As if there was a giant angle-grinder in space that slowly increased its speed. The question assumes that material exists to make this super-disk that doesn't rip apart. What happens as the edges approach light-speed and if it can't, where would the resistance be felt? Could the angle-grinder not produce the torque? Trying to envisage a giant, planet slicing machine. Perhaps the disc could be a lot wider and rotate more slowly.
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6$\begingroup$ Downvotes without comment are unhelpful and unproductive. $\endgroup$– Starfish PrimeJun 3, 2019 at 15:38
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6$\begingroup$ On physics.stackexchange: Rotate a long bar in space and get close to (or even beyond) the speed of light 𝑐 $\endgroup$– AlexanderJun 3, 2019 at 16:02
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3$\begingroup$ I'm voting to close this question as off-topic because this is a (very) simple math problem. $\endgroup$– RonJohnJun 4, 2019 at 1:21
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3$\begingroup$ @RonJohn I disagree that it's a simple math problem-- rotating frames in special relativity are rather involved. But more so than that, I feel like I've seen many questions in the same vein that have not been closed, so I find it weird that this one is. I've actually answered a question about special relativity even simpler than this one asked by one of the VTC-ers themselves! I get that some of these questions might be better suited to other networks but it often feels like the policing of them is wildly inconsistent and has more to do with user rep than question content. $\endgroup$– el duderinoJun 4, 2019 at 12:59
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3$\begingroup$ @RonJohn Yeah, but the point is that even though a simple multiplication would yield a velocity greater than c, such a thing is forbidden by special relativity, so there must be a glitch so that the usual simple multiplication isn't correct. This is indeed the case, and takes you down the decidedly non-trivial rabbit hole of Born rigidity. I think perhaps the title could be changed so that it better reflects the body but I don't think the question is necessarily any worse than many others that have been asked and not closed. $\endgroup$– el duderinoJun 4, 2019 at 13:21
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5 Answers
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This question has actually been well studied throughout the years and is closely related to the Ehrenfest paradox. The answer is no-- it's impossible to spin a giant disk in a way such that its outer rim moves faster than the speed of light. First and foremost, it's a rather basic derivation from the postulates of special relativity to show that no object can ever be accelerated to the speed of light. So, any way you might think of to get around this (within the confines of SR at least) must either mean that special relativity is wrong or something is flawed in your setup. As a general rule of thumb: it's probably the latter.
Here, the problem is that you are assuming that your saw blade is perfectly rigid, but this is not allowed in special relativity. If we had some perfectly rigid object, like say a giant pole, we could use it to send messages faster than light by poking stuff, which is another no-no in special relativity. Now, a rotating object is slightly trickier-- there is a sense in which a rotating object can be rigid in special relativity, although it differs from the classical picture (see this link: Born Rigidity). However, Born rigidity comes with a whole bunch of constraints, one being that the angular velocity stay constant. Clearly this isn't true when you're speeding up the saw, so the blade can't be Born rigid meaning it will be forced to break apart. It's tempting to think that maybe we could get around this with a stronger material, but the point here is that it doesn't matter what the material is. Simply by virtue of living in a universe that obeys special relativity, it can be mathematically proven that what you want to do is impossible.
As one final reason why this is impossible-- consider what your disc actually looks like microscopically. It's a bunch of tiny atoms pulling on each other via the electromagnetic force. But the electromagnetic force, like all forces, only propagates at the speed of light. So, even if you did manage to somehow violate special relativity and rotate your saw fast enough that the circumference went faster than the speed of light, the atoms in it would have no way of holding on to each other and the disc would immediately disintegrate. It doesn't matter how strong of a material you have-- the fundamental nature of our universe prohibits you from getting the edge to move faster than light.
answered Jun 4, 2019 at 2:21
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I won't dare sticking my finger into the relativistic theory of rotating bodies, I will just go with the approximation of linear motion, which is valid for infinitesimal rotations.
We know that, by relativity, the mass of an object moving at velocity v is increased according to Lorentz factor $\gamma=$$1 \over \sqrt{1-v^2/c^2}$.
Therefore, the more the tangential velocity of the disc increases, the more its mass increases, the more becomes difficult to increase its velocity. Moreover, being a rotating body, we also need to take into account the increase in centripetal force, which would sooner or later overcome the resistance of the material.
My guess is that the disc will break way before any relativistic effect can be sensed (ask any turbine manufacturer).
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9$\begingroup$ I've a sneaking suspicion that any material with infinite tensile strength is itself a violation of relativity (in the same way that a magical incompressible rod is). I'm not sure how the untearable disc causes physics to break... maybe you get Weirdness when you try to feed it into a black hole, so the OP is probably doubly out of luck. $\endgroup$ Jun 3, 2019 at 15:54
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10$\begingroup$ @StarfishPrime yeah it would take an infinite amount of bonding energy to have infinite tensile strength. Energy=mass so the object would have infinite mass. So the universe would collapse in on it long before we are at infinite $\endgroup$– AndreyJun 3, 2019 at 16:44
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$\begingroup$ @Andrey ta, that's a nice, straightforward explanation. I'll remember that for future use! $\endgroup$ Jun 3, 2019 at 17:02
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1$\begingroup$ @Andrey Also even if you had some means to magically cancel the gravitational force to prevent collapse this disk would be essentially impossible to accelerate, application of any finite force to an infinite mass would result in an infinitesimal acceleration. Sure technically it still accelerates no such thing as a truly immovable object but you would still have an object that required effectively infinite time to accelerate at which point the distinction becomes rather academic for any practical purposes. $\endgroup$– MttJocyJun 4, 2019 at 10:56
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I was once where you are.
I wondered if it was possible to create a rotating mirrored propeller capable of spinning fast enough to separate particle pairs in the quantum foam in a manner reminiscent of a Quantum Vacuum plasma thruster.
The answer is no. The reason is material stress.
Basically the maximum stress on a rotating solid disc (or arm) scales as the square of both disc radius and orbital velocity. This means that long before the edge of your disc is moving anywhere near C the disc itself is being put under stresses usually reserved for... erm... supernovae? Possibly freshly expanding galaxies.
Either way: it was a while back when I ran the numbers, but suffice it to say that your disc will have broken apart looong before you get up to these kinds of speeds.
answered Jun 3, 2019 at 17:25
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Millisecond pulsars! Gravity waves! Imperial to metric conversions!
60 mile diameter disc. 188 mile circumference. 1000 rotations per second = 188 * 1000 miles / second = 188,000 miles/second
Speed of light = 299 792 458 m / s = 186282 miles / second.
The edge would be going faster than the speed of light; not allowed.
Here is a fine and relevant answer lifted from the astronomy stack, which asks about neutron stars. The one is question rotates at 25% of the speed of light. Go visit and upvote!
The answer cites this wikipedia article https://en.wikipedia.org/wiki/Millisecond_pulsar
with this text
Current theories of neutron star structure and evolution predict that pulsars would break apart if they spun at a rate of c. 1500 rotations per second or more, and that at a rate of above about 1000 rotations per second they would lose energy by gravitational radiation faster than the accretion process would speed them up.
The cool thing for me here "lose energy by gravitational radiation". Huh?? I knew about gravity waves but did not realize that the production of gravity waves would allow a thing to slough energy! https://en.wikipedia.org/wiki/Gravitational_wave
There is the solution for how dark matter sheds accumulated energy on descending into a gravity well - as gravity waves!
OK; the saw blade. It will need to go slower. But you can make it out of a handy millisecond pulsar and still have it go plenty fast.
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$\begingroup$ lol! Great answer, although for the record your millisecond pulsar is exceptionally dangerous to the things around it without trying to use it as a giant buzz saw. $\endgroup$– conmanJun 4, 2019 at 2:05
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1$\begingroup$ Maybe pulsars are giant cutting disks (or the remains thereof) created by some distant civilizations. :) $\endgroup$– vszJun 4, 2019 at 4:13
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$\begingroup$ @vsz But what did they need to cut? That's the real question. $\endgroup$– kikirexJun 4, 2019 at 13:22
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The only way the edge would spin with an angular velocity faster than the speed of light, is if the momentum from the centre (where I assume you're hypothetically applying the force) is transmitted instantaneously to the very edge. However, since there's no such thing as an instantaneous process in reality, there hypothetically needs to be some transmission (usually through the material of the disc).
Imagine, when you were a child, you might have played by spinning a bucket on a rope. You might know that the bucket doesn't immediately align with you spinning at the centre, but lags behind at the start. Then, as the rope is being pulled taut (by to centrifugal force on the bucket), the bucket aligns with your centre.
The same happens to any other material. In the case of inelastic items, the energy we previously required to make the elastic string taut is converted directly into stress on the material's structure itself. For real life turbines, propellers, etc..., the material's strength should hopefully be enough to resist that stress, and keep the edge aligned with the centre from the start.
Now imagine, that instead of a bucket on a rope, you're trying to rotate a (decently sized) lump of playdoh on a dry noodle. You'll know that if you attempt to rotate it quickly from the start, the noodle will break from having to pull the weight attached to its end.
Now, let's examine two scenarios for our hypothetical disc:
- it's made of solid material. As you attempt to rotate the disc faster and faster, the stress on the material increases to resist elastic deformation not only from conferring the momentum to the edge, but also from resisting the edge's centrifugal force! Given the hypothetical size (and mass) of our solid disc, there is no material known to us that would be strong enough to resist such forces. The material structure binding the disc would disintegrate and it would simply crumble, flying to pieces in spirals.
- it's made of non-solids. Basically, it would just turn into a maelstrom. The centre constantly spinning relatively faster and the edges lagging behind, trying to catch up. For a real world example with appropriately large energy, consider the ring of plasma around a quasar.
