Relativistic Energy of Rotating Thin Ring: Quick Qs

[AVentura]

TL;DR Summary

is it just a matter of the tangential velocity?

Quick question about the relativistic energy of a rotating thin ring, hoop or cylinder. Is there any reason why the relativistic energy would be anything different than $E=\gamma_t m_0 c^2$ where $\gamma_t$ depends on the tangential velocity $v_t$ observed by someone at rest with the axis?

Likewise, is the relativistic angular momentum $L = \gamma_t m_0 v_t r$ where r is the radius?

Or is this trickier than it appears? If you spin up a ring does its invariant mass $m_0$ change? (a figure skater who increases her angular velocity by drawing her arms in changes her $m_0$ for example). I realize the radius may change but the observer can easily measure it.

Thanks in advance

 

[PAllen]

With caveats, yes this is correct. First note that modern practice is simply to use m for invariant mass or rest mass, rather than using m₀. Then we assume a rotation at constant angular velocity observed in an inertial frame with the axis of rotation stationary. We don’t care about how the body got to this state, and how it changed in the process of spinning up - we only care about the final state observed in said inertial frame. We note that m is the mass of some disk element locally measured by a device riding with the disk, after spin up. With these assumptions, your formulas are correct.

 

[Ibix]

With caveats, yes this is correct.

Shouldn't the energy associated with internal stresses in the ring come into this somewhere? Or is that accounted for somehow?

 

[pervect]

Summary: is it just a matter of the tangential velocity?

Quick question about the relativistic energy of a rotating thin ring, hoop or cylinder. Is there any reason why the relativistic energy would be anything different than $E=\gamma_t m_0 c^2$ where $\gamma_t$ depends on the tangential velocity $v_t$ observed by someone at rest with the axis?

Likewise, is the relativistic angular momentum $L = \gamma_t m_0 v_t r$ where r is the radius?

Or is this trickier than it appears? If you spin up a ring does its invariant mass $m_0$ change? (a figure skater who increases her angular velocity by drawing her arms in changes her $m_0$ for example). I realize the radius may change but the observer can easily measure it.

Thanks in advance

The hoop is under stress (at least in the hoop frame), so it should properly be modeled with a stress-energy tensor.

Certainly if one imagines a hope of finite modulus of elasticity, energy will be stored the hoop when the hoop ineveitably stretches as it is spun up. This implies that one needs a material model of the hoop to compute it's energy and momentum.

I no longer recall the details, but I'd suggest looking at Egan's webpage http://www.gregegan.net/SCIENCE/Rings/Rings.html. It's not peer-reviewed, unfortuately, but it takes the proper approach of computing the stress-energy tensor of the hoop. The analysis is valid for a specific material model, that Egan calls the "hyper-elastic" model.

 

[PAllen]

Shouldn't the energy associated with internal stresses in the ring come into this somewhere? Or is that accounted for somehow?

It is accounted for by the stipulation that mass of a volume element is measured locally after spin up. It’s relation to prior local mass may be arbitrarily complex, but we don’t care. Further, symmetry implies that it can vary only radially along the disk

 

[PAllen]

The hoop is under stress (at least in the hoop frame), so it should properly be modeled with a stress-energy tensor.

Certainly if one imagines a hope of finite modulus of elasticity, energy will be stored the hoop when the hoop ineveitably stretches as it is spun up. This implies that one needs a material model of the hoop to compute it's energy and momentum.

I no longer recall the details, but I'd suggest looking at Egan's webpage http://www.gregegan.net/SCIENCE/Rings/Rings.html. It's not peer-reviewed, unfortuately, but it takes the proper approach of computing the stress-energy tensor of the hoop. The analysis is valid for a specific material model, that Egan calls the "hyper-elastic" model.

This all looks good at a quick glance, however, I want to note that I was sidestepping all of this by positing measurement of invariant mass of small volume element in equilibrium state after spin up by a momentarily comoving inertial frame. This includes the effects of tension and deformation. Then, the relation of this invariant mass of a volume element to energy and angular momentum in inertial frame of the axis of rotation is as simple as the OP specifies.

 

[AVentura]

Fascinating. If we sum all the invariant masses of small volume elements in the co-moving inertial frames together do we arrive at the proper definition of invariant mass for all observers (moving on the ring or at rest on the axis)? I assume they agree on the total invariant mass and they disagree on the path length around the loop. So they disagree on the density by a factor of $\gamma$?

 

[PAllen]

Fascinating. If we sum all the invariant masses of small volume elements in the co-moving inertial frames together do we arrive at the proper definition of invariant mass for all observers (moving on the ring or at rest on the axis)? I assume they agree on the total invariant mass and they disagree on the path length around the loop. So they disagree on the density by a factor of $\gamma$?

I will try to respond later when I have a bit of time, but briefly, these new questions are quite complex, with counter intuitive answers. Very briefly, while in one inertial frame, energy is additive, invariant mass is not. Further, the invariant mass of a rotating body as a whole is, in some sense, impossible to define. This is because a coherent 'state of the body' at a moment in time does not exist (for a rotating body), so invariant extensive properties are difficult to define.

 

[SiennaTheGr8]

Further, the invariant mass of a rotating body as a whole is, in some sense, impossible to define. This is because a coherent 'state of the body' at a moment in time does not exist (for a rotating body), so invariant extensive properties are difficult to define.

If I'm not mistaken, this is also generally true of an extended body whose mass is changing (the exception being when all energy-transfer between the body and its environment occurs at a single point on the body's exterior).

 

[PAllen]

If I'm not mistaken, this is also generally true of an extended body whose mass is changing (the exception being when all energy-transfer between the body and its environment occurs at a single point on the body's exterior).

Yes, this is true as well.

 

[AVentura]

I will try to respond later when I have a bit of time, but briefly, these new questions are quite complex, with counter intuitive answers. Very briefly, while in one inertial frame, energy is additive, invariant mass is not. Further, the invariant mass of a rotating body as a whole is, in some sense, impossible to define. This is because a coherent 'state of the body' at a moment in time does not exist (for a rotating body), so invariant extensive properties are difficult to define.

That's ok. I almost deleted that question because I thought that might be the case. Thank you everyone for the responses.

 

[pervect]

That's ok. I almost deleted that question because I thought that might be the case. Thank you everyone for the responses.

The very short version is that point particles have an invariant mass, but extended bodies in special relativity have a stress-energy tensor, the same stress-energy tensor that one uses in General relativity.

We could try and talk more about the stress-energy tensor if it's of interest. Of course being familiar with tensors in general is rather highly recommened before talking about the stress-energy tensor in particular.

 

[AVentura]

I need to dig into tensors again. I first tried using a Schaum's outline chapter but I didn't really get it. I had expected to learn it in my physics masters curriculum, but all they ever did was give a handout. A copy of the same Schaum's outline ¯\_(ツ)_/¯ . I never had GR.

 

[pervect]

One can start out the journey into tensors with vector spaces. Vector spaces have the abstract property that they can be multiplied by scalars, and added together. They're typically represented as little arrows in lower dimensional spaces. However, it's tough to visualize little arrows in spaces over three dimensions. For special relativity, we'll need spaces of 4 dimensions, 3 for space and 1 for time.

Next up are dual vector spaces. Dual vector spaces are maps from a vector space to a scalar. The salient properties of dual vector space are that they have same number of dimensions as the original vector space, they are not the same as the original vector space, and the double dual (the dual of the dual) vector space is the same as the original vector space. One might think of vectors as being column vectors in matrix notation, and dual vectors as being row vectors, though the matrix notation doesn't upgrade gracefully into tensor notation.

Hopefully this is all just review. With vectors and their duals defined, a valence m,n tensor is a linear map from m vectors and n dual vectors to a scalar. I won't guarantee that I got the order of m and n correct (assuming that it's standardized across all texts, it may not be).

The metric tensor and it's inverse are important tensors, which can be used to turning a (1,0) tensor into a (0,1) tensor, or vica-versa.

Because of this ease of conversion, it's often useful to look at the sum m+n of tensor, it's total rank, as tensors of the same rank can be mapped between different valences via the metric tensor and it's inverse. Thus, a rank 2 tensor could have a valence of (0,2), (1,1), or (2,0), and appropriate use of the metric tensor and it's inverse can convert between the different valences for the same rank tensor.

The stress energy tensor can be regarded in several different, equivalent, ways. One way that gives a lot of insight into the functionality of the stress energy tensor is to consider it as a rank 2 tensor which is a linear map from an infinitesimal 3-volume element to the energy and momentum that infinitesimal volume element contains. But one needs to discuss how 3-volume elements are represented for this idea to be made useful. This usually requires a rather large detour early in the learning process.

Important precursors to the rank 2 stress energy tensor are the rank 1 tensors of special relativity called 4-vetors. 4-velocities and 4-momentum vectors are particularly useful, with honorable mention going to the number-flux 4-vector, and the volume 1-form (which is the dual of a 4-vector, rather than a 4-vector).