Both s and s^2 are invariant under the Lorentz transformation, so we
should feel free to use which ever we want.
Analogously, in the xy-plane both r and r^2 = x^2 + y^2 are invariant
under rotations. r has the physical meaning "distance from the origin",
so we might think of it as more fundamental. But there are times it's
cleaner to use r^2 and not have to fight with the square root. Then I
don't apologize for starting with r^2.
To expand on your first consideration below: We are used to the concept
of a "time interval", so keeping "interval" to mean "s" makes sense to
me. Just to muddy the waters a bit more, consider that we might take
the unadorned term "interval" to be a convenient abbreviation of
"spacetime interval", which would really need to be "s", not "s^2", so
that for spatially coincident events it would simplify to the familiar
"time interval". Not a watertight case, perhaps, but maybe indicative.
I guess I just don't like the idea of "interval" and "time interval"
having different units.
So my vote is for "interval" to mean s, but still use s^2 most of the
time. In the differential case, ds is the differential interval, but
(ds)^2 gets used more.
"squinterval"? Sure, why not? After all, squinting might help focus
(attention or otherwise). ;-)
Ken
-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu] On
Behalf Of John Denker
Sent: Tuesday, November 15, 2005 4:37 PM
To: PHYS-L@LISTS.NAU.EDU
Subject: variations on spacetime interval
Hi Folks --
Suppose we write
s^2 = t^2 - x^2 - y^2 - z^2
Some authorities say s^2 should be called "the interval".
Other authorities say s itself should be called "the interval".
Does anybody have a good reason why we should prefer one over the
other?
Considerations include:
-- s (not s^2) corresponds more closely to our prosaic
non-relativistic
notion of "time interval"
-- s^2 (not s) is better behaved for most relativistic purposes, since
s^2 will be negative for a spacelike interval, and I would prefer to
deal with a negative s^2 than an imaginary s.
Neither of those considerations seems particularly stronger than the
other, so they pretty much cancel each other out, leaving me as
uncertain as before.
======
Would it help to call s^2 the squinterval (as in squared interval)?
Giving it a one-word name helps me to focus attention on s^2 which
is what I want.
Also coining a new name allows me to duck the question of what the
old name means.
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