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B.2 Spin-spin coupling force

It is straightforward to derive the spin-spin coupling force. The definition of the center of mass does not change the form of the spin-spin coupling force as expected.

We evaluate the following surface integral,

∮ F i = − dS [(− g)tij] + ..., (223 ) 1SS ∂B1 j 10 LL
with ij ij τk,l τ τ[k,i] τj τ[k,j] τi 10[(− g)tLL] = δ 6h 6h [k,l] + 26h 6h ,k + 26h 6h ,k + .... The result is
[ jk jl k l i jk jk i ij jk k jk ki j ] F i = ε6 − 15M-1-M-2-r12r12r12-+ 3M--1-M-2-r12− 3M-1-M-2-r12− 3M-1-M-2-r12 1SS r712 r512 r512 r512 = ε6-1- [15(⃗n ⋅⃗s )(⃗n ⋅⃗s )ni − 3si(⃗n ⋅⃗s ) − 3si(⃗n ⋅⃗s ) − 3ni (⃗s ⋅⃗s )], (224 ) r412 12 1 12 2 12 1 12 2 2 12 1 12 1 2
which perfectly agrees with the previous result (see e.g. [46, 108Jump To The Next Citation Point, 152Jump To The Next Citation Point]).
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"The Post-Newtonian Approximation for Relativistic Compact Binaries"
Toshifumi Futamase and Yousuke Itoh
http://www.livingreviews.org/lrr-2007-2		 Creative Commons License
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