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

Almost a century ago, Einstein used a weak field approximation around Minkowski spacetime 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, $\mathrm{\Lambda }$. To incorporate this fact, Einstein’s celebrated derivation is generalized by replacing Minkowski spacetime with de Sitter spacetime. The investigation is motivated by the fact that, because of the significant differences between the asymptotic structures of Minkowski and de Sitter spacetimes, many of the standard techniques, including the usual $1/r$ expansions, cannot be used for $\mathrm{\Lambda }>0$. Furthermore, since, e.g., the energy carried by gravitational waves is always positive in Minkowski spacetime but can be arbitrarily negative in de Sitter spacetime irrespective of how small $\mathrm{\Lambda }$ is, the limit $\mathrm{\Lambda }\to 0$ can fail to be continuous. Therefore, a priori it is not clear that a small $\mathrm{\Lambda }$ 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.

Figure 1

Left panel: A time-changing quadrupole emitting gravitational waves whose spatial size is uniformly bounded in time. The causal future of such a source covers only the future Poincaré patch $M_{\mathrm{P}}^{+}$ (the upper triangle of the figure). There is no incoming radiation across the past boundary $E^{+}(i^{-})$ of $M_{\mathrm{P}}^{+}$ because we use retarded solutions. The shaded region represents a convenient neighborhood of ${\mathcal{I}}^{+}$ in which perturbations satisfy a homogenous equation and the approximation (3.10), discussed below, holds everywhere. The dashed (red) lines with arrows show the integral curves of the “time translation” Killing field $T^a$ (adapted to the rest frame of the source). Right panel: The rate of change of the quadrupole moment at the point $(-|\overrightarrow{x}|,\overrightarrow{0})$ on the source creates the retarded field at the point $(0,\overrightarrow{x})$ on ${\mathcal{I}}^{+}$. The figure also shows the cosmological foliation $\eta =\text{const}$ and the timelike surfaces $r≔|\overrightarrow{x}|=\text{const}$. As $r$ goes to infinity, the $r=\text{const}$ surfaces approach $E^{+}(i^{-})$. Therefore, in contrast with the situation in Minkowski spacetime, for sufficiently large values of $r$, there is no flux of energy across the $r=\text{const}$ surfaces.