  The positive mass theorem in general relativity states that in an
asymptotically flat spacetime, if the momentum--energy tensor is
divergence-free and satisfies a dominant energy condition, then a total
momentum--energy four-vector can be formed, of which the energy component is
nonnegative. In this paper, we take the wave four-tensor of a plane light wave
in free space as a counterexample to show that there is no guarantee that a
total four-vector can be formed. Thus the theoretical framework for the
positive mass theorem is flawed. In addition, it is also shown as well that the
Lorentz covariance of Dirac wave equation is not compatible with Einstein
mass--energy equivalence.
