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Re: [Phys-L] Fwd: [PTSOS] spacetime simulator: flexible conduit?

The trampoline demo gives the students an unforgettable "feel" for the ideas of mass curving space and curved space telling mass how to move (or in the demo, roll, in the straightest possible way).

I have previously read and appreciated John Denker's thoughtful essay on motion in curved space (and other material on relativity), but I still think the trampoline visualization is helpful, and can "hook" some students in a way that just won't happen with a thought experiment, or a mathematical equation, or strips of paper and darts.

For those who weren't here the last time the topic came up, my summary of the main problem of the trampoline model is: the strength of the force is wrong and the form of its dependence on distance is wrong.

A large mass on the trampoline curves the surface around it, but (I think) not the right amount, not with the correct r-dependence. (But there is some inverse relationship apparent: more warping close to the large mass, less with increasing distance, asymptotically approaching zero.) But "most wrong" is the net force on a marble circling the large mass, which in the simplest model will be mg tan(theta), where mg is the weight of the marble and theta is the local angle the fabric makes to the vertical. (A quick free-body diagram helps: the force of gravity downwards is cancelled by the upward component of the normal force, leaving an inwardly-directed radial component of the normal force to provide the centripetal force that causes the marble to circle the large mass.) Again, more-or-less the right idea (at least right direction) but numerically incorrect.

Hmmm, to quantify the incorrectness we'd really need to model the angle of the fabric as a function of magnitude of the large mass (M) and the radial distance (r) away from its center, plug that angle into F = m g tan(theta) to get a formula giving the "gravitational" force in terms of M, m, and r. In the general relativity weak field approximation the force approaches that postulated by Newton, F = G M m / r^2. Dollars to Donuts the trampoline gives no such thing, nor anything like the full GR result.

But -- the demo is intensely visual, interactive, and I say, may its tribe increase! Things like this get kids interested in physics, may even motivate them to learn the math to get "the rest of the story".

JD, could you explain the statement the prediction would be the same if the trampoline were deformed upward instead of downward? The presenter in the video clip actually put a pole under it and mentioned the parallel to dark energy, causing things to separate instead of attract. Not as good an analogy, since it was only in one spot, but still mildly useful, I thought. In any case it's clear that reverse warping of the surface will model a repulsive rather than an attractive force. I think you're saying that warping spacetime in some direction undetectable by us and orthogonal to the directions we experience would mathematically give the same result whether "up" or "down" in the new direction, whereas the warping of trampoline works differently. But then again, the mathematical treatment of curved space(time) doesn't actually require any hypothetical higher dimension to warp around, so this seems less important to me.

Anyway, I love the video, thanks to bc for forwarding the link. I may have to build one, for the wow value -- and the chance to emphasize that some features are analogous to the explanation of gravity by general relativity, but you can't really get the right answers without learning the mathematical skills.

(And thanks again, JD, for an accessible explanation to which we can send students when they ask for more.)

KC

-----Original Message-----
From: phys-l-bounces@mail.phys-l.org [mailto:phys-l-bounces@mail.phys-
l.org] On Behalf Of John Denker
Sent: Tuesday, July 17, 2012 4:39 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Fwd: [PTSOS] spacetime simulator: flexible conduit?

In the context of
http://www.youtube.com/watch?v=MTY1Kje0yLg

On 07/17/2012 01:03 PM, Bernard Cleyet wrote:
I don't understand how this is a space time simulator? I though it
was simply an (approximate) inverse square model (1/r, potential),
nothing to do w/ special relativity.

(and/or general relativity)

bc needs to go back to school?

BC is being ultra-polite. The trampoline model represents a serious
misconception as to the structure of spacetime and the relationship between
curvature and motion.

Here's a three-sentence proof of what I am saying: Deform the trampoline
upward instead of downward. The intrinsic curvature would be the same
and the geodesics would be the same, so the prediction of real general
relativity would be the same. However the trampoline demo makes
opposite predictions about the motion.

A much more apt model of motion in curved space can be found here:
http://www.av8n.com/physics/geodesics.htm
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