I think my example with 3-velocity serves its purpose, but Bob Sciamanda's example with energy E and momentum P does it much better, because the pseudo-Pythagorean theorem
E^2 - p^2c^2 = Eo^2 (1)
with the rest energy Eo in the energy-momentum space directly echoes the theorem
c^2t^2 - r^2 = c^2to^2 (2)
with the proper time "to" of a clock in Minkowski's space. If, according to John, only "to" has the status of real physical time, then, by the same logics, only "Eo" must have the satus of the object's actual energy. Then no references to quaternions or Clifford algebra can save us from the conclusion that there is no such thing as relativistic energy E and momentum p, and therefore motion is merely an illusion.
Moses Fayngold,
NJIT
-----Original Message-----
From: Forum for Physics Educators on behalf of Bob Sciamanda
Sent: Sun 9/11/2005 5:37 PM
To: PHYS-L@LISTS.NAU.EDU
Cc:
Subject: Re: "moving clock runs slower" (yes)
Apply JD's analogy to the momentum-energy four vector of a particle. In its
proper frame its 3-momentum and 3-velocity are zero. In another inertial
frame, it has non-zero 3-momentum/velocity. This spatial component of the
momentum-energy four vector is merely a projection (a shadow) of the
magnitude invariant 4-vector upon the space axes of the new frame. It must
follow, by the JD analogy, that the particle is not really moving.