23 Global solution in the external spacetime

In the previous sections, we have determined the equation of motion of γ in terms of the metric perturbation; we now complete the first-order solution by determining the metric perturbation. In early derivations of the gravitational self-force (excluding those in Refs. [72Jump To The Next Citation Point, 83Jump To The Next Citation Point]), the first-order external perturbation was simply assumed to be that of a point particle. This was first justified by Gralla and Wald [83Jump To The Next Citation Point]. An earlier argument made by D’Eath [46, 47] (and later used by Rosenthal [155Jump To The Next Citation Point]) provided partial justification but was incomplete [144Jump To The Next Citation Point]. Here, we follow the derivation in Ref. [144Jump To The Next Citation Point], which makes use of the same essential elements as D’Eath’s: the integral formulation of the perturbative Einstein equation and the asymptotically small radius of the tube Γ.

23.1 Integral representation

Suppose we take our buffer-region expansion of (1) h αβ to be valid everywhere in the interior of Γ (in ℳE), rather than just in the buffer region. This is a meaningful supposition in a distributional sense, since the 1∕s singularity in h(1) αβ is locally integrable even at γ. Note that the extension of the buffer-region expansion is not intended to provide an accurate or meaningful approximation in the interior; it is used only as a means of determining the field in the exterior. We can do this because the field values in Ω are entirely determined by the field values on Γ, so using the buffer-region expansion in the interior of Γ leaves the field values in Ω unaltered. Now, given the extension of the buffer-region expansion, it follows from Stokes’ law that the integral over Γ in Eq. (21.8View Equation) can be replaced by a volume integral over the interior of the tube, plus two surface integrals over the “caps” 𝒥cap and Σcap, which fill the “holes” in 𝒥 and Σ, respectively, where they intersect Γ. Schematically, we can write Stokes’ law as ∫ ∫ ∫ ∫ = + − Int(Γ ) 𝒥cap Σcap Γ, where Int(Γ ) is the interior of Γ. This is now valid as a distributional identity. (Note that the “interior” here means the region bounded by Γ ∪ Σcap ∪ 𝒥cap; Int(Γ ) does not refer to the set of interior points in the point-set defined by Γ.) The minus sign in front of the integral over Γ accounts for the fact that the directed surface element in Eq. (21.8View Equation) points into the tube. Because 𝒥cap does not lie in the past of any point in Ω, it does not contribute to the perturbation at x ∈ Ω. Hence, we can rewrite Eq. (21.8View Equation) as

1 ∫ ( ′′ ′ ′ ′ ′) h(α1)β = − --- ∇ μ′ G+αβαβ ∇ μh (1α)′β′ − h (1α)′β′∇ μG+αβα β dV ′ + h(1¯) 4π Σ αβ Int∫(Γ ) -1- ( α′β′ (1) (1) α′β′ + ) ′ (1) = − 4π G αβ E α′β′[h ] − hα′β′E [Gαβ] dV + h¯Σαβ, (23.1 ) Int(Γ )
where (1) h¯Σαβ is the contribution from the spatial surface ¯ Σ := Σ ∪ Σcap, and α′β′ + E [G αβ] denotes the action of the wave-operator on G+αβγ′δ′. Now note that E α′β′[G+αβ] ∝ δ(x,x ′); since x ∕∈ Int(Γ ), this term integrates to zero. Next note that (1) E α′β′[h αβ] vanishes everywhere except at γ. This means that the field at x can be written as
∫ h (1α)β = −-1 lim G+αβα′β′Eα ′β′[h (1)]dV ′ + h(1¯) . (23.2 ) 4π ℛ →0 Σαβ Int(Γ )
Making use of the fact that E αβ[h(1)] = ∂c∂c(1∕s)h(1,−1)+ O(s−2) αβ, along with the identity ∂c∂ (1∕s) = − 4π δ3(xa ) c, where δ3 is a coordinate delta function in Fermi coordinates, we arrive at the desired result
(1) ∫ ( ¯ ¯) (1) hαβ = 2m G+αβα¯¯β 2u ¯αuβ + g¯αβ d¯t + h¯Σαβ. (23.3 ) γ
Therefore, in the region Ω, the leading-order perturbation produced by the asymptotically small body is identical to the field produced by a point particle. At second order, the same method can be used to simplify Eq. (21.5View Equation) by replacing at least part of the integral over Γ with an integral over γ. We will not pursue this simplification here, however.

Gralla and Wald [83Jump To The Next Citation Point] provided an alternative derivation of the same result, using distributional methods to prove that the distributional source for the linearized Einstein equation must be that of a point particle in order for the solution to diverge as 1∕s. One can understand this by considering that the most divergent term in the linearized Einstein tensor is a Laplacian acting on the perturbation, and the Laplacian of 1∕s is a flat-space delta function; the less divergent corrections are due to the curvature of the background, which distorts the flat-space distribution into a covariant curved-spacetime distribution.

23.2 Metric perturbation in Fermi coordinates

Metric perturbation

We have just seen that the solution to the wave equation with a point-mass source is given by

∫ (1) α′ β′ α′β′ ′ (1) hαβ = 2m G αβα′β′(2u u + g )dt + hΣαβ. (23.4 ) γ
One can also obtain this result from Eq. (19.29View Equation) by taking the trace-reversal and making use of the Green’s function identity (16.22View Equation). In this section, we seek an expansion of the perturbation in Fermi coordinates. Following the same steps as in Section 19.2, we arrive at
h(1) = 2m--U ′ ′(2u α′u β′ + gα′β′) + htail(u). (23.5 ) αβ r αβα β αβ
Here, primed indices now refer to the retarded point zα(u) on the world line, r is the retarded radial coordinate at x, and the tail integral is given by
∫ u ∫ t< htail(u) = 2m Vαβα′β′(2u α′uβ′ + gα′β′)dt′ + 2m Gαβα′β′(2u α′u β′ + gα′β′)dt′ + h (1) αβ t< 0 Σαβ ∫ u− = 2m G αβα′β′(2u α′uβ′ + gα′β′)dt′ + h (1Σ)αβ, (23.6 ) 0
where t< is the first time at which the world line enters 𝒩 (x ), and t = 0 denotes the time when it crosses the initial-data surface Σ.

We expand the direct term in (1) hαβ in powers of s using the following: the near-coincidence expansion U αβα′β′ = gα′gβ′(1 + O (s3)) α β; the relationship between r and s, given by Eq. (11.5View Equation); and the coordinate expansion of the parallel-propagators, obtained from the formula α′ α′ 0 α′ a gα = u eα + ea eα, where the retarded tetrad (uα,eαa) can be expanded in terms of s using Eqs. (11.9View Equation), (11.10View Equation), (9.12View Equation), and (9.13View Equation). We expand the tail integral similarly: noting that u = t − s + O (s2), we expand htαaβil(u) about t as htail(t) − s∂ htail(t) + O (s2) αβ t αβ; each term is then expanded using the near-coincidence expansions ′′ ′′ γ′′ ′′ ′′ ′′ V αβα β = g(α gδβ)Rα γ′′β δ′′ + O (s) and ¯β htαaβil(t) = g¯ααg β(hta¯αi¯βl+ shtαa¯βil¯iωi) + O (s2), where barred indices correspond to the point ¯x = z(t), and htail¯ ¯αβ¯γ is given by

∫ t− htail = 2m ∇ G ¯ ′′(2uα′uβ′ + gα′β′)dt′ + h(1) . (23.7 ) ¯α¯β¯γ 0 ¯γ ¯αβαβ Σ ¯α¯β¯γ
This yields the expansion
tail ¯α ¯β tail tail i 2 hαβ(u ) = gαg β(h¯α¯β + sh ¯α¯βiω − 4ms ℰ ¯α¯β) + O (s ). (23.8 )
As with the direct part, the final coordinate expansion is found by expressing ¯α gα in terms of the Fermi tetrad.

Combining the expansions of the direct and tail parts of the perturbation, we arrive at the expansion in Fermi coordinates:

(1) 2m--( 3 i 3 2 ij 15 2 ¯α 1 2 i 5 2 ij) i tail h tt = s 1 + 2saiω + 8s aiajω − 8 s ˙a¯αu + 3s ˙aiω + 6s ℰijω + (1 + 2saiω )h00 +shtailωi + O (s2), (23.9 ) 00i h (1ta) = 4maa − 23msR0iaj ωij + 2ms ℰaiωi − 2ms a˙a + (1 + saiωi)ht0aail+ shta0iailωi + O (s2), (23.10 ) 2m ( ) h (1a)b = ---- 1 − 12saiωi + 38s2aiajωij + 18s2˙a¯αu¯α + 13s2a˙iωi − 16s2ℰijωij δab + 4msaaab s ij tail tail i 2 − 23msRaibj ω − 4ms ℰab + hab + sh abiω + O (s ). (23.11 )
As the final step, each of these terms is decomposed into irreducible STF pieces using the formulas (B.1View Equation), (B.3View Equation), and (B.7View Equation), yielding
(1) 2m-- ˆ(1,0) i [ i ˆ(1,1) i (3- 5 ) ij] 2 h tt = s + A + 3mai ω + s 4maia + A i ω + m 4a⟨iaj⟩ + 3ℰij ˆω + O(s ), (23.12 ) (1) ˆ (1,0) (ˆ (1,1) ˆ(1,1) i j ˆ(1,1) i 2 j ik) 2 h ta = Ca + s B ωa − 2m a˙a + C ai ω + 𝜖a{iD j ω + 3m 𝜖aijℬ kˆω + O (s ), (23.13 ) (1) 2m-- ˆ (1,0) i ˆ(1,0) [ 4 i ˆ(1,1) i 3 ij h ab = s δab + (K − mai ω )δab + H ab + s δab 3maia + K i ω + 4ma ⟨iaj⟩ˆω 5 ij] 4 i 38- ˆ(1,1) i j ˆ(1,1) i − 9m ℰijωˆ} + 3m ℰ⟨aˆωb⟩i + 4ma ⟨aab⟩ − 9 m ℰab + H abi ω + 𝜖i (aIb)j ω ˆ(1,1) 2 +F ⟨a ωb⟩ + O (s ), (23.14 )
where the uppercase hatted tensors are specified in Table 2. Because the STF decomposition is unique, these tensors must be identical to the free functions in Eq. (22.38View Equation); hence, those free functions, comprising a regular, homogenous solution in the buffer region, have been uniquely determined by boundary conditions and waves emitted by the particle in the past.

Table 2: Symmetric trace-free tensors in the first-order metric perturbation in the buffer region, written in terms of the electric-type tidal field ℰ ab, the acceleration a i, and the tail of the perturbation.
ˆ(1,0) tail A(1,0) = h00 ˆCa = ht0aail+ maa ˆK (1,0) = 1δabhtail ˆ (1,0) 3tail ab H ab = h⟨ab⟩ ˆA(a1,1) = ht0a0ila + 2ht0a0ilaa + 23m ˙aa ˆB(1,1) = 1htailδij + 1htailai 3 0ij 3 0i Cˆ(1,1)= htail + 2m ℰab + htailab⟩ ˆa(b1,1) 10⟨abbc⟩ tail tail 0⟨a D a = 2𝜖a (h0bc + h0b ac) Kˆ(a1,1)= 13δbchtabcial+ 23m ˙aa Hˆ(1,1)= htail a(1bc,1) ⟨abc⟩ Fˆa = 35δijhta⟨iila⟩j ˆI(1,1) = 2STF (𝜖ijhtail) ab 3 ab b ⟨ai⟩j

Singular and regular pieces

The Detweiler–Whiting singular field is given by

∫ ′ ′ ′ ′ hSαβ = 2m GSαβα′β′(2u αu β + gα β)dt′. (23.15 )
Using the Hadamard decomposition GSαβα′β′ = 1U αβα′β′δ(σ) − 1V αβα′β′𝜃(σ) 2 2, we can write this as
m ′ ′ ′ ′ m ′′ ′′ ′′′′ hSαβ = --Uα βα′β′(2uα uβ + gα β ) +----U αβα′′β′′(2uα uβ + gα β ) r ∫ v radv ¯α β¯ 1 ¯α¯β ¯ − 2m u V αβ¯α¯β(u u + 2 g )dt, (23.16 )
where primed indices now refer to the retarded point ′ x = z(u); double-primed indices refer to the advanced point x′′ = z(v); barred indices refer to points in the segment of the world line between z(u) and z(v). The first term in Eq. (23.16View Equation) can be read off from the calculation of the retarded field. The other terms are expanded using the identities v = u + 2s + O (s2) and radv = r(1 + 2s2˙aiωi) 3. The final result is
S 2m i [ i 3 ij 5 ij] 2 h tt = ----+ 3mai ω + ms 4aia + 4a⟨iaj⟩ˆω + 3ℰijˆω + O (s), (23.17 ) S s( 2 j ik) 2 hta = s − 2m ˙aa + 3m 𝜖aijℬ kˆω + O (s ), (23.18 ) S 2m i { [ 4 i (3 5 ) ij] 4 i hab = -s-δab − mai ω δab + s δab 3maia + 4ma ⟨iaj⟩ − 9m ℰij ωˆ + 3m ℰ⟨aˆωb⟩i } +4ma ⟨aab⟩ − 389 m ℰab + O (s2). (23.19 )

The regular field could be calculated from the regular Green’s function. But it is more straightforwardly calculated using hR = h(1)− hS αβ αβ αβ. The result is

R ˆ(1,0) ˆ(1,1) i 2 htt = A + sA( i ω + O (s ), ) (23.20 ) hR = Cˆ(1,0)+ s ˆB (1,1)ωa + Cˆ(1,1)ωi + 𝜖aijD ˆ(1,1)ωi + O (s2), (23.21 ) ta a ( ai j ) hR = δ Kˆ(1,0) + Hˆ(1,0)+ s δ ˆK (1,1)ωi + ˆH (1,1)ωi + 𝜖 j ˆI(1,1)ωi + ˆF(1,1)ω + O (s2). (23.22 ) ab ab ab ab i abi i (a b)j ⟨a b⟩

23.3 Equation of motion

With the metric perturbation fully determined, we can now express the self-force in terms of tail integrals. Reading off the components of hR αβ from Table 2 and inserting the results into Eq. (22.84View Equation), we arrive at

μ 1 μν μ ν ( tail tail)|| λ ρ 1 μ ν λρ a(1) = − 2 (g + u u ) 2hνλρ − hλρν |a=0u u + 2m-R νλρu S . (23.23 )
We have now firmly established the results of the point-particle analysis.

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