  One-dimensional topological superconductors are known to host Majorana zero
modes at domain walls terminating the topological phase. Their nonabelian
nature allows for processing quantum information by braiding operations which
are insensitive to local perturbations, making Majorana zero modes a promising
platform for topological quantum computation. Motivated by the ultimate goal of
executing quantum information processing on a finite timescale, we study domain
walls moving at a constant velocity. We exploit an effective Lorentz invariance
of the Hamiltonian to obtain an exact solution of the associated quasiparticle
spectrum and wave functions for arbitrary velocities. Essential features of the
solution have a natural interpretation in terms of the familiar relativistic
effects of Lorentz contraction and time dilation. We find that the Majorana
zero modes remain stable as long as the domain wall moves at subluminal
velocities with respect to the effective speed of light of the system. However,
the Majorana bound state dissolves into a continuous quasiparticle spectrum
once the domain wall propagates at luminal or even superluminal velocities.
This relativistic catastrophe implies that there is an upper limit for possible
braiding frequencies even in a perfectly clean system with an arbitrarily large
topological gap. We also exploit our exact solution to consider domain walls
moving past static impurities present in the system.
