Gravitational wave double copy of radiation from gluon shockwave collisions

In a recent paper, we showed that the gravitational wave spectrum from trans-Planckian shockwave scattering in Einstein gravity is determined by the gravitational Lipatov vertex expressed as the bilinear double copy ${\mathrm{\Gamma }}^{\mu \nu }=\frac{1}{2}{C}^{\mu }{C}^{\nu }-\frac{1}{2}{N}^{\mu }{N}^{\nu }$ where $C^{\mu }$ is the QCD Lipatov vertex and $N^{\mu }$ is the QED soft photon factor. We show here that this result can be directly obtained by careful application of the classical color-kinematic duality to the spectrum obtained in gluon shockwave collisions.