Does GR contradict Mach's principle?

[Ken G]

The FAQ on rotation of the universe contains the following remark:

If you believe wholeheartedly in Mach's principle, then there is no way to test empirically for rotation of the universe as a whole, since there is nothing else for it to be rotating relative to. However, general relativity is not very Machian, and it offers a variety of ways in which an observer inside a sealed laboratory can detect whether the lab is rotating. For example, she can observe the motion of a gyroscope, or measure whether the Sagnac effect is zero. There are alternative theories of gravity, such as Brans-Dicke gravity, that are more Machian than GR,[Brans 1961] and in these theories there is probably no meaningful sense in which the universe could rotate. However, solar-system tests[Bertotti 2003] rule out any significant deviations from GR of the type predicted by Brans-Dicke gravity, so that it appears that the universe really is as non-Machian as GR says it is.

A question that emerges from this is whether or not GR is actually contradictory to Mach's principle, or if it simply isn't built to accommodate it without additional modification. This is a very technical issue, as GR always is, and indeed there are at least two separate versions of "Mach's principle" that come into play, but we can start by simply noting that most people interpret Brans-Dicke theory as "GR plus", by which I mean, it is built on the bones of GR but introduces additional adjustable parameters that can recover GR in certain limits. It is also often said that Brans-Dicke, unlike GR, is built explicitly to accommodate Mach's principle, at the expense of the strong equivalence principle (yet still satisfies Einstein's equivalence principle, which is weaker). Also, it is often said that there is no evidence that the parameters of Brans-Dicke are needed-- experiments are satisfied with taking the limiting versions used in GR.

Given all this, however, it seems an odd claim that GR is inconsistent with Mach's principle. For example, at one point the author of the FAQ asserted that GR is as inconsistent with Mach's principle as 2+2=5 is inconsistent with mathematics. But if Brans-Dicke is consistent with Mach, and if it gives GR in certain parameter limits, then it cannot be true that GR is inconsistent with Mach. Rather, it is often said that GR admits solutions that are non-Machian (the most famous being the Godel metric), but admitting non-Machian solutions is not being inconsistent with Mach-- because we routinely add additional postulates to GR to keep it physical, and Mach's principle could be just another example. If that point is unclear, consider that the Godel metric, a perfectly allowable GR solution, admits closed timelike curves. Yet no one would state that "GR is inconsistent with the impossibility of time travel."

So we are left with the following questions:
1) How can it be said that GR is inconsistent with Mach if GR is a limiting case of a theory (Brans-Dicke) that supports Mach?
2) How do the answers change when the strong and weak Mach's principle are used? (The weak principle is simply that inertia here depends on the prevailing mass distribution, so the mass distribution could be rotating, it would just require that locally inertial frames means rotating with the whole. The strong principle is that the whole cannot be said to rotate, because it has nothing to rotate with respect to. So for example, the Godel metric does not violate the weak principle, but does the strong.)

 

[bcrowell]

But if Brans-Dicke is consistent with Mach, and if it gives GR in certain parameter limits, then it cannot be true that GR is inconsistent with Mach.

Brans-Dicke gravity gets more Machian as you lower the value of $\omega$. (Just to clarify for anyone who might be confused by the context, this $\omega$ is a unitless constant appearing in BD gravity. It's not the same as the rotational velocity of the universe in the thread that this one spun off from.) BD isn't generically consistent with Mach's principle. BD is inconsistent with Mach's principle when the value of $\omega$ is large.

Rather, it is often said[...]

By whom?

[...] that GR admits solutions that are non-Machian (the most famous being the Godel metric), but admitting non-Machian solutions is not being inconsistent with Mach[...]-

It's not just particular solutions of GR that are non-Machian, it's the whole theory. The theory says that if I watch a gyroscope inside a spinning laboratory, I'll see the gyroscope's axis change directions. This is independent of what cosmological solution exists outside the lab's walls. Only in BD gravity with small $\omega$ does the outcome of this experiment depend on distant matter.

[...]-- because we routinely add additional postulates to GR to keep it physical, and Mach's principle could be just another example.

I don't think this is true. What additional postulates do you have in mind? The logical foundation of GR is simply the Einstein field equations. There have been no additional postulates added between 1915 and 2011.

-Ben

 

[Dale]

Brans-Dicke gravity gets more Machian as you lower the value of $\omega$. ... BD is inconsistent with Mach's principle when the value of $\omega$ is large.

I believe that current evidence puts omega at something greater than about 40000.

IMO, it seems to indicate that the universe is non-Machian. Although Mach's principle is philosophically appealing it does not seem to be correct, at least not as far as it is embodied by BD.

 

[bcrowell]

I believe that current evidence puts omega at something greater than about 40000.

IMO, it seems to indicate that the universe is non-Machian. Although Mach's principle is philosophically appealing it does not seem to be correct, at least not as far as it is embodied by BD.

That's my interpretation too.

 

[Ken G]

Brans-Dicke gravity gets more Machian as you lower the value of $\omega$. (Just to clarify for anyone who might be confused by the context, this $\omega$ is a unitless constant appearing in BD gravity. It's not the same as the rotational velocity of the universe in the thread that this one spun off from.) BD isn't generically consistent with Mach's principle. BD is inconsistent with Mach's principle when the value of $\omega$ is large.

2+2=5 is inconsistent with 2+2=4 for any value of 5 different from 4. If BD is inconsistent with Mach's principle, it is so for any value of w different from 0. (And remember, when Brans and Dicke first proposed the idea, they were not imagining w << 1, they were imagining w ~ 1.)

Perhaps we should start by saying more clearly just what is the Mach's principle that you are referring to-- the Einsteinian version is simply "inertia here is created by matter over there." Consider, for example, this paper: http://adsabs.harvard.edu/abs/1981RPPh...44.1151R
in which the abstract states
"Mach's principle, that inertial forces should be generated by the motion of a body relative to the bulk of matter in the universe, is shown to be related to the structure imposed on space-time by dynamical theories. General relativity theory and Mach's principle are both shown to be well supported by observations. Since Mach's principle is not contained in general relativity this leads to a discussion of attempts to derive Machian theories. The most promising of these appears to be a selection rule for solutions of the general relativistic field equations, in which the space-time metric structure is generated by the matter content of the universe only in a well-defined way. "

In other words, GR allows Mach to hold, or not to hold, it depends on the nature of the solutions allowed. Would we say that GR is inconsistent with the impossibility of time travel because the Godel metric allows closed timelike loops? Not a rhetorical question, this requires answering. Raine is certainly claiming that GR allows a subset of solutions that are Machian, refuting the claim that it is inconsistent like 2+2=5.

It's not just particular solutions of GR that are non-Machian, it's the whole theory. The theory says that if I watch a gyroscope inside a spinning laboratory, I'll see the gyroscope's axis change directions. This is independent of what cosmological solution exists outside the lab's walls. Only in BD gravity with small ω does the outcome of this experiment depend on distant matter.

Are you sure? GR is expressed in terms of differential equations-- would you not have to assume a spacetime boundary condition at infinity to do the calculation you propose? How does GR say what that boundary condition should be, with no added postulates?

I don't think this is true. What additional postulates do you have in mind? The logical foundation of GR is simply the Einstein field equations. There have been no additional postulates added between 1915 and 2011.

So GR predicts that time travel is possible then? No paradoxes there? I'm not sure this issue is quite as cut-and-dried as it is being sold, that's all.

 

[yuiop]

I think the main difficulty is that Mach did not clearly define his principle in a mathematical way, so it is difficult to state in any rigorous way whether or not GR or any other theory is consistent with it.

For example Mach's principle suggests that the mass of the universe determines the inertia of an object. By reversing the argument, accelerating a 1kg mass in space and measuring the inertia that should be sufficient information to tell us the mass and possibly the density of the universe, but there are no formulas to refer to.

Does the inertial effect of the mass of the universe in Mach's principle depend on an inverse square distance law like gravity? As far as I know, no one knows because Mach's law is not clearly defined. Is it directional? Does it depend on velocity distributions? Does an object in a large space void have less inertia than similar object near a massive galaxy? Are the inertial giving properties of the majority of mass communicated at the speed of light (and so limited to the visible horizon) or instantaneous? How would you define Mach's principle?

 

[PAllen]

A few observations:

1) Brans-Dicke with constant of 40,000 effectively suppresses its deviation from GR, and makes any Machian character (by the 'traditional definition') not present. It results in the statement that inertial is only determined by distant matter to a small degree, and is mainly absolute.

2) yuiop's point about lack of formal definition of Mach's principle makes these discussions go in circles.

3) Someone who has really tried to put Mach's principle on a formal footing is Julian Barbour, finding that there is more than one way of doing it. One particular formalization is claimed to be consistent with (and lead to) GR:

"Two definitions of Mach's principle are proposed. Both are related to gauge theory, are universal in scope and amount to formulations of causality that take into account the relational nature of position, time, and size. One of them leads directly to general relativity and may have relevance to the problem of creating a quantum theory of gravity. "

http://arxiv.org/abs/1007.3368

 

[yuiop]

In the paper you linked to ...

The specific behaviour identified by Newton as inertial motion could
arise from some causal action of all the masses of the universe; this could
lead to some observable effects different from Newtonian theory: “Newton’s
experiment with the rotating vessel of water simply informs us that the
relative rotation of the bucket with respect to the sides of the vessel produces
no noticeable centrifugal forces ... No one is competent to say how the
experiment would turn out if the sides of the vessel increased in thickness
and mass till they were ultimately several leagues thick”

... Mach's quote suggests that he had in mind that if the bucket was extremely massive, then the outcome of Newton's thought experiment would have been different, implying that the locality of mass is significant in its inertial giving qualities, since even a bucket that was several leagues thick would be insignificant compared to the total mass of the universe. Now it occurs to me that in GR, objects near a massive body such as a black hole experience gravitational time dilation and from the point of view of an observer at infinity object appear to accelerate slower which might/could be interpreted as having greater inertia near massive bodies, giving a Machian "flavour". (I am talking about coordinate acceleration here, rather than proper acceleration.) However, Mach's principle implies that a body infinitely far from other massive bodies would have zero inertia, while GR predicts a "residual inertia", however far a test body is from other massive bodies. Secondly, Mach's principle implies a frame dragging effect of nearby massive bodies and AFAIK, GR is consistent with frame dragging, which can be considered a sort of Machian behaviour.

 

[bcrowell]

Are you sure? GR is expressed in terms of differential equations-- would you not have to assume a spacetime boundary condition at infinity to do the calculation you propose? How does GR say what that boundary condition should be, with no added postulates?

In GR, effects like the motion of a gyroscope in a rotating lab, or the Sagnac effect, are independent of any assumptions about cosmologically distant matter. These are special-relativistic effects, and by the equivalence principle, GR becomes SR locally. Take a look at the equations for the Sagnac effect: http://en.wikipedia.org/wiki/Sagnac_effect They have nothing to do with cosmology, and they can be derived without adding additional postulates to GR.

In BD gravity, this argument fails because BD gravity doesn't obey the strong equivalence principle.

 

[Ken G]

I think the situation bcrowell offered serves perfectly well to establish whether or not we have a Mach's principle-- we have a gyroscope in a lab, and the gyroscope changes direction. If we cannot say that this means the lab is spinning in an absolute sense, and can only say it is spinning with regard to the prevailing mass distribution (say, the CMB rest frame), then we have a Mach's principle. So for example, if we look out from the lab and see an isotropic CMB, yet the gyroscope shows signs of being in a rotating frame, we have falsified Mach.

The problem is, if we simply assert "the CMB frame is not rotating" by fiat, then we can not use cosmological scales to test Mach, we have voided it right from the start by equating a relative frame with an absolute one. So in that case, we'd have to get into the question of just how much rotating mass one needs to "replace" the distant stars as the determiner of the local inertial frame. That sounds like the kind of issue that could be adjudicated by some "omega" parameter, as in Brans-Dicke, but even then we have no idea how much local mass should be needed to overrule the distant mass, so I don't see how any particular choice of omega could be viewed as Machian vs. non-Machian. Rather, it seems to me the omega parameter just tells you, if you want to think of the theory as being Machian, how much mass does it take to establish a local inertial pocket.

So I still conclude that GR can be considered to be Machian, in the sense of "mass there establishes the inertial frame here". With some large omega parameter, you can establish that without resorting to a boundary condition at infinity, you just need a large amount of mass in the environment, but the omega is too large to observe. Alternatively, you can just use infinite omega (so GR), and bring in the Machian character via the boundary condition at infinity (which no one has yet commented on). This also implies that certain types of solutions must be ruled out, the solutions that Raine describes, on the grounds that they are unphysical, like the Godel metric (and importantly, not the metrics used in actual cosmology). Since the Godel metric has closed timelike loops, there may well already be perfectly good reasons to discount it as unphysical-- without contradicting GR.

 

[Ken G]

In GR, effects like the motion of a gyroscope in a rotating lab, or the Sagnac effect, are independent of any assumptions about cosmologically distant matter.

What I'm saying is that to a Machian, they are indeed dependent on assumptions about cosmologically distant matter, but those assumptions are hidden because they are instead expressed, in the actual calculation, via a seemingly innocent boundary condition at infinity. Equivalently, they are expressed by an "obvious" choice of which frame should be Minkowskian.

These are special-relativistic effects, and by the equivalence principle, GR becomes SR locally. Take a look at the equations for the Sagnac effect: http://en.wikipedia.org/wiki/Sagnac_effect They have nothing to do with cosmology, and they can be derived without adding additional postulates to GR.

But there is certainly a hidden assumption in the Sagnac effect-- the assumption that you have a Minkowski space in a given frame. That is imposed by fiat-- GR does not tell you what frame will be the Minkowksi space, so whether or not it is Machian is simply swept under the rug by the equations you mention.

In BD gravity, this argument fails because BD gravity doesn't obey the strong equivalence principle.

That's because BD allows for nearby mass to alter the characteristics of the local inertial frame in a kind of hybrid way, but if your cosmology has a cosmological principle, then you don't see the hybrid character-- the effects of "mass there" can be satisfactorily modeled by a boundary condition at infinity instead of a specific effect on the local inertial frame. The boundary condition tells you what the Minkowski frame is, in a way that you can easily overlook where that assumption snuck in. As such, it obscures the Machian character, making it moot rather than contradicted.

 

[Ken G]

... Mach's quote suggests that he had in mind that if the bucket was extremely massive, then the outcome of Newton's thought experiment would have been different, implying that the locality of mass is significant in its inertial giving qualities, since even a bucket that was several leagues thick would be insignificant compared to the total mass of the universe. Now it occurs to me that in GR, objects near a massive body such as a black hole experience gravitational time dilation and from the point of view of an observer at infinity object appear to accelerate slower which might/could be interpreted as having greater inertia near massive bodies, giving a Machian "flavour".

Yes, at the very least we must give Mach credit for an astonishing insight-- it is certainly true that GR includes gravitomagnetic effects of local mass distributions, and no one but Mach ever expressed the slightest inkling that a theory of gravity should include such effects.

However, Mach's principle implies that a body infinitely far from other massive bodies would have zero inertia, while GR predicts a "residual inertia", however far a test body is from other massive bodies.

I would argue that if all you had in your universe was a single test mass, you'd never have the slightest idea what frame should be Minkowskian for addressing the dynamics of that particle. The inertial observer would never be known in any way but by having all the observers watch the test mass. So we will always need to augment GR with additional postulates, else GR cannot predict the motion that the different observers will see.

 

[Dale]

I think the main difficulty is that Mach did not clearly define his principle in a mathematical way, so it is difficult to state in any rigorous way whether or not GR or any other theory is consistent with it.

I agree completely with this. This is one of the two main reasons that I dislike Mach's principle.

 

[Ken G]

So you don't find it fascinating that Mach was the only person who ever voiced an opinion that a very large rotating mass could induce rotational inertial forces into its vicinity? Would Newton have thought that conceivable?

 

[Dale]

So you don't find it fascinating that ...

More importantly, I don't find it useful (since it is so poorly defined).

 

[Jonathan Scott]

The weird thing is that the rotation and linear frame-dragging effects predicted by GR are highly suggestive of Mach's Principle yet provably incompatible with them, as Einstein was aware.

According to GR, if local masses are rotating or accelerating, then local rest frames are slightly "dragged" in the same way. For example, if you are surrounding by a rotating shell then your gyroscopes will show that a non-rotating local frame is actually slightly rotating in the same direction relative to the fixed stars.

Similarly, if masses near you are all being accelerated in the same direction, then when you locally detect no acceleration, you will find that relative to the fixed stars you are accelerating. Unfortunately, this latter effect is so vanishingly small that it is difficult to conceive of any experiment that could detect it.

The interesting case is when you extend this to the whole universe, as described in the MTW section on the "sum for inertia". If you use plausible order-of-magnitude estimates of the masses and distances of all those masses, and consider what would happen if they were rotating slightly, the effect seems to be that your own frame of reference would rotate in the same direction at the same rate, at least to within a couple of orders of magnitude (depending on whether you count dark matter, etc). This suggests that rotation of the universe might effectively look like rotation of the observer the other way, which is a neat trick.

Even more interesting, if you do this with linear acceleration, you find that when you extend it to the whole universe, you end up with the conclusion that if the whole universe is accelerating relative to an object, then you will require a force on the object proportional to its mass to prevent it from accelerating. This is exactly the same as if the universe were at rest and the object were accelerating, giving F = ma, and provides a neat way of looking at inertia.

However, the frame-dragging expression in GR depends on a sum of terms of the form GM/r for all of the objects in the universe, requiring that sum to be a constant in order for F = ma to be reproduced locally, but clearly the sum of GM/r for everything in the universe (regardless of how it is specifically defined for distant objects) cannot be a constant unless G can vary, as we could just shuffle around local objects to make the sum vary. In GR, G cannot vary, so this means GR cannot be compatible with Mach's principle in this form and does not support this neat explanation of inertia.

 

[Ken G]

More importantly, I don't find it useful (since it is so poorly defined).

Perhaps the use of it comes in finding a clearer definition, as many have done. Einstein found a vague formulation very useful, but now that we have GR, we don't need the vague formulation any more-- to make it useful going forward, we need a more precise definition. That might serve as a guide to the next theory, just as the last one was guided by it.

On the other hand, if Brans-Dicke is ever confirmed by measuring, say, an omega of a million, then Mach's principle will have been useful once again.

 

[Ken G]

If you use plausible order-of-magnitude estimates of the masses and distances of all those masses, and consider what would happen if they were rotating slightly, the effect seems to be that your own frame of reference would rotate in the same direction at the same rate, at least to within a couple of orders of magnitude (depending on whether you count dark matter, etc). This suggests that rotation of the universe might effectively look like rotation of the observer the other way, which is a neat trick.

And that's a pretty darn Machian statement for a theory that contradicts Mach. I'm still not clear on the inevitability of that "contradiction".

However, the frame-dragging expression in GR depends on a sum of terms of the form GM/r for all of the objects in the universe, requiring that sum to be a constant in order for F = ma to be reproduced locally, but clearly the sum of GM/r for everything in the universe (regardless of how it is specifically defined for distant objects) cannot be a constant unless G can vary, as we could just shuffle around local objects to make the sum vary. In GR, G cannot vary, so this means GR cannot be compatible with Mach's principle in this form and does not support this neat explanation of inertia.

But does not the calculation you are suggesting require more than just a frame-dragging correction? Isn't there also going to have to be a boundary condition at infinity that you are frame-dragging with respect to? I'm wondering if we are not making implicit assumptions that we think are "obvious" but which are in fact the true source of the apparent breakdown of Mach's principle.

This is especially true in light of the Big Bang model. The spacetime of our universe can be associated with an interesting history in which every currently empty region was once bustling with mass-energy. That could have easily left a mark on the spatial hyperslice that we are now using for the calculation you describe, causing it to have a different boundary condition than we would use with more Newtonian thinking, rigging the result to miss its true Machian character. I believe that is what the Raine article from above was avoiding by connecting the Machian character of potential GR solutions to the actual cosmological history of the universe.

 

[Dale]

Perhaps the use of it comes in finding a clearer definition, as many have done. Einstein found a vague formulation very useful, but now that we have GR, we don't need the vague formulation any more-- to make it useful going forward, we need a more precise definition.

I wouldn't say "many", and as mentioned above the one that is both clearly formulated and widely recognized is contradicted by observation. So the utility of Machs principle is doubtful at best, IMO.

The only chance it has is that someone sometime in the future might possibly come up with a generally accepted precise formulation which might make it consistent with observation. That possibility is the only reason I wouldn't rule it out quite yet.

 

[Jonathan Scott]

Isn't there also going to have to be a boundary condition at infinity that you are frame-dragging with respect to? I'm wondering if we are not making implicit assumptions that we think are "obvious" but which are in fact the true source of the apparent breakdown of Mach's principle.

The problem is that regardless of how you treat the distant stuff and boundary conditions, if you rearrange some local masses you can theoretically make arbitrary changes to the sum, so the sum cannot remain constant.

 

[Ken G]

The problem is that regardless of how you treat the distant stuff and boundary conditions, if you rearrange some local masses you can theoretically make arbitrary changes to the sum, so the sum cannot remain constant.

So you are confirming that boundary conditions do affect the outcome of the calculation? If so, then there is a simple way to assert the Machian character of GR: for a solution to be physical, it must stem from a Machian boundary condition. Or put differently, every time you choose the boundary conditions you will use to do a GR calculation, you are making an implicit assumption about the prevailing mass distribution. Mach's principle may rest in the details of that implicit assumption, which GR does not tell us-- GR is incomplete, being a differential formulation. The incompleteness of GR is why I dispute the statement that it is categorically incompatible with any particular statement of Mach's principle we would care to use.

 

[Jonathan Scott]

I wouldn't say "many", and as mentioned above the one that is both clearly formulated and widely recognized is contradicted by observation. So the utility of Machs principle is doubtful at best, IMO.

The only chance it has is that someone sometime in the future might possibly come up with a generally accepted precise formulation which might make it consistent with observation. That possibility is the only reason I wouldn't rule it out quite yet.

Theories based on Mach's Principle predict via the "sum for inertia" scheme that G must vary with location, and GR specifically says that it does not.

Experiments confirming GR to very high accuracy in the solar system have sometimes been taken to rule out theories based on Mach's Principle because they apparently prove that G does not vary with location. However, in simple Machian theories the local variation in G has the same effect as the Newtonian potential, and the effective equivalent of the G in GR or Newtonian theory is the G value due to all other masses in the universe except the local source mass, so that value is still effectively a constant. So far we have not been able to study the value of G at locations outside the solar system to sufficient accuracy to prove that G does not vary with location in a Machian way.

Mach's Principle also says that since G depends on the sum of the M/r values, then if that sum varies with time (due to systematic changes in the total mass or average radius) then G should also vary with time. However, no variation of G with time has been detected down to rates similar to the lifetime of the universe, so this strictly limits the possible forms of any viable theory based on Mach's Principle.

 

[Jonathan Scott]

So you are confirming that boundary conditions do affect the outcome of the calculation? If so, then there is a simple way to assert the Machian character of GR: for a solution to be physical, it must stem from a Machian boundary condition. Or put differently, every time you choose the boundary conditions you will use to do a GR calculation, you are making an implicit assumption about the prevailing mass distribution. Mach's principle may rest in the details of that implicit assumption, which GR does not tell us-- GR is incomplete, being a differential formulation. The incompleteness of GR is why I dispute the statement that it is categorically incompatible with any particular statement of Mach's principle we would care to use.

I don't think anyone knows how to do an accurate calculation for the "rest of the universe". However, I think it's reasonable to assume that if local space is a solution of GR then we shuffle some local masses around, this is not going to affect the shape of the distant background, so the fact that the local masses would then have different weights in the frame-dragging expression would mean that the sum must change.

 

[Ken G]

I guess we have yet another kind of issue that different versions of Mach might deal with. I would say that the value m=F/a does not need to involve Mach's principle, that can be inherent to the particle. Mach doesn't require that the value of the mass be determined by mass everywhere else, it requires that the inertial path in spacetime be determined by mass everywhere else, and deviations from that path can be some completely different law (Newton's, basically).

If so, then it is not necessary for "m" to come out the same if we move stuff around, because m is asserted to be the same, it is inherent to the particle. Mach doesn't need to say what happens to a particle when you put a force to it, it needs to say what happens to a particle when you don't put a force to it, and for that we never need any concept of m. The "inertial path" does not require knowing the "inertia" of a particle, and it seems to me the fundamental issue with Mach is the former, not the latter.

So then the question is, how could the "mass elsewhere" determine the inertial path? It would have to be some kind of average of what all that other mass "thinks" should be the inertial path, as if all mass has a "vote" that it is itself inertial, and the weight of that vote is proportional to the mass. (This leads, for example, to the idea that a universe containing only one mass would always have that mass being inertial, and I don't see any evidence that GR is incompatible with that, because GR is incomplete on the matter of boundary conditions). That seems to me to be crux of "Mach's principle"-- that "the motion of the mass there determines what is the inertial path here." So if you move the mass around, there's no problem with that changing the inertial path here, as that's just what moving the mass around is supposed to do. You don't want m to change, but you just assert it is a local invariant.

This I believe is also what people mean when they say that the absence of rotation of the universe is evidence of Mach's principle. By "absence of rotation", I mean that a frame in which the CMB is isotropic is a frame that exhibits isotropic dynamics. The problem is, you could get that two ways-- it could be that the motions of the universal mass determines the frame that is inertial, and there is no absolute inertial frame, or it could be that there is an absolute inertial frame (or set of frames) but the universe simply does not happen to rotate in that frame. The issue can't be resolved cosmologically, except (as Raine points out, http://adsabs.harvard.edu/abs/1975MNRAS.171..507R) to note that the cosmological model failed to refute Mach even when certain GR-allowed versions could have done so. But a true resolution would require studying the frame-dragging and whatnot of a real mass distribution, and seeing if in the limit of enough nearby mass, you can "assume control" of the local inertial paths, to reflect the motion of the nearby mass. So far, I haven't heard anyone say that this is not just what happens in GR.

 

[Jonathan Scott]

GR definitely says that local acceleration or rotation has the effect of dragging frames to make that local effect relatively weaker, and if one extrapolates the effects to the scale of the universe then they appear to account for inertia and rotational effects such as coriolis and centrifugal forces. It would be very neat if this effect did add up exactly, but even Einstein agreed that GR is incompatible with Mach's Principle for multiple reasons.

Even if Mach's Principle does not hold exactly in the expected way, this sum still suggests very strongly that some form of the Whitrow-Randall-Sciama relation should hold, that is:
$$\sum_i \left ( \frac {GM_i}{R_i c^2} \right ) = n$$
for all masses in the universe as seen from any location, for some simple constant n.

If this were true, G would simply be an abbreviation for the relative gravitational effect of the rest of the universe. Brans-Dicke cosmology incorporated this idea, but as formulated it is incompatible with current experiment unless the strength of the Machian component is essentially reduced to zero.

Some people (notably James Woodward of the "Woodward effect") have suggested that ideas based on Mach's Principle require action at a distance. However, I've not seen any clear explanation of why this should be true, and as far as I can see a Machian gravity theory could operate through fields and the shape of space-time just like GR.

 

[Ken G]

GR definitely says that local acceleration or rotation has the effect of dragging frames to make that local effect relatively weaker, and if one extrapolates the effects to the scale of the universe then they appear to account for inertia and rotational effects such as coriolis and centrifugal forces. It would be very neat if this effect did add up exactly, but even Einstein agreed that GR is incompatible with Mach's Principle for multiple reasons.

What's not clear to me is if Einstein was going along with the standard uses of boundary conditions, perhaps not realizing the subtle ways that choosing a boundary condition is tantamount to letting Mach's principle in the back door, even as it seems to be failing to get in the front door. Since we are agreed that GR cannot tell you the inertial trajectories by itself, without some added postulate about the proper boundary condition, how can we really say that GR is or is not compatible with Mach without saying what we are doing with those boundaries? Are we not always imagining a mass distribution embedded in a finite portion of a much larger Minkowskian spacetime ("asymptotically flat"), and do we not have to arbitrarily select the frame of that Minkowski spacetime? Is there any other way to calculate the geodesics? Or if we are doing a cosmological metric, we don't need to assume asymptotic flatness, but we need the cosmological principle instead, and that's about as Machian as they come, but in a kind of concealed way because the inertial frames are the ones that move with the average matter.

If this were true, G would simply be an abbreviation for the relative gravitational effect of the rest of the universe. Brans-Dicke cosmology incorporated this idea, but as formulated it is incompatible with current experiment unless the strength of the Machian component is essentially reduced to zero.

This is my other question-- how do we know how small the Machian component can get and still be Mach's principle? Doesn't a large "omega," say in Brans-Dicke, simply mean that you need a lot of external mass to "impose" its frame onto the local inertial frame? I don't see why that's non-Machian, Mach never said how much external mass you need to establish a concept of relative motion to the local inertial frame.

You are clearly knowledgeable about Brans-Dicke, and the Whitrow-Randall-Sciama relation-- what I don't understand in that relation is it looks like it is trying to be the Newtonian gravitational potential. But making that constant everywhere on global scales is like the cosmological principle, so that's fine, but on smaller scales it would seem to preclude having gravity make things orbit other things and so forth.

 

[Q-reeus]

A list of Mach Principles (as opposed to Mach's Principle): http://en.wikipedia.org/wiki/Mach_principle - #8 seems the most precise and relevant here.
I lean to Ken G's pov, but wish to dispel what is imo a longstanding and totally wrongheaded assumed Machian 'symmetry' re relative rotation - that, in terms of the centripetal forces generated, rotation of the rest of the universe about a fixed object is somehow equivalent to the usually assumed rotation of said object wrt a fixed universe background.
Wikipedia on Mach's Principle (http://en.wikipedia.org/wiki/Mach's_principle) states:
"The idea is that the local motion of a rotating reference frame is determined by the large scale distribution of matter, as exemplified by this anecdote:
You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?
Mach's principle says that this is not a coincidence—that there is a physical law that relates the motion of the distant stars to the local inertial frame. If you see all the stars were whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force..."

Whether Mach himself ever put it just that way or not, it is imo a silly 'optical illusion' perspective. This should be immediately apparent by asking how far out the 'whirling stars' could extend and still be consistent with the apparent speed of such stars being less than c. Clearly this depends on the rotation velocity of the central object, and if it's rim speed approaches c, the 'rest of the universe' consistent with v<c must be very close indeed! Or can someone suggest a super-relativistic transformation rule to apply to said hypervelocity whirling stars? I would suggest there is *never* any sensible symmetry wrt to relative rotation that can *properly* connect to Mach's principle. The correct perspective is that of relative instantaneous motion of each and every constituent 'point' mass wrt to the rest of the universe. If the central rotating object is say a flywheel having constant angular velocity omega, pick any 'point mass' within, moving in a circle of fixed radius r. Then the Machian relative dynamics is that of the rest of the universe undergoing gyratory (*not* rotational) motion - all the distant stars are then uniformly moving with the equal and opposite speed -r*omega (always v<c!), and centripetal acceleration ~ r*(omega)². Sum for all constituent point masses and centripetal force pops out correctly I would strongly suspect, and there seems no contradiction with a proper application of Mach's principle and any sensible metric gravitational theory. No silly hypervelocity to try and skirt by weakly stipulating 'slow rotation' as is often done.
Maybe this is all clearly understood by 'authorities' but I have never seen it put this way. Anyways, while no fan of Julian Barbour's ideas on 'non-time', re Mach's principle he certainly has no trouble finding it in, and in his analysis directly leading to, GR: http://lanl.arxiv.org/abs/1007.3368 http://www.springerlink.com/content/c14021m546l58512/

 

[Ken G]

Interesting post. I think you make a good point about gyratory motion as one way to analyze the frame of reference of a rotating object, and it would be interesting indeed to establish that Mach's principle (of some flavor) is satisfied by that kind of motion. But I'm not sure if we wouldn't still have an issue about rotating frames. Relativity is supposed to be about observers doing physics, so the point you are making brings us right into the question of whether an observer is always a point particle, so that a solid-body-rotating observer is really a kind of amalgamation of individual observer-points undergoing gyratory motion, or if an observer is really allowed to be a physical object with finite extent that can actually spin. If the latter is an allowed concept of an observer, then we must still address Mach's principle from the perspective of a rotating frame. But if relativity in its purest sense really only applies to ideal pointlike observers that don't actually spin but can have spinlike relationships to other pointlike observers, and practical observers are some kind of approximation for whom Mach's principle actually breaks down because it requires a sum over ideal observers, then I can see your point. You would be saying we can still do GR for rotating observers, but it is no longer a fundamental theory, and that's why Mach's principle can break down (if indeed it does, that's still not clear) for rotating observers.

 

[Q-reeus]

Hi Ken - this is the same Ken that encountered 'problems' re moderator 'discretionary policy' a while back in another section, right? Glad you found it in you to stick around!

...Relativity is supposed to be about observers doing physics, so the point you are making brings us right into the question of whether an observer is always a point particle, so that a solid-body-rotating observer is really a kind of amalgamation of individual observer-points undergoing gyratory motion, or if an observer is really allowed to be a physical object with finite extent that can actually spin...

Of course I agree an observer can (and sensibly must) be an extended object. My only point is if one wants the correct dynamics, the amalgam-of-individual-point-masses-each-reacting-individually-to-the-rest-of-the-universe approach is imo the only consistent perspective for reasons given. And yes this distinction evaporates to a certain extent when considering eg extended stationary observer surrounded by a rotating mass shell - but which is not symmetric wrt said observer rotating wrt a fixed mass shell. Rotating frames of reference is a coord transformation thing useful and necessary in general but also not really germaine to this argument.

...But if relativity in its purest sense really only applies to ideal pointlike observers that don't actually spin but can have spinlike relationships to other pointlike observers, and practical observers are some kind of approximation for whom Mach's principle actually breaks down because it requires a sum over ideal observers, then I can see your point. You would be saying we can still do GR for rotating observers, but it is no longer a fundamental theory, and that's why Mach's principle can break down (if indeed it does, that's still not clear) for rotating observers.

In my view Mach's principle is never breaking down here, merely the wrong application of perspective. Whether MP is *totally applicable* under all situations I don't pretend to know, but can't think of any offhand!

 

[Jonathan Scott]

You are clearly knowledgeable about Brans-Dicke, and the Whitrow-Randall-Sciama relation-- what I don't understand in that relation is it looks like it is trying to be the Newtonian gravitational potential.

OK, here's how that bit works. It's a slightly more general version of Sciama's scheme in his "Origin of Inertia" paper, in that it uses an arbitrary constant $n$ instead of the special-relativity-based value of 1 in the W-R-S relation.

Consider the sum of the M/R terms due to all of the masses in the universe as seen from a given location, but split out the local interesting mass as m/r. The overall value of the Machian variable Newtonian gravitational constant $G_N$ then satisfies the W-R-S relation as follows:

$$G_N \left ( \sum_i \frac{M_i}{R_i c^2} + \frac{m}{r c^2} \right ) = n$$
so
$$G_N = \frac{n}{\left ( \sum_i \frac{M_i}{R_i c^2} + \frac{m}{r c^2} \right )}$$
Now consider the local "standard" fixed value of this constant far away from the local object but still in the same vicinity:
$$G_S \left ( \sum_i \frac{M_i}{R_i c^2} \right ) = n$$
so
$$G_S = \frac{n}{\left ( \sum_i \frac{M_i}{R_i c^2} \right )}$$

We can then substitute these in each others' expressions:
$$\frac{G_N}{G_S} = \frac{1} {\left ( 1 + \frac{1}{n} \frac{G_S \, m}{r c^2} \right ) } = \left ( 1 - \frac{1}{n} \frac{G_N \, m}{r c^2} \right )$$

The Newtonian potential (time dilation) factor is then (to first order) given by
$$\left ( \frac{G_N}{G_S} \right )^n \approx \left ( 1 - \frac{G_N \, m}{r c^2} \right ) \approx \frac{1} {\left ( 1 + \frac{G_S \, m}{r c^2} \right ) }$$

To illustrate how this might relate to a theory such as GR where $G$ is a universal constant, consider the Newtonian time dilation in the specific case where $n = 2$.
$$\left ( 1 - \frac{G_N \, m}{r c^2} \right ) = 1 - \frac{G_S \,m/r c^2}{1 + G_S\,m/2rc^2} = \frac{1 - G_S \, m/2rc^2} {1 + G_S \, m/2rc^2}$$
The final expression here is identical to the GR Schwarzschild solution time dilation factor expressed in isotropic coordinates in terms of the standard constant $G$ yet $G_S$ here is only locally constant.

(Please feel free to check my maths, especially for errors in signs and factors of two!)

This illustrates that the experimental proof that the G value used in GR is a constant for purposes of solar system experiments does not on its own rule out Machian theories which involve a value of G that varies with location.

 

[Q-reeus]

...This illustrates that the experimental proof that the G value used in GR is a constant for purposes of solar system experiments does not on its own rule out Machian theories which involve a value of G that varies with location.

Can it be taken here that G_s is specifically applied to both active gravitational mass m_a and passive mass m_p, and that the latter follows from applying WEP - ie m_p = m_i. So there is a chain of logic leading from Mach's principle that originally dealt only with inertia hence inertial mass m_i, and the other two are 'derived' from that?

 

[Jonathan Scott]

Can it be taken here that G_s is specifically applied to both active gravitational mass m_a and passive mass m_p, and that the latter follows from applying WEP - ie m_p = m_i. So there is a chain of logic leading from Mach's principle that originally dealt only with inertia hence inertial mass m_i, and the other two are 'derived' from that?

Sorry, I can't give you a good answer at the moment. I often have difficulty handling these sort of questions nowadays, because once you start talking about ideas which are not part of GR, it's tricky to remember exactly what holds and what doesn't, and it's years since I had time to really think about this area. Also, I've just had Sunday lunch.

I certainly generally assume active and passive mass are the same, and I seem to remember that in Sciama's type of Machian approach, all masses and distances are relative anyway; if you double all mass values in the universe, or all distances, nothing different happens.

 

[Q-reeus]

...Also, I've just had Sunday lunch...

Can't argue with that - hope it was nice.

...I certainly generally assume active and passive mass are the same, and I seem to remember that in Sciama's type of Machian approach, all masses and distances are relative anyway; if you double all mass values in the universe, or all distances, nothing different happens.

Presumably this is relative to some given time-like hyperslice. Not at all sure but had thought Hubble redshift could be interpreted as Mach's principle operating across different eras - ie back then with matter being crowded together, inertia was much greater wrt 'now', hence 'time/light-frequency much slower'. I realize 'expanding space' is the usual interpretation, but just like with ordinary gravitational redshift there can be alternate pov's ('tiring' light or slower clocks in that case).