22 General expansion in the buffer region

We now seek the general solution to the equations of the outer expansion in the buffer region. To perform the expansion, we adopt Fermi coordinates centered about γ and expand for small s. In solving the first-order equations, we will determine a(0)μ; in solving the second-order equations, we will determine a(1)μ, including the self-force on the body. Although we perform this calculation in the Lorenz gauge, the choice of gauge is not essential for our purposes here – the essential aspect is our assumed expansion of the acceleration of the world line γ.

22.1 Metric expansions

The method of matched asymptotic expansions relies on the fact that the inner and outer expansion agree term by term when re-expanded in the buffer region, where πœ€ β‰ͺ s β‰ͺ 1. To illustrate this idea of matching, consider the forms of the two expansions in the buffer region. The inner expansion holds &tidle;s constant (since R ∼ s) while expanding for small πœ€. But if &tidle;s is replaced with its value sβˆ•πœ€, the inner expansion takes the form g αβ = gbody(sβˆ• πœ€) + πœ€H (1)(sβˆ• πœ€) + ⋅⋅⋅ αβ αβ, where each term has a dependence on πœ€ that can be expanded in the limit πœ€ → 0 to arrive at the schematic forms body 2 2 g (sβˆ•πœ€) = 1 ⊕ πœ€βˆ•s ⊕ πœ€ βˆ•s ⊕ ... and (1) 2 πœ€H (sβˆ•πœ€) = s ⊕ πœ€ ⊕ πœ€ βˆ•s ⊕ ⋅⋅⋅, where ⊕ signifies “plus terms of the form” and the expanded quantities can be taken to be components in Fermi coordinates. Here we have preemptively restricted the form of the expansions, since terms such as s2βˆ•πœ€ must vanish because they would have no corresponding terms in the outer expansion. Putting these two expansions together, we arrive at

πœ€- g (bu ffer) = 1 ⊕ s ⊕ s ⊕ πœ€ ⊕ ⋅⋅⋅. (22.1 )
Since this expansion relies on both an expansion at fixed &tidle;s and an expansion at fixed s, it can be expected to be accurate if s β‰ͺ 1 and πœ€ β‰ͺ s – that is, in the buffer region πœ€ β‰ͺ s β‰ͺ 1.

On the other hand, the outer expansion holds s constant (since s is formally of the order of the global external coordinates) while expanding for small πœ€, leading to the form gαβ = g αβ(s) + πœ€h (1α)β(s) + ⋅⋅⋅. But very near the world line, each term in this expansion can be expanded for small s, leading to g = 1 ⊕ s ⊕ s2 ⊕ ⋅⋅⋅ and πœ€h(1) = πœ€βˆ•s ⊕ πœ€ ⊕ πœ€s ⊕ ⋅⋅⋅. (Again, we have restricted this form because terms such as 2 πœ€βˆ•s cannot arise in the inner expansion.) Putting these two expansions together, we arrive at

πœ€ g(buffer) = 1 ⊕ s ⊕ --⊕ πœ€ ⋅⋅⋅. (22.2 ) s
Since this expansion relies on both an expansion at fixed s and an expansion for small s, it can be expected to be accurate in the buffer region πœ€ β‰ͺ s β‰ͺ 1. As we can see, the two buffer-region expansions have an identical form; and because they are expansions of the same exact metric g, they must agree term by term.

One can make use of this fact by first determining the inner and outer expansions as fully as possible, then fixing any unknown functions in them by matching them term by term in the buffer region; this was the route taken in, e.g., Refs. [130Jump To The Next Citation Point, 142Jump To The Next Citation Point, 49Jump To The Next Citation Point, 170]. However, such an approach is complicated by the subtleties of matching in a diffeomorphism-invariant theory, where the inner and outer expansions are generically in different coordinate systems. See Ref. [145Jump To The Next Citation Point] for an analysis of the limitations of this approach as it has typically been implemented. Alternatively, one can take the opposite approach, working in the buffer region first, constraining the forms of the two expansions by making use of their matching, then using the buffer-region information to construct a global solution; this was the route taken in, e.g., Refs. [102, 83Jump To The Next Citation Point, 144Jump To The Next Citation Point]. In general, some mixture of these two approaches can be taken. Our calculation follows Ref. [144Jump To The Next Citation Point]. The only information we take from the inner expansion is its general form, which is characterized by the multipole moments of the body. From this information, we determine the external expansion, and thence the equation of motion of the world line.

Over the course of our calculation, we will find that the external metric perturbation in the buffer region is expressed as the sum of two solutions: one that formally diverges at s = 0 and is entirely determined from a combination of (i) the multipole moments of the internal background metric body gαβ, (ii) the Riemann tensor of the external background g αβ, and (iii) the acceleration of the world line γ; and a second solution that is formally regular at s = 0 and depends on the past history of the body and the initial conditions of the field. At leading order, these two solutions are identified as the Detweiler–Whiting singular and regular fields hS αβ and hR αβ, respectively, and the self-force is determined entirely by hR αβ. Along with the self-force, the acceleration of the world line includes the Papapetrou spin force [138Jump To The Next Citation Point]. This calculation leaves us with the self-force in terms of the the metric perturbation in the neighbourhood of the body. In Section 23, we use the local information from the buffer region to construct a global solution for the metric perturbation, completing the solution of the problem.

22.2 The form of the expansion

Before proceeding, we define some notation. We use the multi-index notation ωL := ωi1 ⋅⋅⋅ωiβ„“ := ωi1⋅⋅⋅iβ„“. Angular brackets denote the STF combination of the enclosed indices, and a tensor bearing a hat is an STF tensor. To accommodate this, we now write the Fermi spatial coordinates as xa, instead of ˆxa as they were written in previous sections. Finally, we define the one-forms tα := ∂αt and xaα := ∂αxa.

One should note that the coordinate transformation x α(t,xa ) between Fermi coordinates and the global coordinates is πœ€-dependent, since Fermi coordinates are tethered to an πœ€-dependent world line. If one were using a regular expansion, then this coordinate transformation would devolve into a background coordinate transformation to a Fermi coordinate system centered on a geodesic world line, combined with a gauge transformation to account for the πœ€-dependence. But in the self-consistent expansion, the transformation is purely a background transformation, because the πœ€-dependence in it is reducible to that of the fixed world line.

Because the dependence on πœ€ in the coordinate transformation cannot be reduced to a gauge transformation, in Fermi coordinates the components gα β of the background metric become πœ€-dependent. This dependence takes the explicit form of factors of the acceleration aμ(t,πœ€) and its derivatives, for which we have assumed the expansion ai(t,πœ€) = a(0)i(t) + a(1)i(t;γ ) + O (πœ€2). There is also an implicit dependence on πœ€ in that the proper time t on the world line depends on πœ€ if written as a function of the global coordinates; but this dependence can be ignored so long as we work consistently with Fermi coordinates.

Of course, even in these πœ€-dependent coordinates, g μν remains the background metric of the outer expansion, and (n) hμν is an exact solution to the wave equation (21.7View Equation). At first order we will, therefore, obtain h (1μ)ν exactly in Fermi coordinates, for arbitrary aμ. However, for some purposes an approximate solution of the wave equation may suffice, in which case we may utilize the expansion of μ a. Substituting that expansion into gμν and (n) h μν yields the buffer-region expansions

a i a (0)i (1) a (1)i 2 gμν(t,x ;a ) = gμν(t,x ;a ) + πœ€gμν(t,x ;a ) + O(πœ€ ) (22.3 ) (n) a i (n) a (0)i hμν (t,x ;a ) = h μν (t,x ;a ) + O (πœ€), (22.4 )
where indices refer to Fermi coordinates, g(μ1)ν is linear in a(1)i and its derivatives, and for future compactness of notation we define (n) a (n) a (0)i h Bμν(t,x ) := h μν (t,x ;a ), where the subscript ‘B’ stands for ‘buffer’. In the case that a(0)i = 0, these expansions will significantly reduce the complexity of calculations in the buffer region. For that reason, we shall use them in solving the second-order wave equation, but we stress that they are simply a means of economizing calculations in Fermi coordinates; they do not play a fundamental role in the formalism, and one could readily do without them.

Now, we merely assume that in the buffer region there exists a smooth coordinate transformation between the local coordinates (T,R, ΘA ) and the Fermi coordinates (t,xa) such that T ∼ t, R ∼ s, and ΘA ∼ πœƒA. The buffer region corresponds to asymptotic infinity s ≫ πœ€ (or &tidle;s ≫ 1) in the internal spacetime. So after re-expressing &tidle;s as sβˆ•πœ€, the internal background metric can be expanded as

∑ ( πœ€)n gbαoβdy(t,&tidle;s,πœƒA ) = -- gbαoβdy(n)(t,πœƒA). (22.5 ) n≥0 s
As mentioned above, since the outer expansion has no negative powers of πœ€, we exclude them from the inner expansion. Furthermore, since gαβ + hαβ = gbαoβdy+ H αβ, we must have body(0) a gαβ = gαβ(x = 0 ), since these are the only terms independent of both πœ€ and s. Thus, noting that a gαβ(x = 0) = ηαβ := diag (− 1,1,1, 1), we can write
πœ€ ( πœ€)2 gbαoβdy(t,&tidle;s,πœƒA) = ηαβ + --gbαodβy(1)(t,πœƒA ) + -- gbBoαdβy(2)(t,πœƒA ) + O (πœ€3βˆ•s3), (22.6 ) s s
implying that the internal background spacetime is asymptotically flat.

We assume that the perturbation H αβ can be similarly expanded in powers of πœ€ at fixed &tidle;s,

A (1) A 2 (2) A 3 H αβ(t,&tidle;s,πœƒ ,πœ€) = πœ€Hαβ (t,s&tidle;, πœƒ ;γ) + πœ€ H αβ(t,&tidle;s,πœƒ ;γ) + O (πœ€ ), (22.7 )
and that each coefficient can be expanded in powers of 1βˆ•&tidle;s = πœ€βˆ•s to yield
2 πœ€H (1)(&tidle;s) = sH (0,1)+ πœ€H (1,0)+ πœ€-H (2,−1)+ O(πœ€3βˆ•s2), (22.8 ) αβ αβ αβ s αβ πœ€2H (2)(&tidle;s) = s2H (0,2)+ πœ€sH (1,1)+ πœ€2H (2,0) + πœ€2lnsH (2,0,ln)+ O (πœ€3βˆ•s), (22.9 ) α(β3) αβ αβ αβ αβ πœ€3H αβ(&tidle;s) = O (πœ€3,πœ€2s,πœ€s2,s3), (22.10 )
where (n,m) H αβ, the coefficient of n πœ€ and m s, is a function of t and A πœƒ (and potentially a functional of γ). Again, the form of this expansion is constrained by the fact that no negative powers of πœ€ can appear in the buffer region. (One might think that terms with negative powers of πœ€ could be allowed in the expansion of gbody αβ if they are exactly canceled by terms in the expansion of Hαβ, but the differing powers of s in the two expansions makes this impossible.) Note that explicit powers of s appear because πœ€&tidle;s = s. Also note that we allow for a logarithmic term at second order in πœ€; this term arises because the retarded time in the internal background includes a logarithmic correction of the form πœ€ lns (e.g., ∗ t − r → t − r in Schwarzschild coordinates). Since we seek solutions to a wave equation, this correction to the characteristic curves induces a corresponding correction to the first-order perturbations.

The expansion of H αβ may or may not hold the acceleration fixed. Regardless of this choice, the general form of the expansion remains valid: incorporating the expansion of the acceleration would merely shuffle terms from one coefficient to another. And since the internal metric body gαβ + H αβ must equal the external metric gαβ + hαβ, the general form of the above expansions of gbody αβ and H αβ completely determines the general form of the external perturbations:

h(1)= 1h(1,−1)+ h(1,0)+ sh(1,1)+ O(s2), (22.11 ) αβ s αβ αβ αβ (2) 1- (2,−2) 1-(2,− 1) (2,0) (2,0,ln) hαβ = s2hαβ + shαβ + hαβ + ln shαβ + O (s ), (22.12 )
where each (n,m) hαβ depends only on t and A πœƒ, along with an implicit functional dependence on γ. If the internal expansion is performed with μ a held fixed, then the internal and external quantities are related order by order: e.g., ∑ (0,m) m H αβ = gαβ, (1,−1) body(1) hαβ = gαβ, and (1,0) (1,0) hαβ = H αβ. Since we are not concerned with determining the internal perturbations, the only such relationship of interest is h (n,−n)= gbody(n) αβ αβ. This equality tells us that the most divergent, s−n piece of the nth-order perturbation (n) h αβ is defined entirely by the nth-order piece of the internal background metric body gαβ, which is the metric of the body if it were isolated.

To obtain a general solution to the Einstein equation, we write each h(n,m) αβ as an expansion in terms of irreducible symmetric trace-free pieces:

h(n,m) = ∑ ˆA(n,m)ˆωL, (22.13 ) tt L β„“∑≥0 ∑ [ ] h(n,m) = ˆB (n,m)ˆω L + ˆC (n,m)ˆωL− 1 + πœ– c ˆD (n,m)ˆωbL−1 , (22.14 ) ta L a aL− 1 ab cL− 1 β„“≥0∑ β„“≥1∑ ∑ [ ] h(n,m) = δab ˆK (n,m)ˆωL + Eˆ(n,m)ˆωabL + Fˆ(n,m )ˆωb⟩L− 1 + πœ–cd(aωˆb )cL−1Gˆ(n,m ) ab β„“≥0 L β„“≥0 L β„“≥1 L−1⟨a dL−1 ∑ [ ] + Hˆ(anb,mL−)2ˆωL −2 + πœ–cd(aˆI(nb),dmL)−2ˆωcL− 2 . (22.15 ) β„“≥2
Here a hat indicates that a tensor is STF with respect to δab, angular brackets ⟨⟩ indicate the STF combination of enclosed indices, parentheses indicate the symmetric combination of enclosed indices, and symbols such as Aˆ(nL,m) are functions of time (and potentially functionals of γ) and are STF in all their indices. Each term in this expansion is linearly independent of all the other terms. All the quantities on the right-hand side are flat-space Cartesian tensors; their indices can be raised or lowered with δab. Refer to Appendix B for more details about this expansion.

Now, since the wave equations (21.4View Equation) and (21.5View Equation) are covariant, they must still hold in the new coordinate system, despite the additional πœ€-dependence. Thus, both equations could be solved for arbitrary acceleration in the buffer region. However, due to the length of the calculations involved, we will instead solve the equations

E αβ[h (1)] = 0, (22.16 ) (0) (2) 2 (0) (1) Eα β[h B ] = 2 δ Rαβ[h ] + O (πœ€), (22.17 )
where E (0)[f ] := E [f ]|| a=a(0) and δ2R (0)[f] := δ2R[f]|| a=a(0). In analogy with the notation used for L(μn), E (μ1ν) [f] and δ2R (μ1)ν[f] would be linear in a μ(1), E (μ2ν) [f] and δ2R (μ2ν)[f] would be linear in μ a(2) and quadratic in μ a(1), and so on. For a function f ∼ 1, (n) L μ [f], E (μnν) [f], and δ2R (nμ)ν [f] correspond to the coefficients of πœ€n in expansions in powers of πœ€. Equation (22.16View Equation) is identical to Eq. (21.4View Equation). Equation (22.17View Equation) follows directly from substituting Eqs. (22.3View Equation) and (22.4View Equation) into Eq. (21.5View Equation); in the buffer region, it captures the dominant behaviour of (2) h αβ, represented by the approximation (2) hB αβ, but it does not capture its full dependence on acceleration. If one desired a global second-order solution, one might need to solve Eq. (21.5View Equation), but for our purpose, which is to determine the first-order acceleration aμ(1), Eq. (22.17View Equation) will suffice.

Unlike the wave equations, the gauge conditions (21.10View Equation) and (21.11View Equation) already incorporate the expansion of the acceleration. As such, they are unmodified by the replacement of the second-order wave equation (21.5View Equation) with its approximation (22.17View Equation). So we can write

[ ] L(μ0)h (1) = 0, (22.18 ) (1)[ (1)] (0)[ (2)] Lμ h = − Lμ hB , (22.19 )
where the first equation is identical to Eq. (21.10View Equation), and the second to Eq. (21.11View Equation). (The second identity holds because (0)[ (2)] (0)[ (2)] L μ hB = L μ h, since (2) h Bαβ differs from (2) hαβ by α a(1) and higher acceleration terms, which are set to zero in L(μ0).) We remind the reader that while this gauge choice is important for finding the external perturbations globally, any other choice would suffice in the buffer region calculation.

In what follows, the reader may safely assume that all calculations are lengthy unless noted otherwise.

22.3 First-order solution in the buffer region

In principle, solving the first-order Einstein equation in the buffer region is straightforward. One need simply substitute the expansion of h(1α)β, given in Eq. (22.11View Equation), into the linearized wave equation (22.16View Equation) and the gauge condition (22.18View Equation). Equating powers of s in the resulting expansions then yields a sequence of equations that can be solved for successively higher-order terms in (1) h αβ. Solving these equations consists primarily of expressing each quantity in its irreducible STF form, using the decompositions (B.3View Equation) and (B.7View Equation); since the terms in this STF decomposition are linearly independent, we can solve each equation term by term. This calculation is aided by the fact that a 0 ∇ α = x α∂a + O(s ), so that, for example, the wave operator E αβ consists of a flat-space Laplacian ∂a ∂a plus corrections of order 1βˆ•s. Appendix B also lists many useful identities, particularly ∂as = ωa := xaβˆ•s, ωa∂ ˆωL = 0 a, and the fact that ˆωL is an eigenvector of the flat-space Laplacian: 2 a L L s ∂ ∂a ˆω = − β„“(β„“ + 1)ˆω.

Summary of results

Before proceeding with the calculation, which consists mostly of tedious and lengthy algebra, we summarize the results. The first-order perturbation (1) hαβ consists of two pieces, which we will eventually identify with the Detweiler–Whiting regular and singular fields. In the buffer-region expansion, the regular field consists entirely of unknowns, which is to be expected since as a free radiation field, it must be provided by boundary data. Only when we consider the global solution, in Section 23, will we express it in terms of a tail integral. On the other hand, the singular field is locally determined, and it is characterized by the body’s monopole moment m. More precisely, it is fully determined by the tidal fields of the external background spacetime and the Arnowitt–Deser–Misner mass of the internal background spacetime gbody αβ. By itself the wave equation does not restrict the behaviour of this monopole moment, but imposing the gauge condition produces the evolution equations

∂tm = 0, ai = 0. (22.20 ) (0)
Hence, at leading order, the body behaves as a test particle, with constant mass and vanishing acceleration.

Order (1,–1)

We now proceed to the details of the calculation. We begin with the most divergent term in the wave equation: the order 3 1βˆ•s, flat-space Laplacian term

1 -∂c ∂ch (1α,β−1)= 0. (22.21 ) s
The tt-component of this equation is
0 = − ∑ β„“(β„“ + 1) ˆA(1,− 1)ˆωL, (22.22 ) L β„“≥0
from which we read off that Aˆ(1,−1) is arbitrary and ˆA (1L,− 1) must vanish for all β„“ ≥ 1. The ta-component is
∑ (1,−1) L ∑ (1,−1) L− 1 0 = − (β„“ + 1)(β„“ + 2 )B ˆL ˆωa − β„“(β„“ − 1 )C ˆaL−1 ˆω β„“≥0 β„“≥1 ∑ ˆ(1,−1) L− 1 − β„“(β„“ + 1)πœ–abcD cL−1 ˆωb , (22.23 ) β„“≥1
from which we read off that Cˆ(a1,−1) is arbitrary and all other coefficients must vanish. Lastly, the ab-component is
∑ ˆ (1,− 1) L ∑ ˆ(1,−1) L 0 = − δab β„“(β„“ + 1)KL ˆω − (β„“ + 2)(β„“ + 3)E L ˆωab β„“≥0 β„“≥0 − ∑ β„“(β„“ + 1) ˆF (1,−1)ωˆ L−1 − ∑ (β„“ + 1)(β„“ + 2)πœ– ˆω cL− 1G ˆ(1,−1) L− 1⟨a b⟩ cd(a b) dL−1 β„“∑≥1 β„“≥1 ∑ − (β„“ − 2)(β„“ − 1)Hˆ(1,−1)ˆωL− 2 − β„“(β„“ − 1)πœ–cd(aˆI(1,−1)ˆωcL− 2, (22.24 ) β„“≥2 abL−2 β„“≥2 b)dL−2
from which we read off that Kˆ(1,−1) and (1,−1) Hˆab are arbitrary and all other coefficients must vanish. Thus, we find that the wave equation constrains h (1,− 1) αβ to be
h(1,− 1) = Aˆ(1,−1)t t + 2Cˆ(1,−1)t xa + (δ Kˆ (1,− 1) + Hˆ(1,−1))xaxb. (22.25 ) αβ α β a (β α) ab ab α β
This is further constrained by the most divergent, 1βˆ•s2 term in the gauge condition, which reads
1 (1,−1) c 1 μν (1,−1) − s2 hαc ω + 2s2ω αη hμν = 0. (22.26 )
From the t-component of this equation, we read off ˆC(a1,− 1) = 0; from the a-component, Kˆ(1,−1) = Aˆ(1,−1) and ˆH (1,−1)= 0 ab. Thus, h (1,− 1) αβ depends only on a single function of time, ˆA(1,− 1). By the definition of the ADM mass, this function (times πœ€) must be twice the mass of the internal background spacetime. Thus, h(1α,β−1) is fully determined to be
h (1,−1)= 2m (t)(tαtβ + δabxaxb ), (22.27 ) αβ α β
where m (t) is defined to be the mass at time t divided by the initial mass πœ€ := m0. (Because the mass will be found to be a constant, m (t) is merely a placeholder; it is identically unity. We could instead set πœ€ equal to unity at the end of the calculation, in which case m would simply be the mass at time t. Obviously, the difference between the two approaches is immaterial.)

Order (1,0)

At the next order, h(1,0) αβ, along with the acceleration of the world line and the time-derivative of the mass, first appears in the Einstein equation. The order 1βˆ•s2 term in the wave equation is

2m ∂c ∂ch (1α,β0) = − --2-acωc (3tαtβ − δabxaαxbβ), (22.28 ) s
where the terms on the right arise from the wave operator acting on s− 1h(α1,β−1). This equation constrains (1,0) h αβ to be
(1,0) (1,0) c htt = Aˆ + 3mac ω , (1,0) ˆ(1,0) hta = C a( , ) (22.29 ) h(1,0)= δ ˆK (1,0) − ma ωc + Hˆ(1,0). ab ab c ab
Substituting this result into the order 1βˆ•s term in the gauge condition, we find
− 4tα∂tm + 4m-a (0a)xaα = 0. (22.30 ) s s
Thus, both the leading-order part of the acceleration and the rate of change of the mass of the body vanish:
∂m--= 0, ai = 0. (22.31 ) ∂t (0)

Order (1,1)

At the next order, h(α1β,1), along with squares and derivatives of the acceleration, first appears in the Einstein equation, and the tidal fields of the external background couple to − 1 (1,−1) s hαβ. The order 1βˆ•s term in the wave equation becomes

( 2) 20m 3m 8m s∂c∂c + -- h (1tt,1) = − ----β„°ijˆωij − ---a⟨iaj⟩ˆωij + ---aiai, (22.32 ) ( s) 3s s s c 2 (1,1) 8m j ik 4m s∂ ∂c + -- h ta = − ---πœ–aijℬ kˆω − ---aΛ™a, (22.33 ) ( s) 3s s c 2- (1,1) 20m-- ij 76m-- 16m-- i 8m-- s∂ ∂c + s h ab = 9s δabβ„°ijˆω − 9s β„°ab − 3s β„° ⟨aˆωb⟩i + s a⟨aab⟩ m ( ) + --δab 83aiai − 3a⟨iaj⟩ˆωij . (22.34 ) s
From the tt-component, we read off that (1,1) ˆAi is arbitrary, Aˆ(1,1) = 4maiai, and Aˆ(1,1)= 5m β„°ij + 3ma ⟨iaj⟩ ij 3 4; from the ta-component, Bˆ(1,1), ˆC (1,1) ij, and Dˆ(1,1) i are arbitrary, (1,1) Cˆi = − 2m aΛ™i, and (1,1) 2 Dˆij = 3m ℬij; from the ab component, (1,1) Kˆi, (1,1) Fˆi, (1,1) Hˆijk, and (1,1) Iˆij are arbitrary, and Kˆ (1,1) = 4maiai 3, ˆK (1,1) = − 5m β„°ij + 3ma ⟨iaj⟩ ij 9 4, Fˆ(1,1)= 4m β„°ij ij 3, and (1,1) 38 Hˆij = − 9-m β„°ij + 4ma ⟨iaj⟩.

Substituting this into the order s0 terms in the gauge condition, we find

0 = (ωi + s∂i)h (1,1) − 1ημν(ω − s∂ )h (1,1)xa − ∂ h(1,0)− 1ημν∂ h(1,0)t αi 2 a a μν α tαt 2 t μν α + 43m β„°ijωˆijω α + 23m β„°aiωixaα, (22.35 )
where the equation is to be evaluated at i i a = a(0) = 0. From the t-component, we read off
( ) ˆ(1,1) 1 ˆ(1,0) ˆ(1,0) B = 6∂t A + 3K . (22.36 )
From the a-component,
( ) Fˆ(a1,1)= 310- Kˆ(a1,1)− ˆA(a1,1)+ ∂tCˆ(a1,0) . (22.37 )
It is understood that both these equations hold only when evaluated at ai = 0.

Thus, the order s component of h(1) αβ is

h (1,1) = 4ma ai + ˆA(1,1)ωi + 5m β„° ωˆij + 3ma a ωˆij, tt i i 3 ij 4 ⟨i j⟩ h (1ta,1) = Bˆ(1,1)ωa − 2m Λ™aa + Cˆ(a1i,1)ωi + πœ–aij ˆD (1j,1)ωi + 23m πœ–aijℬjkˆωik, (1,1) (4 i (1,1) i 5 ij 3 ij) 4 i (22.38 ) h ab = δab 3maia + Kˆi ω − 9m β„°ijˆω + 4ma ⟨iaj⟩ˆω + 3 mβ„° ⟨aˆωb⟩i 38 ˆ (1,1) i j ˆ(1,1) i ˆ(1,1) − 9 m β„°ab + 4ma ⟨aab⟩ + H abi ω + πœ–i (aIb)j ω + F ⟨a ωb⟩.
where ˆB (1,1) and Fˆ(a1,1) are constrained to satisfy Eqs. (22.36View Equation) and (22.37View Equation).

First-order solution

To summarize the results of this section, we have h(α1β)= s−1h(α1β,−1)+ h(1α,β0)+ sh(α1β,1)+ O(s2), where (1,− 1) h αβ is given in Eq. (22.27View Equation), (1,0) hαβ is given in Eq. (22.29View Equation), and (1,1) hαβ is given in Eq. (22.38View Equation). In addition, we have determined that the ADM mass of the internal background spacetime is time-independent, and that the acceleration of the body’s world line vanishes at leading order.

22.4 Second-order solution in the buffer region

Though the calculations are much lengthier, solving the second-order Einstein equation in the buffer region is essentially no different from solving the first. We seek to solve the approximate wave equation (22.17View Equation), along with the gauge condition (22.19View Equation), for the second-order perturbation (2) (2)| hB αβ := h αβ|a=a0; doing so will also, more importantly, determine the acceleration aμ (1). In this calculation, the acceleration is set to i i a = a(0) = 0 everywhere except in the left-hand side of the gauge condition, (1) (1) L μ [h ], which is linear in aμ(1).

Summary of results

We first summarize the results. As at first order, the metric perturbation contains a regular, free radiation field and a singular, bound field; but in addition to these pieces, it also contains terms sourced by the first-order perturbation. Again, the regular field requires boundary data to be fully determined. And again, the singular field is characterized by the multipole moments of the body: the mass dipole Mi of the internal background metric gbody αβ, which measures the shift of the body’s centre of mass relative to the world line; the spin dipole Si of body gαβ, which measures the spin of the body about the world line; and an effective correction δm to the body’s mass. The wave equation by itself imposes no restriction on these quantities, but by imposing the gauge condition we find the evolution equations

m- ˆ(1,0) 5m-- ˆ(1,0) ∂tδm = 3 ∂tA + 6 ∂tK , (22.39 ) ∂ S = 0, (22.40 ) t a (1) (1,1) (1,0) ∂2tMa + β„°abM b = − aa + 12Aˆa − ∂t ˆCa − 1m-Siℬia. (22.41 )
The first of these tells us that the free radiation field created by the body creates a time-varying shift in the body’s mass. We can immediately integrate it to find
[ ] [ ] δm (t) = δm (0) + 16m 2 ˆA(1,0)(t) + 5Kˆ(1,0)(t) − 16m 2Aˆ(1,0)(0) + 5 ˆK (1,0)(0) . (22.42 )
We note that this mass correction is entirely gauge dependent; it could be removed by redefining the time coordinate on the world line. In addition, one could choose to incorporate δm (0) into the leading-order mass m. The second of the equations tells us that the body’s spin is constant at this order; at higher orders, time-dependent corrections to the spin dipole would arise. The last of the equations is the principal result of this section. It tells us that the relationship between the acceleration of the world line and the drift of the body away from it is governed by (i) the local curvature of the background spacetime, as characterized by β„°ab – this is the same term that appears in the geodesic deviation equation – (ii) the coupling of the body’s spin to the local curvature – this is the Papapetrou spin force [138] – and (iii) the free radiation field created by the body – this is the self-force. We identify the world line as the body’s by the condition Mi = 0. If we start with initial conditions Mi (0) = 0 = ∂tMi (0), then the mass dipole remains zero for all times if and only if the world line satisfies the equation
1 a(1a) = 12Aˆ(a1,1)− ∂t ˆC (1a,0) −--Siℬia. (22.43 ) m
This is the equation of motion we sought. It, along with the more general equation containing Mi, will be discussed further in the following section.

Order (2,–2)

We now proceed to the details of the calculation. Substituting the expansion

h(α2β)= 1-h(B2,−α2β)+ 1h (2B,−αβ1) + h(B2,0α)β + ln(s)h(B2,0α,βln)+ O (πœ€,s) (22.44 ) s2 s
and the results for h(α1β) from the previous section into the wave equation and the gauge condition again yields a sequence of equations that can be solved for coefficients of successively higher-order powers (and logarithms) of s. Due to its length, the expansion of the second-order Ricci tensor is given in Appendix A. Note that since the approximate wave equation (22.17View Equation) contains an explicit O (πœ€) correction, h (2α)β will be determined only up to O (πœ€) corrections. For simplicity, we omit these O(πœ€) symbols from the equations in this section; note, however, that these corrections do not effect the gauge condition, as discussed above.

To begin, the most divergent, order 1βˆ•s4 term in the wave equation reads

1 ( ) 4m2 ( ) 4m2 -4 2 + s2∂c∂c h(B2,α−β2)= --4- 7 ˆωab + 43δab xaαxbβ − --4-tαtβ, (22.45 ) s s s
where the right-hand side is the most divergent part of the second-order Ricci tensor, as given in Eq. (A.3View Equation). From the tt-component of this equation, we read off Aˆ(2,−2) = − 2m2 and that ˆA (2a,− 2) is arbitrary. From the ta-component, ˆ(2,−2) B, ˆ(2,−2) C ab, and ˆ (2,− 2) D c are arbitrary. From the ab-component, Kˆ(2,− 2) = 83m2, ˆE (2,−2) = − 7m2, and Kˆ(a2,−2), Fˆ(a2,−2), Hˆ(a2b,−c2), and ˆI(a2b,−2) are arbitrary.

The most divergent, order 3 1βˆ•s terms in the gauge condition similarly involve only (2,−2) hαβ; they read

1-(s∂b − 2ωb )h(2,−2) − -1-η μνxa(s∂a − 2ωa )h(2,−2)= 0. (22.46 ) s3 B αb 2s3 α Bμν
After substituting the results from the wave equation, the t-component of this equation determines that ˆ(2,−2) C ab = 0. The a-component determines that ˆ (2,−2) H abc = 0, ˆ(2,− 2) Iab = 0, and
ˆF (2,−2)= 3Kˆ (2,− 2)− 3Aˆ(2,−2). (22.47 ) a a a
Thus, the order 1βˆ•s2 part of h(α2β) is given by
h (2B,t−t2)= − 2m2 + ˆA (2i,− 2)ωi, (2,−2) (2,−2) ij (2,−2) h Bta = Bˆ ωa + πœ–a ωiDˆj , (22.48 ) (2,−2) ( 8 2 ˆ (2,−2) i) 2 ˆ(2,−2) h Bab = δab 3m + K i ω − 7m ˆωab + F ⟨a ωb⟩,
where ˆ(2,− 2) Fa is given by Eq. (22.47View Equation).

The metric perturbation in this form depends on five free functions of time. However, from calculations in flat spacetime, we know that order πœ€2βˆ•s2 terms in the metric perturbation can be written in terms of two free functions: a mass dipole and a spin dipole. We transform the perturbation into this “canonical” form by performing a gauge transformation (cf. Ref. [45Jump To The Next Citation Point]). The transformation is generated by 1 (2,−2) 1 (2,−2) a ξα = − sBˆ tα − 2sFˆa xα, the effect of which is to remove ˆB (2,−2) and ˆFa(2,−2) from the metric. This transformation is a refinement of the Lorenz gauge. (Effects at higher order in πœ€ and s will be automatically incorporated into the higher-order perturbations.) The condition (2,−2) (2,− 2) (2,−2) Fˆa − 3Kˆa + 3Aˆa = 0 then becomes Kˆ(a2,−2) = Aˆ(2a,−2). The remaining two functions are related to the ADM momenta of the internal spacetime:

ˆ(2,−2) ˆ (2,−2) Ai = 2Mi, D i = 2Si, (22.49 )
where M i is such that ∂M t i is proportional to the ADM linear momentum of the internal spacetime, and Si is the ADM angular momentum. Mi is a mass dipole term; it is what would result from a transformation xa → xa + M aβˆ•m applied to the 1βˆ•s term in (1) hαβ. Si is a spin dipole term. Thus, the order 1βˆ•s2 part of h(2) Bαβ reads
(2,− 2) 2 i h Btt = − 2m + 2Mi ω , h (2,− 2)= 2πœ–aijωiSj, (22.50 ) B(2ta,− 2) ( ) h Bab = δab 83m2 + 2Mi ωi − 7m2 ˆωab.

Order (2,–1)

At the next order, 1βˆ•s3, because the acceleration is set to zero, h(2,− 2) B αβ does not contribute to (0) (2) E μν [h ], and (1,−1) hB αβ does not contribute to 2 (0) (1) δ Rμν[h ]. The wave equation hence reads

1- c (2,−1) 2- 2 (0,− 3) [ (1)] s∂ ∂ch Bαβ = s3δ Rα β h , (22.51 )
where 2 (0,− 3)[ (1)] δ Rαβ h is given in Eqs. (A.4View Equation)–(A.6View Equation). The tt-component of this equation implies s2∂c∂ch (2,− 1)= 6m Hˆ(1,0)ˆωij Btt ij, from which we read off that Aˆ(2,−1) is arbitrary and Aˆ(2,−1)= − m Hˆ(1,0) ij ij. The ta-component implies 2 c (2,−1) (1,0) i s ∂ ∂chBta = 6m ˆCi ˆωa, from which we read off (2,−1) (1,0) ˆB i = − m Cˆi and that Cˆ(2a,−1) is arbitrary. The ab-component implies
( ) s2∂c∂ch (2,− 1) = 6m Aˆ(1,0) + ˆK (1,0) ˆωab − 12m Hˆ(1,0)ˆωb⟩i + 2m δab ˆH (1,0)ˆωij, (22.52 ) Bab i⟨a ij
from which we read off that Kˆ(2,−1) is arbitrary, ˆK (2,−1)= − 1m ˆH (1,0) ij 3 ij, ˆE(2,− 1) = − m ˆA (1,0) − m ˆK (1,0), ˆ(2,−1) ˆ (1,0) F ab = 2m H ab, and ˆ(2,−1) H ab is arbitrary. This restricts (2,−1) hαβ to the form
(2,−1) (2,−1) (1,0) ij h Btt = Aˆ − mHˆij ˆω , (2,−1) ˆ(1,0) i ˆ(2,−1) h Bta = − m(C i ˆω a + C a , ) ( ) (22.53 ) h (2,−1)= δ ˆK (2,−1) − 1m ˆH (1,0)ˆωij − m ˆA(1,0) + ˆK (1,0) ˆω + 2m ˆH (1,0)ˆω i + Hˆ(2,− 1). Bab ab 3 ij ab i⟨a b⟩ ab

We next substitute (2,−2) hB αβ and (2,− 1) h Bαβ into the order 2 1βˆ•s terms in the gauge condition. The t-component becomes

1-( ˆ(1,0) ˆ(2,−1)) i s2 4m Ci + 12 ∂tMi + 3C i ω = 0, (22.54 )
from which we read off
ˆC(i2,− 1) = − 4∂tMi − 43m ˆC(i1,0). (22.55 )
And the a-component becomes
( ) ( ) 0 = − 43m ˆA (1,0) − 43m Kˆ(1,0) − 12 ˆA(2,−1) + 12Kˆ(2,−1) ωa + 23m Hˆ(a1i,0)− Hˆ(a2i,− 1) ωi − 2πœ–ijaωi∂tSj, (22.56 )
from which we read off
( ) ˆA (2,− 1) = Kˆ(2,−1) − 8m ˆA(1,0) + ˆK (1,0) , (22.57 ) 3 Hˆ (2,− 1) = 2m Hˆ(1,0), (22.58 ) ij 3 ij
and that the angular momentum of the internal background is constant at leading order:
∂Si = 0. (22.59 ) t

Thus, the order 1βˆ•s term in h(B2α)β is given by

( ) (2,−1) ˆ(2,− 1) 8 ˆ (1,0) ˆ (1,0) ˆ(1,0) ij h Btt = K − 3m A + K − m H ij ˆω , (2,−1) ˆ(1,0) i 4 ˆ(2,−1) (22.60 ) h Bta = − m(C i ˆω a − 4 ∂tMi − 3m C) i (, ) h (2,−1)= δ ˆK (2,−1) − 1m ˆH (1,0)ˆωij − m ˆA(1,0) + ˆK (1,0) ˆω + 2m ˆH (1,0)ˆω i + 2m ˆH (1,0). Bab ab 3 ij ab i⟨a b⟩ 3 ab
Note that the undetermined function Kˆ(2,−1) appears in precisely the form of a mass monopole. The value of this function will never be determined (though its time-dependence will be). This ambiguity arises because the mass m that we have defined is the mass of the internal background spacetime, which is based on the inner limit that holds πœ€βˆ•R fixed. A term of the form πœ€2βˆ•R appears as a perturbation of this background, even when, as in this case, it is part of the mass monopole of the body. This is equivalent to the ambiguity in any expansion in one’s choice of small parameter: one could expand in powers of πœ€, or one could expand in powers of 2 πœ€ + πœ€, and so on. It is also equivalent to the ambiguity in defining the mass of a non-isolated body; whether the “mass” of the body is taken to be m or m + 1Kˆ(2,−1) 2 is a matter of taste. As we shall discover, the time-dependent part of (2,−1) Kˆ is constructed from the tail terms in the first-order metric perturbation. Hence, the ambiguity in the definition of the mass is, at least in part, equivalent to whether or not one chooses to include the free gravitational field induced by the body in what one calls its mass. (In fact, any order πœ€ incoming radiation, not just that originally produced by the body, will contribute to this effective mass.) In any case, we will define the “correction” to the mass as δm := 12Kˆ(2,−1).

Order (2,0,ln)

We next move to the order ln(s)βˆ•s2 terms in the wave equation, and the order ln (s)βˆ•s terms in the gauge condition, which read

c (2,0,ln) ln s∂ ∂ch Bαβ = 0, (22.61 ) ( b (2,0,ln) 1 μν a (2,0,ln)) ln s ∂ hBαb − 2η x α∂ahBμν = 0. (22.62 )
From this we determine
(2,0,ln) (2,0,ln) (2,0,ln) a (2,0,ln) (2,0,ln) a b hBαβ = Aˆ tαtβ + 2 ˆC a t(βx α) + (δab ˆK + ˆH ab )xαx β. (22.63 )

Finally, we arrive at the order 1βˆ•s2 terms in the wave equation. At this order, the body’s tidal moments become coupled to those of the external background. The equation reads

1 ( ) 2 [ ] ∂c∂ch (2B,α0)β + -- h(B2,0α,βln)+ E&tidle;αβ = --δ2R (0α,β−2) h(B1) , (22.64 ) s2 s2
where &tidle;Eαβ comprises the contributions from h (2B,α−β2) and h(B2,α−β1), given in Eqs. (A.10View Equation), (A.15View Equation), and (A.21View Equation). The contribution from the second-order Ricci tensor is given in Eqs. (A.7View Equation) – (A.9View Equation).

Foregoing the details, after some algebra we can read off the solution

h(B2t,0t)= ˆA(2,0) + Aˆ(i2,0)ωi + Aˆ(i2j,0)ˆωij + Aˆ(2ij,k0)ˆωijk (22.65 ) ( ) h(B2t,0a)= ˆB(2,0)ωa + ˆB (2ij,0)ˆωaij + ˆC(a2,0)+ ˆC(a2i,0)ˆωai + πœ–abc ˆD (2c,0)ωb + ˆD (2ci,0)ˆωbi + ˆD (2ci,j0)ˆωbij (22.66 ) ( ) h(B2a,0b)= δab Kˆ(2,0) + Kˆ(i2,0)ωi + ˆK (2ij,k0)ˆωijk + ˆE (2i,0)ˆωabi + ˆE(i2j,0)ˆωabij + Fˆ(2,0)ˆωb⟩ + ˆF (2,0)ˆωb⟩i ⟨a i⟨a +Fˆ(2,0)ˆωb⟩ij + πœ–cd(aˆωb)ci ˆG (2di,0)+ ˆH (2ab,0) + ˆH (a2,bi0)ωi + πœ–cd(aIˆ(2,0)ωc, (22.67 ) ij⟨a b)d
where each one of the STF tensors is listed in Table 1.

In solving Eq. (22.64View Equation), we also find that the logarithmic term in the expansion becomes uniquely determined:

(2,0,ln) 16 2 a b hBαβ = − 15m β„°abxαx β. (22.68 )
This term arises because the sources in the wave equation (22.64View Equation) contain a term ∝ β„°ab, which cannot be equated to any term in c (2,0) ∂ ∂chBab. Thus, the wave equation cannot be satisfied without including a logarithmic term.

Table 1: Symmetric trace-free tensors appearing in the order πœ€2s0 part of the metric perturbation in the buffer region around the body. Each tensor is a function of the proper time t on the world line γ, and each is STF with respect to the Euclidean metric δij.
Aˆ(2,0) is arbitrary ˆ(2,0) 2 4 j 1 j 7 ˆ(1,1) 3 ˆ(1,1) 4 ˆ(1,0) A i = − ∂tMi − 5S ℬji + 3M β„°ji − 5m A i − 5m K i + 5m ∂tCi Aˆ(2ij,0) = − 73m2 β„°ij Aˆ(2,0) = − 2S ℬ + 5M β„° − 1m ˆH (1,1) ij(2k,0) ⟨i(j1,k0⟩) 3 ⟨i jk⟩ 2 ijk ˆB = m ∂(tKˆ ) ˆB (2,0) = 1 2M lℬk − 5Slβ„°k πœ–j)kl − 1m Cˆ(1,1) ij(2,0) 9 (i (i 2 ij ˆC i is( arbitrary ) ( ) ˆC (2,0) = 2 Slβ„° k− 14M lℬk πœ– − m 6 ˆC(1,1)− ∂ Hˆ(1,0) ij (i 15 (i j)lk 5 ij t ij ˆD (2i,0) = 15 (6M jℬij − 7Sjβ„°ij) + 2m Dˆ(i1,1) ˆ (2,0) 10 2 D ij(2,0) = 3 m ℬij ˆD ijk = 13S ⟨iβ„°jk⟩ + 23M ⟨iℬjk⟩ ˆK (2,0) = 2δm ˆ (2,0) 2 4 j 5 j 13 ˆ(1,1) 9 ˆ(1,1) 16- ˆ(1,0) K i(2,0) = − ∂tMi − 5S ℬij − 9M β„°ij + 1(15,m1) A i + 5m K i − 15m ∂tC i ˆK ijk = − 59M ⟨iβ„°jk⟩ + 29S ⟨iℬjk⟩ − 16m ˆH ijk ˆE (2,0) = -2M iβ„° + 1Sjℬ + -1m ∂ ˆC (1,0)− 9-m ˆK (1,1)− 11m Aˆ(1,1) i(2,0) 175 2 ij 5 ij 10 t i 20 i 20 i ˆE ij = 5m β„°ij ˆFi(2,0) = 184M jβ„°ij + 72Sjℬij + 46m ∂t ˆC(i1,0)− 28m ˆA(i1,1)+ 18m ˆKi(1,1) ˆ (2,0) 752 25 25 25 25 Fij(2,0) = 4m β„°ij (1,1) ˆFijk = 43M ⟨iβ„°jk⟩ − 43S ⟨iℬjk⟩ + m ˆH ijk ˆG (2,0) = − 4πœ– β„°kM l − 2πœ– ℬk Sl + 1m ˆI(1,1) ij(2,0) 9 lk(i j) 9 lk(i j) 2 ij ˆH ij is arbitrary ˆH (2,0) = 58M ⟨iβ„°jk⟩ − 28S⟨iℬjk⟩ + 2m ˆH (1,1) ˆij(2k,0) 15104 k 15l 112 5k iljk8 ˆ(1,1) Iij = − 45 πœ–lk(iβ„° j)M − 45 πœ–lk(iℬ j)S + 5m Iij

Gauge condition

We now move to the final equation in the buffer region: the order 1βˆ•s gauge condition. This condition will determine the acceleration aα(1). At this order, (1) hαβ first contributes to Eq. (22.19View Equation):

L (1,−1)[h(1)] = 4m--a(1)xa. (22.69 ) α s a α
The contribution from h(2) Bαβ is most easily calculated by making use of Eqs. (B.24View Equation) and (B.25View Equation). After some algebra, we find that the t-component of the gauge condition reduces to
4 4m (1,0) 10m (1,0) 0 = − --∂tδm + ---∂t ˆA + ----∂tKˆ , (22.70 ) s 3s 3s
and the a-component reduces to
4-2 4m-- (1) 4- i 4- i 2m--ˆ(1,1) 4m-- ˆ(1,0) 0 = s∂tMa + s aa + sβ„°aiM + sℬaiS − s Aa + s ∂tC a . (22.71 )
After removing common factors, these equations become Eqs. (22.39View Equation) and (22.41View Equation). We remind the reader that these equations are valid only when evaluated at aa(t) = aa(0)(t) = 0, except in the term (1) 4ma a βˆ•s that arose from (1) (1) L α [h ].

Second-order solution

We have now completed our calculation in the buffer region. In summary, the second-order perturbation in the buffer region is given by h (2α)β = s−2h(B2,−αβ2) + s−1h(B2,−αβ1) + h(B2α,0β)+ ln(s)h(B2,0α,lβn)+ O (πœ€,s), where (2,− 2) h Bαβ is given in Eq. (22.50View Equation), (2,−1) hBαβ in Eq. (22.60View Equation), (2,0) hB αβ in Eq. (22.65View Equation), and (2,0,ln) hBαβ in Eq. (22.68View Equation). In addition, we have found evolution equations for an effective correction to the body’s mass, given by Eq. (22.39View Equation), and mass and spin dipoles, given by Eqs. (22.59View Equation) and (22.71View Equation).

22.5 The equation of motion

Master equation of motion

Eq. (22.71View Equation) is the principal result of our calculation. After simplification, it reads

∂2M + β„° M b = − a(1)+ 1 ˆA(1,1)− ∂ Cˆ(1,0)− 1-S ℬi. (22.72 ) t a ab a 2 a t a m i a
We recall that M a is the body’s mass dipole moment, β„° ab and ℬ ab are components of the Riemann tensor of the background spacetime evaluated on the world line, (1) a a is the first-order acceleration of the world line, Sa is the body’s spin angular momentum, and (1,1) ˆAa, (1,0) Cˆa are vector fields on the world line that have yet to be determined. The equation is formulated in Fermi normal coordinates.

Eq. (22.72View Equation) is a type of master equation of motion, describing the position of the body relative to a world line of unspecified (though small) acceleration, in terms of the metric perturbation on the world line, the tidal fields of the spacetime it lies in, and the spin of the body. It contains two types of accelerations: ∂2tMi and ai(1). The first type is the second time derivative of the body’s mass dipole moment (or the first derivative of its ADM linear momentum), as measured in a frame centered on the world line γ. The second type is the covariant acceleration of the world line through the external spacetime. In other words, ∂2tMi measures the acceleration of the body’s centre of mass relative to the centre of the coordinate system, while ai measures the acceleration of the coordinate system itself. As discussed in Section 21, our aim is to identify the world line as that of the body, and we do so via the condition that the mass dipole vanishes for all times, meaning that the body is centered on the world line for all times. If we start with initial conditions Mi(0) = 0 = ∂tMi (0), then the mass dipole remains zero for all times if and only if the world line satisfies the equation

a(1) = 1Aˆ(1,1)− ∂ ˆC (1,0) − -1S ℬi . (22.73 ) a 2 a t a m i a
This equation of motion contains two types of terms: a Papapetrou spin force, given by − S ℬi i a, which arises due to the coupling of the body’s spin to the local magnetic-type tidal field of the external spacetime; and a self-force, arising from homogenous terms in the wave equation. Note that the right-hand side of this equation is to be evaluated at aμ = a(0)μ = 0, and that it would contain an antidamping term − 113 m Λ™aμ [92, 93, 150Jump To The Next Citation Point] if we had not assumed that the acceleration possesses an expansion of the form given in Eq. (21.9View Equation).

In our self-consistent approach, we began with the aim of identifying γ by the condition that the body must be centered about it for all time. However, we could have begun with a regular expansion, in which the world line is taken to be the remnant γ(0) of the body in the outer limit of πœ€ → 0 with only xμ fixed. In that case the acceleration of the world line would necessarily be πœ€-independent, so i a(0) would be the full acceleration of (0) γ. Hence, when we found ai = 0 (0), we would have identified the world line as a geodesic, and there would be no corrections i a(n) for n > 0. We would then have arrived at the equation of motion

∂2tMa + β„°abM b = 1ˆA(a1,1)− ∂tCˆ(a1,0)− 1-Siℬia. (22.74 ) 2 m
This equation of motion was first derived by Gralla and Wald [83Jump To The Next Citation Point] (although they phrased their expansion in terms of an explicit expansion of the world line, with a deviation vector on γ(0), rather than the mass dipole, measuring the correction to the motion). It describes the drift of the body away from the reference geodesic (0) γ. This drift is driven partially by the local curvature of the background, as seen in the geodesic-deviation term β„°abM b, and by the coupling between the body’s spin and the local curvature. It is also driven by the self-force, as seen in the terms containing Aˆ(1a,1) and ˆC(a1,0), but unlike in the self-consistent equation, the fields that produce the self-force are generated by a geodesic past history (plus free propagation from initial data) rather than by the corrected motion.

Although perfectly valid, such an equation is of limited use. If the external background is curved, then Mi has meaning only if the body is “close” to the world line. Thus, ∂2Mi t is a meaningful acceleration only for a short time, since Mi will generically grow large as the body drifts away from the reference world line. On that short timescale of validity, the deviation vector defined by i M accurately points from γ (0) to a “corrected” world line γ; that world line, the approximate equation of motion of which is given in Eq. (22.73View Equation), accurately tracks the motion of the body. After a short time, when the mass dipole grows large and the regular expansion scheme begins to break down, the deviation vector will no longer correctly point to the corrected world line. Errors will also accumulate in the field itself, because it is being sourced by the geodesic, rather than corrected, motion.

The self-consistent equation of motion appears to be more robust, and offers a much wider range of validity. Furthermore, even beyond the above step, where we had the option to choose to set either M i or i a(0) to zero, the self-consistent expansion continues to contain within it the regular expansion. Starting from the solution in the self-consistent expansion, one can recover the regular expansion, and its equation of motion (22.74View Equation), simply by assuming an expansion for the world line and following the usual steps of deriving the geodesic deviation equation.

Detweiler–Whiting decomposition

Regardless of which equation of motion we opt to use, we have now completed the derivation of the gravitational self-force, in the sense that, given the metric perturbation in the neighbourhood of the body, the self-force is uniquely determined by irreducible pieces of that perturbation. Explicitly, the terms that appear in the self-force are given by

3 ∫ Aˆ(a1,1)= --- ωah (t1,t1)d Ω, (22.75 ) 4π Cˆ(1,0)= h (1,0). (22.76 ) a ta
This is all that is needed to incorporate the motion of the body into a dynamical system that can be numerically evolved; at each timestep, one simply needs to calculate the field near the world line and decompose it into irreducible pieces in order to determine the acceleration of the body. The remaining difficulty is to actually determine the field at each timestep. In the next section, we will use the formal integral representation of the solution to determine the metric perturbation at the location of the body in terms of a tail integral.

However, before doing so, we emphasize some important features of the self-force and the field near the body. First, note that the first-order external field (1) hαβ splits into two distinct pieces. There is the singular piece S hαβ, given by

S 2m--{ 3 i 2 i 2(3 5 ) ij} 2 htt = s 1 + 2saiω + 2s aia + s 8a⟨iaj⟩ + 6β„°ij ωˆ + O(s ) (22.77 ) S 2 j ik 2 h ta = − 2ms Λ™aa + 3ms πœ–aijℬ kˆω + O(s ) (22.78 ) S 2m--{ [ 1 i 2 2 i 2 (3 -5 ) ij] 2 h ab = s δab 1 − 2saiω + 3s aia + s 8a⟨iaj⟩ − 18β„°ij ˆω + 2s a⟨aab⟩ 19 2 2 2 i } 2 − -9 s β„°ab + 3s β„°βŸ¨aˆωb⟩i + O (s ). (22.79 )
This field is a solution to the homogenous wave equation for s > 0, but it is divergent at s = 0. It is the generalization of the 1βˆ•s Newtonian field of the body, as perturbed by the tidal fields of the external spacetime gαβ. Comparing with results to be derived below in Section 23.2, we find that it is precisely the Detweiler–Whiting singular field for a point mass.

Next, there is the Detweiler–Whiting regular field R (1) S hαβ = h αβ − hαβ, given by

hRtt = Aˆ(1,0) + sAˆ(i1,1)ωi + O (s2), (22.80 ) R (1,0) ( (1,1) (1,1) i j (1,1) i) 2 hta = Cˆa + s ˆB ωa + Cˆai ω + πœ–aiDˆj ω + O (s ), (22.81 ) ( ) hRab = δabKˆ(1,0) + Hˆ(a1b,0)+ s δab ˆK (1i,1)ωi + Hˆ(a1,bi1)ωi + πœ–ij(aˆI(b1)j,1)ωi + ˆF(⟨1a,1)ωb⟩ + O(s2). (22.82 )
This field is a solution to the homogeneous wave equation even at s = 0. It is a free radiation field in the neighbourhood of the body. And it contains all the free functions in the buffer-region expansion.

Now, the acceleration of the body is given by

(1) 1 R R 1- i aa = 2∂ah tt − ∂thta − m Siℬa, (22.83 )
which we can rewrite as
α 1( αδ α δ) ( R R ) || β γ -1-- α β γδ a(1) = − 2 g + u u 2h δβ;γ − hβγ;δ a=0u u + 2m R βγδu S , (22.84 )
where γδ γ δ cdj S := ecedπœ– Sj. In other words, up to order 2 πœ€ errors, a body with order πœ€ or smaller spin (i.e., one for which γδ S = 0), moves on a geodesic of a spacetime R gαβ + πœ€hαβ, where R hαβ is a free radiation field in the neighbourhood of the body; a local observer would measure the “background spacetime,” in which the body is in free fall, to have the metric gαβ + πœ€hRαβ instead of gαβ. If we performed a transformation into Fermi coordinates in gαβ + πœ€hR αβ, the metric would contain no acceleration term, and it would take the simple form of a smooth background plus a singular perturbation. Hence, the Detweiler–Whiting axiom is a consequence, rather than an assumption, of our derivation, and we have recovered precisely the picture it provides in the point particle case. In the electromagnetic and scalar cases, Harte has shown that this result is quite general: even for a finite extended body, the field it produces can be split into a homogeneous field [89Jump To The Next Citation Point, 90Jump To The Next Citation Point, 91Jump To The Next Citation Point] that exerts a direct force on the body, and a nonhomogeneous field that exerts only an indirect force by altering the body’s multipole moments. His results should be generalizable to the gravitational case as well.

22.6 The effect of a gauge transformation on the force

We now turn to the question of how the world line transforms under a gauge transformation. We begin with the equation of motion (22.41View Equation), presented again here:

2 i (1) 1- i [1 ˆ(1,1) ˆ(1,0)] ∂tMa + β„°aiM = − a a − m ℬaiS + 2A a − ∂tC a aμ=0 . (22.85 )
Setting Mi = 0, we derive the first-order acceleration of γ, given in Eq. (22.73View Equation). If, for simplicity, we neglect the Papapetrou spin term, then that acceleration is given by
( ∫ ) (1) 3-- ωa- (1) (1) aa = lsim→0 4π 2shtt dΩ − ∂thta ∫ ( ) = lim -3- 1∂ h(1)− ∂ h(1) ωiω dΩ, (22.86 ) s→0 4π 2 i tt t ti a
where it is understood that explicit appearances of the acceleration are to be set to zero on the right-hand side. The first equality follows directly from Eq. (22.73View Equation) and the definitions of ˆ(1,1) A a and ˆ(1,0) Ca. The second equality follows from the STF decomposition of h(1α)β and the integral identities (B.26View Equation) – (B.28View Equation). We could also readily derive the form of the force given by the Quinn–Wald method of regularization: -1 ∫ (1 (1) (1)) lims →0 4π 2∂ah tt − ∂th ta dΩ. However, in order to derive a gauge-invariant equation of motion, we shall use the form in Eq. (22.86View Equation).

Suppose that we had not chosen a world line for which the mass dipole vanishes, but instead had chosen some “nearby” world line. Then Eq. (22.85View Equation) provides the relationship between the acceleration of that world line, the mass dipole relative to it, and the first-order metric perturbations (we again neglect spin for simplicity). The mass dipole is given by Mi = 3-lims→0 ∫ s2h(2)ωid Ω 8π tt, which has the covariant form

∫ -3- α 2 (2) μ ν M α ′ = 8π lsi→m0 gα′ω αs hμνu u dΩ, (22.87 )
where a primed index corresponds to a point on the world line. Note that the parallel propagator does not interfere with the angle-averaging, because in Fermi coordinates, gαβ′ = δαβ + O(πœ€,s2). One can also rewrite the first-order-metric-perturbation terms in Eq. (22.85View Equation) using the form given in Eq. (22.86View Equation). We then have Eq. (22.85View Equation) in the covariant form
∫ ( ) -3- α D2-- β 2 (2) μ ν || 8 π lis→m0 gα′ gαβdτ2 + β„°α β ω s hμν u u dΩ a=a(0) ∫ ( ) | = − 3m--lim gα′ 2h(1) − h (1) uμuνω βdΩ | − ma (1′) . (22.88 ) 8π s→0 α βμ;ν μν;β α a=a(0) α

Now consider a gauge transformation generated by (1)α 1 2 (2)α πœ€ξ [γ] + 2πœ€ ξ [γ] + ⋅⋅⋅, where (1)α ξ is bounded as s → 0, and ξ(2)α diverges as 1βˆ•s. More specifically, we assume the expansions ξ(1)α = ξ(1,0)α(t,πœƒA) + O (s ) and ξ(1)α = 1ξ(2,− 1)α(t,πœƒA) + O (1) s. (The dependence on γ appears in the form of dependence on proper time t. Other dependences could appear, but it would not affect the result.) This transformation preserves the presumed form of the outer expansion, both in powers of πœ€ and in powers of s. The metric perturbations transform as

(1) (1) (1) hμν → hμν + 2ξ(μ;ν), (22.89 ) (2) (2) (2) (1) ρ (1) ρ ρ (1) ρ (1) ρ (1) hμν → hμν + ξ(μ;ν) + h μν;ρξ(1) + 2hρ(μξ(1);ν) + ξ(1)ξ(μ;ν)ρ + ξ(1);μξρ;ν + ξ(1);(μξν);&#x03
Using the results for (1) hαβ, the effect of this transformation on (2) htt is given by
2m h(t2t)→ h(t2t)− --2-ωiξ(i1) + O (s −1). (22.91 ) s
The order 1βˆ•s2 term arises from h(μ1)ν;ρξρ(1) in the gauge transformation. On the right-hand side of Eq. (22.88View Equation), the metric-perturbation terms transform as
( D2 ) (2h (1β)μ;ν − h(μ1ν);β)u μuνωβ → (2h(β1μ);ν − h(1μ)ν;β)uμu νωβ + 2ω β gγβ--2-+ β„° γβ ξ(γ1). (22.9 dτ
The only remaining term in the equation is ma α (1). If we extend the acceleration off the world line in any smooth manner, then it defines a vector field that transforms as a → a + πœ€£ a + ⋅⋅⋅ α α ξ(1) α. Since α a(0) = 0, this means that α α a(1) → a(1) – it is invariant under a gauge transformation. It is important to note that this statement applies to the acceleration on the original world line; it does not imply that the acceleration of the body itself is gauge-invariant.

From these results, we find that the left- and right-hand sides of Eq. (22.88View Equation) transform in the same way:

3 ∫ ( D2 ) LHS/RHS → LHS/RHS − ---lim gαα′ωαβ gγβ----+ β„° γβ ξ(γ1)dΩ. (22.93 ) 4π s→0 dτ 2
Therefore, Eq. (22.88View Equation) provides a gauge-invariant relationship between the acceleration of a chosen fixed world line, the mass dipole of the body relative to that world line, and the first-order metric perturbations. So suppose that we begin in the Lorenz gauge, and we choose the fixed world line γ such that the mass dipole vanishes relative to it. Then in some other gauge, the mass dipole will no longer vanish relative to γ, and we must adopt a different, nearby fixed world line γ′. If the mass dipole is to vanish relative to γ ′, then the acceleration of that new world line must be given by a α = πœ€aα + o(πœ€) (1), where
∫ a(1)′ = − 3m--lim gα′(2h (1) − h(1) )uμuνω βdΩ.|| . (22.94 ) α 8π s→0 α βμ;ν μν;β α a=a(0)
Hence, this is a covariant and gauge-invariant form of the first-order acceleration. By that we mean the equation is valid in any gauge, not that the value of the acceleration is the same in every gauge. Under a gauge transformation, a new fixed world line is adopted, and the value of the acceleration on it is related to that on the old world line according to Eq. (22.93View Equation). In the particular case that ξ(1) μ has no angle-dependence on the world line, this relationship reduces to
( 2 β ) a(1)α = a(1)α − (g α+ uαu ) D--ξ(1)+ Rβ u μξν uρ , (22.95 ) new old β β dτ2 μνρ (1)
as first derived by Barack and Ori [17]. (Here we’ve replaced the tidal field with its expression in terms of the Riemann tensor to more transparently agree with equations in the literature.) An argument of this form was first presented by Gralla [81] for the case of a regular expansion, and was extended to the case of a self-consistent expansion in Ref. [145Jump To The Next Citation Point].
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