TL;DR

relatavistic gradient across rotating object

Gedankenexperimenten: take any object and rotate it. Lets use a disc for simplicity, but by extension the question applies to any relative motion. The center of the disc is stationary (has zero translational velocity). The edge is moving relative to the center and moving relative to the environment.

In the real world the disc will come apart at some high rpm. But lets pretend it doesn't. Spin it fast. The edge approaches c. The center is still stationary. Clearly there is a relativisitic gradient from the center to the edge, with the edge doing all that fun compression-in-direction-of-motion and mass increase and time slowing. While the core tracks the ambient space-time.

Questions: what does this do to the disc? More importantly, what implications does this have for contiguity of any given object? When does an object cease to be a singular object and become a collection of separate objects? What is the granularity of such a separation?

By extension, any contiguous object (say, you hand while typing) experiences relativistic motion (one finger types a letter while the adjacent finger is hovering). Yes, these are slow enough to ignore the relativistic issues. But ignoring does not mean they are not occurring. My point is to suggest we are constantly tearing at space-time fabric by existing. Not sure what that implies, though, so I"m asking. Dont' let normalcy bias prevent insight!

Last bit of the experiment: add a second disc placed next to but not touching the first, but counter-rotate it relative to the first. Clearly the space-time gets torn a-la frame-twisting around black holes. But what else happens?

I'm a PhD but not of physics, so please act as if I were an intelligent but uneducated newbie. Explain in simple terms rather than equations...such may be useful or may simply obscure the event.

Thanks!

For a better starting point you might try Taylor and Wheeler’s “Spacetime Physics”- the first edition is available free online.

TL;DR: relatavistic gradient across rotating object

Clearly the space-time gets torn a-la frame-twisting around black holes

IIRC from the last time this was discussed here, a lump of concrete the size of a washing machine spinning at millions of revolutions per second (if it could somehow be held together) might just produce enough frame dragging type effects to be detectable by our most sensitive equipment. The experiment would require the annual energy consumption of a small country, assuming perfect frictionless operation.

If you build it non-spinning to then spin it up, then using only very rigid elements will eventually not work. The circumferential elements need to be elastic or extendable (telescopic) enough to compensate their length contraction.

If you build it already spinning from very rigid elements, then you need to account for the non-Euclidean ratio between circumference to diameter of more than pi. So you have to order more/longer circumferential elements, than you would need for a non-spinning station of the same diameter.

And you cannot keep clocks at different radial positions of the station synchronized, as the more central clocks will run faster.

Consider a GPS satellite clock sending an electromagnetic signal that encodes the time shown by itself at the departure (say ##t_1##). It then sends an encoded signal later when it reads ##t_2##.

The clock at rest on the geoid on the earth receives both the signals and calculates the difference ##t_2 - t_1## from the values encoded in the received signals. It then compare this difference against the difference ##t_2' - t_1'## between its own local readings at the reception of those signals.

The clock rates match if the above differences are the same.

Back to Born rigidity.

Born proposed a simple notion of rigid motion in special relativity. In lay language, Born rigid motion is any motion that leaves the distance between all pairs of points on a body constant as it moves.

A wiki reference on the topic: https://en.wikipedia.org/wiki/Born_rigidity

Erhenfest, using Born's original notion of rigidity, showed that it was not possible in the frameork of special relativity to make a disk start or stop rotating without violating Born's original criterion for rigid motion.

That's not the end of the story, but to go further will likely require much more than a high school background. One general possibilities include - using a more relaxed notion of what it means for an object to be rigid, notions of which are mentioned in the wiki article I cite.

Several weaker substitutes have also been proposed as rigidity conditions, such as by Noether (1909) or Born (1910) himself.

A modern alternative was given by Epp, Mann & McGrath.

Other approaches might involve keeping Born's original notion of rigidity, but specifying a material model as to how things stretch. Greg Egan took this approach in a non-peer reviewed paper, which has apparently changed since I last looked at it, at https://www.gregegan.net/SCIENCE/Rings/Rings.html. It's also not high school level, I'd put it at the graduate level.

Objects in classical Newtonian physics deform all the time, and their behavior can be analyzed. But even without relativity, this would likely be college level rather than high school level. Mechanical engineers deal with this all the time. I'm not aware of what texts they use, and their approach is different enough where you may not get enough leverage out of it for it to help you understand the relativistic formulation.

To summarize. Simple notions of mechanics based on idealized rigid bodies simply cannot handle the relativistic rotating disk. I believe you will need some notion of how to handle objects that can deform to handle the problem. This may leave you feeling sad if your entire understanding of physics is based on rigid bodies which are unable to deform. The solution is easy to state but difficult to cary out - learn enough physics to handle bodies that are not necessarily rigid, bodies that are able to deform. This will probably be beyond the high school level, though. The typical approach to the continuum problem involves partial differential equations, where the "rigid body" approach gives ordinary differential equations.

A very simple example of the continuum problem, one that might be approachable at a high school level, would be to understand how, when you push on one end of a bar, the bar does not instantaneously move in a rigid manner, but how a wave propages through the bar at a characteristic speed of the material, a lot like a wave travelling along a slinky. The slinky analogy is particuarly fun and useful because the speed of propagation of a wave through even the most rigid materials (often called the speed of sound in the material) is MUCH less than the speed of light. I'm not sure even the "wave equation" is truly at high school level, though, because it requires some understanding of partial differential equations :(.

Putting a plug in for my previous point, I'd argue that a knowledge of partial differential equations is important in formulating problems of this sort, which I would argue puts it out of the high school level. So, though it's a different problem, it does illustrate the sorts of math that are needed to address it.