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On 5/31/19 1:09 PM, David Bowman wrote:
in order for the analogy to work properly the tape needs to be
*carefully* laid down on the surface locally straight/even with no
bunching/wrinkling or stretching/tearing of the tape material as it
is laid down. If the surface has significant curvature over a
length scale comparable to the width of the tape and the tape *can't*
be laid down without local distortion, then the tape needs to be
laid down in a way that any unavoidable stretching or binding is
laterally symmetric across the centerline of the tape.
Yup.
(But maybe this caveat is included in one of JD's lurid details.)
It is now :-)
There were hints about this scattered at various places in
the document, but just now I clarified them and collected
them into their own section:
https://www.av8n.com/physics/geodesics.htm#sec-guiding
Remarkably enough, even without running any risk of crumpling,
you can exhibit quite a bit of non-Euclidean geometry, such as
geodesics that start out parallel but don't stay parallel.
To address this point, I added a whole new section about
geodesics on a cone:
https://www.av8n.com/physics/geodesics.htm#sec-cone
It emphasizes that all physics is local, and cone-space is
flat /almost everywhere/ ... but that little bit of non-
flatness has remarkably far-reaching effects.
These experiments are easy to do, even at the elementary-
school level. Masking tape and cardboard. Yet if you think
about things properly, you come away with an understanding
of what "straightness" means and what "curved space" means,
at a level that escapes most physics grad students.
It's one thing to study the equation of geodesic deviation;
it's something else entirely to watch tapes wrap themselves
around a cone.
=========================
Tangential philosophical and pedagogical remark:
There is a fine art to /not/ answering the question that was
asked. Politicians are infamous for dodging questions by
giving useless non-responsive pseudo-answers.
However, it doesn't have to be that way. Sometimes you can
give a response that is /more/ useful than a direct answer
would have been.
For example, it is fairly common to get questions about
gravitational waves, coming from people who don't have the
slightest clue about waves or about gravitation. In that
case it makes sense to get out the cardboard and masking
tape and teach them a few things about geodesics and curved
spacetime. That doesn't answer the question, but it's true
and interesting, and it's a step in the right direction.
I like to say:
I'm not Superman, and you're not either. I can't teach
you how to leap tall buildings in a single bound. I
can however show you where the stairwell is. We can
climb to the top step by step.
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