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Sorry, this is a long one, please be patient and thoroughly read it to understand it before answering.

Consider an object rotating along its length. The longer the object is doing the rotation the faster the tangential speed of the ends of the object would be. For example, if the length of a person’s hand to elbow is $0.5m$, and the elbow swings $90°$ in a second, then the tangential speed of the hand would be $[(90°/360°)(0.5(2)) π] m/s$. If the hand is holding a stick $1.5m$ long, then the tangential speed of the far end of the stick would be $[(90°/360°)(2(0.5+1.5))π] m/s$.

Now assume that there is an object that weighs the same as human hair, but is rigid, and does not bend or break. If the object is $382,000km$ long, and spinning along its midpoint at $90°/s$, then each end of the object would reach a tangential speed of $300,000 km/s$, which is the speed of light. If the object stretches any longer than at a point it would no longer be able to spin, because the angular speed of each end would exceed the speed of light. At this point, it would seem that if the object is freely floating in space, then its only motion possible would be along the x or y-axis, and unable to tilt or spin.

One may argue that such an object is impossible to exist, but an easy calculation suggests otherwise. If the material is similar to human hair, then it would only weigh $20,600 kg$; or if steel rebar, then about $236,000,000 kg$. To put it in perspective, $382,000 km$ is about the distance from Earth to the moon, and a fully loaded large container ship is about $220,000,000 kg$. Therefore such an object would be difficult, but not impossible to produce.

At this point, we reach the end of the thought experiment. If such an object exists and is freely floating in space, what motion would it take, and what would happen if someone tried to spin it?

Qmechanic
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Lida Zheng
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2 Answers2

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what would happen if someone tried to spin it?

It would bend and/or stretch.

In the early days of relativity Born did a very careful analysis of rigid body motion. A structure as long as you describe would naturally be extremely floppy. However, in principle, instead of simply twisting it in the middle you could try to spin gently by applying a large number of small forces distributed appropriately all along the length of the structure.

Born was able to prove that no matter how gently you produce angular acceleration there is unavoidably some internal strain. This is not part of dynamics, but an inherent part of relativistic kinematics. Angular acceleration unavoidably involves internal strain.

As you accelerate the tangential velocity arbitrarily close to $c$ the strains become arbitrarily high. So not only does it bend/stretch, but it will unavoidably fail. Regardless of whether it is made from hair, steel, or unobtanium.

Dale
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To see why the thought experiment fails, think about the endpoints of the rotating rod. According to special relativity, it takes an infinite amount of energy to accelerate a massive particle up to the speed of light. So it would take an infinite amount of energy to accelerate the atoms at the endpoints of your rod up to the speed of light. In other words, no matter how much energy you put into angularly accelerating this rod, it will never be enough for the endpoints to reach the speed of light.

You can also think of the centripetal force required. The atoms at the endpoints of the rod would have infinite momentum if they travel at the speed of light. This requires an infinite centripetal force to keep these atoms moving along a circular path. That is, no matter what material you choose for your rod, it will eventually bend or break before its endpoints reach the speed of light.

Aiden
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