The ponderomotive four-momentum

Using classical mechanics and classical field theory, we confirm that the ponderomotive momentum exists as a counterpart of the ponderomotive energy to form the ponderomotive four-momentum when an electron interacts with a light plane wave. As a solution to the equations of motion, a time-dependent four-momentum on the electron (rest) mass shell is obtained in a coordinate-free form. By taking out the quiver motion, we find that the nonoscillating part of the four-momentum is not on the electron mass shell. We show mathematically that the decomposition of the nonoscillating four-momentum into an on-mass-shell four-momentum and a lightlike four-momentum along the light-propagation direction is unique. The time-component of the Lightlike four-momentum is exactly the ponderomotive energy, while the space-component is the ponderomotive momentum. We discuss the critical role of the ponderomotive four-momentum in light-electron scattering processes due to the lightlike property and the uniqueness of the decomposition of a four-momentum. Two pieces of experimental evidence of the ponderomotive momentum are identified in the discussion.