Is there Lorentz contraction in accelerated motion?

Question

The usual Lorentz contraction is deduced for the case of a rod moving with linear uniform motion relative to an inertial frame but by a kind of "continuity reasons" it is hard to believe that no similar contraction happens in the case of accelerated motion and that's why I ask whether there is some generalization of the Lorentz contraction theorem from uniform to variated motion.

Such a generalization would be particularly welcome in the discussion of the relativistic rotating disk, where it has been argued by some physicists that Ehrenfest's argument [P. Ehrenfest, “Gleichförmige Rotation starrer Körper und Relativitätstheorie”, Phys. Zeitschrift 10, 919-919 (1909). English translation, “Uniform rotation of rigid bodies and the theory of relativity”, available in the web at Wikisource] is not well founded because it rests on an application of the Lorentz contraction theorem in a situation where that theorem doesn't apply.

Two examples of such objections against Ehrenfest paper are by Varićak, in the very beginning of the polemics concerning “Ehrenfest paradox” (V. Varićak, “Zum Ehrenfestschen Paradoxon”, Physikalische Zeitschrift 12, 169 (1911). Wikisource translation, “On Ehrenfest's Paradox”), and, much later, by Asher Peres [A. Peres, “Relativity in rotating frames” (2004). arXiv:gr-qc/0401043v1]

Answer 1

To get the insight you need for this kind of question, I think it is best to consider the geometric presentation of special relativity which is offered by spacetime diagrams (Minkowski diagrams). In these diagrams you don't have to worry about Lorentz contraction or time dilation in the first instance; you simply plot the worldlines of particles. Such a plot would be four-dimensional if you could draw in 4 dimensions, but to get the idea it is sufficient to restrict to two spatial dimensions and one temporal one and then the diagram is 3-dimensional.

Where does Lorentz contraction come in? It is all in the way we now may choose to interpret spacetime as a set of spatial surfaces, one after the other. The central point is that one can legitimately pick any set of spatial surfaces as telling us what is the situation from one moment in time to the next, as long as they are all mutually parallel and all spacelike. Spacelike means the spatial surface is a set of events which can in principle be deemed to be simultaneous (in an appropriately chosen inertial frame). For such events no signals pass from one to the other because it would require fast-than-light signaling.

Having got that image in mind (I mean the image of a fixed spacetime, with worldlines of particles, and a set of parallel spacelike planes one on top the other) I can tell you what Lorentz contraction is. If we take a given spatial surface then the worldlines intersect that surface where they pass through it. Lorentz contraction is the observation that the size and shape of this intersection will depend on the surface you picked. In particular, it depends on the slope of the surface. In other words, observers in different inertial frames find that a given set of worldlines intersects their spatial surface in a way which depends on the slope of that surface and therefore on the relative motion between themselves and the particles whose worldlines we are considering.

That relative motion can be of any kind, including acceleration.

Answer 2

Let us define the instantaneous speed of such rod that is undergoing uniform proper acceleration to be the speed of such inertial observer that observes every part of the rod to be at rest.

We know the aforementioned observer measures the length of the rod to be the rest length of the rod.

And now we just transform that length to whatever inertial frame we want. Now we know the length of that accelerating rod in any inertial frame.

Same definition works with constant coordinate acceleration of every part of the rod too.