Martin,
I can't think of a correct response. It should be possible to understand a lot of stuff about nature and the universe without what you said (basically going for a MS in astrophysics.)
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You got me thinking about teaching physics.
I watched a bit of Walter Lewin:
http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/
It's free on YouTube. Do you know if there is something comparable for Highschool students?
Good HS physics teachers do a lot of good in the world (a good one maybe more than some PhDs do).
You are asking about the fact that geometry is changeable. Distances can expand and contract in response to matter, or to some initial event that plays out. But if this can happen, how do we define distance?
Could an understanding of this (in a modest intuitive way) be incorporated in HS physics or at College Physics level? Since people seem interested in it. I wonder.
At large scale at least, geometry is dynamic--it responds and changes. Why should we expect otherwise? Everything else does. We have no right to insist that geometry should rigid and immutable, nothing else is.
Does this recognition have to wait until one is in graduate school?
Let's say you have done an undergrad physics major Bachelors degree, but you find that you don't have a great affinity for textbook learning. It seems unmotivated and unintuitive to you. Imagine yourself, or someone, like this. Say you want an intuitive understanding of some stuff that comes up in cosmology---early universe stuff, expansion, accelerated expansion---what can you do?
It's all in the Friedmann equations (two: the main one and the acceleration one) and they are just simple ordinary differential equations governing the scalefactor a(t).
And the scalefactor is just a simple factor that plugs into the metric--the distance function that contains the idea of geometry.
why shouldn't---if young people are interested in it---why shouldn't this be made accessible at the HS or freshman level, as an optional unit of some general physics course?
The question bugs me. I really don't know how to respond.
You get Euclid geometry in HS tenth grade, I think. When you are 15 years old.
Suppose they TOLD you at that point that the stiff version is beautiful but only approximately right. That geometry really isn't static.
The beautiful Euclid version is what comes out when gravity is weak enough to be neglected and when the effects of some intense initial conditions have dissipated and spread out enough that they too can be neglected (except over very long distances).
Suppose they told you when you were 15 that it wasn't exact (in very strong gravity or over very large distances) and promised that in two years---if you learned the static Euclid version---and then took calculus as HS Junior---you could choose a 3 week Senior elective, or 6-week elective, and learn about the dynamic Friedmann version.
Would this make any difference. Would people grow up being less puzzled about expansion cosmology?
when it was still available to move onto "higher" algebra. Luckily my Calc III/Linear Algebra instructor has taken a random day or two do discuss things like hyperbolic geometry (he teaches ARML-level geometry at a math summer camp). I suppose students probably would be able to understand the changes in metric(the way the conversion factor "plugs-in" reminds me of the Jacobian right now) if we were taught it but I'm still not sure where we would fit in the math understanding first
