Usual spin waves in 3-D antiferromagnets are well described as transverse magnetic fluctuations in linear spin-wave theory. Here we investigated the non-linear response which leads to longitudinal fluctuations, as shown in an experiment with polarized neutrons on MnF2 at temperatures well below the Neel temperature TN = 67K. It is found that the dynamic magnetic response due to two-magnon creation or annihilation is separated by a gap centered near the spin wave frequency from the central peak corresponding to neutron-magnon scattering (creation of one magnon and annihilation of another). The longitudinal energy spectrum extends to about twice the frequency of the zone boundary modes. This tail at high energies is fairly independent of momentum transfer. The observed longitudinal spectra are in qualitative agreement with the theory for two-magnon processes and are determined for large energy transfers w by the density of states D(w/2).