Imagine a large disk floating in space. It’s, say, 100 km across. We attach rocket engines to it halfway out from the center and begin spinning it.
What happens as the edge approaches lightspeed?
Assume that the disk is sturdy enough to withstand the tremendous stresses involved.
Actually, this is (sort of) a very classic thought-experiment. Basically, someone on the outside of the disk measures a radius R and a circumference C exactly as for the non-rotating disk. Where it gets interesting is for an observer on the disk (co-rotating).
So, we know he’s going to feel a force outwards (try taking a corner hard in your car). Also, when he measures the circumference it’s in the direction he’s moving, so his measurement gives c < C (exactly how far less depends on how fast the point on the edge is moving. However, the radial direction is perpendicular to his direction of motion, so it doesn’t change at all. The nonrotating observer calculates C/R = 2pi, but the corotating observer measures c/R < 2pi.
So, how can the ratio of circumference to radius be other than 2pi? If the space is curved! The corotating observer sees a curvature, and it’s directly connected to feeling the force. If instead of the force being due to inertia from rotation it were from, say, gravity, he’d measure the exact same curvature: the deviation of circle measurements above.
I see. So the observer on the edge of the disk basically experiences an increasing acceleration toward the center of the disk as its speed increases. And being in a constantly accelerating reference frame is just like being in a gravity well.
A couple of follow-on questions:
The rotating observer measures r & c when the disk is stationary, then again when the disk is spinning. Does r get bigger while c remains constant? Or the other way around? Or is the question meaningless because the rotating observer’s measuring stick is changing as well?
What about mass and time effects?
What will happen? The sun will go nova for any the planet where govenment seriiously tries to have this idea implemented. “The universe itself resists such things.”
That’s according to a story by Larry Niven, written 26 years ago: “Rotating Cylinders and the possibility of global casuality violation”, available in the collection Convergent Series.
Slight correction: the acceleration is towards the edge of the disk.
Really it’s easiest to just compare two observers of the same situation. The corotating and stationary observers measure the same value for the radius, but different values for the circumference.
This also gets complicated. Really, the well-known effects of special relativity apply to “inertial” or “unaccelerated” frames, while the two observers here are accelerated relative to each other. To really have an idea what’s going on would require working the whole thing out in general relativity.
The story itself based on mathematician Frank J. Tipler’s “thought experiment” published in Physical Review under a title of the same name. (Niven gives no explaination for why Tipler, the editors of Physical Review, Tulane University, and indeed, the planet Earth weren’t destroyed in a causality reinforcement event as related in the story.)
Aside from the fact that Tipler is as mad as a march hare, Tipler’s concept fails to take into account that the centrifugal forces developed in a cylinder of any real material moving as such rotational velocities would rend it into component atoms, even if it didn’t collapse upon itself. (One also has to flare the ends to avoid massive differential effects from the endpoints, further compounding the problem.) One could utilize compact supermassive objects such as a large spinning singularity or (assuming they really exist) a rapidly rotating cosmic string for the same purpose; the differential tides, however, would shear any normal object to slivers.
Note that the increase in mass also results in a secondary phenomenon called frame-dragging; to compensate for the inertial variance spacetime is warped about the spinning object, resulting in a path inscribed about it that (for an observer traveling about it in the rotating direction) is somewhat less than 2*pi. This is an example of a “spacelike” curve through time; in essence, you get from point A to point B in less time than an object moving at the same apparent speed (according to a relatively objective observer) would calculate you moving. Since the curve is a closed spacelike curve–eventually you “return” to your startpoint–this doesn’t necessarily result in causality violation (which may or may not be a concern anyway) and there doesn’t appear to be any way this could be used to travel in the reverse arrow direction regardless; the best you could to is race ahead of your opponent in hopes of getting the upper hand, only to find out that they’ve developed warfare technologies that make your meson torpeodos obsolete.
Time travel is futile, all the more so when it is introduced as an ill-conceived attempt at conflict resolution.
Stranger
Acceleration in a rotating disk due to the rotational velocity is towards the center: the velocity vector continually changes, and the change is in a direction toward the center of rotation. An observer “standing” on the outer circumference of the disk might feel like a force is pushing him outwards (I think this is what you’re alluding to), but that’s a reaction to the disk accelerating him inwards.
I’m having trouble grasping this bit. I thought two people in different reference frames would each perceive their frame as being the “normal” one while pointing to the other guy and saying they are the ones moving.
But with the above you suggest a method whereby one can tell they are in an accelerated reference frame because their math stops working. I mean, if BOTH observers measure the same value for the radius on paper they should both get the same result for the circumference. Yet when one guy goes out and does the measurments he finds they do not agree with fundamental geometry.
I thought Relativity allowed the math to work to regardless of how a person is moving…hence things like time slowing down and contraction of space and so on. How is this different?
Thinking on my post above some more I am guessing the “difference” is the person on the rotating disk is in a constant state of acceleration which allows one observer (the one being accelerated) to know something is different about his state of affairs versus a “stationary” observer.
That’s the way it was always described to me in my physics classes. Mathochist is there possibly something zut and I are missing here?
Well, maybe we’re all getting hung up on what acceleration we mean. The observer is being accelerated towards the center, but feels an acceleration (an “inertial force”) towards the edge. Basically, it’s like he feels gravity pulling outward, not inwards.
Yeah, that’s pretty much it. Special relativity doesn’t really cover this case.
Einstein himself wanted to talk about the “principle of covariance” rather than “relativity”. Yes, what two observers moving (at constant velocities) relative to each other measure different values for lengths and times, but that’s really not what’s important.
What’s important is this: as the velocities change (vary) the measurements change predictably (vary with, or “covary”). In fact, things like the “spacetime interval” (the analogue of 3-dimensional distance) don’t change at all – they’re invariant. Special relativity talks about what stays the same and how things change when observers are related to each other by what are called “Lorentz transformations”, which contain rotations as well as “boosts”, which relate two observers moving at constant velocities relative to each other. To handle any change of coordinates (not just Lorentz transformations), you need a principle of general covariance – general relativity. What this thought-experiment indicates is that the inertial forces (and so any force) show up as curvatures in spacetime.
I think what you mean is, if the rotating observer drops a rock, he’ll observe it accelerating outwards. If the observer is fixed to the disk, however, he will not regard himself as accelerating at all, in his reference frame: He will describe the gravity-like centrifugal force and whatever force is holding him still (say, normal force from the floor) cancelling out to 0, resulting in no acceleration.
And contrary to popular belief, it is actually possible to treat such problems in special relativity, without all the machinery of general relativity. The trick is that you need a continuum of reference frames, and you’re continually shifting from one to another. It’s nothing that a bit of integral calculus can’t handle, though I confess that I avoid calculus whenever practical.
Better stated than I managed to.
This is true, but I find it too complicated to bring calculus in. It seems (to me) to obscure the underlying conceptual picture that moving to GR provides. Analogously, one could describe a planet’s orbit by a series of circular epicycles, but it’s far from evident from that picture that when you sum the whole series you get the ellipse that a more modern view makes clear from the start.