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Suppose that in addition to the photon we also have a
particle.
Suppose the particle is moving with instantaneous 4-velocity v.
Let's calculate the invariant quantity P•v. This is
easy, because P•v is just m times U•v and therefore
P•v is zero.
Now it turns out that the P•v is equal to the energy>of the photon as measured by an observer comoving with
We also used some basic laws of physics and
mathematics (such as the axiom that says zero times x is
zero for all x).
So we must reject the hypothesis. Unless you believe
that photons never have any energy, and/or you want to start
repealing the axioms of arithmetic, then you'd better not
accept the idea that P = m U for photons.