GW190412: measuring a black-hole recoil direction through higher-order gravitational-wave modes

General relativity predicts that gravitational waves (GWs) carry linear momentum. Consequently, the remnant black hole of a black-hole merger can inherit a recoil velocity or ``kick'' of crucial implications in, e.g, black-hole formation scenarios. While the kick magnitude is determined by the mass ratio and spins of the source, estimating its direction requires a measurement of the two orientation angles of the source. While the orbital inclination angle is commonly reported in GW observations, the azimuthal one has been to date ignored. We show how the presence of more than one GW emission mode allows constraining this angle and, consequently, determines the kick direction in a real GW event. %We show that the higher-order mode content GW190412 enables the determination of both these angles and, consequently, the kick direction. We analyse the GW190412 signal, which contains higher-order modes, with a numerical-relativity surrogate waveform model for black-hole mergers. We find that while GW190412 is barely informative about the kick magnitude, we can constrain its direction. This forms angles $\theta_{KL}^{\text{-100M}}=28^{+23}_{-11}\,\deg$ with the orbital angular momentum defined at a reference time $t_{\rm ref}=-100\,M$ before merger (being preferentially kicked upwards), $\theta_{KN}=37^{+15}_{-12}\,\deg$ with the line-of-sight and $\phi_{KN}^{\text{-100M}}=46^{+32}_{-41}\,\deg$ with the projection of the latter onto the former, all at the $68\%$ credible level. We briefly discuss the potential application of this type of measurement for multi-messenger observations of black-hole mergers occurring in Active Galactic Nuclei.