Lorentzian regularization and the problem of point-like particles in general relativity

TL;DR

The paper tackles the problem of defining a Lorentz-covariant regularization of the self-field of point-like particles in general relativity within post-Newtonian expansions, addressing the breakdown of traditional Hadamard regularization at high orders. It constructs a Lorentzian Hadamard regularization by performing the regularization in the instantaneous rest frame of a particle after a Lorentz boost, and introduces delta- and delta-like pseudo-functions to describe the point-particle sources. A key result is an explicit first-order ($O(1/c^2)$) expression for the regularized functional $[F]_{1}$ that preserves Lorentz invariance up to at least 3PN, along with a consistent stress-energy tensor for point particles, derived from a variational principle and yielding geodesic-like equations on a regularized metric. This framework enables a consistent two-body analysis in harmonic coordinates at high post-Newtonian orders and provides a physically natural model for point masses in GR with applications to gravitational-wave source modeling.

Abstract

The two purposes of the paper are (1) to present a regularization of the self-field of point-like particles, based on Hadamard's concept of ``partie finie'', that permits in principle to maintain the Lorentz covariance of a relativistic field theory, (2) to use this regularization for defining a model of stress-energy tensor that describes point-particles in post-Newtonian expansions (e.g. 3PN) of general relativity. We consider specifically the case of a system of two point-particles. We first perform a Lorentz transformation of the system's variables which carries one of the particles to its rest frame, next implement the Hadamard regularization within that frame, and finally come back to the original variables with the help of the inverse Lorentz transformation. The Lorentzian regularization is defined in this way up to any order in the relativistic parameter 1/c^2. Following a previous work of ours, we then construct the delta-pseudo-functions associated with this regularization. Using an action principle, we derive the stress-energy tensor, made of delta-pseudo-functions, of point-like particles. The equations of motion take the same form as the geodesic equations of test particles on a fixed background, but the role of the background is now played by the regularized metric.