A Consistency Condition for the Double Series Approximation Method

TL;DR

This paper analyzes the double series approximation method for gravitational radiation, deriving a condition to prevent singularities and testing its validity up to fourth order for quadrupole sources, highlighting limitations in physical interpretation.

Contribution

It introduces a necessary condition for the consistency of the double series approximation method and evaluates its implications for physical interpretation of solutions.

Findings

Derived a condition to avoid singularities in the double series method.

Verified the condition up to fourth order for quadrupole sources.

Found that imposing the condition complicates physical interpretation.

Abstract

The double series approximation method of Bonnor is a means for examining the gravitational radiation from an axisymmetric isolated source that undergoes a finite period of oscillation. It involves an expansion of the metric as a double Taylor series. Here we examine the integration procedure that is used to form an algorithmic solution to the field equations and point out the possibility of the expansion method breaking down and predicting a singularity along the axis of symmetry. We derive a condition on the solutions obtained by the double series method that must be satisfied to avoid this singularity. We then consider a source with only a quadrupole moment and verify that to fourth order in each of the expansion parameters, this condition is satisfied. This is a reassuring test of the consistency of the expansion procedure. We do, however, find that the imposition of this condition makes a physical interpretation of any but the lowest order solutions very difficult. The most obvious decomposition of the solution into a series of independent physical effects is shown not to be valid.