Asymptotics with a positive cosmological constant: III. The quadrupole formula

TL;DR

This paper generalizes Einstein's gravitational wave energy formula to include a positive cosmological constant, revealing new features but concluding that corrections are negligible for current astrophysical observations.

Contribution

It extends Einstein's quadrupole formula to de Sitter space-time, accounting for a positive cosmological constant and analyzing its effects on gravitational wave energy calculations.

Findings

Corrections due to small $\Lambda$ are negligible for current sources.

Standard techniques like $1/r$ expansions are not applicable in de Sitter space.

The energy carried by gravitational waves can be negative in de Sitter space, unlike Minkowski.

Abstract

Almost a century ago, Einstein used a weak field approximation around Minkowski space-time to calculate the energy carried away by gravitational waves emitted by a time changing mass-quadrupole. However, by now there is strong observational evidence for a positive cosmological constant, $\Lambda$. To incorporate this fact, Einstein's celebrated derivation is generalized by replacing Minkowski space-time with de Sitter space-time. The investigation is motivated by the fact that, because of the significant differences between the asymptotic structures of Minkowski and de Sitter space-times, many of the standard techniques, including the standard $1/r$ expansions, can not be used for $\Lambda >0$. Furthermore since, e.g., the energy carried by gravitational waves is always positive in Minkowski space-time but can be arbitrarily negative in de Sitter space-time \emph{irrespective of how small $\Lambda$ is}, the limit $\Lambda\to 0$ can fail to be continuous. Therefore, a priori it is not clear that a small $\Lambda$ would introduce only negligible corrections to Einstein's formula. We show that, while even a tiny cosmological constant does introduce qualitatively new features, in the end, corrections to Einstein's formula are negligible for astrophysical sources currently under consideration by gravitational wave observatories.