Recent content by PAllen

Well I can think of a recipe for building one:wink: Just start accumulating cold iron filings in one place, continue till you’ve added about 2 to 3 solar masses. Voila, it will collapse to a black hole.

Metric theories of gravity all obey all but the strongest form of equivalence principle. So, no Newton-Cartan theory could not accommodate this. However, conventional Newton’s gravity clearly could, in which case gravity would not be a fictitious force. It could be measured comparing free fall...

Elevator on the surface of earth is not inertial because the surface of the earth is exerting a real force on the bottom of the elevator. An inertial frame is one with no external force applied.

I should add that there are different versions of the equivalence principle. For something called the strong equivalence principle, general relativity (a specific metric theory of gravity) is the only known theory consistent with it. There are other metric gravity theories that satisfy weaker...

On the other hand, a particle model is, itself, an approximation. If you believe QFT, there are only continuous fields at the root. And if you consider particles as blobs of continuum, then the issues of the continuum model arise in the same way. I think it is perfectly ok to consider the...

I want to add a few more observations about this topic. I think the key point is that there is nothing wrong or paradoxical about the OP result when properly understood. Comparing the continuum case to "what about a particle on the rim" neglects that with a continuum model, the total mass all...

It seems obvious to me that the finite particle approach can be made to exceed any energy bound, based on extrapolation from my post #73. For me, my posts #70 and #73 answer all intuitive issues around the continuum model.

And one more thing… the issue of transitioning to a continuum is a problem for the radial direction, but not the tangential. So the following mixed approach captures the expected intuition. Consider a finite set of one dimensional rings, with mass for each chosen per constant linear density per...

I think there is no nice way to allow a continuum model to capture the notion that there is no upper bound to particle energy before c is reached. To do this, you would need to have nonzero rest mass exactly at the rim. With standard functions, this is impossible. Thus, to get a continuum model...

One further way of looking at this is that transitioning from a ring of point particles to a smooth annulus adds width to the particles. The amount of width depends on the chosen particle mass and chosen density. For any such width, the maximum particle speed is capped at a specific value below...

Also note that the constraint of constant density per a local, comoving observer, means adding particles to outer rings as they are sped up. And, for the outermost ring, the number of high speed particles you add, plus their KE, can exceed any bound before the smoothed rim velocity reaches c...

I think I have the key notion for the finite to continuum transition that explains the counterintuitive continuum result. The uniform density model implies that the total rest mass in any annulus stays constant during spin up. An annulus can be modeled as a ring of particles. The spin up process...

But the uniform COM frame density follows necessarily from any simple (i.e. not including stress energy of holding the disc together) model of spinning up a disc. Note, there is not attempt to maintain rigidity during this process. I think the core of the intuitive disconnect must be connected...

I don’t think the improper integral is the issue at all. In some sense, you never need to get there. Intuition suggests that any chosen energy should be exceeded before anything actually reaches c. The math shows otherwise.

Note that if we change the mass assumption to be constant density in each element's local comoving frame, we get for an arbitrary annulus: $$E=\frac{\pi\rho}{\omega^2}\left(\ln(\omega^2 r_1^{~2}-1)-\ln(\omega^2 r_2^{~2}-1)\right)$$ This, of course, diverges if the outer edge speed approaches c...