Covers some of the reasons why various formulations of Mach's Principle are incompatible with GR.
Some of that was a little over my head (and I only gave it a cursory reading at the moment anyway, may revisit later), but I believe their formulation "Mach3" is the closest to my understanding of Mach's principle (and which they say is compatible with GR and the Lense-Thirring effect specifically).
More specifically I would say my version of Mach's principle derives via their "Mach0" from a general principle of relativity (lower-case, not GR specifically): all spatial and temporal relations are relative. So me spinning around in the universe and the universe spinning around me are the same state of affairs. That leads to the Mach0 observation: somehow spinning the universe around me pulls my arms away from my body. That raises the question of why that happens, and since the conventional explanation of my arms pulling away from my body when I spin is "inertia", that indicates that the rest of the universe somehow influences inertia here, which is their Mach3. The rest is details: how does the rest of the universe influence inertia here, in a way that pulls my arms away from my body when it spins around me, that being an equivalent description of me spinning around?
gmalivuk wrote:Why would accelerating *slightly* in the *same* direction as the already-spinning thing result in feeling like it was spinning at an *equal* rotational velocity in the *opposite* direction?
It's not that one causes the other, its that they both happen from the same cause: the second ring is having its frame of reference pulled around by the first. Lets think of the spinning shell instead of the rings for ease of illustration: You're floating in the lotus position in zero G inside a massive hollow shell of matter (in our universe, for simplicity). It starts to spin (relative to the distant stars) along the axis of your spine in what you would call a counterclockwise motion, so the wall in front of you is rushing to your left. It drags your frame of reference around with it, so what feels like "straight ahead" to you keeps drifting to your left, so you keep feeling like you are turning to the right. (A gyroscope would feel pulled to the left, for example, like it would if you tried to turn right). You are slowly pulled into the rotating reference frame yourself however, diminishing the feeling that you are turning right, and the feeling that you are turning to your right ceases entirely when you catch up to the spinning shell of matter and no longer are turning right relative to it. I'm not certain that could actually happen in our universe though, as there is the mass of the rest of the universe trying to keep you in its frame; I don't think you would ever be pulled completely into the spinning shell's frame, and so would always feel like you were turning slightly to the right, as the massive shell spun rapidly to your left. (Max, can you please confirm I have this part correct, since you spoke of this scenario earlier?)
Thinking on it further, I'm not certain what to expect in such a circumstance if outside the shell is an otherwise empty universe, because the component of your inertia received from the rest of the universe, if we did this in our universe, would be resisting the shell's dragging of your frame of reference, as above; but without the rest of the universe, perhaps you would be pulled along by the shell all the more readily? In which case, if that is the case, with the spinning rings example you gave, spinning one ring might well pull the other ring right along with it in an exaggerated form of frame dragging (instead of the tiny amount I expected earlier), there being no rest of the universe to counter-drag the other ring's frame; the magnitude of the frame-dragging effect is strictly the same, but it is now the sole source of (or influence on) inertia, not acting against the mass of the rest of the universe, so it accomplishes much more unimpeded. In which case, in order to really establish relative rotation between them, one ring would have to thrust one way and, to counteract being pulled along with it, the other ring would have to thrust the other way, in which case as you would expect both would experience acceleration.
(Intuitively I imagine that even with the frame-dragging effects greatly exaggerated by having no rest of the universe to compete with, there would be some lag and dally between the motion of the first ring and the second, like how the rotating shell scenario would work in our universe, though lessened. Giving that a little though, it seems like if we are using thrusters to spin one ring up, we have some propellant mass now floating off in our universe to account for -- the rest of our universe is no longer completely empty -- and that that may account for the expected lag between the two rings. If instead one ring pushed directly off the other by some mechanism, so there was no propellant mass floating off to account for, then of course the third law of motion would result in both rings spinning equally in opposite directions and experiencing equal centrifugal force etc.)
It would take very little thrust to spin up one ring (because it would have so little inertia, with so little mass in the universe around it) and so people on it would feel very little "acceleration" as it spun up relative to the other ring. Meanwhile the other ring would be (ever so slightly) pulled along by it and so feel (to the tiny amount they feel anything) that it was spinning the opposite direction.
Acceleration in a rotating frame is a matter of rotational velocity and radius, so the amount of acceleration they'd experience would be independent of how much mass they have.
That's why I put "acceleration" in quotes there. We never directly feel
acceleration per se; we feel a force of some kind, we feel weight. That force is our mass being accelerated. I can tell that the airplane I'm in just got pushed upward by turbulence because my chair pressed harder into my ass. We normally presume our mass is fixed and so the force we feel is proportional to the amount we are being accelerated. But if our mass was less, then being literally accelerated (velocity changing as measured visually or such) by the same amount would not "feel" like the same amount of "acceleration", because we would in effect weigh less in the acceleration-induced "artificial gravity", feel less force against us, as it would take less force to accelerate us the same amount. F=ma and all that.
And again, why would being pulled along with the other ring result in feeling like they were moving in the opposite direction? Why would frame dragging, which is a teeny tiny effect, result in their feeling like they're moving at the exact same velocity in the opposite direction as the ring that actually had thrust applied to it?
I believe I've covered these both in my explanation and reexamination above.