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Re: [Phys-L] step by step: curved space; was: gravitational waves



On 04/02/2016 08:29 PM, Ludwik Kowalski wrote:
I am trying to explain gravitational waves qualitatively, to people
like educated laymen, first year college students, etc.

Possibly constructive suggestion: Grab some masking tape
(aka painter's tape) and experiment with laying out geodesics:
https://www.av8n.com/physics/geodesics.htm

This works at every level from 5th grade through graduate
school ... including lay persons. It's interesting as well
as educational. It demystifies the idea of straight lines
in a curved space.

Beware that on the web and in "popular" books, very nearly all
of the diagrams of curved space in the context of gravitation
are wrong. Not even close. They bear no discernible relationship
to the right answer. In contrast, with the masking tape you
have some chance of getting the right answer.

If the student complains that I'm not answering the question
about waves, I say that a gravitational wave is a wave and
it's gravitational. If that's a good enough answer, fine,
that's my answer. On the other hand, if you want to understand
anything more than that, you need to have a clue about how
gravitation works. That involves curved spacetime. The first
question should be, in /what direction/ is it curved? If you
can't answer that, you don't have a strong enough foundation
on which to build any real understanding. If you want a bucket
of words that don't mean anything, you can go to the web and
download as much of that as you like, but if you want to
actually understand something we have to take it one step
at a time.

Note that understanding curvature has lots of applications
other than general relativity. A familiar example is
tailoring and dressmaking; a tuck here and a dart there
can change the overall shape of the garment, in rather
non-obvious ways. Also navigation and geodesy. Also there
are mechanisms that fold and unfold using origami-like tricks.
Not to mention protein folding.

==================

Here's another possibly-constructive suggestion. This is
too hardcore for the introductory physics course, but it's
perfectly doable for junior or senior physics majors.

The goal is to create a spreadsheet to integrate the geodesic
equation to map out geodesics on a sphere, using polar coordinates.
The deliverable should be a graph something like this:
https://www.av8n.com/physics/geodesics.htm#fig-geodesics-sphere

Anybody who wants to use visual python instead of a spreadsheet
is welcome to do so.

One advantage here is that the number of components in the
messiest equation is only 2×2×2 ... instead of 4×4×4 for general
relativity. And only three of the components are nonzero.

It could take you a week to figure out what all the symbols
mean, but once you do that, the calculation is little more
than an exercise in bookkeeping. It's a calculation that
ordinary mortals can do. Relatively speaking, it's incomparably
easier than typical general-relativity calculations.

Also, the graph provides a nice way to check the work. If
you get the equations wrong, the graph will look terrible.
There are other possible checks, too.

The details are worked out at
https://www.av8n.com/physics/geodesics.htm#sec-geodesic-equation

That includes a link to my spreadsheet.