Relativistic Limits on a Rotating Rigid Bar

[Surlethe]

Given a rigid bar, what sort of limitations does relativity impose on its length? Presumably, the linear velocity of a point on the bar cannot exceed c. I'm unsure how to derive the maximum possible length of a bar, given an initial rotational velocity. Furthermore, what sort of wierd effects can be expected out toward the tip of the bar? At which point does the rotational acceleration from a constant torque on the bar max, and then start falling due to the increasing force required as the bar reaches relativistic velocities?

These questions have been bugging me for a while, and I'm afraid I have no idea how to quantitatively approach the answers. Help would be much appreciated.

[Darth Wong]

This is an absurd question. The bar's tensile strength would give out long, long before relativistic effects become an issue.

[darthdavid]

Well what if you assume that it was made out of Unobtanium and thus had the strength to do what he's talking about?

[Son of the Suns]

Well what if you assume that it was made out of Unobtanium and thus had the strength to do what he's talking about?

What the heck is that?

[wolveraptor]

Unobtainium is used to refer to any Sci-fi wank material that can do impossible things, like be made into a portable suit that can take you to the center of a star, or be used to build Imperial palaces so massive that they throw the planet off it's very orbit *cackles*

[Darth Wong]

So in other words, I am being asked what is a realistic science-based prediction for a scientifically impossible scenario?

[kheegster]

So in other words, I am being asked what is a realistic science-based prediction for a scientifically impossible scenario?

Relativity and quantum mech is replete with such thought experiments or 'gedanken' which involve physically impossible hypothetical situations, which are nevertheless still capable of shedding light on the physics.

[Zornhau]

Actually, I've always wondered about this one too. I shall watch this thread with interest.

[Xenophobe3691]

Well, the linear velocity of any point on the bar would be Velocity == (Radius)*(Angular velocity). So the fastest the bar could spin would be C divided by the radius, no?

[Il Saggiatore]

Given a rigid bar, what sort of limitations does relativity impose on its length? Presumably, the linear velocity of a point on the bar cannot exceed c. I'm unsure how to derive the maximum possible length of a bar, given an initial rotational velocity. Furthermore, what sort of wierd effects can be expected out toward the tip of the bar? At which point does the rotational acceleration from a constant torque on the bar max, and then start falling due to the increasing force required as the bar reaches relativistic velocities?

These questions have been bugging me for a while, and I'm afraid I have no idea how to quantitatively approach the answers. Help would be much appreciated.

First, how is a rigid bar defined in a purely hypothetical way?
Within classical mechanics, a rigid body is defined as a system of point-like masses whose distances are constant in time.
This can be extended to a body with continuous mass, but it is not necessary here.

In classical mechanics the distance between two points in space does not depend on the frame of reference.
This is no longer the case in relativity.
So, the classical definition of rigid bar breaks down in relativity: as a nfriend of mine used to say "there are no rigid bodies in relativity".

From a more practical point of view, the particles of material are kept together by electomagnetic interaction.
At relativistic velocities, the interactions that give rise to the bonds between particles would be significantly delayed.
Assuming that the particles of your bar still interact strongly enough, the particles farther from the cenetr would lag behind in the rotation, and the bar would end up deformed (looking like a spiral).

But realistically, the bar would break down well before relativistic speeds are reached.

[Surlethe]

Thanks for all the replies.

[Kuroneko]

If internal stress and self-gravity [*] are completely neglected, the question becomes simple. If the bar is accelerated, then a Rindler horizon forms at a distance of c²/a; this can be deduced by a variety of means (a fairly straightforward method is to exploit the velocity four vector by taking the rotational matrix, substitute imaginary coordinates, as per the difference between Euclidean and Minkowski metrics, and set this equal to the standard Lorentz transformation matrix). Obviously, if the bar does not fit within this length, then some portions of it are out of causal contact with the rest, and the bar must break here if it did not do so at less strenuous conditions. This is, of course, the case of one-dimensional acceleration, but in actuality the result generalizes. To see that this is true for rotational acceleration as well, one can substitute spatial rotational displacement into ds² = dx²+dy²-dt², using the original rotational matrix this time. After some arduous algebraic gymanistics with differentials, the end result will be singular at r²ω² = c², where ω is the rotational speed and r² = x²+y², and so r = c²/(rω²) = c²/a is the upper bound once again. As for interesting effects, I think the event horizon at the tip is interesting... however, your last question is confused.

[*] A disclaimer: this represents the zero-mass, unbounded-strength limit; it does not represent real bars; but then, this is what was asked for. Self-gravity is an important issue even for supernaturally strong bars, since internal stress will contribute to this, collapsing the bar into a black hole past a certain limit. Thus, the zero-mass limit is simply an excuse to ignore general relativity.