We predict a 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 of an arbitrary profile. 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. The position of the shape excitation on a Josephson vortex acts like a “minute hand” showing the time in the rest frame associated with the vortex. Remarkably, at some conditions, the shape wave can carry negative energy: a vortex with the shape excitation can have less energy than the same vortex without it.