Relation between matched-expansions and Harte effective metrics beyond linear order

Determine how the effective metric constructed from matched asymptotic expansions relates to Harte’s effective metric beyond linear perturbative order.

Background

Matched asymptotic expansions separate the metric into singular self-fields and regular external fields. The background metric together with the regular fields defines a smooth effective metric in which the small body moves as a test body, at least through the perturbative orders established in the cited literature.

Harte’s framework likewise introduces an effective metric by removing the body’s self-field from the full physical metric. The paper states that the relationship between these two effective-metric constructions has not been established beyond linear order, leaving open whether and how the two approaches are equivalent at higher perturbative orders.

References

Although it is not known how this effective metric relates to Harte's beyond linear order, the body is known to move as a test body in it (at least at low orders in $$), and $h{R(1)}_{\alpha\beta}$ is the Detweiler-Whiting regular field in both the matched-expansions and Harte approach.

— Point particles in general relativity: beyond linear perturbation theory  (2609.10521 - Musaeus et al., 9 Sep 2026) in Section 1, “Perspectives on point particles”