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6 Third Post-Newtonian Mass-Energy Relation

It was found that the direct-integration part does not play any role in the evaluation of the evolution equation of τ PA at 3 PN order. Thus we use the same method as in the evaluation of the 2.5 PN equations of motion. Evaluating the surface integrals in Equation (110View Equation), we obtain the evolution equation of P τA as

dP-τ1Θ 2m1m2-- dτ = − ε r2 [4(⃗n12 ⋅⃗v1) − 3(⃗n12 ⋅⃗v2)] 12 [ + ε4m1m2-- − 9(⃗n ⋅⃗v )3 + 1-v2(⃗n ⋅⃗v ) + 6(⃗n ⋅⃗v )(⃗n ⋅⃗v )2 r212 2 12 2 2 1 12 2 12 1 12 2 2 ⃗ 2 2 − 2v1(⃗n12 ⋅⃗v1) + 4(⃗v1 ⋅⃗v2)(⃗n12 ⋅V ) + 5v2(⃗n12 ⋅⃗v2) − 4v2(⃗n12 ⋅⃗v1) ] m1- m2- + r (− 4(⃗n12 ⋅⃗v2) + 6(⃗n12 ⋅⃗v1)) + r (− 10(⃗n12 ⋅⃗v1) + 11 (⃗n12 ⋅⃗v2)) [ ( 12 ) (12 ) + ε6m1m2-- − 3v4 + 2v2v2 + 4v4 (⃗n ⋅⃗v ) + 5-v4+ 3v2v2 + 7v4 (⃗n ⋅⃗v ) r12 2 1 1 2 2 12 1 8 1 2 1 2 2 12 2 ( 2 2) ( 2 2) + 2v1 + 4v2 (⃗n12 ⋅⃗v1)(⃗v1 ⋅⃗v2) − 2v1( + 8v2 (⃗n12)⋅⃗v2)(⃗v1 ⋅⃗v2) ( 2 2) 2 3-2 2 3 + 3v1 + 12v2 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) − 4v1 + 12v2 (⃗n12 ⋅⃗v2) 2 2 3 + 2(⃗n12 ⋅⃗v2)(⃗v1 ⋅⃗v2) − 6(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) (⃗v1 ⋅ ⃗v2) + 6(⃗n12 ⋅⃗v2) (⃗v1 ⋅⃗v2) 15- 4 45- 5 − 2 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) − 8 (⃗n12 ⋅⃗v2) { ( ) + m1- − 42v21 − 117-v22 (⃗n12 ⋅⃗v1) + 60 (⃗n12 ⋅⃗v1)3 r12 4 ( 137 37 ) 297 + ---v21 +---v22 (⃗n12 ⋅⃗v2) + ---(⃗n12 ⋅⃗v1)(⃗v⋅⃗v2) 4 2 4 219- 2 − 4 (⃗n12 ⋅⃗v2)(⃗v⋅⃗v2) − 151 (⃗n12 ⋅⃗v1) (⃗n12 ⋅ ⃗v2) } + 109(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)2 − 23(⃗n12 ⋅⃗v2)3 + m2-{ − (13v2 + 18v2) (⃗n ⋅⃗v ) + (17v2 + 25v2) (⃗n ⋅⃗v ) r12 1 2 12 1 1 2 12 2 + 26 (⃗n ⋅⃗v )(⃗v ⋅ ⃗v ) − 28 (⃗n ⋅⃗v )(⃗v ⋅⃗v ) + 2(⃗n ⋅⃗v )2(⃗n ⋅⃗v ) 12 1 1 2 2 12 2 13} 2 12 1 12 2 + 16 (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) − 20(⃗n12 ⋅⃗v2) m2 ( 33 13 ) m m ( 35 17 ) + -21 ---(⃗n12 ⋅⃗v1) −---(⃗n12 ⋅⃗v2) − --12--2 --(⃗n12 ⋅⃗v1) + --(⃗n12 ⋅⃗v2) r12( 4 2 ) ] r12 4 4 m22 23 + r2- − 12(⃗n12 ⋅ ⃗v1) +-2-(⃗n12 ⋅⃗v2) 12 + 𝒪 (ε7). (147 )
Remarkably, we can integrate Equation (147View Equation) functionally:
( ) P τ1Θ = m1 1 + ε22Γ 1 + ε44Γ 1 + ε66Γ 1 + 𝒪 (ε7), (148 )
with
Γ = 1-v2+ 3m2-, (149 ) 2 1 2 1 r12 3m 2m 7m 4m 3 7m2 5m m 4Γ 1 = −---2(⃗n12 ⋅⃗v2)2 + ---2v22 + ---2v21 − ---2(⃗v1 ⋅⃗v2) + -v41 + --22 − ---12-2, (150 ) 2r12 r12 2r12 r12 8 2r12 2r12
and
m21m2 21m1m22 5m32 5 6 6Γ 1 = ---3--+ ----3---+ --3- + --v 1 2r12( 4r12 2r12 16 ) m22 45-2 19- 2 1- 2 2 + r212 4 v1 + 2 v2 + 2(⃗n12 ⋅⃗v1) − 19 (⃗v1 ⋅⃗v2) − (⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2) − 3(⃗n12 ⋅⃗v2) ( ) + m1m2-- 43-v2+ 53-v2− 69(⃗n12 ⋅⃗v1)2− 53(⃗v1 ⋅⃗v2)+ 85(⃗n12 ⋅⃗v1)(⃗n12 ⋅⃗v2)− 69(⃗n12 ⋅⃗v2)2 r212 8 1 8 2 8 4 4 8 m ( 33 3 7 + --2 --v41 + --v21v22 + v42 − 6v21(⃗v1 ⋅⃗v2) − 4v22(⃗v1 ⋅⃗v2) −-v21(⃗n12 ⋅⃗v2)2 r12 8 2 4 ) 5- 2 2 2 2 9- 4 − 2 v2(⃗n12 ⋅⃗v2) + 2(⃗v1 ⋅⃗v2) + 2(⃗n12 ⋅⃗v2) (⃗v1 ⋅⃗v2) + 8(⃗n12 ⋅⃗v2) . (151 )
Equation (148View Equation) together with Equations (149View Equation, 150View Equation, 151View Equation) gives the 3 PN order mass-energy relation.

 6.1 Meaning of τ P AΘ

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