  It has long been known that a free electron in an intense plane-wave field
has a mass shell that differs from the usual free-electron mass shell, with a
form that implies that an intensity-dependent increase in mass occurs. It has
been an enticing, but elusive goal to observe this mass shift. Many schemes
have been proposed by which a definitive measurement may be made, and some
claims of success exist, but these tests are not conclusive. It is shown here
that the intense-field mass shell is not the result of a change in mass.
Rather, it is a consequence of the potential energy that a charged particle
must possess in the presence of a plane-wave field. When the effects of this
potential are incorporated in a properly covariant form, the mass shift no
longer appears and kinematical relations are conventional. If the plane-wave
pulse is sufficiently long to allow the electron to exit the field
adiabatically, then there is no alteration at all of the mass shell expression.
Other aspects of the role played by the ponderomotive 4-potential are examined.
It is also shown that the putative "relativistic mass" of the electron is
illusory when confronted with covariance requirements. Both "mass increases" of
the electron are thereby discredited by fundamental principles.
