜀 = 0. This singular limit can be made regular by rescaling time as well, such that T&tidle; = (T − T0)∕𝜀, and then rescaling the entire metric by a conformal factor 1∕𝜀2. In order to arrive at a global-in-time inner expansion, rather than a different expansion at each time T0, we forgo this extra step. We do, however, make an equivalent assumption, which is that the metric body gαβ and its perturbations are quasistatic (evolving only on timescales ∼ 1). Both approaches are equivalent to assuming that the exact metric contains no high-frequency oscillations occurring on the body’s natural timescale ∼ 𝜀. In other words, the body is assumed to be in equilibrium. If we did not make this assumption, high-frequency oscillations could propagate throughout the external spacetime, invalidating our external expansion.

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. [144Jump To The Next Citation Point] 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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