shape of a relativistic wheel

[RJBeery]

I'm less interested in the visual ramifications, specifically whether 4 spokes or 6 spokes are able to be seen over the wall. I write it off as a psychological illusion...
Read this excerpt:

...consider a train carriage that moves with constant speed along a straight track, past us who stand on the platform watching it. Light that leaves the back of the carriage will arrive at our eyes coincident with light from the side of the carriage; and as well, the carriage is Lorentz contracted. At first glance this seems to ensure that the carriage will look rotated, because we are used to seeing just such an effect from rotations in everyday (nonrelativistic) life. But clearly, on second glance, we will realise the carriage is not rotated, because we notice that its wheels are still attached to the track, which has not changed its aspect (since it's at rest relative to us). So there will be a psychological effect of the carriage looking somewhat rotated, but we will quickly notice that this is only an imperfect optical illusion.

The gravity issue, though, remains. We ALL agree that frame-dependent gravity would not occur, but it's fascinating to watch the discussion between Tach VS The World, where both sides claim the other "owns" the paradox, and is therefore wrong...I think it's safe to say that we all agree it wouldn't happen but no one can currently explain why.

 

[Tach]

I'm less interested in the visual ramifications, specifically whether 4 spokes or 6 spokes are able to be seen over the wall.

BUT! This is what this thread is all about. Look at the OP.

I write it off as a psychological illusion...
Read this excerpt:

It isn't a "psychological illusion", it goes to the core of generating realistic images through a process called raytracing.

The gravity issue, though, remains. We ALL agree that frame-dependent gravity would not occur,

But, the more knowledgeable people than you (przyk, to be more specific) claims that there is "matter accumulation" in the upper half of the wheel. So, according to such claims, an asymmetry in spacetime curvature would result. Of course, when he's pressed , he denies any such effect.

but it's fascinating to watch the discussion between Tach VS The World, where both sides claim the other "owns" the paradox, and is therefore wrong...I think it's safe to say that we all agree it wouldn't happen but no one can currently explain why.

The answer is quite simple, there isn't any "matter accumulation" in the upper half of the wheel.

 

[RJBeery]

Tach: I maintain that we're all in agreement, the paradox would not occur, so please stop trying to claim that Przyk is wrong on this basis. The disagreement over "where" the midline resides is irrelevant because whether we consider the MASS ACCUMULATION or the MIDLINE to be above the fence, the apparent gravitational imbalance is the same. The question is how does the framework avoid this?

 

[Tach]

Thanks for the link. I'd forgotten about that thread. Classic Tach.
You should read it again, and the link you used in your first post...

So, what is your point?That I have introduced you to this website about a year ago?

...especially where it describes this image (visual appearance of the rolling wheel in the ground frame):

(movie)
...and this image (diagram of the rolling wheel at an instant in a) the axle frame, and b) the ground frame):

You didn't seem to think it absurd then.

I never said it was absurd. A year ago we were discussing the number of revolutions and the correct equation of the ellipse.
Now we are discussing the number of visible spokes. This part of the picture seems wrong. New subject.
Look at the rotating-in-place wheel (in green). Contrast with the rolling wheel (in blue). Do you see a contradiction? The spokes are curved downwards for the rotating one and upwards for the rolling one. How can that be? At times you can see only 3 spokes above the midline for the green wheel while you are always seeing 6 for the blue wheel. Do you see the anomaly? When the wheels overlap their spokes point in OPPOSITE directions (one upward, the other one downward). So, you don't think this is absurd? What makes you so sure that the website authors are correct?
We must first agree on the equations of the spokes, I don't think yours is correct. This and this and this is how the spokes should look like. Try plotting yours and you'll find out that they are not correct. I posted my derivation, I suggest that you redo yours.

 

[przyk]

This is the first time that either of you admits it. I explained it to przyk since post 60 (bottom of the page), he never admitted it.

No, I made it clear in [POST=2849161]post #123[/POST] that I did acknowledge it and have been well aware of it since my first posts in this thread. In fact the relativity of simultaneity effect I described in [POST=2849803]post #59[/POST] is exactly what makes the separation "frame-dependent" in the sense it is. You will of course notice that post #59 precedes post #60.

Ironically, I also accused you of failing to read and respond to what people actually say (as opposed to what you imagine they said or would have found it convenient for them to say) in both posts.

 

[Tach]

No, I made it clear in [POST=2848989]post #123[/POST] that I did acknowledge it and have been well aware of it since my first posts in this thread. In fact the relativity of simultaneity effect I described in [POST=2848643]post #59[/POST] is exactly what makes the separation "frame-dependent" in the sense it is. You will of course notice that post #59 precedes post #60.

You will also notice that I disagree with your explanation of how "matter moves between the A and B halves":

I don't think this is a valid explanation, especially the bit about length contraction generating higher density in the upper half.
A much simpler proof is that the midline in the ground frame is no longer horizontal but inclined (due to relativistic aberration), so it no longer coincides with the axis of symmetry of the ellipse.

Ironically, I also accused you of failing to read and respond to what people actually say (as opposed to what you imagine they said or would have found it convenient for them to say) in the very same post.

Actually I responded to your exact claim and I showed my exact cause of disagreement with it.

 

[przyk]

Actually I responded to your exact claim and I showed my exact cause of disagreement with it.

No, you just said you disagreed with it and failed to provide any elaboration, even when [POST=2848722]specifically asked[/POST] for it:

I don't think this is a valid explanation, especially the bit about length contraction generating higher density in the upper half.

So specifically what's wrong with it? Also the bit about length contraction at the end wasn't the main explanation.

You never replied to that. You just left it at an unjustified denial.

Of course, this has no bearing on the point of my last post, and as usual you are desperately grasping at straws. In [POST=2849629]post #138[/POST] you say I never "admitted" that the distinction between the two halves was frame-dependent when I clearly was in [POST=2849161]Post #123[/POST], and [POST=2848643]post #59[/POST] is evidence I was well aware of this right from the beginning, regardless of what you think of the reasoning I employed.

 

[przyk]

You will also notice that I disagree with your explanation of how "matter moves between the A and B halves"

Addendum: A and B were points on the wheel, not halves. There's evidence you didn't even understand the argument you were dismissing.

 

[RJBeery]

This thread has turned into a exercise in debating skills rather than a search for enlightenment. The only thing I've learned from it I taught myself, which was that the KE change due to translational movement of a rotating body is equivalent to changing its axis of rotation.

 

[przyk]

So, according to such claims, an asymmetry in spacetime curvature would result.

Derivation or reference please. On what basis do you think this?

 

[Tach]

Derivation or reference please. On what basis do you think this?

On the basis that you claimed in post 52 "transfer of matter". Look, until you will put up any math to back up your claims, I will ignore your posts.

 

[przyk]

This thread has turned into a exercise in debating skills rather than a search for enlightenment. The only thing I've learned from it I taught myself, which was that the KE change due to translational movement of a rotating body is equivalent to changing its axis of rotation.

I'm not sure I see where the problem here. Earlier you said:

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?

Maybe the point of the reply I gave to that wasn't clear. Your reasoning above has the built in assumption that gravitation increases with momentum. I don't know any good reason to believe this should be true in general, and in fact I feel pretty safe deducing just the opposite from this scenario.

 

[przyk]

On the basis that you claimed in post 52 "transfer of matter". Look, until you will put up any math to back up your claims, I will ignore your posts.

What the hell is wrong with you? The point under contention is the assumption you are making, that a matter distribution asymmetry in the wheel would necessarily lead to a "paradoxical" asymmetry in the gravitational field. For this, you need to be able to provide support. And you need to be able to properly qualify that "paradoxical" I put in quotes too: it is obvious that there will be an asymmetry in the gravitational field anyway because of the (horizonal symmetry breaking) frame-dragging in the axle frame. You need to be able to show that the "gravitational effects" don't just consist in an asymmetry but one that is incompatible with the frame-dragging in the rest frame. So once again, derivation or reference please.

 

[RJBeery]

Maybe the point of the reply I gave to that wasn't clear. Your reasoning above has the built in assumption that gravitation increases with momentum. I don't know any good reason to believe this should be true in general, and in fact I feel pretty safe deducing just the opposite from this scenario.

Yes, I am making the presumption that "energy" is non-discriminatory. Your response, and Tach's as well, is that this must not be the case apparently because of the paradox...?
This link supports what I'm saying...it doesn't matter "how" energy sufficient to form a black hole is put into an object...momentum suffices, apparently, which baffles me.

 

[przyk]

Yes, I am making the presumption that "energy" is non-discriminatory. Your response, and Tach's as well, is that this must not be the case apparently because of the paradox...?

I'm arguing that, and I know that the gravitational field also depends on momentum independently anyway (the dependence is included explicitly in the Einstein field equation). Tach isn't: he's making the same presumption as you and concluding that the matter and energy distribution in the wheel can't be asymmetric in the first place.

But we're both making deductions taking that there is no paradox as a premise, if that's your point. And it's clear a priori that there isn't a paradox. It's not just a matter of "lots of clever people have studied GR in the past and nobody has noticed a paradox like this before, so there probably isn't one". The way GR is formulated rigorously ensures that no paradox like this is possible. Specifically, the way the Einstein field equation

$$
R_{\mu\nu} \,-\, \frac{1}{2}R g_{\mu\nu} \,=\, 8 \pi G T_{\mu\nu}
$$​

is formulated (in terms of relativistic tensors) ensures that the left and right hand sides transform in exactly the same way under any arbitrary coordinate transformation.

So the situation right from the beginning is that we have a hand-wavy argument suggesting there's a paradox, and a mathematically rigorous way of seeing that there can't be.

This link supports what I'm saying...it doesn't matter "how" energy sufficient to form a black hole is put into an object...momentum suffices, apparently, which baffles me.

My own reasoning tells me the conclusion in your link is wrong. The general consensus seems to be on my side, if the Usenet physics FAQ is any indication.

 

[Tach]

What the hell is wrong with you? The point under contention is the assumption you are making, that a matter distribution asymmetry in the wheel would necessarily lead to a "paradoxical" asymmetry in the gravitational field.

Err, a spherically symmetrical body creates a spherically symmetrical gravitational field. A non-spherically symmetrical body (with "more matter accumulated on one side" generates a non-symmetrical gravitational field. Can you extrapolate the above to 2 dimensions? No? Too bad.

The main point is that there is NO "matter distribution asymmetry". I asked you multiple time for a mathematical proof of your claim of "matter distribution asymmetry", until you provide one , we have nothing to discuss.
You have claimed* with no proof to back it up that there is "more matter on the upper side of the ellipse", until you provide mathematical proof for your claim, I have nothing to discuss with you.


I do have an interesting subject to discuss with Pete, who, unlike you, doesn't shy away from backing up his claims with math.


*

This isn't the case here: the Lorentz transformation does not map the upper half of the wheel in one frame uniquely to the upper half of the wheel in another. It doesn't even preserve the amount of matter in each half: 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.

I can disprove the redlined sentence , before I do it, I am asking you to provide your proof. This is the claim that you made in post 52 that it is at the origin of our disagreement. Provide your proof (since you made the claim) and I will provide my counter-proof.

 

[przyk]

The point is that there is NO "matter distribution asymmetry".

You are dodging the issue. You are concluding this because you think it would imply supposedly problematic "gravitational effects". Can you justify that or not? If so, do it. If not, retract the argument.

Whether SR really does imply that most of the mass will be in the top half of a rolling wheel is an entirely seperate issue.

 

[Tach]

You are dodging the issue.

No, you have been dodging the issue since post 52, where you made the unsubstantiated claim that got the argument started. The post above tells you what you need to do in order to get the ball rolling.

 

[przyk]

Err, a spherically symmetrical body creates a spherically symmetrical gravitational field. A non-spherically symmetrical body (with "more matter accumulated on one side" generates a non-symmetrical gravitational field. Can you extrapolate the above to 2 dimensions? No? Too bad.

*Sigh*. You're so predictable:

The point under contention is the assumption you are making, that a matter distribution asymmetry in the wheel would necessarily lead to a "paradoxical" asymmetry in the gravitational field. For this, you need to be able to provide support. And you need to be able to properly qualify that "paradoxical" I put in quotes too: it is obvious that there will be an asymmetry in the gravitational field anyway because of the (horizonal symmetry breaking) frame-dragging in the axle frame. You need to be able to show that the "gravitational effects" don't just consist in an asymmetry but one that is incompatible with the frame-dragging in the rest frame. So once again, derivation or reference please.

In short, I've already explained why it isn't enough just to point out that there's an asymmetry.

I can disprove the redlined sentence

So do it.

before I do it

Why delay? Supposedly you've already done the calculation. What are you waiting for?

Why do we keep having this problem with you?

 

[Tach]

What are you waiting for?

For you to prove your false claim from post 52 that started the argument. You started it, you go first. AFTER you do it, I'll rebut your proof. If you cannot provide a mathematical proof, just admit it. Very simple.