Gravitational waves from isolated systems: Surprising consequences of a positive cosmological constant

TL;DR

The principal difficulties of Einstein's 1918 quadrupole formula are explained and it is shown that it is possible to overcome them in the weak field limit.

Abstract

There is a deep tension between the well-developed theory of gravitational waves from isolated systems and the presence of a positive cosmological constant $Λ$, however tiny. In particular, even the post-Newtonian quadrupole formula, derived by Einstein in 1918, has not been generalized to include a positive $Λ$. We first explain the principal difficulties and then show that it is possible to overcome them in the weak field limit. These results also provide concrete hints for constructing the $Λ>0$ generalization of the Bondi-Sachs framework for full, non-linear general relativity.

Figures (1)

• Figure 1: The rate of change of quadrupole moments at the point $(-|\vec{x}|, \vec{0})$ on the source creates the retarded field at the point $(0, \vec{x})$ on $\mathcal{I}^{+}$. The figure also shows the cosmological foliation $\eta= {\rm const}$ and the time-like surfaces $r={\rm const}$. As $r$ goes to infinity, the $r:= |\vec{x}| ={\rm const}$ surfaces approach $E^{+}(i^{-})$. Therefore, in contrast with the situation in Minkowski space-time, for sufficiently large values of $r$, there is no flux of energy across the $r={\rm const}$ surfaces.