The Einstein equations and multipole moments at null infinity

TL;DR

This paper explores the structure of vacuum spacetimes near null infinity, deriving metric expansions in Bondi-Sachs form, and investigates how multipole moments influence the metric, especially in stationary cases and Kerr-like solutions.

Contribution

It introduces a recursive method to determine metric coefficients from initial data and multipole moments, extending understanding of stationary and non-stationary vacuum solutions.

Findings

Mass dipole can be non-zero when mass vanishes.

Stationary metrics may deviate from Kerr-like behavior due to multipole moments.

Derived an approximate Bondi-Sachs form of the Kerr metric including quadrupole moments.

Abstract

We consider vacuum metrics admitting conformal compactification which is smooth up to the scri $\mathscr{I^+}$. We write metric in the Bondi-Sachs form and expand it into power series in the inverse affine distance $1/r$. Like in the case of the luminosity distance, given the news tensor and initial data for a part of metric the Einstein equations define coefficients of the series in a recursive way. This is also true in the stationary case however now the news tensor vanishes and the role of initial data is taken by multipole moments which are equivalent to moments of Thorne. We find an approximate form of metric and show that in the case of vanishing mass the mass dipole may be different from zero. Then the known result about the Kerr like behaviour of a stationary metric is violated. Finally we find an approximate (up to the quadrupole moment) Bondi-Sachs form of the Kerr metric.