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Dear John,
Do you plan on posting notes regarding the use of bivectors for dealing
with coordinate singularities for motion confined to the sphere as
mentioned in your tangential remark? I would be very interested in this.
Any leads on how to most easily learn about the use of Clifford algebra to
solve mechanics problems would be greatly appreciated.
Best,
Francois
P.S. I quite enjoy your online notes. They are very clear and helpful.
Thank you.
On Sat, Jun 8, 2019 at 9:20 PM John Denker via Phys-l <
phys-l@mail.phys-l.org> wrote:
Hi Folks --provides
I recently added a new major section to my screed on geodesics in curved
space.
https://www.av8n.com/physics/geodesics.htm#sec-printable
You can create a real 3D paper cone, with preprinted geodesics, by
printing out a suitable template and then cutting and gluing. This
“hands on” demonstrations of several interesting concepts about life in athey
non-Euclidean space, such as the fact that geodesics that start out
parallel do not remain parallel, and may eventually cross, even though
are absolutely straight.is
The cone is a valuable pedagogical stepping stone, because even though it
is a non-Euclidean space, it is as nearly Euclidean as it could possibly
be. The cone has zero intrinsic curvature everywhere except at the apex.
The only way for denizens of the cone-world to discover that their space
not completely flat is to do an experiment that somehow circumnavigatesthe
apex.set.
I provide PDF files with templates you can print out, to make it easy to
create your own models. Actually I'm not sure that "model" is the right
word; it's an actual /instance/ of a non-Euclidean space, not merely an
analogy. (You can of course use it as a model for other curved spaces.)
If you have access to a glue stick, scissors, and a printer, you're all
watch
It is one thing to write down the equation of geodesic deviation, but it
is something else to look at actual geodesics on an actual cone, and
how they wrap around.flatness.
The question arises, what's the appropriate grade level? I don't really
know. A fifth-grader can assemble a cone in about five minutes, and can
learn some valuable lessons by staring at it. Meanwhile, physics grad
students can also learn some valuable lessons. To fully understand what
the model is trying to tell you takes a lot longer than five minutes.
It is important to learn the correct lessons from all this:
*) Fact: We have two straight lines (A and B) that start out parallel
but do not remain parallel. They may eventually cross. By the same token,
lines that start out non-parallel may cross each other twice.
*) Fact: This happens even though the space is absolutely flat
everywhere in the neighborhood of line A, and also absolutely flat
everywhere in the neighborhood of line B.
*) Do not let this undermine your notion of straightness. The lines
really are straight. Also do not let it undermine your notion of
The space really does have zero intrinsic curvature in the neighborhoodof
each of the lines of interest.crosses
*) The notion of parallel that you learned in high-school geometry
needs to be updated. The idea that parallel lines are everywhere parallel
only works in a Euclidean space. You can formulate a reliable local
definition of parallelism using a Schild’s Ladder construction. This
requires constructing straight lines and measuring distances, as we now
discuss.
*) The concept of distance is nontrivial. You cannot expect some
magical oracle to tell you the distance between point a (on line A) and
point b (on line B). If you want to know the distance, you have to run a
measuring tape from a to b, or perform some similar experiment that
the space between the two points. The operational definition of distancegeodesics
therefore depends on curvature in regions between the two lines, even if
the lines themselves never actually touch the high-curvature regions.
*) The cone is not a good model of gravitation. The fact that
partially wrap themselves around the cone is not a good model of orbitsin
that wrap themselves around a planet. A vastly better model is presented
section 3.of
The fact that geodesics are sensitive to curvature elsewhere has direct
application to astronomy and cosmology: You can measure the mass of a
planet without ever setting foot on the surface. Just put a satellite in
orbit around it, and apply Kepler’s laws. You can even measure the mass
a black hole this way, which is significant because the hole doesn’thave a
surface that you could even imagine setting foot on.
===========
This is fairly new and probably infested with typos or worse.
Let me know if you have questions or suggestions.
_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l
--
Professor of Earth System Science
Rm 3224 Croul Hall
University of California, Irvine
_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l