As David Bowman pointed out, what I wrote on 05/20/2016 11:08 AM
was not quite right. Here it is again, with details repaired.
The main conclusions are unchanged.
Here's an example: Suppose we write a particle's energy and
momentum as
E = m cosh θ [1]
p = m sinh θ
which is just what we would get by considering [E,p] as a
four-vector and rotating it by an angle θ in the xt plane
(in units where c=1). You can't have the cosh contribution
without the sinh contribution, so equation [1] is one idea,
not two. It's just trigonometry.
The angle θ is sometimes called the rapidity. For tiny
angles (but not otherwise), it is equal to the velocity v.
More generally, v = tanh θ. Velocity is related to rapidity
in the same way that slope is related to angle.
We can expand the trig functions using Taylor series in
the neighborhood of θ=0:
E = m + 1/2 m θ^2 + 1/24 m θ^4 ....
p = m θ + 1/6 m θ^3 ....
where the coefficients are just the factorials that one
expects in a Taylor series. If we write it in terms of
velocity, the higher coefficients are slightly different:
E = m + 1/2 m v^2 + 9/24 m v^4 ....
p = m v + 3/6 m v^3 ....
In either case, I have to ask, why are the first-order and
second-order terms considered "non-relativistic" while the
higher terms -- and the zeroth-order term -- are considered
"relativistic"?
Relativity only seems weird when people refuse to look at the
familiar non-weird terms.