On a choice of the Bondi radial coordinate and news function for the axisymmetric two-body problem

TL;DR

This paper proposes a coordinate choice in the Bondi formulation for the axisymmetric two-body problem that incorporates physical parameters from the start, leading to a new expression for radiated energy that aligns with recent non-linear results.

Contribution

It introduces a coordinate condition linked to the Newtonian potential, integrating source parameters into the asymptotic solution and providing a new interpretation of the Bondi news function.

Findings

Derived an asymptotic solution incorporating source parameters.

Proposed a new expression for radiated energy consistent with non-linear calculations.

Obtained closed-form solutions matching the Schwarzschild metric with time.

Abstract

In the Bondi formulation of the axisymmetric vacuum Einstein equations, we argue that the ``surface area'' coordinate condition determining the ``radial'' coordinate can be considered as part of the initial data and should be chosen in a way that gives information about the physical problem whose solution is sought. For the two-body problem, we choose this coordinate by imposing a condition that allows it to be interpreted, near infinity, as the (inverse of the) Newtonian potential. In this way, two quantities that specify the problem -- the separation of the two particles and their mass ratio -- enter the equations from the very beginning. The asymptotic solution (near infinity) is obtained and a natural identification of the Bondi "news function" in terms of the source parameters is suggested, leading to an expression for the radiated energy that differs from the standard quadrupole formula but agrees with recent non-linear calculations. When the free function of time describing the separation of the two particles is chosen so as to make the new expression agree with the classical result, closed-form analytic expressions are obtained, the resulting metric approaching the Schwarzschild solution with time. As all physical quantities are defined with respect to the flat metric at infinity, the physical interpretation of this solution depends strongly on how these definitions are extended to the near-zone and, in particular, how the "time" function in the near-zone is related to Bondi's null coordinate.