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The title and abstract read: Gravitational radiation in black-hole collisions at the speed of light. I. Perturbation treatment of the axisymmetric collision P. D. D'Eath Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England P. N. Payne 59, The Avenue, Nedlands 6009, Western Australia (Received 4 February 1992) In this and the two following papers II and III we study the axisymmetric collision of two black holes at the speed of light, with a view to understanding the more realistic collision of two black holes with a large but finite incoming Lorentz factor y. The curved radiative region of the space-time, produced after the two incoming impulsive plane-fronted shock waves have collided, is treated using perturbation theory, following earlier work by Curtis and Chapman. The collision is viewed in a frame to which a large Lorentz boost has been applied, giving a strong shock with energy v off which a weak shock with energy 2 < v scatters. This yields a singular perturbation problem, in which the Einstein