### 6.1 Meaning of

In this section we explain the meaning of . First of all, we expand in an series the
four-velocity of star normalized as , where . The result is
The field should be evaluated somehow at . This is a formal series since the metric derived via the
point particle description diverges at .
Now let us regularize this equation with the Hadamard partie finie regularization (see [88, 143] and for
example, [26, 30] in the literature of the post-Newtonian approximation). Consider a function which
can be expanded around in the form

Then the Hadamard partie finie at of the function is defined by
For example, by this procedure becomes (see Equation (130))

for star . In the above equation, means that we regularize the quantity at star by the
Hadamard partie finie. Evaluating Equation (152) and by this procedure, then comparing the
result to Equation (148) combined with Equations (149, 150, 151), we find at least up to 3 PN order:
It is important that even after regularizing all the divergent terms, there remains a nonlinear effect.
Equation (156) is natural. And note that we have never assumed this relation in advance. This relation has
been derived by solving the evolution equation for functionally. In this regard, it is worth
mentioning that the naturality of Equation (156) supports the use of the Hadamard partie finie
regularization (or any regularization procedures if we can reproduce Equation (156) with them)
to derive the 3 PN mass-energy relation to deal with divergences when one uses Dirac delta
distributions.
Finally, we note that up to 3 PN order

is satisfied if we use the Hadamard partie finie regularization explained above.