Rotations in Special Relativity

[fliptomato]

Greetings--I think I've confused myself about rotational motion in special relativity. Suppose you had a cog-shaped object in space that you caused to rotate by shining a focused beam of light onto its side.

Classically, if we treated the light as discrete photons carrying some momentum, the cog would spin faster and faster to some arbitrarily fast angular velocity. (Also, the cog would be pushed back.)

However, we know that the velocity of any part of the cog cannot increase past c, so this imposes a maximum angular velocity. Hence there is a contradiction. How would we treat this system in special relativity?

Thanks,
Flip

 

[Garth]

First of all, the cog or any material object would actually fly apart before relativistic rotational velocities were achieved. However, as a thought experiment this is similar to the question of a relativistic rocket that maintained a constant thrust.

In the inertial frame of reference of an observer the total energy, and hence relativistic inertial mass, of the cog/rocket, would increase as the velocity of c was approached. This would make it harder to accelerate and the rotational velocity/linear velocity of the cog/rocket would asymptotically approach c but not reach or exceed it.

There is no contradiction. I hope this helps Flip.

Garth

 

[fliptomato]

Thanks very much for the reply, Garth. It makes sense that there should be some asymptotic angular velocity, but is there a simple way to compute it? Suppose that I'm sitting on a heavy asteroid in free space (so that I'm not pushed backwards appreciably when I shine the laser). In my reference frame, I'm shooting a laser at the cog, causing it to spin.
In the limit where the linear momentum of the cog is negligible (i.e. suppose all of the force from the laser is going towards making the cog spin--I guess this is the limit of a massive cog with a small moment of inertia), I can calculate the power radiated on the cog. Now do I just apply energy conservation? (How do I know this rotational is lorentz covariant?)

What about in a more realistic scenario where the cog, in addition to spinning, is pushed back linearly? Is this still straightforward to solve?

Thanks,
Flip

 

[pervect]

Thanks very much for the reply, Garth. It makes sense that there should be some asymptotic angular velocity, but is there a simple way to compute it? Suppose that I'm sitting on a heavy asteroid in free space (so that I'm not pushed backwards appreciably when I shine the laser). In my reference frame, I'm shooting a laser at the cog, causing it to spin.
In the limit where the linear momentum of the cog is negligible (i.e. suppose all of the force from the laser is going towards making the cog spin--I guess this is the limit of a massive cog with a small moment of inertia), I can calculate the power radiated on the cog. Now do I just apply energy conservation? (How do I know this rotational is lorentz covariant?)
What about in a more realistic scenario where the cog, in addition to spinning, is pushed back linearly? Is this still straightforward to solve?
Thanks,
Flip

This is not going to be straightforwards to solve, because there isn't a GR formula for the energy of the disk that you can look up.

What we can say is that stresses and strains in the disk are going to contribute to its mass. (There are actually several different concepts of mass in GR, here the most convenient one is the Komarr mass which assumes the metric is static).

So to solve the problem you'd need to find the complete relativistic stress-energy tensor of the rotating disk, then solve Einstein's equations. The first is very hard, and requires some assumptions about the material that the disk is made of (which brings back the question that any realistic material will fly apart, so you need an exact specification of an unobtainable material, being careful that your unobtainable material doesn't violate any of the laws of physics).

The second (solving Einstein's equations) is even harder (only a few exact solutions are known) - except as an approximation, where you can do a sort of series expansion (this goes back to being only very hard).


Probably the easiest case to do is to consider what happens if you feed a laser beam into a rotating black hole "off-center". In this case the black hole will gain energy, angular momentum, and linear momentum from the laser beam. If you don't want to worry about linear momentum, feed it two laser beams with counter-balancing linear momentum, but both of which contribute angular momentum.

There are then some fairly simple formulas for the total mass (i.e. energy) of such a black hole that you can look up in terms of its energy and its irreducible mass.

for instance MTW, pg 890

M^2 = M_ir^2 + S^2/M_ir^2

(for an uncharged rotating black hole).

Here M is the mass of the black hole, M_ir is the "irreducible mass" of the black hole. It is the mass that the black hole would have if you stopped it from spinning, via reversible proceses. Reversible processes are processes that leave the area of the event horizon unchanged - the area of the event horizon can be identified with the entropy of the black hole. S is the angular momentum of the black hole.

All of these quantities are in geometric units (i.e. the speed of light is set equal to 1, and so is G, the gravitational constant).

This equation shows you that you can consider the angular momentum of a black hole as contributing to it's mass, but that it is not an additive process.

 

[masudr]

Err...isn't the asymptotic angular velocity such that no infinitesimal portion of the disk has speed equal to or greater than c?