BRS: degenerate cases of Born-rigidity

[Chris Hillman]

More on the alleged counterexample

Ben and all:

The confusions here have to do with elementary curve theory, nothing hard, but a bit frustrating so we all need to try to be patient.

George, what you did is correct, but I don't think it helps: we really need to use an affine parameter, and if we can find an affine parameter for a timelike curve, it's trivial to turn it into an arc length parameter, which is what we really want to use.

Ben, trying to define a congruence not as a family of parameterized curves but as a family of unparameterized curves is generally a bad idea:

• the theory requires us to work with the unit vector field underlying a congruence of proper time parameterized (or arc length parameterized) curves, so if the curves are not presented as parameterized curves, we will need to parameterize them and then convert that to an arc length parameterization,
• even in simple examples, it can be quite difficult to find explicitly an arc length parameterization or proper time parameterization

Example: consider the euclidean plane curve
$$y = \sqrt{1-x^2}$$
To parameterize it by some w (not neccessarily an affine parameter!) is easy:
$$x = w, \; y = \sqrt{1-w^2}$$
Here, w is an unknown function of the arc length parameter s. By definition of arc length in E^2 (cartesian chart, obviously), we have
$$1 = \left( \frac{dx}{ds} \right)^2 + \left( \frac{dy}{ds} \right)^2 = \dot{w}^2 + \frac{\dot{w}^2 \, w^2}{1-w^2} = \frac{\dot{w}^2}{1-w^2}$$
which gives $w = \sin(s)$. So our arc length parameterization is
$$x = \sin(s), \; y = \cos(s)$$
Now try the same procedure in E^{1,1} (with appropriate signature change) with
$$x = k \, \exp(t)$$
You can find the ODE for the naive parameter in terms of the arc length parameter, but it is not easy to find its solution in closed form, agreed?

Ben, can you try to come up with an example of whatever you are trying to illustrate which is given as an explicit congruence of proper time parameterized curves? Or if you can clearly restate what you were trying to show, maybe I can come up with an example myself.

 

[George Jones]

I seem to be under the weather and interest

I you hope you feel better Chris. I, too, am very under the weather right now. Thursday, I stayed home from work to take care of my sick daughter. Thursday and Friday, my wife was mildly sick. Wife and daughter are now much better, but I'm quite sick.

I have done some calculations, though. I think that I understand Ben's example and point, and I think that I can frame his point in terms of the expansion tensor for his example, parametrization issues notwithstanding. I don't have the energy or concentration right now either to check my calculation, or to type in the latex. Maybe tomorrow or Tuesday.

Also, some people who have the math background have been known to misapply or misinterpret the fancy mathematics.

I don't have near the math background that Chris does, but I did take many more pure math courses than most North American physics students take. I hope that I haven't misinterpreted or misapplied "fancy mathematics" to Ben's example.

My motivation for going back and really digging into GR recently was that I had never been satisfied with the level of conceptual understanding I'd achieved in the one-semester graduate course I took. As a grad student, there were a lot of times when I needed to get my field theory and GR problem sets done, so I just cranked out the calculations, without feeling good about really understanding in detail what they *meant*.[/I]

I didn't have the opportunity as a physics student (undergrad and grad) to take any GR courses, but I had somewhat similar experiences in my grad field theory courses.

 

[bcrowell]

Ben, can you try to come up with an example of whatever you are trying to illustrate which is given as an explicit congruence of proper time parameterized curves?

I don't think that's necessary or relevant for the present purpose.

 

[Chris Hillman]

Those were the days

I think that I understand Ben's example and point, and I think that I can frame his point in terms of the expansion tensor for his example, parametrization issues notwithstanding. I don't have the energy or concentration right now either to check my calculation, or to type in the latex. Maybe tomorrow or Tuesday.

OK, let's pick this up again when (hopefully) we are both feeling more alert/energetic.

I hope that I haven't misinterpreted or misapplied "fancy mathematics" to Ben's example.

No need to worry, I think But my so far unfinished thread would illustrate how to conduct a "reality check" by studying simple but nontrivial examples where intuitive expectations should be reliable, especially in a Newtonian limit.

I didn't have the opportunity as a physics student (undergrad and grad) to take any GR courses, but I had somewhat similar experiences in my grad field theory courses.

I've said this before, but I have no coursework in formal physics. Closest I got was attending an informal seminar conducted by a famous function theorist on quantum mechanics. I suggested that von Neumann made a serious mistake by founding the theory upon unitary operators instead of projective operators and was laughed down. Later I learned I have reinvented a well known and respectable (but apparently not entirely successful) idea for fixing up various well known problems.

I don't think that's necessary or relevant for the present purpose.

So did you lose interest in notions of rigidity? I was planning to get to that in the thread "BRS: Timelike Congruences", eventually.