# A Einstein, Mach's Principle, and the Speed of Light Limit

#### e2m2a

I am confused about Einstein's thinking. I understand when he formulated his general theory of relativity, he wanted to incorporate 3 foundations for his theory: The relativity principle, the equivalence principle, and Mach's Principle. He believed that inertia and weight were essentially the same phenomenon, that inertia was a function of the surrounding mass of the universe. Spin a rock tied to a string, he believed the centrifugal force tugging on your fingers was essentially a gravitational effect caused by the rotation of the rock relative to the cosmic mass of the univers. Here is my confusion. All of his professional life he insisted that nothing-- light, physical causality,etc. could ever exceed the speed of light. So why would he even attempt to implement Mach's principle, knowing that no effect, wave, field,etc. could travel faster than c?

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#### Paul Colby

Gold Member
I've always thought that the way the distribution of mass and energy in the universe determines the local metric in GR is consistent at least in spirit with Mach's principle. I've always been told by people that know that this view is simply incorrect. So, if one were to spin up the distant stars respecting the speed of light and all, wouldn't frame dragging eventually effect the water in a initially non-rotating Newton's pail? Whatever....

#### e2m2a

Again, I'm not sure what Einstein was thinking. If the local metric is already determined by the total cosmic mass-energy distribution of the universe, then was he thinking that acceleration with respect to the local metric pre-determined by distant mass-energy is what actually causes the inertial force, not acceleration with respect to the distant mass-energy distribution? I don't know. Can someone throw some more light on this problem, specifically, the speed of light restriction and causality due to distant mass-energy. Inertial reactions are instantaneous reactions, so something local must be causing it. Therefore, how could Einstein try to weave Mach's principle into GR without overstepping the light speed restriction? Remember, Einstein rediculed spooky-action-at-a-distance in quantum mechanics pertaining to quantum entanglement.

#### haushofer

Excellent question!

I think a lot of people believe Einstein rejected the notion of an ether as soon as he published his special theory, but this is not the case. E.g., in 1920 he lectured in Leiden where he stated that "if inertia determines the accelerations of masses far away, this implies an action at a distance, which is unacceptable for a modern physicist". As such he proposed to reincarnate the idea of an ether. See e.g. Walter Isaacson's biography, chapter 14. As I understand it now, in that lecture he proposed to do so because his theory of GR is not Machian; you will feel inertial forces when you rotate in an empty universe, as you can check by transforming the geodesic equation in an empty universe for an inertial observer by going to rotating coordinates. So whereas Newton proposed that one rotates wrt absolute space, Einstein proposed that you will rotate wrt the metric field (which is a substructure of spacetime), which in some sense serves as an "ether" (although it is not the same concept as before!).

#### Paul Colby

Gold Member
I don't know. Can someone throw some more light on this problem, specifically, the speed of light restriction and causality due to distant mass-energy.
I certainly can't comment on what Einstein was thinking or even grasp why one should care so much. In the limit that light might shine both ways, how is it that GR is not an answer to the above question? In EM the local fields are determined by the global sources in a causal way, why is GR different? The Wiki discussion throws up Godel's rotating universe as some grand counter example. That one can find some special solution of the GR field equations that tweak poor Mach in a way he wouldn't be able to understand doesn't surprise. Mach didn't understand either relativity theory given the time period when he was writing. People tend to speak about these things like the progenitor of a theory has some choice in how the theory turns out? The facts drive the (successful) theories not the will of researcher.

#### e2m2a

Excellent question!

I think a lot of people believe Einstein rejected the notion of an ether as soon as he published his special theory, but this is not the case. E.g., in 1920 he lectured in Leiden where he stated that "if inertia determines the accelerations of masses far away, this implies an action at a distance, which is unacceptable for a modern physicist". As such he proposed to reincarnate the idea of an ether. See e.g. Walter Isaacson's biography, chapter 14. As I understand it now, in that lecture he proposed to do so because his theory of GR is not Machian; you will feel inertial forces when you rotate in an empty universe, as you can check by transforming the geodesic equation in an empty universe for an inertial observer by going to rotating coordinates. So whereas Newton proposed that one rotates wrt absolute space, Einstein proposed that you will rotate wrt the metric field (which is a substructure of spacetime), which in some sense serves as an "ether" (although it is not the same concept as before!).

Thank you for the response. I will have to investigate deeper into his papers on the Einstein Paper's website on his ideas concerning an ether.

#### phyti

There is a simple scenario that counters Mach's idea.
Two spheres separated by space, spinning in a common plane, with opposite and equal rotations.
If the reciprocal effect was due to the remaining mass of the universe spinning with the opposite rotation, the net rotation would be zero.
The spheres would still experience centrifugal effects.

#### Paul Colby

Gold Member
There is a simple scenario that counters Mach's idea.
Two spheres separated by space, spinning in a common plane, with opposite and equal rotations.
If the reciprocal effect was due to the remaining mass of the universe spinning with the opposite rotation, the net rotation would be zero.
The spheres would still experience centrifugal effects.
Does this assume that the two spheres are the only mass in the universe? It would seem required to counter Mach's idea.

#### phyti

Mach would attribute equivalent centrifugal effects of a rotating mass m
if the universal mass (M-m) rotated in the opposite direction and m had no rotation.
Two masses m1 and m2, with equal rates of rotation but opposite directions, would not work per his principle.
There is also the distinction between allowed physical behavior of matter, and perception.
The distant stars rotating at speeds >c is perception, or reality confined to the mind.

#### Paul Colby

Gold Member
Two masses m1 and m2, with equal rates of rotation but opposite directions, would not work per his principle.
Agreed. Is this due to Mach predating the relativities? If Mach was aware and understood the relativities wouldn't GR have address Mach's philosophical stand?

#### PeterDonis

Mentor
if one were to spin up the distant stars respecting the speed of light and all, wouldn't frame dragging eventually effect the water in a initially non-rotating Newton's pail?
According to the equations of GR, yes (with some caveats related to what "spin up the distant stars" would actually mean). I suggest checking out Cuifolini and Wheeler's book Gravitation and Inertia. It goes into this general issue in excruciating detail.

#### Elemental

According to the equations of GR, yes (with some caveats related to what "spin up the distant stars" would actually mean). I suggest checking out Cuifolini and Wheeler's book Gravitation and Inertia. It goes into this general issue in excruciating detail.
Ciufolini and Wheeler give an interesting analogy from electrodynamics in chapter 7 of their book. If a charged body is suddenly accelerated, its electric field lines are deformed from radial into a roughly spherical shell at large distances r = ct from the region of the charge's acceleration, where c the speed of light and t the elapsed time from the acceleration event (see Figure 7.1). The field within this shell will tend to drag oppositely charge particles in the same direction as the original body's acceleration. Importantly, since the deformed field lines form a roughly spherical shell, this effect falls off as 1/r instead of 1/r2, consistent with the known characteristics of the radiative component of an electromagnetic field. This makes it plausible that extremely distant charges have an important local influence, because if charge distribution is on a large scale uniform, the quantity of charge in a shell or radius r is proportional to r2, so the effect of distant charges can predominate.

Now look at it the other way around: the body is stationary while all other charges in the universe accelerate. The radiative component of the all those other charges drag the body along with them, similar to Mach's principle.

The problem of course is this effect is not instantaneous (if you stick with retarded potentials) while inertia is. Where I'm puzzled is, the author's do not seem to rely on this analogy to support their account of Mach's principle. They cite Ellis, Sciama, and Fermi saying the charge "does not see" those other distant charges, instead it "feels them." But here is where I get lost, because I'm not sure what it means to "feel" distant charges but not "see" them. Somehow Ciufolini and Wheeler attribute inertia to the distribution of other gravitating systems on a space-like hypersurface where there is no possibility of any radiative effect being involved, but I haven't been able to follow their reasoning.

#### PeterDonis

Mentor
The problem of course is this effect is not instantaneous (if you stick with retarded potentials) while inertia is.
But that's because, in the analogy you described, the distant charge is accelerated, while the charge sitting right next to us, the observer, is not. So of course it's going to take time for the effects of the distant charge accelerating to propagate to us.

In the case of inertia, it's the other way around. The distant masses in the universe are just sitting there, but the mass right next to us, the observer, is accelerated, and resists the acceleration--meaning, it requires a force to accelerate it. Since the force is being applied right here, and the acceleration is right here, the resistance is also right here, so we observe it immediately. The effect of the distant masses is not their effect "right now" but their effect one light-travel time ago, but since the distant masses aren't accelerating, they're just sitting there, it looks the same to us as if it were their effect "right now", because there's no change in the distant masses that has to propagate to us.

#### Elemental

Thanks, PeterDonis. Agree that if the distant masses didn't accelerate, then their retarded effects do the job. But then you can't reverse the situation and say (based on the analogy) that there is no difference whether the local body accelerates or the rest of the universe accelerates instead (as you could for different inertial frames in special relativity).

#### PeterDonis

Mentor
then you can't reverse the situation and say (based on the analogy) that there is no difference whether the local body accelerates or the rest of the universe accelerates instead (as you could for different inertial frames in special relativity).
You're confusing coordinate acceleration and proper acceleration. In the scenario I was talking about, the distant masses have zero proper acceleration: they are in free fall. It's the local object that has nonzero proper acceleration: it feels weight. That's true regardless of whether we choose a frame in which the local object is at rest and distant masses "accelerate" (in the coordinate sense), or if we choose a frame in which the distant masses are at rest and the local object has coordinate acceleration as well as proper acceleration.

#### PeterDonis

Mentor
But then you can't reverse the situation and say (based on the analogy) that there is no difference whether the local body accelerates or the rest of the universe accelerates instead (as you could for different inertial frames in special relativity).
To put this another way: if some distant mass had a rocket attached to it and the rocket were turned on, we would not feel the effects of that on inertia here instantaneously. The effects would have to propagate to us. They would propagate as gravitational waves, just as the effects of accelerating a charge propagate as electromagnetic waves.

#### Povel

In the case of inertia, it's the other way around. The distant masses in the universe are just sitting there, but the mass right next to us, the observer, is accelerated, and resists the acceleration--meaning, it requires a force to accelerate it. Since the force is being applied right here, and the acceleration is right here, the resistance is also right here, so we observe it immediately. The effect of the distant masses is not their effect "right now" but their effect one light-travel time ago, but since the distant masses aren't accelerating, they're just sitting there, it looks the same to us as if it were their effect "right now", because there's no change in the distant masses that has to propagate to us.

Isn't the point behind Mach's principle (at least some versions of it) to posit the "relativity" of acceleration (without distinguishing between proper or coordinate)?

What I mean is, one should not be able to tell whether it is the observer or the distant masses that are accelerating, because the effect of such masses on a observer "resisting" their frame-dragging effect in the latter case would generate the sort of inertial forces that are associated with proper acceleration (i.e. , the water in Newton's bucket rising).
If this is the case, I'd expect that the local observer mass would also generate a counter dragging effect on the distant masses. In the frame were distant masses are not accelerating, then, the accelerated observer would produce miniscule "inertial forces", so they would also feel "weight".

By the way, regarding Wheeler and Ciufolini's book, their quote concerning the instantaneous effect of inertia referenced by Elemental is the following:

[..]
The sign of the force therefore is such as to drag the particle in question along with all the other particles in the "universe." In other words, we ourselves have to exert "externally" a force forward if we want to hold the particle in question "stationary" against the tide of particles accelerating backward.
We can translate this result back into a statement about what happens in the frame of reference at rest with the other particles. There also, a force is required to accelerate the particle relative to all the others. But that force is not new in the world. It is the familiar force required to impart to a particle of mass m an acceleration a in the familiar frame of reference in which the faraway matter of the universe is at rest, F = ma.

[..]

However, we know that gravitation, like electromagnetism and every other primordial force, must propagate at the characteristic speed c. The elementary sum in equation (7.1.4) for the coefficient of inertia envisages a radiative interaction between particle and particle which propagates instantaneously. But how can stars at distance of 10^9 and 10^10 lt-yr respond to the acceleration of a test particle here and now in such a way as to react back upon this test particle at this very moment? This difficulty alone should be sufficient cause for dropping the elementary formulation of equation (7.1.4) and to observe that one would, at least, need to use retarded potentials.

I might be misreading this, but it seems to me they are pointing out the fact that there is indeed some problem with the istantaneous character of inertia if it is indeed a gravitational radiative interaction.

#### PeterDonis

Mentor
Isn't the point behind Mach's principle (at least some versions of it) to posit the "relativity" of acceleration (without distinguishing between proper or coordinate)?
Not with the parenthetical comment you make, no. Proper acceleration is a direct, local observable; it isn't "relative". Mach's principle does not deny that at all.

The point of Mach's principle, at least as I see it (there is not general agreement among physicists on exactly what "Mach's principle" is or what it means), is to try to put together a satisfactory view of why objects feel proper acceleration in certain states of motion but not others, without hypothesizing some fixed background structure of spacetime that does not depend on the actual matter and energy content of the universe. But saying that the proper acceleration an object feels can be explained, ultimately, by the distribution of matter and energy in the universe is not at all the same as saying that proper acceleration is "relative".

one should not be able to tell whether it is the observer or the distant masses that are accelerating
Here you are talking about coordinate acceleration, which is relative in GR. But for proper acceleration, what you say here is obviously false: the observer can easily tell whether or not he has proper acceleration by using an accelerometer, or asking himself whether he feels weight.

If this is the case, I'd expect that the local observer mass would also generate a counter dragging effect on the distant masses.
But not the distant masses "now"--the distant masses one light travel time from now. So a local observer mass on Earth now won't generate a counter dragging effect on distant masses in the Andromeda galaxy for two million years. Any counter dragging effect of the Earth on the Andromeda galaxy now came from Earth two million years ago. We have no way of measuring any such effects directly.

But it is still possible to measure the frame dragging effect of local masses on nearby masses--that's what Gravity Probe B did. Cuifolini and Wheeler talk about that.

it seems to me they are pointing out the fact that there is indeed some problem with the istantaneous character of inertia if it is indeed a gravitational radiative interaction
They are pointing out that you can't view our feeling of inertia here and now as being due to the effect of distant masses now; you have to view it as being due to the effect of distant masses one light travel time ago. That's what their reference to "retarded potentials" means.

#### Povel

Not with the parenthetical comment you make, no. Proper acceleration is a direct, local observable; it isn't "relative". Mach's principle does not deny that at all.

The point of Mach's principle, at least as I see it (there is not general agreement among physicists on exactly what "Mach's principle" is or what it means), is to try to put together a satisfactory view of why objects feel proper acceleration in certain states of motion but not others, without hypothesizing some fixed background structure of spacetime that does not depend on the actual matter and energy content of the universe. But saying that the proper acceleration an object feels can be explained, ultimately, by the distribution of matter and energy in the universe is not at all the same as saying that proper acceleration is "relative".
I expressed myself incorrectly. With "relative" I meant exactly the content of your last sentence, that is objects feel proper acceleration due to the effect of the distribution of mass and energy in the universe.

But not the distant masses "now"--the distant masses one light travel time from now. So a local observer mass on Earth now won't generate a counter dragging effect on distant masses in the Andromeda galaxy for two million years. Any counter dragging effect of the Earth on the Andromeda galaxy now came from Earth two million years ago. We have no way of measuring any such effects directly.

But it is still possible to measure the frame dragging effect of local masses on nearby masses--that's what Gravity Probe B did. Cuifolini and Wheeler talk about that.
I agree fully, there must be a time delay between observer and distant universe and viceversa.
To me, however, this rises some question related on how the interaction travels.

Surely linearly accelerating an observer creates gravitational waves, and linearly accelerating the entire matter-energy content of the universe relative to the observer generates a (much much stronger) gravitational wave that travels toward the observer (assuming coordinate acceleration is what is needed for their appearance*), but there are situations where I don't see how any sort of wave could be generated, even though the interaction must be obviously time-delayed.

For example, consider the case of a gyroscope sitting inside a massive spherical shell with some non-negligible radius embedded in asymptotically flat spacetime (a much more simplified situation conceptually than a "rotating universe") .
The solution close to the gyroscope for a non-rotating shell is well known; the solution for a steadily rotating shell is also well known (this is the setup for Lense-Thirring effect).
I can then imagine an initially non-rotating shell undergoing a period of acceleration that ends up with a shell rotating with some constant angular velocity.
The gyroscope axis will initially be aligned with some distant star, then a $\Delta t$ after the shell starts accelerating it will start to precess more and more until the shell velocity becomes steady. At this point, after a time delay proportional to the shell radius, the gyroscope precession will also stabilize.
Is the above picture correct?
If it is so then, since the effect is time-delayed, it seems like a radiative interaction should be at play.
But if the shell is perfectly spherical it cannot generate any quadrupolar gravitational wave, so what sort of radiation is involved here?

(If all the above is considered excessively off topic, feel free to remove it, I'll open another thread)

They are pointing out that you can't view our feeling of inertia here and now as being due to the effect of distant masses now; you have to view it as being due to the effect of distant masses one light travel time ago. That's what their reference to "retarded potentials" means.
Ok, I did notice they referenced retarded potentials, but I have some lingering doubt sabout the picture they portray.
If inertia is a gravitational radiative interaction, as Wheeler and Ciufolini conclude, it must involve real radiation, even though it is the result of coordinate acceleration*. But if so, how does the matter along the past lightcone "know" when to radiate the appropriate field to produce the right force at the right time on the locally accelerated body?

*As far as I can tell, the consensus is that a charge falling in a gravitational field would be seen to radiate by an oberver on the ground, but not by an observer falling along with it. So the only thing that counts for radiation to be detected is coordinate acceleration. I suppose it should be the same with gravity.

#### PeterDonis

Mentor
To me, however, this rises some question related on how the interaction travels.
"Travel" can mean any of a number of things. Gravitational waves is one, but it's not the only possible way of thinking of how an interaction involving the geometry of spacetime would "travel".

Surely linearly accelerating an observer creates gravitational waves
No, it doesn't. There is actually an exact solution known for this case, called the "Kinnersley photon rocket", which is known to have no gravitational waves anywhere.

Is the above picture correct?
It seems so to me, but I don't think any exact solution is known for this case.

if the shell is perfectly spherical it cannot generate any quadrupolar gravitational wave
If the shell is perfectly spherical before it starts spinning, it won't stay perfectly spherical; that's impossible. It has to deform, and the deformation, since it's due to a time-varying rate of spin, will produce (I think) a time-varying quadrupole moment and thereby cause GW emission.

If inertia is a gravitational radiative interaction, as Wheeler and Ciufolini conclude, it must involve real radiation, even though it is the result of coordinate acceleration
Inertia is not the result of coordinate acceleration. It's the result of proper acceleration. More precisely, "inertia" in the sense of "whatever it is that makes you feel weight" is the result of proper acceleration.

As far as I can tell, the consensus is that a charge falling in a gravitational field would be seen to radiate by an oberver on the ground, but not by an observer falling along with it. So the only thing that counts for radiation to be detected is coordinate acceleration.
First, gravity is not the same as electromagnetism, although there are similarities. Gravity obeys the equivalence principle; electromagnetism does not. And it's the equivalence principle that lets us view gravity as an effect of spacetime geometry. Gravitational waves are waves of changing spacetime geometry, so you should not expect analogies with EM like the one you are making here to hold.

Second, the EM radiation seen by the observer on the ground from a free-falling charge is due to the observer's proper acceleration, not the charge's coordinate acceleration. The observer free-falling with the charge sees no radiation because he has zero proper acceleration.

#### Povel

"Travel" can mean any of a number of things. Gravitational waves is one, but it's not the only possible way of thinking of how an interaction involving the geometry of spacetime would "travel".
So there are ways, different from gravitational radiation, in which the influence of some distant mass can propagate to us in a causality-respecting way? For example?

No, it doesn't. There is actually an exact solution known for this case, called the "Kinnersley photon rocket", which is known to have no gravitational waves anywhere.
So, if I'm intepreting it correctly, it consist of a point particle emitting "null fluid/ photon radiation" and accelerating due to the recoil. This system does not emit gravitational waves because the contribution from the point particle mass and the photon cancel each other, at least if the anisotropy of the photon flux is purely dipolar. (https://arxiv.org/abs/gr-qc/9412063)

(EDIT: In a later paper Bonnor found out that a generalized Kinnersley photon rocket does not radiate even if the photon flux contains quadrupole terms (https://iopscience.iop.org/article/10.1088/0264-9381/13/2/015). However this has been challenged by other treatments of the problem, for example https://arxiv.org/pdf/gr-qc/9711001.pdf . This is really weird)

Does this constitute a problem for the picture that was described above of dragging and counter dragging between the local observer and the distant universe?
If the dragging and counter dragging interaction is radiative it should involve gravitational waves presumably. However this photon rocket does not emit any, even though it is properly accelerated, and thus feels inertia.

It seems so to me, but I don't think any exact solution is known for this case.
I see. It would be very interesting if such exact solution could be found. I suppose this situation might still be studied numerically, right?

If the shell is perfectly spherical before it starts spinning, it won't stay perfectly spherical; that's impossible. It has to deform, and the deformation, since it's due to a time-varying rate of spin, will produce (I think) a time-varying quadrupole moment and thereby cause GW emission.
Indeed, it makes sense. I didn't thought about the shell deformation.

EDIT: I thought a bit more on this, but I still see some issue.

In principle, I can make the shell of whatever material I want.
(Since Lense-Thirring scales with angular momentum, and so on the moment of inertia, for any given final angular velocity , if I make the shell out of, say, carbon nanotubes I will presumably need to make it bigger to obtain the same dragging effect I would obtain if it was made of, for example, steel.)

During the speed up phase, then, the two shells will deform into an oblate spheroid by a different amount, and consequently, it would seem like the amount of gravitational waves they produce is different, even though the final dragging effect on the gyroscope is the same by construction.
Even though I cannot make a perfectly rigid shell, it seems like I can arbitrary reduce its deformation to a tiny amount.

Inertia is not the result of coordinate acceleration. It's the result of proper acceleration. More precisely, "inertia" in the sense of "whatever it is that makes you feel weight" is the result of proper acceleration.
With "it is the result of coordinate acceleration" I was referring to "real radiation". In light of your latest post I suppose that the correct thing to say would be that radiation is the result of proper acceleration of either the oberserver mass or the distant masses or both.

But even so, I can't make sense on how does the matter along the past lightcone "know" when to radiate the appropriate field to produce the right force at the right time on the locally accelerated body. How does this work?

First, gravity is not the same as electromagnetism, although there are similarities. Gravity obeys the equivalence principle; electromagnetism does not. And it's the equivalence principle that lets us view gravity as an effect of spacetime geometry. Gravitational waves are waves of changing spacetime geometry, so you should not expect analogies with EM like the one you are making here to hold.

Second, the EM radiation seen by the observer on the ground from a free-falling charge is due to the observer's proper acceleration, not the charge's coordinate acceleration. The observer free-falling with the charge sees no radiation because he has zero proper acceleration.
I didn't mean to stretch the similarity between the gravitational and electromagnetic case beyond their limit. I was thinking more about a linearized gravity - EM comparison.
In this limit, it should be possible to make at least certain comparisons regarding radiation emission, right?

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#### FactChecker

Gold Member
2018 Award
So there are ways, different from gravitational radiation, in which the influence of some distant mass can propagate to us in a causality-respecting way? For example?
Information. The light from distant stars and special events certainly cause astronomers to stay up at night.

#### Povel

Information. The light from distant stars and special events certainly cause astronomers to stay up at night.
Of course, but what I meant was specifically how an interaction involving the geometry of spacetime could "travel" to us aside from propagating through gravitational waves.

I know Wheeler and Ciufolini talk about "instantaneous" elliptical constraints equation applied on a space-like hypersurface, but this looks to me more like a convenient way to sidestep a complicated retarded integration along the past lightcone which is nonetheless physically analogous to it (because what's physically happening involves a time-delayed interaction).

#### PeterDonis

Mentor
So there are ways, different from gravitational radiation, in which the influence of some distant mass can propagate to us in a causality-respecting way? For example?
Consider the static field of a planet like the Earth. The influence of the Earth at some point distant from it is not due to the Earth "now"--it's due to the Earth in the past light cone of that point. But because the field is static, there are no gravitational waves. So if you think of the influence of the Earth at a given point as being due to the Earth in the past light cone of that point, you have to think of that influence "propagating" somehow from the Earth in the past light cone to the point, without there being any gravitational waves.

Note that the same issue arises in EM: the Coulomb field of a point charge does not instantaneously affect another charge at a distance; the influence of one charge on another is due to the first charge in the past light cone of the second. But there are no EM waves present if the EM field is just the static Coulomb field.

#### PeterDonis

Mentor
If the dragging and counter dragging interaction is radiative
But it might not be. You can have distant masses affecting the spacetime geometry here and now without gravitational waves propagating. I gave an example in post #24.

I suppose this situation might still be studied numerically, right?
Yes.

During the speed up phase, then, the two shells will deform into an oblate spheroid by a different amount, and consequently, it would seem like the amount of gravitational waves they produce is different
No, because the gravitational wave emission also varies with things like the density and moment of inertia of the shell. If the GWs are what is propagating the change in precession rate (from zero to some nonzero value) to the gyroscope, then if the change in precession is the same, the GWs must be the same.

Even though I cannot make a perfectly rigid shell, it seems like I can arbitrary reduce its deformation to a tiny amount.
No, you can't, because the final deformation is due to a change in spacetime geometry that is necessarily finite if the final angular velocity is finite. The only way to make the deformation arbitrarily small would be to make the angular velocity arbitrarily small.

"Einstein, Mach's Principle, and the Speed of Light Limit"

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