"In thought experiments in special relativity, when a rocket engine pushes on the rear of a spacecraft, or a tether is dragged behind an accelerating craft, assuming that these extended objects respond to the forces on them instantly and without deformation can lead to confusion and apparent paradoxes. The idealisation of a rigid body, which can often be useful in Newtonian physics, is generally no longer applicable, since it corresponds to an assumption of an infinite speed of sound within the body, in violation of relativistic causality.
In many cases, a qualitative understanding that perfectly rigid motion will be the exception rather than the rule in special relativity is enough to avoid the pitfalls. At other times, though, it can be useful to have a simple, quantitative model that deals with elastic deformation in a way that is consistent with special relativity.
The purpose of this page is to describe such a model of relativistic elasticity. We will restrict ourselves to the simplest possible situation: one-dimensional hyperelastic bodies. By “hyperelastic”, we mean the body will obey Hooke's law, which states that any change in length of a portion of the body (compared to its unstretched, or “relaxed” length) will be proportional to the tension to which it is subjected. For any real material, Hooke's law is just an approximation which holds over a limited range of tension and compression. However, for the sake of producing a (reasonably) mathematically tractable model, we will treat it as an exact law which holds up to the point where the material breaks."
5 out of 5
http://www.gregegan.net/SCIENCE/Rindler/SimpleElasticity.html
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