It was found that the direct-integration part does not play any role in the evaluation of the evolution equation of 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 (110), we obtain the evolution equation of as

Remarkably, we can integrate Equation (147) functionally: with and Equation (148) together with Equations (149, 150, 151) gives the 3 PN order mass-energy relation.

6.1 Meaning of

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