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6.1 Meaning of P τ AΘ

In this section we explain the meaning of τ PA Θ. First of all, we expand in an ε series the four-velocity of star A normalized as μ ν −2 gμνu AuA = − ε, where i τ i uA = uAv A. The result is
[ ] τ 2 1 2 1 ττ u A = 1 + ε 2-vA + 44h [ ] 4 1- ττ 1- k -3- ττ2 5- ττ 2 τ k 3-4 + ε 4 6h + 44h k − 32 (4h ) + 8 4h vA − 4h kvA + 8vA 1[ ] + ε5--7h ττ + 5hkk 4[ 6 1- ττ 1- k -1- ττ k 3-- ττ ττ --7- ττ 3 1- τk τ + ε 4 8h + 46h k + 16 4h 4h k − 164h 6h + 128 (4h ) + 44h 4h k − 6h τkvkA − 14h ττ4h τkvkA + 14hklvkAvlA + 14hkkv2A + 5-(4hττ)2v2A 4 2 8 ] 64 5- ττ 2 3- τ k 2 27- ττ 4 5--6 + 86h vA − 2 4h kvAvA + 324h vA + 16vA 7 + 𝒪 (ε ). (152 )
The field should be evaluated somehow at ⃗z A. This is a formal series since the metric derived via the point particle description diverges at ⃗zA.

Now let us regularize this equation with the Hadamard partie finie regularization (see [88, 143] and for example, [26Jump To The Next Citation Point, 30] in the literature of the post-Newtonian approximation). Consider a function f(⃗x) which can be expanded around ⃗z1 in the form

∑ 1 f (⃗r1) = -pfˆ(N⃗1 ). (153 ) p=p0 r1 p
Then the Hadamard partie finie at ⃗z1 of the function f(⃗x ) is defined by
∮ dΩn ----1fˆ(⃗n1 ). (154 ) 4π 0

For example, by this procedure hττ becomes (see Equation (130View Equation))

τ [ 2 ] [hττ]H = 4ε4P-2 + ε6 − 2 m2-{(⃗n ⋅⃗v )2 − v2} − 16m1m2-- + 7m-2 + 𝒪 (ε7) (155 ) 1 r12 r12 12 2 2 r212 r212
for star 1. In the above equation, H [f]A means that we regularize the quantity f at star A by the Hadamard partie finie. Evaluating Equation (152View Equation) and √ --- H [ − g]A by this procedure, then comparing the result to Equation (148View Equation) combined with Equations (149View Equation, 150View Equation, 151View Equation), we find at least up to 3 PN order:
√ --- PτAΘ = mA [ − gu τA]HA. (156 )
It is important that even after regularizing all the divergent terms, there remains a nonlinear effect. Equation (156View Equation) is natural. And note that we have never assumed this relation in advance. This relation has been derived by solving the evolution equation for P τAΘ functionally. In this regard, it is worth mentioning that the naturality of Equation (156View Equation) supports the use of the Hadamard partie finie regularization (or any regularization procedures if we can reproduce Equation (156View Equation) with them) to derive the 3 PN mass-energy relation to deal with divergences when one uses Dirac delta distributions11.

Finally, we note that up to 3 PN order

--- --- [√ − gu τ]H = [√ − g ]H [uτ ]H + 𝒪 (ε7) (157 ) A A A A A
is satisfied if we use the Hadamard partie finie regularization explained above.


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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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