Peter Christensen wrote:
To be very short: I'm trying with vectors like this: (c*"proper
time",x,y,z) and (m*c,p_x,p_y,p_z) instead of the usual representation
with (c*t,x,y,z) and (E/c,p_x,p_y,p_z).

Make sure they are indeed vectors. Arbitrary collections of four
quantities, even with an obvious relationship, need not be the
components of vectors.

That is, one must be able to find a basis, and if one changes the basis
and the components accordingly, the vector itself should remain unchanged.

For your choice, m*c never changes for an object when you use a basis
related to inertial frame A or B; but p_x, p_y, and p_z can all change
with a change in basis from A to B. The vector itself cannot possibly
depend on which basis one projects it upon, so I don't see how your
4-tuples can possibly be the components of a vector.

Moreover, using proper time as a component, it cannot possibly be
invariant. That completely undermines the entire mathematical structure.
Ditto for mass.


Tom Roberts