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

TL;DR

This paper introduces a Lorentz-invariant regularization method for point-like particles in general relativity, enabling consistent post-Newtonian expansions and stress-energy tensor definitions using Hadamard's partie finie concept.

Contribution

It develops a Lorentzian regularization technique for point-particles in GR that preserves covariance and facilitates higher-order post-Newtonian calculations.

Findings

Regularization maintains Lorentz covariance up to any order in 1/c^2.

Derived stress-energy tensor using delta-pseudo-functions.

Equations of motion resemble geodesic equations on a regularized metric.

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.