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Talk:General theory of relativity (view source)
Revision as of 20:37, November 14, 2009
, 20:37, November 14, 2009→Idea for presenting this: new section
[[User:PatrickD|PatrickD]] 20:31, 13 November 2009 (EST)
[[User:PatrickD|PatrickD]] 20:31, 13 November 2009 (EST)
== Idea for presenting this ==
The General Relativity seems to be the place to be this weekend!!!!!
I've done some thinking about how we can explain, semi-adequately, what is going on, at an acceptable educational level. We all know that adequate explanations for lay people just don't exist. But we're making an effort.
Here's an idea that I had: Put something in, probably in front of your sections on T_mu_nu and G_mu_nu, about fictitious forces, and how curved coordinate systems, and curved manifolds, lead to these forces.
First, some kind of spacetime diagram, completely flat, with x going straight left to right, t going straight up, and some events labeled "me, now" at the origin, "me, 5 minutes from now" up the page, "My neighbor's house, now" to the right, and a diagonal line for my neighbor walking to my house and arriving in 5 minutes. An explanation that these are "world lines". Mine goes straight up, and my neighbor's goes diagonally. We show coordinate "graph paper" lines, all rectilinear.
Next, figure 2: a diagram showing the frame of reference of a car traveling uniformly down the street. The formerly vertical lines are now slanted. My neighbor appears to be walking faster in my frame of reference, and he arrives at my house at x=-5 or whatever. He is now behind me, because my car moved.
Point out that a "flat coordinate system" is one in which the coordinate lines are straight. That means they make boxes that are either parallelograms or rectangles. They don't have to be rectangles. The coordinate system of figure 2 is still flat. Point out also, that both the stationary observer in front of my house (figure 1) and the observer in the car (figure 2) don't feel any fictitious forces. They are in inertial frames of reference.
Now do figure 3. This is a car that starts out at rest at t=0, and accelerates. We show the formerly vertical coordinate lines as being curved. (They're parabolas, I believe.) This is a "curved coordinate system", characterized by curved coordinate lines. (You and I know that it's not that simple -- curved? What does that mean? It means they aren't geodesics. We can't get into that. It involves Christoffel symbols. We just draw curved lines and leave it at that.) We also point out that someone in the accelerating car ''feels a [fictitious] force'', even though he is "at rest" in his coordinate system.
Then we state the fundamental principle: ''If you are in a curved coordinate system, you will feel fictitious forces.'' A fictitious force is traditionally something that arises from some kind of acceleration, such as an accelerating vehicle, or a centrifugal force, or a Coriolis force.
Then we state the ''equivalence principle'' -- gravity perhaps can be explained as a fictitious force. We can point out that this principle had sort of been known for a long time before Einstein.
Now let's look at curved coordinate systems. Put up a piece of polar graph paper. This is a curved coordinate system. But, no problem -- it's a flat piece of paper. The ''manifold'' (we're using that term, right?) is flat; it just happens to have a curved coordinate system. This is equivalent the the accelerating car. By pressing down on the accelerator of the car, we intentionally put ourselves into a curved coordinate system, but the manifold itself is flat. The observer standing on the sidewalk can attest to that.
Now let's look at curved ''manifolds''. We say, in a hand-wavey sort of way, that there are "curved manifolds", and that the surface of a sphere is an example of a curved 2-dimensional manifold. And we state (we can't possibly prove it, or even rigorously define the terms, at this level of exposition) that ''curved manifolds have only curved coordinate systems''. There is no flat coordinate system on the surface of the sphere. Flat manifolds, like the spacetime diagrams of figures 1, 2, and 3, have both flat and curved coordinate systems. People living in the curved coordinate systems will feel fictitious forces.
We make some hand-wavey statement that the curvature of a manifold arises from somethng called the "metric tensor", "Riemann's tensor", "Ricci's tensor", and "Einstein's tensor". If we know the metric tensor, we can tell whether the manifold is curved. If it is (that is, Einstein's tensor is nonzero), then the manifold is curved, all possible coordinate systems are curved, and there is no global coordinate system that is free of fictitious forces. (Of course, we can make a coordinate system that is locally flat -- just jump down an elevator shaft.)
Then we say that the essence of General Relativity, and its explanation of gravity, is that gravity appears to be a fictitious force for which ''no flat coordinate system exists'', and hence it arises from the ''curvature of the manifold itself'', as given by Einstein's tensor. Which is the left side of the equation.
I really think this will be better than the exhibits of balls rolling around on curved surfaces that we see in science museums. A lot of handwaving, but we just may be able to do this a bit better than most elementary treatments.
[[User:PatrickD|PatrickD]] 15:37, 14 November 2009 (EST)