Then we're all agreed that there is nothing unusual or inexplicable about the situations you describe.

Neat. Now you're into general relativity.

Grant Hutchison

The core of your paradox appears to be Lorentz contraction. This is the same thing that the train-and-tunnel and the pole-vaulter paradoxes rests on. The fact that your paradox takes place on a circular track while the other two take place in a linear setting is irrelevant, so I still don't think there's anything that really differentiates your paradox from those two.

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The resolution to the train/tunnel and pole/barn "paradoxes" is just a disagreement about simultaneity between the front and back of the train/pole. It's simply resolved with SR.
By wrapping the thing around a track, so the front and back of the train are next to each other, vergentbill has closed the door on that one. If front and back are at the same place at the same time, we can't unlimber simultaneity to help resolve the paradox.
There's no problem for an outside observer, who sees a one-lightyear train on a one-lightyear track.
But we're going to need GR, I think, to find out how long the train-driver thinks his train is, and how long he thinks the track is: the resolution will be in there, somewhere. It's a nice variant of Ehrenfest's paradox.

Grant Hutchison

Oh I see; thank you for the explanation.

Pun intentional?

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I'm tempted to claim it, since it's so neat. But it was just dumb luck.

Grant Hutchison

There is one big thing about the Rocket and Rope "paradox" that needs to be said. There no mysterious additional force of Lorentz contraction at work here. It's just geometry. If you accelerate something, you must apply a force. In Newton, if we have rigid rod and accelerate it and require that it remain rigid "in its own frame", then all points on that rod accelerate at the same rate. An observer riding at any point feels the same force.

However, with Minkowski space-time, if we wish that rod to remain rigid in its own accelerating frame, the points at the rear must feel a greater force, and must accelerate faster according to an inertial observer that points toward the front.

In Rindler, the proper acceleration felt by observers accelerating along the x axis is given simply by c^2/d, where 'd' is the distance to the horizon (just like with a black hole, the force to remain stationary goes to infinity at the horizon). In geometric units, it is simply 1/d, illustrating that an acceleration there is just 1/distance. Proper acceleration is "simply" the rate of change of the tangent vector, a dimensionless quantity, per unit "arc length" (c*proper time) along the world line. The pseudo-tide, the gradient in the proper acceleration felt, is then -c^2/d^2, which translates to
-a^2/c^2 for a given 'a'. If you're accelerating at 1g, there is a tide toward the rear of g^2/c^2. Not much at all.

Thinking in these geometric ways will give you a leg up to appreciate full GR. Proper acceleration is a type of "path curvature".

This is really very simple, although it seems odd at first. If we wish something to remain rigid while we accelerate it, we have to accelerate different parts at different rates. That wouldn't happen in Newton, but it does in SR. The different points of the object have to follow slightly different world lines in order for them to remain stationary according to their own rulers.

-Richard

The train is not 1 ly long "in the frame of the train", because there is no frame of the train. There is only a frame of someone on the train, and note that such a person would see other parts of the train as being in motion, and being length contracted. Also note that motion in a circle is actually a lot harder than motion along a line, because of the inclusion of rotation and the need to use a different metric as a result. I will agree with two points you've made though:
1) few people recognize that length contraction in some cases requires internal stresses, and in others does not. Indeed, it was rumored that the "theory group at CERN" generally did not understand this, see posts by John Mendenhall and that rope and spaceship puzzle Grant referred to. (I'll bet the "critique" of past work mentioned in the abstract Grant cited gets into this, there's a lot of confusion on this matter.) Basically, as I see it, you need stresses to get contraction if the object itself is accelerated, but not if the observer is accelerated instead of the object. I suspect many physicists would not think that statement is correct, but I believe it is, as does Grant. (I think his objection was in your claim that physics itself doesn't understand this, rather than if you had said "many physicists" don't, as I do not view the latter statement as controversial based on my experience.)
2) rotation is a whole lot harder than linear motion in special relativity, and the issue of whether or not a perfectly rigid object (i.e., an object that can exert arbitrary stresses in the frame of an infinitesmal section of its length to avoid changing that length) is capable of rotation is sometimes debated. publius posts the deepest stuff on this, I have a hard time understanding all the subtleties but I think a flat rigid object cannot be rotated without warping its shape into the third dimension. As for a train, I suspect the motion of other parts of the train might compensate for the stationary reference frame of any observer on the train, such that the paradox is resolved. Or another way to say that, the train would require internal stresses as it accelerated to its final speed, and those stresses should adequately account for its length as analyzed by any observer.

A thought experiment is supposed to help a person develop constructive reasoning and such- But you seem to have applied more science into an analogy than actually exists. As if you are trying to outthink Einstein.

NOw outthinking Einstein is just fine- and we all might aspire to try to prove the Old Man wrong.

However, in doing so- you must also understand all the science involved and apply each in proper respective manner.

Going in circles about a thought experiment isn't going to prove squat. Whip out some hardcore math and let's REALLY figure this thing out.

I may be "deep" only in the sense of "piled high and deep". Of all the "paradoxes" of relativity, Ehrenfest's takes the cake. The Twin "Paradox" might be the 9th grade level, the Rope and Rocket might be junior varsity, etc, etc, but Ehrenfest's is at the Pro level.

The long and short of it is that the circumference of a rotating circle is "longer" by a factor of 1/gamma -- the geometry for those stationary (rotating and revolving) observers is hyperbolic. But, it's a doosey to get there.

See the Wiki article on Born coordinates. That is written by one Chris Hillman, a regular GR contributor at Physics Forum, who is pretty good. I don't know if we can confer "high priest" status upon him, but if he hasn't been officially annointed, its only a matter of time. His mathematical level is very advanced and he throws it out like it's grammar school stuff, but one can get a general picture if you're familiar with the basic concepts....sometimes. That's the problem with a lot of this stuff. Those who understand it have a lot of trouble explaining it, since the math is just so advanced. If you want to understand this stuff, you're just going to have to roll up your sleeves.

But here is the problem as simply and basically as I can put it according to my understanding. "Langevin observers", the name for the rotating and revolving observers "stationary" relative to the disc have "vorticity" along their world lines. That's related to rotation, but involves a little more than that.

There is a theorem about vorticity which states that "spatial hyperslices" for such observers cannot be consistenly defined globally. That means that "space" at constant time just can't be defined. And this is the root of all the problems trying to understand Ehrenfest. Space just don't work. I get the sense of frustration reading Hillman and others that so many experts who ought to know better still get tripped up on this.

Now, while space cannot be consistently defined globally, the problematic behavior goes to zero *locally*. So, in a small neigborhood on the rotating disc, a group of observers could do geometric experiments and they would conclude their local space is hyperbolic, and so a small angular chunk along a circumference would have C > Theta* R by the inertial observer's SR gamma factor.

That local spatial metric is called the "Langevin-Landau-Lifshi-tz" metric (hyphen to avoid the profanity filter -- that Landau and Lifshi-tz get around, their names are on a lot of important stuff).

The rub is the "global spatial geometry" of a rotating disc is not even Riemannian, much less Euclidean. Apparently, you can work with non-Riemannian geometries, but you loose a lot of "common sense". For example, if you and I were "far apart" on the disc, and agreed on some method of measuring distance, it wouldn't be symmetric, it wouldn't "commute" so to speak.

That is the distance from A to B can different from the distance from B to A, no matter how we try to define it. That is just nuts, and that's what
non-Riemannian gets into.

Piled high and deep, indeed. Seriously, this is a humdinger and it's that vorticity and no consistent global space problem at work.

-Richard

This is really throwing me for a loop. Could someone confirm/explain this a bit more please?

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You should start by checking out the recent thread on ropes and spaceships that Grant mentioned, but I can give you a brief summary. First of all, it's clear that if you accelerate, you will perceive Lorentz contraction of other objects that would not be accompanied by any stresses in the reference frame of the object-- you can't affect the object physically by accelerating yourself. So I presume your problem arises if it is the object that is accelerated. Now the issue is, accelerated as measured by whom? We can imagine a constant acceleration in the frame of the object itself, or we can say the acceleration all along the object is the same for the stationary outside observer. If the former, there's no need for internal stresses, but there's also no paradox-- the outside observer will see the different parts of the object as accelerating for a different amount of time and with differing magnitudes, because the acceleration will have everywhere started at the same "now" for the outside observer, but will have concluded at different "nows", yet must achieve the same final speed (involved here is the crucial simultaneity effect in SR, without which nothing can be understood). If the latter, i.e. if the acceleration is the same for the outside observer and begins and ends at a consistent time for that observer, then the train cannot maintain the same length in its own frame, it will have been stretched, so there must be internal stresses to maintain its length (for the train) or to achieve the contraction (for the outside observer). That is again because the internal observers will accumulate a discrepancy in the meaning of "now" as you move along the train, and so additional stresses are required to maintain a constant length for the train in that frame (i.e., the train is assumed to be rigid, and rigidity requires internal stresses in general).

Ah, ok, I think I have it now (I've said that a lot in this thread, haven't I? ). My problem was that I got stuck at a shallow interpretation of your statement, so I thought you were simply saying "observer accelerating = no stress, object accelerating = stress". Let's double-check to see if I really do have it:

Observer acclerating = no stress
Object accelerating = no stress if acceleration is uniform across all parts of object (as measured in reference frame of object), stress if acceleration is not uniform across all parts of object (as measured in reference frame of object)

Is that about right?

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Yes, that's it. You just have to add the question, which of the above applies if the object is to maintain a fixed length in its own frame (by which I mean, if you populate the objects with little observers, and each has a short ruler which they lay end to end, then their rulers will always span the object). This is what we mean by "rigid", when there's no real gravity around (I have no idea what rigid means in the context of real gravity, and even rotation makes it a fuzzy concept-- see publius' remarks above.) The point is, to be rigid, the object cannot be seen as uniformly accelerating from the point of view of a stationary external observer. And if it's rigid, and you ask the internal observers, they will all report the same accelerations, but they will not agree on how the other observers are accelerating, because they think the other observers on the same rigid object have time running slower everywhere behind them, and faster everywhere ahead of them. They think the other rulers on the same rigid object are also messed with-- how that interacts with the need for stresses, I'm going to take a walk and think about. Maybe the problem is that there is no "frame of the rigid object" when the object accelerates, that is simply not a valid local frame.

This is the upshot of the rocket and rope puzzle. If two identical rockets, in a line, start out a distance D apart, and accelerate identically from the point of view of an external stationary observer, then they will always be separated by D in the external inertial frame-- just by the nature of the fundamental symmetry of translation. The questions that may be asked are:
1) what is the separation as measured/calculated by either rocket
2) what is the number of rulers that may be laid end to end between them, say by observers on more identical rockets that span the interval between them
3) what acceleration does each rocket think the other rocket is undergoing?
The easiest way to do any calculation of this nature is to do it in the inertial frame and then transform only at the end to any other frame. Then we can work in a global reference frame, with no ficticious forces. We know the answer in that frame: d(t) = a*t is the location of the rear rocket, and d(t) = a*t + D is the location of the forward rocket. But in the frame of the rear rocket, the acceleration of the forward rocket is larger than a, because time is running faster for them (via the equivalence principle). So the distance between rockets increases with time, in that frame. This must be made consistent with D being constant in the inertial frame, but note the inertial frame clock is running slower and slower from the point of view of the rockets, and also note that the concept of "now" for the locations of the two rockets is getting skewed in the inertial frame relative to the frame of either rocket. The best way to do this mathematically is with a metric (as publius will tell you), but one can still imagine it physically by saying that "now" for the rear rocket corresponds to a time for the forward rocket that is sooner than what would be simultaneous in the inertial frame, so this dramatically shortens the distance between rockets from the point of view of the rear rocket-- it explains why the inertial observer thinks D is constant even though the inertial observer's ruler is short, and the distance is actually more than D, from the perspective of the rear rocket.

But to understand the internal stresses in a train or rope between the rockets, we need to look at all those little rulers laid end to end between the rockets, and moving with the rockets. There the need for stress comes from the fact that the rear rocket thinks the forward rocket has an acceleration greater than its own, as measured in the rear rocket's concept of length and time, and it also thinks that all the rulers ahead of it are shortened by the acceleration (to keep the speed of light constant as time speeds up). If you put those together, you conclude that the same number of rulers could not be laid end to end between the rockets-- not without some internal stress that is pulling back on the forward rocket (i.e., "the rope breaks").

What do things look like in the front rocket's frame of reference? Would an observer there see that the rear rocket has a smaller acceleration, therefore the distance between the rockets increasing and thus the rope also breaking in this frame of reference? Or would this observer see something else?

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Yes, that's right, the way I have it scored.

Yes. The rope breaking or not breaking is an "invariant", something that all frames of reference must agree on. Different frames may explain how and why it happens in different ways, but they agree on the invariant.

From the perspective of either accelerating rocket, the other one has a relative acceleration, and the rope must break. From the inertial observer's perspective, the rope is Lorentz contracting, but the distance between the rockets remain the same, so the rope has to break.

The perspective of the accelerating rockets, globally, is a non-inertial frame of reference, and the familiar rules of SR don't apply. Well, there's some fuss that can be made about "SR vs GR". Space-time itself is flat, but the accelerating observer coordinates are "curved". It's the same thing conceptually as using polar coordinates vs Cartesian in the plane. Same space, just curved coordinates.

"The familiar rules of SR" is equivalent to using cartesian, "straight" coordinates. Non-inertial coordinates just involve extra stuff that are not there in Minkowski coordinates.

From this accelerating frame, called a Rindler frame sometimes, you have a non-Minkowski metric, which, via the Equivalence Principle living and breathing, you are free to say is a massive gravitational field. To remain stationary, you have to feel a force.

Now, it turns out there is gradient of that force to remain stationary. That is, things "higher" than you "weigh less", don't have to accelerate as hard, while things behind must "weigh" more, and have to accelerate harder to remain stationary.

That gradient, that psuedo tidal force, is simply -g^2/c^2, where g is the local acceleration you feel.

Now, it turns out that acceleration, as I'm using it here, but haven't explicity made clear (sigh ), is proper acceleration, which means, well, "force you feel where you are". This is an invariant, and all observers can calculate it and come up with same value.

So, if we're watching something rigid accelerate, we know the tail is accelerating faster and feeling a slightly greater force than the nose.

-Richard