Ibix said:

TL;DR: Off the back of yesterday's thread about relativistic discs I tried to work out the KE of a rotating disc and got an odd result.

I remember looking at relativistic hoops and disks a long time ago, but I no longer recall most of the details. I do recall Greg Egan had a treatment at https://www.gregegan.net/SCIENCE/Rings/Rings.html using a "hyper-elastic" material model. This may be significantly harder than what you want to do.

Veary basic suggestions from what I do recall:

Analyze a hoop, it's easier. I believe Sagittarius A-Star is on the correct track when he mentions that the rest mass of the disk depends on w because of the internal stresses.

Compute the stress energy tensor, including the tension terms. It's probably best to come up with a stress energy in the lab frame.

You'll need the stress energy tensor to formulate an analysis, because that's the only covariant entity. Without that concept, you won't be able to formulate the relativistic and covariant equations.

In lay language - pressure (or in this case stress) affects inertia.

I think the only nonzero terms in the stress-energy tensor for the thin hoop will be ##T^{tt}, T^{t\theta},T^{\theta\theta}##, which correspond to energy density, momentum and/or angular/momentum density, and tangential stress. Radial stress ##T^{rr}## needs to vanish at the boundary for a thin hoop, and if the hoop is thin, it's constant through the hoop, so it can be set to zero.

What's the principle that allows you to solve for the tension terms? It's the fact that ##\nabla_a T^{ab}=0##. That's the covariant replacement for what you might do with F=ma in a Newtonian analysis.

I already mentioned the boundary conditions, I think.

You'll probably want to use both coordinate basis and an orthonormal basis. The difference is that the vector pointing in the ##\theta## direction in the coordinate basis, ##\partial / \partial_{\theta}##, does not have a unit length. I'd write ##T^{\theta\theta}## for the coordinate basis and ##T^{\hat{\theta}\hat{\theta}}## for the orthonormal basis.

If you insist on analyzing a disk, you'll have to some how figure how the stress that holds the disk together splits between radial and tangential stress, rather than must making the radial stress vanish.

Once you have the stress energy tensor in the lab frame, you can convert it to the "rest" frame of the disk.

Beware of the weak energy condition. If the stress exceeds the density, you'll get a negative energy density in the rest frame - this is probably unphysical.

Anyway, even in the simplest case, it's quite involved.

I recall weird things did happen with the rotating ring, there was an upper bound on the angular momentum in Egan's hyperelastic version, and when the derivative of angular momentum with rotation speed vanished, the model basically failed.