Christodoulou's nonlinear gravitational-wave memory: Evaluation in the quadrupole approximation

Christodoulou has found a new nonlinear contribution to the net change in the wave form caused by the passage of a burst of gravity waves ("memory of the burst"). We argue that this effect is nothing but the gravitational wave form generated by the stress energy in the burst itself. We derive an explicit formula for this effect in terms of a retarded-time integral of products of time derivatives of wave-zone gravitational wave forms. The resulting effect corresponds in size to a correction 2.5 post-Newtonian orders [$O\left({\left(\frac{\mathrm{Gm}}{r{c}^2}\right)}^{\frac{5}{2}}\right)=O\left({\left(\frac{v}{c}\right)}^5\right)$] beyond the quadrupole approximation, and is therefore negligible for all but the most relativistic of systems. For gravitational bremsstrahlung from two stars moving at 3000 km ${\mathrm{s}}^{-1\mathrm{}}$, the effect is much less than ${10}^{-10\mathrm{}}$ of the usual linear quadrupole wave form, while for a system of coalescing binary compact objects we estimate that the effect is of order ${10}^{-1\mathrm{}}$ for two neutron stars.