shape of a relativistic wheel

[Tach]

Anyway...

Are we all agreed that in the moving frame, the portion of the wheel above the level of the axle has more kinetic energy than the portion of the wheel below the level of the axle?

This is irrelevant. Does the "half" above the transformed of the horizontal axis passing through the axle have more kinetic and/or total energy than the "half" below it?
Think about it, if it did, then you would be able to detect gravitational effects that are motion (speed) dependent , effects that are NOT present in the frame comoving with the axle.

 

[Tach]

...
But the implication here is that the momentum density and/or momentum flux must be reduced in the upper semi-circle in such a way that the total gravitation of each half remains the same?

The discussion is not about ant momentum density or "momentum flux", whatever that may be.

It still doesn't sit well with me.

Then you would be able to detect gravitational effects that are motion (speed) dependent , none of these effects being present in the frame comoving with the axle.

 

[Pete]

No. There is no mention of aberration in the whole thread except in my latest posts. If you think otherwise, please point out the post.

:bugeye: Are you serious?

From the opening post:

Next to the observer is a very bright light source. This light source illuminates the strip of paper and the half of the wheel. But, the wheel is rotating so that it looks lorentz contracted:

See the distorted wheel? That's a diagram of a relativistic rolling wheel at one instant in the ground frame.

Also posts 9,10,11,12,27, 33 (and maybe more, I stopped there), where DRZion was arguing that the above picture was an accurate representation of the rolling wheel, and you repeatedly denied it.

And let's not forget posts 31, 32, and 58, where I showed that any point on the rim spends more time above the level of the axle than below, and suggested ways to demonstrate that the spokes are in fact curved as in the opening post.

Also pryzk, from post 52 where he pointed out that "in the frame in which the wheel is rolling and the ground is at rest, most of its mass is actually concentrated in the top half." (a claim which you again denied.)

Sure it is, this is how you separate the "upper" half from the lower "half". The two points where the midline intersects the ellipse determine the limits of your two line integrals for calculating E or KE or whatever.

Which is exactly what przyk did in the ground frame, using the ground frame midline. You said it was wrong.
You seem to be suggesting that he should have used the axle frame midline.
Why?

 

[Tach]

Well, I suspect it's still relevant to the argument you're having with RJBeery, but whatever. It's not worth splitting hairs.


Sorry, mistake in the tex.

sin\frac{-\omega^2Rx_1'}{c^2}
...needs brackets...
sin(\frac{-\omega^2Rx_1'}{c^2})

Fix'd, thanks.

Yes, looks right now.
Except that a problem surfaces: if you put up a "fence" of height R , more than half the spokes will be visible above the "fence". This doesn't seem right. This is where this thread started from. So, something is terribly wrong with this "picture".

 

[Tach]

:bugeye: Are you serious?

From the opening post:


See the distorted wheel? That's a diagram of a relativistic rolling wheel at one instant in the ground frame.

Also posts 9,10,11,12,27, 33 (and maybe more, I stopped there), where DRZion was arguing that the above picture was an accurate representation of the rolling wheel, and you repeatedly denied it.

How can I deny it when I introduced you to this whole concept long ago, in the thread we debated the number of turns.
How can I deny it when I gave you the equation of the ellipse and the equation of the spokes and the equation of the speed components (necessary in calculating the tangential speed)?
What I did disagree with is the fact that the spokes may be distributed unevenly in the ground frame.

And let's not forget posts 31, 32, and 58, where I showed that any point on the rim spends more time above the level of the axle than below, and suggested ways to demonstrate that the spokes are in fact curved as in the opening post.

....and you ended up with a correct proof after I flagged down your errors. What does have anything to do with the fact that I pointed out that the midline is aberrated?

Also pryzk, from post 52 where he pointed out that "in the frame in which the wheel is rolling and the ground is at rest, most of its mass is actually concentrated in the top half." (a claim which you again denied.)

...because there is no PROOF to that. Just some armwaving and no math. Neither of you calculated the mass of the "upper" half. We can't even agree what "upper" half is. Put up a calculation and we'll talk.

Which is exactly what przyk did in the ground frame, using the ground frame midline. You said it was wrong.

przyk did no calculation, he showed a generic formula that proves nothing.

You seem to be suggesting that he should have used the axle frame midline.
Why?

That is false. For the third time, you need to use the transformed (aberrated) line, i.e. the inclined line that is the transformation of the horizontal line passing through the axle.

 

[Pete]

But the implication here is that the momentum density and/or momentum flux must be reduced in the upper semi-circle in such a way that the total gravitation of each half remains the same? It still doesn't sit well with me.

I don't have anywhere near enough of a grasp on GR to trust my "sits well" instincts, but I'll share some ideas that you might find interesting...


I have a vague notion that when you're figuring the gravity of some system of masses, you have to use a reference frame in which the centre of mass of that system is at rest, perhaps averaged over time.

I also have a vague notion that it's acceleration more than kinetic energy that is important, and that gravity waves are somehow involved...


But I can't distinguish these vague notions from complete bullshit. They're just my mental toys.

 

[Pete]

This is irrelevant. Does the "half" above the transformed of the horizontal axis passing through the axle have more kinetic and/or total energy than the "half" below it?

Ummm... since when was the discussion about the transformed horizontal axis? I suspect you're misunderstanding people.

But I think the answer is still yes.
In the ground frame, the upper half of the wheel's mass (ie the portion above a transformed horizontal axis) has higher velocity than the lower half of the wheel's mass...

Equal mass, higher velocity -> higher KE, right?

Maybe if you posted the integral you said you did earlier?

Think about it, if it did, then you would be able to detect gravitational effects that are motion (speed) dependent , effects that are NOT present in the frame comoving with the axle.

I have no idea. That's GR spookiness, isn't it.

But... Actually, I'm not sure there would be anything paradoxical about the upper half having more gravitational pull in the moving frame.
What kind of effects did you have in mind?

 

[Tach]

I don't have anywhere near enough of a grasp on GR to trust my "sits well" instincts, but I'll share some ideas that you might find interesting...


I have a vague notion that when you're figuring the gravity of some system of masses, you have to use a reference frame in which the centre of mass of that system is at rest, perhaps averaged over time.

I also have a vague notion that it's acceleration more than kinetic energy that is important, and that gravity waves are somehow involved...


But I can't distinguish these vague notions from complete bullshit. They're just my mental toys.

It must be BS. Think this way:

-there are no gravitational effects in the frame of the axle

-if you run quickly with respect to the axle, the gravitational effects, whatever they may be , show up

-the effects increase with the relative speed between the observer and the axle

 

[przyk]

I wasn't wrong at all.

You were wrong to say [POST=2847612]this[/POST]:

NO, the "relativistic mass" DOES NOT intervene in ANY gravitational force or, for that reason, in ANYTHING to do with theory of gravitation. So, it is a non issue.

especially when you later followed it with [POST=2847707]this[/POST]:

Energy is known to gravitate.

given the relationship between energy and relativistic mass. So I find your assertion that you weren't "wrong at all" just a tad disingenuous.

It depends how you define "top" and "bottom"

The part of the wheel above the axle (i.e. above half its height) in whatever reference frame is being considered. Both because this seems the most natural meaning of top and bottom "half" and because this is what I would have assumed RJBeery was talking about in the [POST=2847579]post that started this[/POST].

did you miss the fact that the midline separating the two "halves" is frame dependent, i.e. it is aberrated in the moving frame?

I wasn't considering or referring to any kind of illusion whatsoever. Just the Lorentz boosted rotating wheel.

 

[Tach]

Ummm... since when was the discussion about the transformed horizontal axis?

Since I introduced it as the explanation of the two "halves" actually exhibiting the same E.

I suspect you're misunderstanding people.

Based on your posts, it is the other way around.

But I think the answer is still yes.
In the ground frame, the upper half of the wheel's mass (ie the portion above a transformed horizontal axis) has higher velocity than the lower half of the wheel's mass...

Equal mass, higher velocity -> higher KE, right?

No mathematical proof of that. So far, just armwaving.

I have no idea. That's GR spookiness, isn't it.

Really? This is in the same class as the claims that "running very fast wrt a gravitating body will make it collapse into a black hole. The faster you run, the quicker is collapses."

 

[przyk]

But the implication here is that the momentum density and/or momentum flux must be reduced in the upper semi-circle in such a way that the total gravitation of each half remains the same? It still doesn't sit well with me.

Why are you assuming that more momentum in GR necessarily means more gravitation? This is the point I'm making: unlike the classical F = Gmm'/r[sup]2[/sup] where it's easy to see that more mass means more gravity, the relation in GR isn't so straightforward. I completed two courses in GR at university and despite this I have no real intuition for what the gravitational field of a mass with a certain energy and momentum should look like. The only way I'd know how to do it would be to work things out in the "rest" frame and apply a boost (though the concept of a Lorentz boost becomes a bit fuzzier in curved spacetime), and guess from this that momentum can actually reduce gravitation.

 

[Tach]

You were wrong to say [POST=2847612]this[/POST]:

especially when you later followed it with [POST=2847707]this[/POST]:

given the relationship between energy and relativistic mass. So I find your assertion that you weren't "wrong at all" just a tad disingenuous.

I don't know why you persist. Relativistic mass does not contribute to any gravitational effects. If you think otherwise, it would be nice for you to give a textbook reference.
Making a tenuous connection between relativistic mass and total energy and jumping from there to the stress-energy tensor (that does intervene in EFE), doesn't really cut it.

The part of the wheel above the axle (i.e. above half its height) in whatever reference frame is being considered. Both because this seems the most natural meaning of top and bottom "half" and because this is what I would have assumed RJBeery was talking about in the [POST=2847579]post that started this[/POST].

But you know fully well that "above the axle" is a frame-dependent notion.

I wasn't considering or referring to any kind of illusion whatsoever. Just the Lorentz boosted rotating wheel.

This is not an "illusion", this is a FACT: the line separating the two "halves" is frame-dependent. Do you disagree with this?

 

[przyk]

...because there is no PROOF to that. Just some armwaving and no math. Neither of you calculated the mass of the "upper" half. We can't even agree what "upper" half is. Put up a calculation and we'll talk.

Why don't you do it first, since you claim to have already done it?

przyk did no calculation, he showed a generic formula that proves nothing.

Actually it proves quite a bit. If you choose a parameterisation of the perimeter of the wheel in the ground frame where the parameter x measures distance in a homogenous manner (e.g. it's a distance in metres), the lower half of the wheel is 0 < x < L, and the upper half is L < x < 2L, then you can express the energy difference between the two halves as

$$
\Delta E \,=\, \int_{0}^{L} \mathrm{d}x \, \bigl( \rho(x + L) \, k[v(x+L)] \,-\, \rho(x) \, k[v(x)] \bigr) \,.
$$​

It's an inescapable conclusion that $$\Delta E > 0$$. The only way you could argue against this would either be to deny that

1. $$\rho \, k(v)$$ is larger in the top half than in the bottom half, or
2. the integral of a positive function is necessarily positive.

So if you think my conclusion is wrong, exactly which of these two points do you take issue with?

 

[Tach]

Consider the worldlines of the rim ends of opposite spokes, horizontal at t'=0:

$$x_1' = -R.cos(\omega t') \\
y_1'-R = R.sin(\omega t')$$

$$x2' = R.cos(\omega t') \\
y_2'-R = -R.sin(\omega t')$$

Transforming and solving for y at t=0, we get:
$$y_1-R = R sin(\frac{-\omega^2Rx_1'}{c^2}) \\
y_2-R = -R sin(\frac{-\omega^2Rx_2'}{c^2})$$

I can't get an exact expression for x1' and x2' in terms of t, but I think it's clear that at t=0:
$$-R < x_1' < 0 \\
0 < x_2' < R$$

This means that at t=0:
$$y_1-R > 0 \\
y_2-R > 0$$

Unfortunately, not always true. You can easily have :

$$\frac{-\omega^2Rx_1'}{c^2}< \pi$$

and

$$\frac{\omega^2Rx_2'}{c^2}> \pi$$

 

[Tach]

If you choose a parameterisation of the perimeter of the wheel in the ground frame where the parameter x measures distance in a homogenous manner (e.g. it's a distance in metres), the lower half of the wheel is 0 < x < L, and the upper half is L < x < 2L,

The point is that, with the exception of the axle frame, the two "halves" do NOT have the same length, so you cannot use the above parametrization.

So if you think my conclusion is wrong, exactly which of these two points do you take issue with?

1. Your parametrization is false.
2. The physical effects your approach is implying are false. Think this way:

-there are no gravitational effects in the frame of the axle

-if you run quickly with respect to the axle, the gravitational effects, whatever they may be , show up

-the effects increase with the relative speed between the observer and the axle

 

[przyk]

I don't know why you persist. Relativistic mass does not contribute to any gravitational effects. If you think otherwise, it would be nice for you to give a textbook reference.

Now you're back peddling. Earlier when I said (emphasis added):

In that sense you were perfectly correct when you said the relativistic mass was irrelevant (to the general characteristics of the gravitational field around a dense mass). I'm just saying that you can't deny that the relativistic mass is relevant in GR in exactly the same way and to exactly the same extent as the total energy is, because apart from a factor of c[sup]2[/sup] they're exactly the same thing.

to all of this, you responded "OK".

But you know fully well that "above the axle" is a frame-dependent notion.

And...? That just makes "top half" and "bottom half" frame dependent notions. That doesn't stop them being what the problem RJ posed was about.

 

[Tach]

Now you're back peddling. Earlier when I said (emphasis added):

to all of this, you responded "OK".

I responded "OK" to the part were you said I was correct. I proved your connection between relativistic mass and kinetic energy wrong. I still don't understand why you insist on making the unsupported connection relativistic mass->kinetic energy->total energy->gravitation when you fully know that it is the stress-energy tensor that enters the EFEs.

And...? That just makes "top half" and "bottom half" frame dependent notions. That doesn't stop them being what the problem RJ posed was about.

And it makes you attempt at calculating the energies associated with the "top" and "bottom" "halves" of the ellipse invalid since your parametrization of the ellipse is invalid.

In addition, the physical effects your approach is implying are false. Think this way:

-there are no gravitational effects in the frame of the axle

-if you run quickly with respect to the axle, the gravitational effects, whatever they may be , show up

-the effects increase with the relative speed between the observer and the axle

 

[przyk]

1. Your parametrization is false.

Given that RJBeery originally posed the problem way back in post #13, I think that's for RJBeery to say, not you. So RJBeery: am I considering the right problem or not?

2. The physical effects your approach is implying are false. Think this way:

-there are no gravitational effects in the frame of the axle

Yes there are. Frame-dragging, for instance.

 

[Tach]

Given that RJBeery originally posed the problem way back in post #13, I think that's for RJBeery to say, not you. So RJBeery: am I considering the right problem or not?

This is a first, you seeking RJBerry's approval.

Yes there are. Frame-dragging, for instance.

From a wheel? Detectable? LOL, you are grasping at straws now. I knew that you were going to go there.

 

[przyk]

I responded "OK" to the part were you said I was correct.

So you just didn't read the whole of what you were replying to then? That would explain a few things.

I proved your connection between relativistic mass and kinetic energy wrong.

No you didn't. You just denied it.

I still don't understand why you insist on making the unsupported connection relativistic mass->kinetic energy->total energy->gravitation when you fully know that it is the stress-energy tensor that enters the EFEs.

I don't know what you're talking about. I only need the fully supported connection "relativistic mass -> total energy" and your admission that energy produces gravitational effects.

And it makes you attempt at calculating the energies associated with the "top" and "bottom" "halves" of the ellipse invalid since your parametrization of the ellipse is invalid.

Invalid how specifically?