2013 edits to the sci.physics.faq Rotating Rigid Disk in Relativity

[pervect]

2013 edits to the sci.physics.faq "Rotating Rigid Disk in Relativity"

I was just reading the sci.physics.faq http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html on the rotating disk.

I was really dissapointed by the recent (2013) editorial comments. I felt that they totally missed the point - the split of space-time that the editor assumes is possible is, unfortunately, not possible :-(. (See for instance http://arxiv.org/abs/gr-qc/9805089, Tartaglia's paper published in "Foundations of Physics".

I'm not quite sure who to complain to, however - it would be nice if the FAQ could be salvaged. I would actually agree that the splitting of space-time is the fundamental issue, but this fundamental issue is not well treated by the FAQ :(.

 

[PeterDonis]

I hadn't noticed the 2013 edit to this page, but I've read it now and I don't like it either. In addition to the point about the split of spacetime, I also think the "speed of light" issue is not correctly handled: the rotating frame *does* in fact have a limited spatial range, precisely because objects "at rest" in the frame (i.e., with constant "spatial" coordinates) outside some finite radius would be traveling on spacelike worldlines rather than timelike ones. Even Wikipedia gets this right in its discussion of Born Coordinates.

As for who to complain to, the index page for the FAQ has Don Koks' email; he's apparently the current maintainer, and also the one who wrote the 2013 update, so he seems like a good point of contact.

 

[PAllen]

Well, for coordinates, there is nothing wrong with a coordinate whose direction changes from timelike to null, to spacelike; or for coordinate bases not to be orthogonal. Rotating coordinates work fine as general coordinates. If you require various attributes of a 'frame', then, of course, they have limited range.

 

[PeterDonis]

Well, for coordinates, there is nothing wrong with a coordinate whose direction changes from timelike to null, to spacelike; or for coordinate bases not to be orthogonal. Rotating coordinates work fine as general coordinates. If you require various attributes of a 'frame', then, of course, they have limited range.

Good point, a more precise way of stating the point I was trying to make is that the rotating chart itself can cover the entire spacetime, but integral curves of its "time" coordinate will only be timelike for a limited range. But the FAQ entry, as revised, gives the impression that the latter is not the case.

[Edit: Btw, the Wikipedia page I linked to on Born coordinates claims that the Born chart is only valid for $0 < r < 1 / \omega$, when it really should only say that the "rotating" frame field is only valid for that range. As far as I can tell, the Born chart line element they give is well-defined for $0 < r < \infty$; the $g_{tt}$ metric coefficient becomes zero at $r = 1 / \omega$, but the metric as a whole is not singular there.]

 

[Dale]

the rotating frame *does* in fact have a limited spatial range, precisely because objects "at rest" in the frame (i.e., with constant "spatial" coordinates) outside some finite radius would be traveling on spacelike worldlines rather than timelike ones.

Hmm, I am not sure about that. There is no singularity or other mathematical feature which makes the coordinate system intractable. You simply have your dt coordinate basis vector becoming a spacelike vector and your dθ becoming a timelike vector. But there is nothing preventing you from using a timelike vector named dθ and a spacelike vector named dt even though it is a little strange.

EDIT: unless you are referring to the frame field tetrad rather than the coordinates. A distinction I blur too often.

 

[PeterDonis]

I am not sure about that.

I should have been clearer about what I meant by "the rotating frame": I meant the frame field whose timelike basis vector is tangent to the integral curves of the chart's $t$ coordinate. Obviously such a frame field is only possible where the integral curves of $t$ are in fact timelike, i.e., for $0 < r < 1 / \omega$. But that frame field is certainly not the only possible one.

You simply have your dt coordinate basis vector becoming a spacelike vector and your dθ becoming a timelike vector.

No, you don't. For $r > \omega$, all four basis vectors of the Born chart are spacelike. Timelike vectors will be of the form $a \partial_t - b \partial_{\phi}$, where $a$ and $b$ are both positive.

 

[Dale]

I should have been clearer about what I meant by "the rotating frame": I meant the frame field whose timelike basis vector is tangent to the integral curves of the chart's $t$ coordinate.

Thanks for the clarification. I agree with that. I am guilty of equating the frame with the coordinates.

No, you don't. For $r > \omega$, all four basis vectors of the Born chart are spacelike. Timelike vectors will be of the form $a \partial_t - b \partial_{\phi}$, where $a$ and $b$ are both positive.

Interesting, I had not worked through it enough to realize that. For some reason I had assumed that all coordinate bases involved either 3 spacelike and 1 timelike vector or 2 spacelike and 2 null vectors. Now I wonder if there are any restrictions or if all combinations are possible.

 

[George Jones]

Now I wonder if there are any restrictions or if all combinations are possible.

Let $\left\{ x^\mu \right\}$ be inertial coordinates for Minkowski spacetime. Define new coordinates by $X^1 = x^0 + x^1$, $X^2 = x^0 - x^1$, $X^3 = x^0 + x^2$, $X^4 = x^0 + x^3$.

I am in a coffeeshop and rushing to catch my bus, but I think these coodinates are linearly independent and all lightlike.

 

[PeterDonis]

Now I wonder if there are any restrictions or if all combinations are possible.

The combinations I've seen are:

* 1 timelike, 3 spacelike;

* 2 null, 2 spacelike;

* 4 spacelike (in addition to the Born chart for $r > \omega$, other examples are the Painleve and Eddington-Finkelstein coordinates on Schwarzschild spacetime inside the horizon);

* 4 null (I've never seen this in actual use, but examples have been given in several threads in this forum--including the post George Jones just made in this thread).

Since the coordinates have to be linearly independent, I'm not sure more than one can be timelike, and I'm not sure that null and timelike coordinates can be combined in a single chart. But those are the only potential restrictions I can think of.

 

[George Jones]

Since the coordinates have to be linearly independent, I'm not sure more than one can be timelike

It is not possible to have more than one orthogonal timelike coordinate, but it is certainly possible to have four linearly independent timelike coordinates. For example, if $\left\{ \bf{e}_\mu \right\}$ are vectors tangent to inertial coordinate lines, then consider vectors $\bf{E}_0 = \bf{e}_0$, $\bf{E}_1 = 2\bf{e}_0 + \bf{e}_1$, $\bf{E}_2 = 2\bf{e}_0 + \bf{e}_2$, $\bf{E}_3 = 2\bf{e}_0 + \bf{e}_3$, and construct coordinates whose coordinate lines have these independent vectors as tangents.

I'm not sure that null and timelike coordinates can be combined in a single chart.

Similarly, all combinations are possible.

 

[D H]

This is the same person who mangled the sci.physics.faq relativistic mass page over a decade ago. I doubt complaining would be of much help.

Addendum:
Here's a link to the current relativistic mass page:
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

Here's what it looked like before his first edit:
http://web.archive.org/web/20030212...edu/home/baez/physics/Relativity/SR/mass.html

 

[George Jones]

This is the same person who mangled the sci.physics.faq relativistic mass page over a decade ago.

After prodding by pmb

 

[D H]

After prodding by pmb

Neither of whom is a working physicist.

 

[Doc Al]

After prodding by pmb

Exactly.

 

[PeterDonis]

It is not possible to have more than one orthogonal timelike coordinate, but it is certainly possible to have four linearly independent timelike coordinates.

Oof. I was afraid you were going to say that.

Similarly, all combinations are possible.

It looks like I need to re-train my intuitions in this area. I'll start by working through your example of 4 timelike vectors, and then try to construct a 2 timelike-2 null vector chart using that one as a starting point.

 

[bcrowell]

Maybe I'm just dumb, but I don't understand what idea he's even trying to express in the added note.

He alludes to WP, but he doesn't say what WP article he has in mind. It doesn't seem to be this one http://en.wikipedia.org/wiki/Ehrenfest_paradox , since I can't find any material that resembles his characterization.

Maybe the meta-issue here is why anyone should pay attention to something labeled "sci.physics.faq." Usenet is dead, and therefore there is no longer any social structure available that would allow people to discuss ongoing changes to a usenet faq and make sure they're right.