BRS: degenerate cases of Born-rigidity

[bcrowell]

I've been having too much fun on PF this weekend, and now I need to grade lab notebooks (ugh), but I thought I'd post the general outlines of some thoughts I had while out running today. This has to do with how to use congruences to talk about the degenerate cases.

Let a k-congruence be a congruence in a manifold with signature (k,1). We can embed a k-congruence in a manifold with spatial dimension m>k. We can also define a notion of extending such an embedding by adding more world-lines to the collection, such that every point on a world-line in the embedded congruence is surrounded by an open neighborhood where the m-congruence is defined.

Let's describe a k-congruence embedded in a higher-dimensional spacetime as intrinsically rigid (i-rigid) if the k-congruence is Born rigid, and say that it's extrinsically rigid (e-rigid) if there exists an extension of the embedding such that the extension is Born rigid. When m=3, I think e-rigidity holds exactly under the conditions given by the Herglotz-Noether theorem.

I decided I wanted more vivid names for the examples, so I'm now referring to the angularly accelerating ruler as the "baseball bat," and the wriggling filament as the "snake."

The bat and the snake are both i-rigid, but neither is e-rigid.

One thing that i-rigidity is sort of good enough for is giving a rigorous definition of the measuring rods that Einstein refers to all over the place in his early writings. I-rigidity gives you a measuring chain, not a measuring rod, but that's probably good enough. The usual way of handling this is to assume that the rod deforms when you accelerate it, but returns to its equilibrium shape when you stop accelerating it. This doesn't quite work in general, because it means that you can't use such a rod to measure some dimension of an object while that object is undergoing acceleration. Born-rigidity of a space-filling rod is too strict, because such a rod can't be put freely into any desired orientation -- in general, you can't even verify that one space-filling rod is the same length as another space-filling rod, because you may not be able to match their orientations without violating the H-N theorem.

The letter "C" with angular deceleration about its center isn't e-rigid for all time, but it's e-rigid everywhere in the past of some spacelike hypersurface that has in it the event where the tips intrude on one another. This is a reasonably useful way of dealing with the self-intersection issue within the framework of congruences.

There are some seemingly silly physical consequences. For example, say we have a zero-thickness disk that is rotating at constant angular velocity. If the disk's center accelerates in its own plane, we don't even need an embedding, and it's clearly not rigid, by the H-N theorem. If it accelerates along its own axis, it's i-rigid but not e-rigid. What this means is that an e-rigid object can have infinite inertia along some axes, but finite inertia along others -- i.e., its mass isn't even a scalar!

Neither i-rigidity nor e-rigidity is really the right physical notion. Basically i-rigidity is too permissive and e-rigidity too lax. We want to be able to distinguish between the snake and the bat. I think there is probably some other useful notion that can be defined, which I'll call n-rigidity, according to which the bat is n-rigid and the snake isn't. If we want to define such a thing using congruence methods, then the usual expansion, shear, and vorticity tensors exhaust all the possibilities that can be written down in terms of the velocity field and its derivative. However, in the case of an embedding we have a new vector available, which is the normal to the world-sheet (or more than one normal, if m is greater than k by 2 or more). I think it's probably possible to cook up something tensorial that describes n-rigidity by taking the contraction of the shear with the normal vector, or doing something with the covariant derivative of the normal vector, or something along those lines.

-Ben

 

[Chris Hillman]

Ben, as often happens with new ideas, your definitions are not yet precise, I think, and it is not yet clear (at least not to me) that your Post #31 makes sense either physically or mathematically. But if you are on to something valid, I'd like to know!

I appreciate that time is an issue, and clarifying/fixing these problems might take time, so I guess we'll all need to be patient... maybe pick this up next weekend?

To clarify what you really have in mind (both to me and to yourself), I think it would be really helpful for you to write down some valid explicit examples. Do you see why your alleged counterexample didn't even define a congruence?

(A timelike congruence consists of parameterized timelike curves which do not intersect in U such that precisely one curve passes through each event in U, i.e. they are the integral curves of a suitable timelike vector field on U. Sometimes it is convenient to assume the curves are proper time parameterized, and some authors make that assumption without spelling it out, so be careful.)

Re

www.physicsforums.com/showthread.php?t=430381

I tend to doubt you will get better answers from non-SA/Ms, but in any case, FWIW:

Another good reference for the definitions is section 6.1.1 in Stephani et al., Exact Solutions of Einstein's Field Equations, second ed., Cambridge University Press, 2003.

Am I right in thinking that Wald's reason for restricting to geodesic congruences is that under these circumstances he gets the simpler expressions shown above, rather than the more complex ones that Hawking gives?

Yes. But for charged particles in an EM field, or fluid elements in a fluid with nonzero pressure or stress, &c., the accelleration vector will usually vanish so you need the extra term.

The definition of the spatial metric would clearly have to have the + sign flipped if you were using the +--- signature (since the purpose of the term is to punch the time-time component out of the metric).

Right,
<br /> h_{ab} = g_{ab} + u_a \, u_b<br />
is just the projection tensor, which might look more familiar as the projection operator
<br /> {h^a}_b = {g^a}_b + u^a \, u_b = {\delta^a}_b + u^a \, u_b<br />
which projects to the spatial hyperplane element orthogonal to \vec{u}. This is just linear algebra; Halmos, Finite Dimensional Vector Spaces, has a superb discussion of projections with and without a notion of orthogonality. Algebraically, projection operators are idempotent and in a euclidean inner product space they are represented by symmetric matrices, while in a Lorentzian inner product space, they are self-adjoint using the Lorentzian adjoint, which is slightly different from the euclidean adjoint.

Would any other signs have to be changed for +---, like the sign in the definition of the shear?

Unfortunately, various authors choose various signatures (but -+++ is best for comparing E^3 and E^{1,3}), various normalizations for the quadratic invariants of the shear and vorticity tensors, and some authors, such as Lauritzen, even flip the sign of the expansion tensor. So I hesitate to say "no" for fear of creating some further misunderstanding.

So let me say this: a good way of making a "reality check" that your sign conventions are not awful is to compute your quantities for a judiciously chosen suite of examples for which it is clear what the correct answers should be. If I ever get to the proposed BRS, I plan to provide such examples.

There are currently too many threads related to expansion/vorticity to follow easily, unfortunately, but regarding the issue implicitly raised in "differing definitions of expansion, shear, and vorticity"

www.physicsforums.com/showthread.php?t=430381

I plan to explain why Wald (and some other authors) can get away with dropping the projection tensor from their definitions, provided one applies them only to timelike geodesic congruences. But we want to consider non-geodesic congruences too (for example, the Langevin and Rindler congruences).

Gawk, I can see that trying to organize a comparision of notation in some standard books will be one of the things I'd have to do in the proposed BRS on timelike congruences.

 

[bcrowell]

Hi, Chris,

Re your #30, it looks like we're doing a great job of confusing one another!

Here you are, I think, claiming to give an example of a non-rigid congruence in Minkowski vacuum, which contains a "degenerate" configuration of world lines corresponding to a Born rigid object.

Right.

But what you wrote does not actually define a congruence of parameterized curves in Minkowski vacuum!

Here you lost me right away. I wrote x=x_oe^t, y=const, z=const. Are you saying this doesn't define a congruence at all?? Why not? My understanding of a congruence is that it's simply a set of nonintersecting, timelike world-lines whose union covers some open set of a manifold. I think mine satisfies all of those criteria on the open set defined by t<0. (For t>0, the world-lines become spacelike.) Are you saying that it doesn't satisfy all those criteria, or are you using some other definition of a congruence...?

To fix up the counterexample, you need to first of all define a congruence with nonzero expansion tensor, and then to find a one or two dimensional configuration of world lines belonging to the congruence which you can show are "rigid" according to some definition you'll need to explain.

The definition is the one I gave in #1. The one-dimensional configuration of world-lines is the one I gave in #27, "a line segment lying at rest in the y-z plane," as clarified in #29: "Say the ruler's world lines are of the form x=0, y=k, z=0, where 0 \le k \le 1. Then for all k we have a world line of the form given above, with x_o=0."

And still seeking clarification: "degenerate" means a selection of world lines from a timelike congruence which can be interpreted as the "world sheet" of a one or two dimensional object, right?

Right.

-Ben

 

[Chris Hillman]

Almost there!; more later...

Hi, Ben, glad you haven't given up on me yet, since I think we are making progress.

Congruence: a family of nonintersecting smooth curves filling up some domain (open neighborhood) U in some smooth manifold M, i.e. through every point in U passes precisely one curve in the family.

(But I think you already know that.)

Very briefly: I now see that I was expecting you to write down proper time parameterized curves forming a congruence. Reason first: easiest way to define a congruence of (proper time parameterized) timelike curves is generally to write down a timelike unit vector field \vec{X} and say "take the integral curves of \vec{X}". Particularly if you are looking ahead to extending said unit vector field to a frame field. Reason second: the formalism of the decomposition of a congruence of timelike curves works with timelike unit vector fields. Reason third: there is no reason third, I just got carried away.

However, it is true that one should always be able to convert a family of curves given by equations relating the coordinates into a family of proper time parameterized curves. (But I think you know that also.)

I plead exhaustion but I'll try to check your alleged counterexample more carefully when I have more energy.

<gentlehint>

If you can possibly install xfig, I find that pictures really help in preventing "preventable misunderstandings", which is particularly useful when participants are tired or rushed! You might have noticed that I have been trying to provide some sketches to illustrate my own BRS posts.

</gentlehint>

More later...

 

[George Jones]

I am going to abuse notation and suppress two dimensions for much of this post. I hope Chris doesn't faint.

Ben has defined a bunch of curves parametrized by t,

t\mapsto \left( t\left( t\right) ,x\left( t\right) \right) =\left( t,k\exp \left( t\right) \right).

The tangent vectors V to events on these curves are given by

V=\frac{d}{dt}\left( t,k\exp \left( t\right) \right) =\left( 1,k\exp \left( t\right) \right) =\left( 1,x\right) ,

or

V=\frac{d}{dt}=\partial _{t}+x\partial _{x}.

where x_{0} has been changed to k to avoid confusion with index notation.

At each event \left( t,k\exp \left( t\right) \right), draw a vector with components \left( 1,k\exp \left( t\right) \right). Ben's curves are the integral curves of the vector field V.

Now find curves parametrized by proper time \tau that have the same images as Ben's curves. First, normalize the vector field V, to find the tangent vectors (4-velocities) of the new curves, which will then be worldlines.

V^{a}V_{a}=-1+x^{2}=-1+k^{2}\exp \left( 2t\right) ,

where 0&lt;x&lt;1 restricts to a region where V is timelike. Define

X=\frac{V}{\sqrt{-V^{a}V_{a}}}=\left( \frac{1}{\sqrt{1-x^{2}}},\frac{k\exp \left( t\right) }{\sqrt{1-x^{2}}}\right) =\left( \frac{1}{\sqrt{1-x^{2}}},\frac{x}{\sqrt{1-x^{2}}}\right) ,

or

X=\frac{d}{d\tau }=\frac{dt}{d\tau }\frac{d}{dt}=\frac{dt}{d\tau }V=\frac{1}{\sqrt{1-x^{2}}}\left( \partial _{t}+x\partial _{x}\right).

Clearly, X^{a}X_{a}=-1.

Now find the integral curves of X; find curves

\tau \mapsto \left( t\left( \tau \right) ,x\left( \tau \right) \right)

that have X as tangent vector,

<br /> \begin{equation*}<br /> \begin{split}<br /> X&amp;=\left( \frac{dt}{d\tau },\frac{dx}{d\tau }\right) \\<br /> \left( \frac{1}{\sqrt{1-x^{2}}},\frac{x}{\sqrt{1-x^{2}}}\right) &amp;=\left( \frac{dt}{d\tau },\frac{dx}{d\tau }\right) ,<br /> \end{split}<br /> \end{eqaution*}<br />

so,

\frac{dt}{d\tau }=\frac{1}{\sqrt{1-x^{2}}},\qquad \frac{dx}{d\tau }=\frac{x}{\sqrt{1-x^{2}}}.

This gives

\frac{dx}{dt}=\frac{\frac{dx}{d\tau }}{\frac{dt}{d\tau }}=x,

and integrating

\int \frac{dx}{x}=\int dt

over appropriate intervals produces (the images of) Ben's original curves. Integrating

\frac{dx}{d\tau }=\frac{x}{\sqrt{1-x^{2}}}

will give \tau as a function of x (and thus as a function of [/itex]t.<br /> <br /> I should go and calculate an orthonormal frame, 4-acceleration, vorticity, expansion, but I&#039;m a little bit winded, and I have to do some work that puts bread on the table.

 

[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
<br /> y = \sqrt{1-x^2}<br />
To parameterize it by some w (not neccessarily an affine parameter!) is easy:
<br /> x = w, \; y = \sqrt{1-w^2}<br />
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
<br /> 1 = \left( \frac{dx}{ds} \right)^2 + \left( \frac{dy}{ds} \right)^2<br /> = \dot{w}^2 + \frac{\dot{w}^2 \, w^2}{1-w^2}<br /> = \frac{\dot{w}^2}{1-w^2}<br />
which gives w = \sin(s). So our arc length parameterization is
<br /> x = \sin(s), \; y = \cos(s)<br />
Now try the same procedure in E^{1,1} (with appropriate signature change) with
<br /> x = k \, \exp(t)<br />
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.