I Relativity's Effect on Long Rod Shape and Speed

[saddlestone-man]

TL;DR Summary

What would a pivoted long rod look like if one end was moving at close to c?

Hello All

I pivot a long rigid rod at one quarter its length and gradually accelerate the tip of the short end to close to the speed of light.

An observer is standing some distance away from the mechanism, so that he/she can see the whole length of the rod.

What would the shape of the rod look like (in particular the part of the rod on the other side of the pivot), and at what speed would it be moving at various distances from the pivot?

best regards ... Stef

 

[A.T.]

What would the shape of the rod look like (in particular the part of the rod on the other side of the pivot), and at what speed would it be moving at various distances from the pivot?

To get numbers, you need numbers: Material properties of the rod, exact relaxed shape and forces applied.

But this won't be a simple calculation, even in Newtonian mechanics.

 

[Orodruin]

Just to add: there is no such thing as ”rigid rod” in the relativistic limit. In particular, the wave speed in the material, which depends on the material properties, much be smaller than c.

 

[Halc]

A rigid object cannot be rotated at all, so there must necessarily be some strain somewhere. So I guess the object just rolls up like a measuring tape from the torque being applied to it somewhere.
The far end is obviously not going to move at 3c, so there's no way a linear object can have the kind of rotation described in the OP. It's like saying I build a space elevator, but don't quit until it reaches the orbit of Neptune. Not going to happen no matter the tensile strength of the elevator.

 

[russ_watters]

If you eliminate the acceleration the problem gets a lot easier. You don't have to worry about bending or materials stresses, so you can safely assume a rigid rod. So, the OP's question can be re-stated: if you rotate a rod about an axis at constant speed such that it's tip is moving at .9c (relative to the stationary center), how does the rod appear to someone in the center? E.g., does it appear bent? I would think yes, but mostly due to light propagation delay, not Relativity.

 

[russ_watters]

A rigid object cannot be rotated at all...

Why not?

 

[A.T.]

If you eliminate the acceleration the problem gets a lot easier.

But the OP specifically asks about that:

...and gradually accelerate the tip of the short end to close to the speed of light.

 

[russ_watters]

But the OP specifically asks about that:

1. I don't agree with your interpretation. I think the acceleration is just how we get to close to the speed of light and it's the end result (being near the speed of light) that is important to the OP.

2. Even if I did, being helpful to the OP sometimes means adjusting the OP's scenario to make it work, and then asking the OP if that helps/is what they were after.

3. Broader, the first pass at scenarios like this rarely requires bringing-up materials properties -- though people often cite it as a reason a question can't be answered. It would be more helpful to start as simple as possible, setting aside constraints until it is clear they are absolutely necessary.

 

[Halc]

If you eliminate the acceleration the problem gets a lot easier.

True. A perfectly rigid rod can (in principle, not in practice) be manufactured already spinning so there is no relativistic deformation of the material that results from any change in angular spin rate.

So, the OP's question can be re-stated: if you rotate a rod about an axis at constant speed such that it's tip is moving at .9c (relative to the stationary center), how does the rod appear to someone in the center.

Two things: The OP states that the short end is moving near c, meaning the far end is moving at nearly 3c, which cannot happen, so the scenario is impossible.

So a normal object (not rigid, say a pair of equal masses connected by strong string) is already moving at nearly c at the tips. The OP asks about a distant observer. It's pretty easy if the observer is very distant and located on the axis of rotation, in which case the spinning object always appears straight.

Any other point of view and the lines curve based on different times it takes for light to reach you. The difference is only due to unequal light-travel times from the various points which are not equidistant.

E.g., does it appear bent? I would think yes, but mostly due to light propagation delay, not Relativity.

Agree, the view from the center is a ~ shape.

Relative to the frame of an observer that is moving relative to the pivot point, the object is actually such a shape and doesn't just appear that way. This does not violate the premise of the rigidity of the object.

Similarly, an accelerating rigid object is only moving uniformly (all parts at the same speed) in the frame in which it is stationary. In any other frame with motion along the line of acceleartion, the different parts of the rigid obect are moving at different speeds.
Note that as @ Orodruin above points out above, such a rigid object cannot be accelerated by application of a point force (say from the engines at the rear) due to speed of sound limitations. Thus the object must be accelerated by proportional distributed and coordinated force. Only then is strain on the object eliminated, preventing the infinitely rigid (and thus infinitely brittle) object from shattering.

 

[pervect]

Why not?

A rigid rod can't change it's state of rotation, this is shown by the Herglotz-Noether theorem. See for instance https://en.wikipedia.org/wiki/Born_rigidity, the current version of which reads in part:

Born rigidity is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics.

The concept was introduced by Max Born (1909),[1][2] who gave a detailed description of the case of constant proper acceleration which he called hyperbolic motion. When subsequent authors such as Paul Ehrenfest (1909)[3] tried to incorporate rotational motions as well, it became clear that Born rigidity is a very restrictive sense of rigidity, leading to the Herglotz–Noether theorem, according to which there are severe restrictions on rotational Born rigid motions. It was formulated by Gustav Herglotz (1909, who classified all forms of rotational motions)[4] and in a less general way by Fritz Noether (1909).[5] As a result, Born (1910)[6] and others gave alternative, less restrictive definitions of rigidity.

As far as the fine points go - a rigid rod, rigid according to the original Borne formulation, can rotate, but it can't change it's state of rotation and still be rigid. And the problem as stated seems to demand that the later.

Alternate formulations of what "rigidity" might mean do exist, as the Wikipedia reminds me, but I don't think they have been widely accepted. At any rate, to answer the question, one would need to specify in detail what particular notion of "rigidity" one was using in order to solve the problem, and it'd have to be a different notion of rigidity than the original Born formulation.

We could into more detail on the Born formulation if there is interest, but it might be better in a different thread, it seems too far removed from the original poster's question.

 

[A.T.]

Even if I did, being helpful to the OP sometimes means adjusting the OP's scenario to make it work

OK, but that adjustment basically removes the relativistic effects that the title asks about, as you noted yourself:

E.g., does it appear bent? I would think yes, but mostly due to light propagation delay, not Relativity.

But maybe that's what the OP means.

 

[PAllen]

Well, there are some 'easy' answers to variations of the OP. Before this, it is worth noting that all real materials can almost be viewed as being made of jello for the purposes of interaction at relativistic speeds. This reflects the very low speed of sound in a material compared to light. Thus, rephrase the question as "if I have a rod of jello held in place 1/4 up its length and push rapidly on one end, what happens?". I leave it to the reader to picture this, and this is a valid first order description of the reality.

Another answer is if the rod is undergoing a Born rigid rotation, and the longer end is moving at .999 c (in the frame of the pivot), then ... what? The shorter end is moving at .333 c and there is nothing that can change about this without violating rigidity. Note that no sophisticated math is needed to show this is a valid Born rigid motion. You have a situation where no feature of the system changes in the frame of the pivot or as momentarily described in the MCIF of any point of the rod. This is sufficient to establish Born rigidity.

There is no Born rigid motion where the shorter segment is moving e.g. at .4c, in the pivot frame. This is not surprising, because there are many such limits with Born rigid motion. For example, uniform linear acceleration of (one end of) a Rod is born rigid motion. However, if one posits that the leading edge is accelerating at 1 g, then the maximum length of the rod is about 1 light year, as a matter of principle, and the gee force on the trailing edge approaches infinite as the rod length approaches 1 light year.