  We predict a new class of excitations propagating along a Josephson vortex in
two-dimensional Josephson junctions. These excitations are associated with the
distortion of a Josephson vortex line and have an analogy with shear waves in
solid mechanics. Their shapes can have an arbitrary profile, which is retained
when propagating. We derive a universal analytical expression for the energy of
arbitrary shape excitations, investigate their influence on the dynamics of a
vortex line, and discuss conditions where such excitations can be created.
Finally, we show that such excitations play the role of a clock for a
relativistically-moving Josephson vortex and suggest an experiment to measure a
time dilation effect analogous to that in special relativity.
