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*From*: John Denker <jsd@av8n.com>*Date*: Tue, 26 Mar 2013 13:53:32 -0700

Hi Folks --

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.

This is discussed with more detail (including diagrams) in a

new section I just wrote for my "Welcome to Spacetime" screed:

http://www.av8n.com/physics/spacetime-welcome.htm#sec-trig-identities

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]?

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