A Second-order expansions of the Ricci tensor

We present here various expansions used in solving the second-order Einstein equation in Section 22.4. We require an expansion of the second-order Ricci tensor 2 δ R αβ, defined by
2 1 μν ( ) 1 μν 1 μ ;ν δ R αβ[h ] = − 2γ ;ν( 2h μ(α;β) − h αβ;μ + 4h ;α)hμν;β + 2h β (hμα;ν − h να;μ) − 12h μν 2hμ(α;β)ν − hαβ;μν − h μν;αβ , (A.1 )
where μν γ is the trace-reversed metric perturbation, and an expansion of a certain piece of (2) Eμν[h ]. Specifically, we require an expansion of (0) δ2R αβ[h (1)] in powers of the Fermi radial coordinate s, where for a function f, 2 (0) δ R αβ[f ] consists of 2 δ R αβ[f] with the acceleration μ a set to zero. We write
2 (0) (1) 1 2 (0,−4)[ (1)] 1 2 (0,− 3)[ (1)] 1 2 (0,−2)[ (1)] δ R αβ[h ] = -4δ R αβ h + -3δ R αβ h + -2δ R αβ h + O (1∕s), (A.2 ) s s s
where the second superscript index in parentheses denotes the power of s. Making use of the expansion of h (1α)β, obtained by setting the acceleration to zero in the results for h(α1β) found in Section 22.3, one finds
(2,−4)[ ] ( ) δ2R αβ h(1) = 2m2 7ˆωab + 43δab xaαxbβ − 2m2t αtβ, (A.3 )
and
δ2R (2,− 3)[h(1)] = 3m Hˆ(1,0)ˆωij, (A.4 ) tt [ ] ij δ2R (2ta,− 3) h(1) = 3m Cˆ(1i,0)ˆωia, (A.5 ) 2 (2,− 3)[ (1)] ( (1,0) (1,0)) (1,0) i (1,0) ij δ R ab h = 3m Aˆ + Kˆ ˆωab − 6m Hˆi⟨a ˆωb⟩ + m δabHˆij ˆω , (A.6 )
and
(2,−2)[ ] (1,1) (1,1) (1,1) δ2R tt h(1) = − 230m2 ℰijωˆij + 3m Hˆijk ˆωijk + 75m Aˆi ωi + 35m Kˆi ωi 4 ˆ (1,0) i − 5m ∂tC i ω , ( ) (A.7 ) 2 (2,−2)[ (1)] ˆ (1,0) ˆ(1,1) ij 6ˆ(1,1) ˆ(1,0) i δ R ta h = − m ∂tK ωa + 3m C ij ˆωa + m 5Cai − ∂tH ai ω ij ˆ(1,1) 4 2 k ij +2m 𝜖(a D i ωj + 3m 𝜖aikℬjωˆ , ) (A.8 ) δ2R (2,−2)[h(1)] = δ m 16∂ Cˆ(1,0)− 13Aˆ(1,1)− 9ˆK (1,1) ωi ab ab ( 15 t i 15 i 5 i) 50 2 ij ˆ(1,1) ijk 14- 2 ij + δab − 9 m ℰijωˆ + m H ijk ˆω − 3 m ℰijˆωab ( 33ˆ(1,1) 27-ˆ (1,1) 3 ˆ (1,0)) i +m 10Ai + 10K i − 5∂tC i ˆωab ( 28 (1,1) 18 (1,1) 46 (1,0)) +m 25 ˆA⟨a − 25-ˆK⟨a − 25∂t ˆC⟨a ˆωb⟩ 8 2 i (1,1) ij jk (1,1) − 3m ℰi⟨aˆωb⟩ − 6m ˆHij⟨a ˆωb⟩ + 3m 𝜖ij(aˆωb) Iˆik 2- 2 2 ˆ (1,1) i 8 i ˆ(1,1) j + 45m ℰab − 5m Habi ω + 5m 𝜖 j(aIb)i ω . (A.9 )

Next, we require an analogous expansion of (0)[-1 (2,−2) 1 (2,−1)] Eαβ r2h + rh, where (0) E αβ[f] is defined for any f by setting the acceleration to zero in E αβ[f]. The coefficients of the 4 1∕r and 3 1∕r terms in this expansion can be found in Section 22.4; the coefficient of 1∕r2 will be given here. For compactness, we define this coefficient to be E&tidle;αβ. The tt-component of this quantity is given by

&tidle;Ett = 2∂2tMi ωi + 85Sjℬijωi − 23M jℰijωi + 823 m2 ℰijˆωij + 24S ⟨iℬjk⟩ˆωijk − 20M ⟨iℰjk⟩ˆωijk. (A.10 )
The ta-component is given by
( ) ( ) E&tidle;ta = 4145𝜖aijM kℬjkωi − 215 11Siℰ jk + 18M iℬj 𝜖ijaωk + 125 41Sj ℰka − 10M jℬka 𝜖ijkωi ( j j) ikl j i kl 68 2 j ik +4𝜖aij S ℰkl + 2Mk ℬ l ˆω + 4𝜖ij⟨kℰl⟩S ˆωa + 3 m 𝜖aijℬ kˆω . (A.11 )
This can be decomposed into irreducible STF pieces via the identities
𝜖aijSiℰkj= Siℰj(k𝜖a)ij + 12𝜖akjSiℰji (A.12 ) [ ] 𝜖aj⟨iℰkl⟩Sj = STF 𝜖jalS⟨iℰjk⟩ − 23δalSpℰj(i𝜖k)jp (A.13 ) ikl [ ] 𝜖aj⟨iMl ℬk⟩j = STF 𝜖jalM ⟨iℬjk⟩ + 1δalM pℬj 𝜖k)jp , (A.14 ) ikl 3 (i
which follow from Eqs. (B.3View Equation) and (B.7View Equation), and which lead to
( ) &tidle;E = 2𝜖 6M kℬj − 7Sk ℰj ωi + 4(2M lℬk − 5Slℰ k)𝜖 ˆω ij ta 5 aij k k 3 (i (i j)kl a + (4Sj ℰk − 56M jℬk )𝜖 ωi + 4𝜖 l(S ℰ + 2M ℬ ) ˆωijk + 68m2 𝜖 ℬj ˆωik. (A.15 ) (a 15 (a i)jk ai ⟨j kl⟩ ⟨j kl⟩ 3 aij k
The ab-component is given by
[( ) ] &tidle;Eab = 563 m2 ℰijˆωabij + 5425m2 ℰab − δab 2∂2tMi + 85Sjℬij + 190M jℰij ωi + 1090m2 ℰijˆωij (20 8 ) ijk 8- i 8- i 56 2 i − δab 3 M ⟨iℰjk⟩ − 3S ⟨iℬjk⟩ ˆω + 15M ⟨aℰb⟩iω + 15M ℰi⟨aωb⟩ + 3 m ℰi⟨aˆωb⟩ +16Mi ℰj⟨aωˆb ⟩ij − 32S⟨aℬb⟩iωi + -4(10Siℬab + 27Mi ℰab) ωi 16 i 5 j l 1ik5m 16 j il k + -3 S ℬi⟨aωb⟩ − 8 𝜖ij⟨a𝜖b⟩klS ℬ mωˆ + 15𝜖ij⟨a𝜖b⟩klS ℬ ω . (A.16 )
Again, this can be decomposed, using the identities
S ℬ = S ℬ + STF 1𝜖 j𝜖 ℬ lSk + -1δ ℬ Sj, (A.17 ) ⟨a b⟩i ⟨a bi⟩ ab 3 ai kl(b j) 10 i⟨a b⟩j S ℬ = S ℬ − STF 2𝜖 j𝜖 ℬ lSk + 3δ ℬ Sj, (A.18 ) i ab ⟨a bi⟩ ab 3 ai kl(b j) 5 i⟨a b⟩j 𝜖ij⟨a𝜖b⟩klSjℬil = STF 𝜖akjSlℬi 𝜖 − 1δk⟨aℬb⟩iSi, (A.19 ) ab ( (jb)il 2 ) j l 1 l j p -3 j SiTkFm 𝜖ij⟨a𝜖b⟩klS ℬ m = STikFm STaFb 2δaiS⟨bℬkm ⟩ + 3δai𝜖 bkS ℬ(l𝜖m)jp − 10δaiδbkℬmjS , (A.20 )
which lead to
&tidle; [( 2 4 j 5 j ) i (10 4 ) ijk] Eab = − 2δab ∂ tMi + 5S ℬi(j + 9M ℰij ω + ) 3 M ⟨iℰjk⟩ − 3S⟨iℬjk⟩ ˆω − 1090δabm2 ℰijˆωij + 15 8M jℰij + 12Sj ℬij ˆωabi + 536m2 ℰijˆωabij -4 ( j j ) 56 2 i + 75 92M (ℰj⟨a + 108S ℬj⟨a) ωb ⟩ + 3 m ℰi(⟨aˆωb⟩ ) +16 STF Miℰj⟨a − Siℬj⟨a ˆωb⟩ij − 83𝜖pq⟨j 2ℰk⟩pMq + ℬk ⟩pSq 𝜖ki(aˆωb)ij aij ( ) + 16m2 ℰab + 4- 29M ⟨aℰbi⟩ − 14S ⟨aℬbi⟩ ωi 1156 j15i pq ( ) − 45 STaFb 𝜖ai ω 𝜖 (b 13ℰj)qMp + 14ℬj)qSp . (A.21 )

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