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Re: [Phys-L] Acceleration and spacetime paths in the twin "paradox"

On 1/12/21 3:52 AM, Antti Savinainen via Phys-l wrote:

a friend of mine pointed out prof. Sean Carroll's explanation of the twin
"paradox": <>

He states that acceleration is not the reason for the time difference
between the twins; instead, the reason is in different paths in space-time.
However, he continues and says that taking a different path requires
acceleration in this scenario. This seems valid to me. What do you think?

All this was motivated by a philosopher of physics called Maudlin, who
claimed that Feynman's explanation on the twin "paradox" in terms of
acceleration was completely wrong. Here is a rebuttal of that claim:

It's possible for two or more things to be true at the same time,
in a way that is *not* cumulative.

I could describe a situation as six florins or half-a-dozen guilders.
It's not the sum of the two; it's just two different ways of describing
the same thing.

Half a pound of garbanzos or eight ounces of chickpeas.

Closer to physics, we can say that the lift of a wing is well described
by Bernoulli's equation, and we can say that it is well described by
Newton's laws. It's not the sum of the two; you don't get a little bit
from Bernoulli and a little bit more from Newton. And it's not one or
the other. It's entirely both, because Bernoulli's equation is a corollary
of Newton's laws (subject to mild conditions).

A) It is entirely appropriate to talk about the traveling twins in terms of
different paths. If we take identical cars and drive from point A to point
B by different paths, the odometer readings will be different. The distance
between A and B as measured by a ruler is always the same, but the path-length
as measured by an odometer depends on the path.

Clocks are more like odometers than rulers. The elapsed time depends on the
path. This is counterintuitive for non-experts, because the path-dependence
is weak under ordinary terrestrial conditions, but the fact remains: clocks
are more like odometers than rulers.

Of course it's not sufficient to say the paths differ; it depends on details
of /how/ the paths differ.

B) It's also appropriate to discuss how acceleration contributes.

The killer argument here is gravitational redshift. You can have two twins
on paths that are /parallel/ (in the sense that the distance is not changing
but the elapsed time will be different if one of them is deeper in the
gravitational potential. The acceleration here is significant just as it
is for the standard twin scenario with the zig-zag path.

It's not "just" acceleration, as discussed in the next item.

C) It's also appropriate to discuss it in terms of breakdown of simultaneity
at a distance.

It's never "just" acceleration. It's important that the twin who accelerates
is /far away/ when he accelerates.

In pedagogical terms, this is super-important. In my experience, breakdown
of simultaneity at a distance is the thing that people learn most slowly and
get wrong most often, when learning relativity. The Lorentz transformation
can be represented by a matrix. It is common to find people who have some
clue about the diagonal elements (time dilation and length contraction,
roughly speaking) but no clue about the off-diagonal elements (breakdown of
simultaneity at a distance).

That's particularly sad, because the off-diagonal element is first order in
small quantities, while the diagonal elements are second order.

Thinking about it as a /rotation/ in the xt plane helps a lot. I cannot
imagine why anybody would approach the subject in any other way.