Rotations in general relativity?

[hawkingfan]

In special relativity, the mass of an object increases as the speed approaches c. Geometrically, this can be interpreted as a pure result of the relativistic length contraction. As the length (and volume) of a point-like particle becomes smaller, the field lines are forced together and the divergence of the gravitational vector field approaches negative infinity as v approaches c. Therefore, the object appears to be more massive to objects at rest. (more field lines pass through) And they appear to be the same mass for objects that travel along with them because the lengths decrease at the same rate so that less field lines pass through them. (making the mass appear unchanged)

The above paragraph can correctly explain relativistic mass in special relativity but what about general relativity and rotations. The kinetic energy of a rotating system is proportional to the square of it's angular speed with the moment inertia being the analogue of the mass which appears in the KE equation for translational motion. This must mean that there's an upper limit to the angular speed attainable by an object. It must have an asymptotic angular speed such that no particle in the system is traveling faster than c. Similarly, it's moment of inertia must somehow increase to compensate for the added energy.

Therefore either, the mass of the system must increase or it's radius must increase, or both must happen simultaneously to compensate for the increasing rotational KE. If it's radius increases, then all other particles will be further away from the center of mass of the system and this will make the outermost particles have greater speeds approaching c even more than expected. (so this must not happen if we are trying to prevent it from exceeding c) This must mean that it's mass is the predominant quantity which increases with rotational kinetic energy. How will it's field strength increase in the reference frames of particles at rest? Does it correspondingly contract it's length in the direction opposite to the angular velocity vector to lump the field lines together in the same way?

But wouldn't this cause a rotating system to fly apart by becoming too elongated. I know that spin distorts the shape of the Earth making it oblate but isn't this due more to classical mechanical effects. (where the weight and normal force aren't aligned so the object must spin with a certain centripetal acceleration proportional to the square of the velocity at that point) Another example of this classical phenomena would be the way in which water changes it's shape whenever you create whirlpools with your hand inside of a cylindrical container. The volume of the water remains constant and does not decrease like it should if it were a form of Lorentz contraction.

I really don't know the answer to this so can someone please help with this. It's not really making much sense. I know that the mass or gravitational field strength of a rotating system must increase in some way due to the equivalence of mass and energy. I am not yet mathematically adept for general relativity so can anybody who knows tell me the real effect of rotating a system of particles to angular speeds that approach an upper limit.

 

[hawkingfan]

Wow, still no answers. OK, in simple terms, how would the mass of an object increase if we rotate it to relativistic angular speeds?

 

[djy]

Wow, still no answers. OK, in simple terms, how would the mass of an object increase if we rotate it to relativistic angular speeds?

The relativistic energy is given by,

$E = \frac{m_0 c^2}{\sqrt{1 - v^2/c^2}}$.

Note that this equation is the same regardless of whether the motion is uniform or accelerated, so it applies perfectly well to a rotating object.

As the fastest parts of the rotating object approach $c$, the energy (and relativistic mass) go to infinity.

 

[bcrowell]

Hawkingfan, your original analysis was essentially right...did you just want confirmation that it was correct?

Some relatively minor points:

-A rigid body can't undergo angular acceleration in relativity.

-The source term in the Einstein field equations is not a scalar mass, it's the stress-energy tensor. Therefore it's not quite right to assume that you can just ascribe a greater mass to the rotating object, but it is approximately right.

-Today, most people use the convention that relativistic mass is invariant. This is why djy refers to energy rather than mass. In the modern convention, we'd say that the particles that make up the rotating body have their normal mass, but their kinetic energy is equivalent to some mass, and contributes to gravitational fields.

 

[TCS]

In special relativity, the mass of an object increases as the speed approaches c. Geometrically, this can be interpreted as a pure result of the relativistic length contraction. As the length (and volume) of a point-like particle becomes smaller, the field lines are forced together and the divergence of the gravitational vector field approaches negative infinity as v approaches c. Therefore, the object appears to be more massive to objects at rest. (more field lines pass through) And they appear to be the same mass for objects that travel along with them because the lengths decrease at the same rate so that less field lines pass through them. (making the mass appear unchanged)

The above paragraph can correctly explain relativistic mass in special relativity but what about general relativity and rotations. The kinetic energy of a rotating system is proportional to the square of it's angular speed with the moment inertia being the analogue of the mass which appears in the KE equation for translational motion. This must mean that there's an upper limit to the angular speed attainable by an object. It must have an asymptotic angular speed such that no particle in the system is traveling faster than c. Similarly, it's moment of inertia must somehow increase to compensate for the added energy.

Therefore either, the mass of the system must increase or it's radius must increase, or both must happen simultaneously to compensate for the increasing rotational KE. If it's radius increases, then all other particles will be further away from the center of mass of the system and this will make the outermost particles have greater speeds approaching c even more than expected. (so this must not happen if we are trying to prevent it from exceeding c) This must mean that it's mass is the predominant quantity which increases with rotational kinetic energy. How will it's field strength increase in the reference frames of particles at rest? Does it correspondingly contract it's length in the direction opposite to the angular velocity vector to lump the field lines together in the same way?

But wouldn't this cause a rotating system to fly apart by becoming too elongated. I know that spin distorts the shape of the Earth making it oblate but isn't this due more to classical mechanical effects. (where the weight and normal force aren't aligned so the object must spin with a certain centripetal acceleration proportional to the square of the velocity at that point) Another example of this classical phenomena would be the way in which water changes it's shape whenever you create whirlpools with your hand inside of a cylindrical container. The volume of the water remains constant and does not decrease like it should if it were a form of Lorentz contraction.

I really don't know the answer to this so can someone please help with this. It's not really making much sense. I know that the mass or gravitational field strength of a rotating system must increase in some way due to the equivalence of mass and energy. I am not yet mathematically adept for general relativity so can anybody who knows tell me the real effect of rotating a system of particles to angular speeds that approach an upper limit.

The space in which the object is rotating will be compressed and it's mass will increase. However, what happens to the object will depend upon it's material properties like elasticity and fracture strength.

I think that most real world objects will fly apart long before they reach relatvisitc speeds.

 

[Passionflower]

-A rigid body can't undergo angular acceleration in relativity.

While it is true that in flat spacetime one cannot change angular acceleration while maintaining rigidity is it true in GR?

But are you saying that it is impossible in curved spacetimes?

 

[hawkingfan]

Yes, I was searching for a confirmation that my analysis was correct.

I have another question about rotations. You say that any spherical material that rotates will fly apart at a sufficiently high angular speed. That may be the case if it rotates in one of the 3 Cartesian planes. But what would happen if it rotated in all 3 Cartesian planes simultaneously and the rotational kinetic energy was evenly distributed over the three degrees of freedom? Based on the classical Newtonian analysis of the sphere, it would not need to become an oblate ellipsoid in order to ensure that all 3 vectors sum to zero. (these 3 vectors being the weight toward the center of mass, the normal force at the surface and the product of the mass and the centripetal acceleration vector) This is very difficult to visualize in mind.