## 8 Regularization of the Field of Point Particles

Our aim is to compute the metric (and its gradient needed in the equations of motion) at the 3PN order
for a system of two point-like particles. A priori one is not allowed to use directly the metric
expressions (115), as they have been derived under the assumption of a continuous (smooth) matter
distribution. Applying them to a system of point particles, we find that most of the integrals become
divergent at the location of the particles, i.e. when or , where and denote
the two trajectories. Consequently, we must supplement the calculation by a prescription for how to remove
the “infinite part” of these integrals. At this stage different choices for a “self-field” regularization
(which will take care of the infinite self-field of point particles) are possible. In this section we
review:
- Hadamard’s self-field regularization, which has proved to be very convenient for doing practical
computations (in particular, by computer), but suffers from the important drawback of yielding
some ambiguity parameters, which cannot be determined within this regularization, at the 3PN
order;
- Dimensional self-field regularization, an extremely powerful regularization which is free of any
ambiguities (at least up to the 3PN level), and permits therefore to uniquely fix the values
of the ambiguity parameters coming from Hadamard’s regularization. However, dimensional
regularization has not yet been implemented to the present problem in the general case (i.e. for
an arbitrary space dimension ).

The why and how the final results are unique and independent of the employed self-field regularization (in
agreement with the physical expectation) stems from the effacing principle of general relativity [81] -
namely that the internal structure of the compact bodies makes a contribution only at the formal 5PN
approximation. However, we shall review several alternative computations, independent of the self-field
regularization, which confirm the end results.