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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.
This is easy indeed, but wrong. You can write P*v = mU*v, but you
cannot conclude from this that P*v = 0 even though m = 0. Such
conclusion is based on unspoken assumption that U*v IS FINITE. But
precisely this assumption is wrong here. Even though the norm |U| =
1, its COMPONENTS are infinite for the photon (they are all
proportional to the Lorentz factor - find its value for the speed of
This is one of the beautiful features of Relativity