A complete measurement of a black-hole recoil 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, for example, 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 scientific potential of the azimuthal one has not been exploited so far. Here we show how the presence of more than one GW emission mode allows one to constrain this angle and, consequently, the kick direction of a real GW event. We analyse the GW190412 signal, which contains higher-order modes, with a numerical relativity surrogate waveform model for black-hole mergers. We rule out kick magnitudes below the typical escape velocity of dense globular clusters v_esc ≈ 50 km s⁻¹ with a Bayes factor of ~21 (or ~95% probability). The kick forms angles \({\theta }_{{\mathrm{KL}}}^{-100M}=3{2}_{-14}^{+35}\,\text{deg}\) with the orbital angular momentum defined at a reference time t_ref = −100 M before merger (with M denoting the system mass in geometric units), \({\theta }_{{{KN}}}=4{4}_{-17}^{+19}\,\text{deg}\) with the line of sight and \({\phi }_{{{KN}}}^{-100M}=6{9}_{-38}^{+33}\,\text{deg}\) with the projection of the latter onto the former, all quoted at a 90% credible level. We anticipate that complete characterization of black-hole recoils will aid in evaluating candidate multi-messenger observations of black-hole mergers in active galactic nuclei, by testing the consistency of observed signals with proposed electromagnetic emission mechanisms.