Quadrupole-quadrupole gravitational waves

This paper investigates the nonlinear self-interaction of quadrupole gravitational waves generated by an isolated system. The vacuum Einstein field equations are integrated in the region exterior to the system by means of a post-Minkowskian algorithm. Specializing to the quadrupole-quadrupole interaction (at the quadratic nonlinear order), we recover the known results concerning the non-local modification of the ADM mass-energy of the system accounting for the emission of quadrupole waves, and the non-local memory effect due to the re-radiation by the stress-energy distribution of linear waves. We then compute all the local (instantaneous) terms which are associated, in the quadrupole-quadrupole metric, with the latter non-local effects. Expanding the metric at large distances from the system, we obtain the corresponding radiation-field observables, including all non-local and transient contributions. This permits, notably, the completion of the observable quadrupole moment at the post-Newtonian order.