Re: Proper quantities in SR

Ken S. Tucker wrote:
I think Tom is unfamiliar with practical
applications.

I think Ken does not know me at all.


It's common sense that two
worldlines synchronized at a point of
divergence must, at a later time converge
to compare measurements, and that process
requires relative +/- accelerations.

Except that in GR it is easily possible for two geodesics to diverge and converge later. Geodesics, of course, have identically zero 4-acceleration.

Bottom line: one must be more careful in using terminology. Tucker must mean some sort of coordinate accelerations, without warning the reader (perhaps that's what he means by "relative +/- accelerations", a peculiar phrase indeed).


Suppose two observer's "A" and "B" diverge
from relative rest from P0 at which point
they synchronized their watches, and in the
future diverge and come to rest together at
P1, to find their watches differ in elapsed
time.

OK. This is not two geodesic paths, but it is a perfectly reasonable scenario -- they must accelerate apart and then decelerate together (with the acceleration/deceleration split arbitrarily between them).


The spacetime interval between P0 and P1
is invariant and equal for all frames.

This is plain and simply not true. The "spacetime interval" between a pair of points is PATH DEPENDENT. The above scenario clearly shows that. <shrug>

Why is this? -- because one must INTEGRATE the interval, and that requires a PATH. In general, different paths have different integrated intervals. Note that for timelike objects like this we normally say "elapsed proper time" rather than "integrated spacetime interval".


Clearly the elapsed measured time is NOT
invariant, and thus a wrist watch does NOT
keep proper time, that's where Tom and most
other amateurs are confused.

YOU are confused. Each observer's watch measured the elapsed proper time along THAT OBSERVER'S PATH. But elapsed proper time can differ for different paths -- this is after all the whole basis of the "twin scenario" in SR. In GR it is no different, but there are many more variations in paths possible. In this case the two observers measured different elapsed proper times for their different paths, but each of those values is itself an invariant. <shrug>

Remember the mantra: coordinate time is integrable but not invariant; proper time is invariant but not integrable.

(here "integrable" essentially means independent of path)


Incidentally, I wonder if Feynman advocated
taking all paths that "A" and "B" could
follow between P0 and P1 such that the
spacetime integrated interval "s" would
be a constant and an invariant.

I have no idea what you mean by "Feynman advocated". But I can state that in Feynman's path integral approach to quantum field theory, to evaluate the two-point propagator in configuration space ALL paths between the two points must be integrated over, without regard for the value of the "integrated interval" -- indeed if one restricted oneself to only paths for which that interval is constant one would obtain no quantum interference at all [#]!.

[#] One needs a few additional caveats for this, but they
all apply to a free particle of constant mass. Also one
normally computes in the momentum representation....


Tom Roberts
.

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