Suppose we know where a certain particle is initially, and we
want to know where it will be a short time later. We want a
relativistically-correct answer.
Ask yourself which approach you would prefer:
1) Start with "distance = rate * time" in spacetime, i.e.:
ΔR = u Δτ [1]
where R is the position, u is the four-velocity and τ is the
proper time.
Derive various more-complicated expressions if/when the
situation warrants. Maybe even derive equation [2] (below),
but emphasizing that it is equivalent to equation [1], and
emphasizing that the first factor in equation [2] is just a
messy way of writing (1/γ) i.e. (1/gamma) ... understandable
in terms of trigonometric identities.
2) Start with the following expression:
1 p_xyz
ΔR_xyz = ----------------------- (-------) Δt [2]
[ p_xyz ] m
√ [ 1 + (-------)^2 ]
[ m c ]
In a later chapter, work backwards to obtain equation [1].
3) Start with equation [2] and just leave it out there as if
it were the whole story, without ever mentioning equation [1],
and indeed without ever mentioning spacetime, four-vectors,
or any of that.
===========================================
Obviously I have an opinion about this. Equation [2] does
have some advantages, but some of the alleged advantages are
illusory ... especially given that every "scientific" pocket
calculator made in the last 20 or 30 years can do hyperbolic
trig functions just as easily as it can do square roots.
Equally obviously, opinions differ. There are still plenty of
folks who profess to take a vigorously "modern" approach yet
still avoid spacetime and four-vectors like the plague.
Can anybody explain the attraction of approach (2) or (3)?
I don't get it, but I'm willing to listen.
I am particularly confused by the argument that says equation [1]
is "too complicated". Really? Compared to what? Compared to
equation [2]?