I was reading Landau & Lifshitz's Classical Theory of Fields and I noticed that they mention that an extended rigid body isn't "relativistically correct".
For example, if you consider a rigid rod and apply a torque at one end, by definition of being rigid, the whole body must start rotating at the same instant. But the information about the force being applied cannot travel faster than the speed of light.
The book hasn't mentioned how to resolve this paradox, and after thinking about it for a while, I'm wondering if one can even define a rigid body in a relativistically correct manner.
To summarize: How does one define a rigid body in special relativity?
Answer
In reality there's no such thing as a perfect rigid body. There will always be a delay in the motion propagating along the body.
Under "normal" conditions you don't notice this delay as it's infinitesimal when compared to the size of everyday objects you interact with.
However if you had a rod several light years long (assuming that's at all possible) the delay in transmitting the motion (rotation or translation) along the body would mean that the far end wouldn't move instantaneously, but only at some time later consistent with the information not travelling faster than light.
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