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

#### PeterDonis

Mentor
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
It doesn't have to "know" anything. The causality is forward, not backward. The locally accelerated body "knows" how much proper acceleration to feel based on the local spacetime geometry. In other words, the local spacetime geometry contains all of the information about the path curvature of every possible local worldline, and "path curvature" is the same as "proper acceleration".

#### PeterDonis

Mentor
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?
Not the ones you're trying to make. Linearized gravity is still not the same as EM, because, as I said, gravity obeys the equivalence principle and EM does not. Also, the lowest order EM radiation is dipole while the lowest order gravitational radiation is quadrupole (because gravity is spin-2 while EM is spin-1).

#### Povel

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.

Ok, I see how this works for static situations, but the discussions over Mach's principle usually deal also with acceleration, both proper and coordinate, of either the local observer mass or the distant universe/"cosmic shell".
I don't see how it is possible to evade the need of employing some sort of radiative interaction in such circumstances.

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 guess one would truly need an exact/numerical solution to properly discuss this, but if the shell is being accelerated by some external mean and the local observer is resisting it, and this situation is postulated to be both kinematically and dynamically analogous to simply have the local observer accelerate with respect to the rest frame of the shell, as Wheeler and Ciufolini seem to suggest, wouldn't this mean that the interaction must be radiative, since the local observer has the freedom to undergo all sort of arbitrary cycles of acceleration?

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.
I agree with the the second sentence of course.

Regarding the first, however, I did consider those parameters in my example. Maybe I can refine it a bit:

Consider two nearly spherical shells, made of different materials, one more dense than the other but not necessarily with an higher tensile strength (i.e, steel and carbon nanotubes).
The shell made of the stronger material is built to have an oblatness greater than the other shell, so that when the two shell are spun to the same final angular velocity $\omega$ their deformation will make their moment of inertia the same. The density along their surface will also change due to stress, but by a different amount.
According to Lense-Thirring formula, then, their dragging effect on a gyroscope in their center will be the same once they reach and keep their final angular velocity.
(Of course this formula is a weak-field approximation for a spherical spinning body, but I suppose it should keep being valid if the oblatness is kept small)

The gravitational wave emission of the shells depends on the time variation of their quadrupole moment (actually on the second time derivative of the quadrupole moment).
During the spin-up phase the shells density and moments of inertia change, but not by the same amount in general. This would seem to indicate that the time-rate of change of their quadrupole moment is not equal.
Is there something incorrect in this picture?

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.
Ok, rigorously I cannot do that "arbitrary", but by simply changing the material I can surely reduce the deformation by orders of magnitude for any final $\omega$ by relying on the tensile strength of the material.

The locally accelerated body "knows" how much proper acceleration to feel based on the local spacetime geometry. In other words, the local spacetime geometry contains all of the information about the path curvature of every possible local worldline, and "path curvature" is the same as "proper acceleration".
Sure, I agree with all of this, but if Mach's principle is about why certain states of motions are associated to proper acceleration and not others I don't think it is enough to stipulate that such information is contained in local space-time geometry/gravitational field without also providing a mechanism by which the local spacetime/gravitational field enforces such distinction.

At the end of the section of Gravitation and Inertia I quoted, Wheeler and Ciufolini wrote the following:

"Therefore, what can we conclude from this analysis? First, nothing provable. Without using a sound theory of gravitation, we cannot expect a soundly founded theory of gravitational radiative reaction. Second, we have translated the Machian idea that "inertia here arises from mass-energy there" from a vague idea to a concrete suggestion with two components: first, the mechanism of coupling is not some new feature of nature but long-known gravitation and, second, it is the radiative or acceleration-proportional component of this interaction that counts. Finally, we recognize that we cannot go further in translating these intimations and suspicions into mathematically sound conclusions until we make use of a properly relativistic theory of gravitation at our disposition for the doing of it. "
Even though they are very cautious with the EM-GR analogy, they seem to accept the suggestion that the above mentioned enforcing is obtained through a radiative interaction, and that inertia is in fact a sort of "radiative reaction". (I'd also like to mention the fact that Sciama, their target in this section for his use of the EM-GR analogy, had already put his ideas in a properly general relativistic form in 1964, in a little known paper. The radiativeness is still there, see here).

To return to my question, then, if inertia is enforced by such radiative interaction with local space-time, it would seem like, in the rest frame of the locally accelerated observer, such radiation must have departed from the distant masses billions of years before the observer started accelerating. (In the meanwhile, I found a peer-reviewed paper by Grøn that claim this, see here , page 1726). This looks weird.

I guess it would be possible to claim that this is just an artifact of the chosen frame of reference and that the observer is really just somehow interacting non-radiatively with local-spacetime, but if so the nature of such interaction is left obscure (everyone who studied this seems to talk about a radiative interaction of some sort) and one cannot claim that all frames of reference give a coherent dynamical description of reality, because in this case there would be no artifacts, only equally real physical descriptions.

Not the ones you're trying to make. Linearized gravity is still not the same as EM, because, as I said, gravity obeys the equivalence principle and EM does not. Also, the lowest order EM radiation is dipole while the lowest order gravitational radiation is quadrupole (because gravity is spin-2 while EM is spin-1).
I know about these differences, but do they imply that the situation for a falling mass as seen from an observer on Earth is different than the case of a falling EM charge? Doesn't the observer standing on Earth see the mass emitting GW, while an observer falling along sees none?
I've read that one way to calculate the energy carried by GWs is to use pseudo-tensors, and these have the characteristic of being local but frame dependent, so that they vanish in free-falling frames. To me, this sounds very similar to the result in EM case.

#### PeterDonis

Mentor
the discussions over Mach's principle usually deal also with acceleration, both proper and coordinate, of either the local observer mass or the distant universe/"cosmic shell"
The discussions almost always deal with one of two cases:

(1) Linear proper acceleration of a test object. This doesn't require radiative interaction because the spacetime geometry itself can be static; only the motion of the test object is not. So the proper acceleration felt by the test object can simply be attributed to the local spacetime geometry.

(2) Proper acceleration due to rotation. This is the most commonly discussed case because of the seemingly "absolute" nature of the rotation: the rotating object (considered as a test object) feels proper acceleration, but it seems like if we adopt a frame where it's the distant universe that's rotating, not the object, then the distant univers should be feeling the acceleration, not the object--which is contrary to observation. However, this case can still be modeled as stationary (i.e., rotation rate not changing with time), which again means no radiative interaction is required: again, the spacetime geometry does not have to change, so the proper acceleration of the rotating object can again simply be attributed to the local spacetime geometry.

More general non-stationary cases still follow the principle that proper acceleration is due to the local spacetime geometry, but now accounting for the local spacetime geometry might indeed require some kind of gravitational radiation to be propagating from a source in the past light cone.

if the shell is being accelerated by some external mean and the local observer is resisting it, and this situation is postulated to be both kinematically and dynamically analogous to simply have the local observer accelerate with respect to the rest frame of the shell, as Wheeler and Ciufolini seem to suggest
I don't think that's what Cuifolini and Wheeler are suggesting. If there is some external source of energy acting on the shell, it's acting on the shell, not the local observer: that's an invariant and doesn't change if you change frames. More generally, if the local observer is in free fall, with zero proper acceleration, while the shell has nonzero proper acceleration, that's an invariant and doesn't change if you change frames.

Much of the discussion around Mach's principle fails to recognize that invariants like these are ways in which the situations are not "kinematically and dynamically analogous" if you change frames.

The shell made of the stronger material is built to have an oblatness greater than the other shell
Then the shells aren't identical before they are spun up. You've basically adjusted the original (before spin-up) composition of each shell to ensure that they have the same frame dragging effect after spin-up.

The gravitational wave emission of the shells depends on the time variation of their quadrupole moment (actually on the second time derivative of the quadrupole moment).
Actually on the third time derivative of the quadrupole moment.

During the spin-up phase the shells density and moments of inertia change, but not by the same amount in general. This would seem to indicate that the time-rate of change of their quadrupole moment is not equal.
Sure, because the shells start out in different states but they have to end up in the same state, at least as far as frame dragging is concerned. Note that the time it takes to spin up the shells to the same final state will, in general, be different, because the shells' initial states are different. And for the final frame dragging effect to be the same, all that has to be true is that the total integrated change in the shells is the same; it doesn't have to be true that they change in exactly the same way at every instant, which is what would be necessary for their GW emissions to be identical.

if Mach's principle is about why certain states of motions are associated to proper acceleration and not others I don't think it is enough to stipulate that such information is contained in local space-time geometry/gravitational field without also providing a mechanism by which the local spacetime/gravitational field enforces such distinction
The local spacetime geometry is the mechanism. Specifying the local spacetime geometry is equivalent to specifying the proper acceleration for every possible state of motion in the local spacetime. They are the same thing.

The discussions over Mach's principle have centered on how the local spacetime geometry gets determined by the matter/energy content of the rest of the universe. General Relativity gives a solution to that: the Einstein Field Equation.

they seem to accept the suggestion that the above mentioned enforcing is obtained through a radiative interaction, and that inertia is in fact a sort of "radiative reaction".
I don't have my copy of the book handy to look at the context, so I'll have to defer comment on this until I can dig it up. (Similarly, I haven't read that Sciama paper before, only his earlier one from the 1950s where he uses a theory of gravity that is admittedly not GR, and is only supposed to be an initial investigation. So I'll have to defer comment on that until I can read the paper.)

if inertia is enforced by such radiative interaction with local space-time, it would seem like, in the rest frame of the locally accelerated observer, such radiation must have departed from the distant masses billions of years before the observer started accelerating.
Yes, this is just another way of saying that the local spacetime geometry is determined by the matter/energy present in the past light cone, and that propagating gravitational waves are what carry information about changes in the matter/energy distribution in the past light cone to here and now.

the nature of such interaction is left obscure (everyone who studied this seems to talk about a radiative interaction of some sort)
Yes; I don't think classical GR itself can be interpreted as having a "radiative interaction" for all inertia. Most of the commonly used spacetimes in GR (Schwarzschild, Kerr, FRW) have no gravitational waves present, but objects still have inertia. I personally have no problem with this, but some people do.

Doesn't the observer standing on Earth see the mass emitting GW, while an observer falling along sees none?
If the object is free-falling radially, I think neither observer will see GWs being emitted. For cases with angular motion involved, I'm not sure; I would have to look at the detailed math (and I don't know if there is any analysis of this case in the literature).

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