Is the Sagnac effect a reliable measure of rotation in Kerr space-time?

[RiccardoVen]

Hi,
I've just started Gravitation and Inertia by Wheeler and Ciufolini. This is really a sort of trasure to me, since I was looking for something about it since when I read Einstein's relativity papers, involving Mach's principle.
The main point is really simple "Inertia here arises from mass there". I've not read carefully yet the Cauchy problem for GR, but it's depicted here everything, if "well" prepared, raises to a deterministic behaviour with GR ( which makes sense to me ).
My main ( subtle ) problem is about Mach's principle. Of course I can see it can be really well encompassed by GR, since we could think about "far stars influence" as carved within the spacetime curvature.
So, from here we can see Inertia ( or locally intertial frames, as pointed out many times in the book ) is actually influenced by the fixed stars.
So, I got two problems arising here:

1) Inertia is also defined to be the property of matter to keep its state of motion ( sorry if this is a bit rough or not matematically better defined so far ), so its propery to fight against changing in motion. This is also used in inertial frame definition, i.e. for Newton a frame in which law of inertia is valid, or for GR in which a frame is free floating.
So my doubt is: let's imagine a universe in which there are no far fixed stars. Does this mean there would be no inertia at all? Better: does a universe in which there's a just a particle ( or a couple of, in order to have an observer as well ) this particle would not keep its state of motion?
I can't believe about this, so I guess something is wrong in my reasoning above.

2) does this mean, all in all, everything is gravity? I mean, even inertia is due to gravity, from this picture.

Sorry for my noob's questions as always.

Regards thanks

 

[bcrowell]

So my doubt is: let's imagine a universe in which there are no far fixed stars. Does this mean there would be no inertia at all? Better: does a universe in which there's a just a particle ( or a couple of, in order to have an observer as well ) this particle would not keep its state of motion?
I can't believe about this, so I guess something is wrong in my reasoning above.

What you're asking about is called Mach's principle. You may find this useful:

http://physics.stackexchange.com/questions/5483/is-machs-principle-wrong

 

[WannabeNewton]

Hey Riccardo. As an aside, "Gravitation and Inertia"-Wheeler and Ciufolini is one of my most favorite GR texts as well :)

Now you should know that Mach's principle is not a rigorously formulated physical principle let alone a tenable one in GR. Inertia is, albeit to a lesser extent, also not a precisely defined term across the literature. This makes threads about Mach's principle and inertia hard to entertain as there is very little rigorous terminology and physics being employed from the outset.

(1) The point Wheeler is trying to make is, inertial frames are not determined just by the local gravitational field but also by the boundary conditions imposed on the space-time e.g. at spatial infinity. This is clear from the standpoint of PDEs. It is the choice of boundary conditions that determines the status of Mach's principle. In Kerr space-time one has asymptotically flat boundary conditions and a rotating central mass and as a result one has a clear violation of Mach's principle in which the local standard of non-rotation relative to coincident gyroscopes (the absolute rotation) is different from the global standard of non-rotation relative to spatial infinity (relative to the distant stars).

(2) All fields in space-time are affected by gravity since they all have a canonical energy-momentum tensor coupling to the metric but apart from that I can't make sense of your statement "everything is gravity".

 

[RiccardoVen]

What you're asking about is called Mach's principle. You may find this useful:

http://physics.stackexchange.com/questions/5483/is-machs-principle-wrong

Thanks for that link, I will read it carefully.
If it's not a problem to you, may you eventually tell me your opinion about it, please?
I've ready many post from you, and mainly the one in which you were defining an intertial frame in a slightly different way than others ( i.e. this one https://www.physicsforums.com/showthread.php?t=437784 )
I guess this definition may eventually involve a personal different view about inertia and Mach's principle as well.

 

[RiccardoVen]

Hey Riccardo. As an aside, "Gravitation and Inertia"-Wheeler and Ciufolini is one of my most favorite GR texts as well :)

Hi WBN,
yes I know you suggested me about it. Incidentally ( and luckily ) I had already purchased a copy of it and then I started reading it a bit.

Now you should know that Mach's principle is not a rigorously formulated physical principle let alone a tenable one in GR. Inertia is, albeit to a lesser extent, also not a precisely defined term across the literature. This makes threads about Mach's principle and inertia hard to entertain as there is very little rigorous terminology and physics being employed from the outset.

Yes I was a bit aware about it.

(1) The point Wheeler is trying to make is, inertial frames are not determined just by the local gravitational field but also by the boundary conditions imposed on the space-time e.g. at spatial infinity. This is clear from the standpoint of PDEs. It is the choice of boundary conditions that determines the status of Mach's principle. In Kerr space-time one has asymptotically flat boundary conditions and a rotating central mass and as a result one has a clear violation of Mach's principle in which the local standard of non-rotation relative to coincident gyroscopes (the absolute rotation) is different from the global standard of non-rotation relative to spatial infinity (relative to the distant stars).

I've still to understand a bit better about gyroscopes and, hopefully, Wheeler's book will help about it. So if Mach's principle is all in all determined by boundary conditions fixing a priori, it means my question about the alone particle is a bit meaningless, I guess.
I mean, with just one particle there would not be any stress energy tensor at all due to distant stars. Does this mean no Inertia would be present, in that case?

(2) All fields in space-time are affected by gravity since they all have a canonical energy-momentum tensor coupling to the metric but apart from that I can't make sense of your statement "everything is gravity".

Yes, I've bit abused terms in that sentence, sorry for that. Of course I really meant what you stated above. Neverthelss, I'm still not able to figure out a "straightforward" derivation of Inertia from spacetime.
I've still to work out it deeper, probably. I mean, equivalence principle is stating about the equivalence of gravitational field and an inertial one ( and the equivalence of respective masses as well, indeed ). But I'm not yet able to derive directly in mind a mathematical model of Inertia from it.

 

[bcrowell]

Thanks for that link, I will read it carefully.
If it's not a problem to you, may you eventually tell me your opinion about it, please?

I wrote one of the answers there.

 

[WannabeNewton]

I mean, with just one particle there would not be any stress energy tensor at all due to distant stars. Does this mean no Inertia would be present, in that case?

Mach's principle states that acceleration and rotation are relative, particularly with respect to the distant stars. This is clearly not true in GR wherein acceleration and rotation are absolute and one can talk about the acceleration and rotation of a lone body in an otherwise empty universe without any need for distant stars.

In GR acceleration of a body is given (without any reference to the distant stars in any form whatsoever) by $a = \nabla_u u$ and is measured simply by a comoving accelerometer. Rotation of a Lorentz frame is given by $F_u e_{\alpha} = \omega_{\alpha}{}{}^{\beta}e_{\beta}$ where $F_u$ is the Fermi derivative, $e_{\alpha}$ is a Lorentz frame, and $\omega_{\alpha\beta}$ is its rotation and is measured by a comoving compass of inertia. If the Lorentz frame is attached to a world-line belonging to a time-like Killing field $\xi$ then the absolute rotation is simply $\Omega \propto \xi^{\flat}\wedge d\xi^{\flat}$ which ties back to the above comment about the disagreement between the local rotation $\Omega$ and rotation relative to spatial infinity in Kerr space-time.

 

[RiccardoVen]

Mach's principle states that acceleration and rotation are relative, particularly with respect to the distant stars. This is clearly not true in GR wherein acceleration and rotation are absolute and one can talk about the acceleration and rotation of a lone body in an otherwise empty universe without any need for distant stars.

In GR acceleration of a body is given (without any reference to the distant stars in any form whatsoever) by $a = \nabla_u u$ and is measured simply by a comoving accelerometer. Rotation of a Lorentz frame is given by $F_u e_{\alpha} = \omega_{\alpha}{}{}^{\beta}e_{\beta}$ where $F_u$ is the Fermi derivative, $e_{\alpha}$ is a Lorentz frame, and $\omega_{\alpha\beta}$ is its rotation and is measured by a comoving compass of inertia. If the Lorentz frame is attached to a world-line belonging to a time-like Killing field $\xi$ then the absolute rotation is simply $\Omega \propto \xi^{\flat}\wedge d\xi^{\flat}$ which ties back to the above comment about the disagreement between the local rotation $\Omega$ and rotation relative to spatial infinity in Kerr space-time.

Thanks WBN, I've still to reach that level on knowledge about killing vectors and similar in GR. But I will use this answer for future as reference for sure.
My question was less math oriented, and was focused more on definition of Inertia. I was wondering if without distant stars, an alone body would keep its motion.
I mean, what it's exactly the Inertia? May be this question is more "Newton" oriented than "GR" oriented.
But in mind this is the reasoning ( and I found no answer yet on books ): does Inertia really depends from distant stars presence? Is there anything from distant stars which forces a body to keep its motion even when no acceleration is present.
I know ( now ) the answer from a GR POV. But probably my doubt is about Mach's principle, so I should solve doubts about it first.

 

[WannabeNewton]

I was wondering if without distant stars, an alone body would keep its motion.

Again this depends on the theory being discussed. Mach's principle is not a theory of physics, it is a principle which can be incorporated into physical theories. If one sticks to the standard Newtonian and general relativistic frameworks then yes a particle with no forces acting on it will continue in uniform motion until acted upon by another force, regardless of whether or not there are distant stars. This is made very evident in GR with the advent of Minkowski space-time as a solution to the EFEs. Minkowski space-time is entirely empty as far as mass-energy sources go but in it we can have test observers who can measure their accelerations, or lack thereof, using local instruments and local instruments alone. The same goes for rotation. In GR there is nothing stopping you from using global or quasi-local measures of rotation such as rotation relative to spatial infinity or rotation as measured by counter-propagating light signals in closed circuits (Sagnac effect) but in GR these will not agree in general and will also in general disagree with the local and "absolute" measure of rotation using comoving gyroscopes.

I mean, what it's exactly the Inertia?

It is the resistance to change in motion when subject to forces.

But in mind this is the reasoning ( and I found no answer yet on books ): does Inertia really depends from distant stars presence? Is there anything from distant stars which forces a body to keep its motion even when no acceleration is present.
I know ( now ) the answer from a GR POV. But probably my doubt is about Mach's principle, so I should solve doubts about it first.

As stated above Mach's principle is not a physical theory, it is a component of theories so in order to assess whether or not its tenets are present in nature one must first adopt a valid physical theory and proceed from there. If one adopts GR as the theory of space-time upon which Mach's principle is to be assessed in the sense you are interested in then the answer is: no, one does not require the distant stars for such things.

 

[RiccardoVen]

As stated above Mach's principle is not a physical theory, it is a component of theories so in order to assess whether or not its tenets are present in nature one must first adopt a valid physical theory and proceed from there. If one adopts GR as the theory of space-time upon which Mach's principle is to be assessed in the sense you are interested in then the answer is: no, one does not require the distant stars for such things.

I will to quote directly the piece of Wheeler's book I'm referring to, and which is making a bit of trouble you saw above.
Page 4, bottom:

"Let us bring out the main idea on what we may call poor man's language. Inertia here, in the sense of local inertial frames, that is the grip of spacetime here on mass here, is fully defined by the geometry, the curvature, the structure of spacetime here ( my note: so far, so good ).
The geometry, here, however, has to fit smoothly to the geometry of the immediate surroundings; those domains onto their surroundings; and so on, all the way around the great curve of space. Moreover, the geometry in each local region responds in its curvature to the mass in that region. Therefore every bit of momentum-energy, wherever located, makes its influence felt on the geometry of space thourghout the whole universe--and felt, thus, on inertia right here."

It seems to me, from this words, I can see Inertia here is due to every bit of momentum-energy in the universe, i.e. even distant stars. If I had to continue this reasoning I would say:

"...and so, without that emergy-mass distribution, there wouldn't be any inertia at all, felt here."
And also, I can deduce inertia is here, just because some mass-energy is bending spacetime somewhere else.
So the curvature is actually causing inertia. My real and final question: is this the only thing causing it?
I'm not speculating around it, my question is not phylosophical. I'm just trying to understand if "the great curve of space" mentioned in the book, is actually causing inertia here.
So: without that great curve of space, if a force was acting on a particle, would this latter react the same, opposing to the force?

Hope my doubts are a bit more clear, thanks.

 

[WannabeNewton]

"...and so, without that emergy-mass distribution, there wouldn't be any inertia at all, felt here."

This is not a valid conclusion from the quoted passage. The passage simply states that energy-momentum throughout a given space-time contributes to the influences on the local inertial frames at any given event in a very loose sense through the influences said energy-momentum has on nearby regions of space-time. It never claims that a non-vanishing energy-momentum is necessary in order to define local inertial frames and indeed it isn't as evidenced by the existence of inertial frames in Minkowski space-time.

 

[Bill_K]

As stated above Mach's principle is not a physical theory, it is a component of theories so in order to assess whether or not its tenets are present in nature one must first adopt a valid physical theory and proceed from there. If one adopts GR as the theory of space-time upon which Mach's principle is to be assessed in the sense you are interested in then the answer is: no, one does not require the distant stars for such things.

It never ceases to amaze me, the psychological appeal that Mach's Principle has. I think it's partially because it's NOT a theory - there are no equations to master, and yet you're left with the feeling that now you really understand something about the universe! It's almost astrological - the idea that the "distant stars" exert a local influence. It's certainly in conflict with causality.

Mach's Principle says, "The local inertial frame is determined by the presence and motion of distant objects."
General Relativity says, "The local inertial frame is affected by the presence and motion of nearby objects."

 

[WannabeNewton]

Mach's Principle says, "The local inertial frame is determined by the presence and motion of distant objects."
General Relativity says, "The local inertial frame is affected by the presence and motion of nearby objects."

That's a very elegant way to put it Bill! And I agree wholly with your previous sentiments. The lack of any actual mathematical formulation of the principle leads to all kinds of misinterpretations and even reinterpretations of its original formulation by Mach, as is made evident through a plethora of statements in Wheeler and Ciufolini's text.

I also don't get why a principle formulated by a person like Mach has managed to carry such weight even to the present day but that's another story!

:)

 

[RiccardoVen]

The lack of any actual mathematical formulation of the principle leads to all kinds of misinterpretations and even reinterpretations of its original formulation by Mach, as is made evident through a plethora of statements in Wheeler and Ciufolini's text.

So, actually you agree roughly with me that book contains sentences which could be easily misunderstood, in a way. Probably I should have a stronger background ( physical, non math-oriented ) on GR before fully appreciating it.

 

[WannabeNewton]

So, actually you agree roughly with me that book contains sentences which could be easily misunderstood, in a way. Probably I should have a stronger background ( physical, non math-oriented ) on GR before fully appreciating it.

I think even then it will still lead to misunderstandings because the wordings and explanations in that book tend to be very cryptic and unclear as is rather characteristic of Wheeler.

 

[UltrafastPED]

I also don't get why a principle formulated by a person like Mach has managed to carry such weight even to the present day but that's another story!

It's because Mach was a respected physicist/philosopher in his day, and Einstein was much taken by his approach. But Einstein eventually abandoned Mach's principle ... for the good reason that GR was inconsistent with it!

This is discussed in his scientific biography, "Subtle is the Lord ...".

 

[RiccardoVen]

I think even then it will still lead to misunderstandings because the wordings and explanations in that book tend to be very cryptic and unclear as is rather characteristic of Wheeler.

Personally, but this is just my opionion of course, I really like his style, and this is the style I'd like to had in my past from my teachers.
That said, I continued in reading some pages around the one I quoted. In those ones, he really stated things in a clear way, at least to me. In following the final goal to define "poor man's inertia" and to support the main topic of the whole book, i.e. "inertia here arises from mass there", he does some estimations about how much mass there could influence inertia here.
After some very rough and intuitive computations, he's estimating ( if I remember correctly, I've not the book here while writing ), on average, a mass far 10^28m has a chance to influence inertia here.
Again, continuining with this mood, he does a menthal experiment in which tries to put a gyroscope in plain free space. This gyroscope, from what he wrote, ( again I quote roughly here ) is able to align to "distant stars" even "if it's really cloudy and it is not able to see them" ( he also defined the local gyroscope as a sort of compass for inertia ).

So, as you stated, some sentences are quite loosy and foggy. Nevertheless it's very clear in what it states:

"inertia here araises from mass there, and that mass could be, on average, far 10^28m to contribute to inertia here".

I'm sure it will become more clear going on with the book, but if I'd had to take this chapter without having posted this theread, I would conclude "the great curve spacetime" is actually the cause of inertia. And for supporting it, he puts in game some real "non local" reasoning, which clash with local properties of GR ( at least as my concern ).

How those will be merged and melt with GR, I will discover going on with it.

EDIT: please don't let me be misunderstood. I'm not stating Mach's principle is correct or trying to resume old fashioned theories. I'm just trying to resolve this clashing between what Wheeler is saying in the very first pages of this book and GR. I'm just trying to understand. Not trying to create newage theories or so.

 

[A.T.]

It never ceases to amaze me, the psychological appeal that Mach's Principle has.

I think it's appealing, because it is an attempt to take relativity one step further and apply it more consequently, which would have a certain elegance, if it would actually work out. But elegance it not the main criteria and it is eventually one step too far.

the idea that the "distant stars" exert a local influence. It's certainly in conflict with causality.

That's hard to say without an actual quantitative theory. It doesn’t have to be an instantaneous influence over large distance, just an artifact of the formation of the universe, when it all was much closer together. Of course this experiment is not easily reproducible, so even if there was math it might be not falsifiable experimentally. The core question is this:

Is it just a coincidence that the local measurement of rotation (based on inertia, Sagnac) matches the global measurement of rotation (based on light from distant stars) ?

The answer might be yes, but it seems like weird coincidence, hence the appeal of ideas that seek a connection.

 

[WannabeNewton]

I think statements like "inertia here arises from mass there" are far too vague to be of any use in understanding GR, even though such statements are abound in Wheeler's book. For example the interior of a thin spherically symmetric shell of mass is just Minkowski space-time as if the massive shell wasn't even there (the only way to distinguish between the shell being there and the shell not being there, even though the interior is Minkowski space-time in both cases, is through global experiments that connect to the Schwarzschild solution exterior to the shell e.g. by drilling a small hole in the shell and having light signals come through it). On the other hand if the massive shell is made to rotate uniformly about an axis through its center then a gyroscope inside the shell will precess relative to spatial infinity so "inertia here arises from mass there" is just far too imprecise a statement to even be of use in interpreting the nature of general relativity.

 

[RiccardoVen]

I think it's appealing, because it is an attempt to take relativity one step further and apply it more consequently, which would have a certain elegance, if it would actually work out. But elegance it not the main criteria and it is eventually one step too far.That's hard to say without an actual quantitative theory. It doesn’t have to be an instantaneous influence over large distance, just an artifact of the formation of the universe, when it all was much closer together. Of course this experiment is not easily reproducible, so even if there was math it might be not falsifiable experimentally. The core question is this:

Is it just a coincidence that the local measurement of rotation (based on inertia, Sagnac) matches the global measurement of rotation (based on light from distant stars) ?

The answer might be yes, but it seems like weird coincidence, hence the appeal of ideas that seek a connection.

Hi A.T., good to hear from you.
So basically, I can assume Wheeler is somehow supporting this idea in his book? I mean, he's not stating there's any instant "signal" from distant stars ( mass there ) to local mass ( here ). He states inertia here is due to mass here, but mass here is influenced to a little neighbours and so on and so forth up to the "large scale".
As I stated above, I'm not supporting any version of it. I'm just trying to understand a bit better the whole thing.

 

[RiccardoVen]

I think statements like "inertia here arises from mass there" are far too vague to be of any use in understanding GR, even though such statements are abound in Wheeler's book. For example the interior of a thin spherically symmetric shell of mass is just Minkowski space-time as if the massive shell wasn't even there (the only way to distinguish between the shell being there and the shell not being there, even though the interior is Minkowski space-time in both cases, is through global experiments that connect to the Schwarzschild solution exterior to the shell e.g. by drilling a small hole in the shell and having light signals come through it). On the other hand if the massive shell is made to rotate uniformly about an axis through its center then a gyroscope inside the shell will precess relative to spatial infinity so "inertia here arises from mass there" is just far too imprecise a statement to even be of use in interpreting the nature of general relativity.

Yes I agree with you this is yet another "Wheeler sentence" ( like the therm Black Holes, or mass tells to space how to bend, etc etc ) and it's some sense vague and foggy.
Nevetheless, as stated, Wheeler is supporting this statement with really precise extimations, so it seems to me it's more than a loosy sentence, in his mind.
Probably the whole book in centered around this concept, but as always Wheleer did in his books, he usually support idea with rough intuitive sketches and extimations on raw paper.

Again, I hope the remaining part of the book will offer to me more insights to it ( and I guess so, since you told me this is one of your favorite ones, despite Wheeler "marketing sentences" )

 

[Bill_K]

The core question is this:

Is it just a coincidence that the local measurement of rotation (based on inertia, Sagnac) matches the global measurement of rotation (based on light from distant stars) ?

But they don't match. That ancient incantation is at the heart of the Mach Myth, isn't it, and the fact is plain and simple: they do not match.

 

[A.T.]

But they don't match.

Interesting. How much is the difference and against what exactly was the local rotation compared?

 

[Bill_K]

As I said above,

General Relativity says, "The local inertial frame is affected by the presence and motion of nearby objects."

The effects of GR (and SR) on the local inertial frame are numerous. Several of them apply to a satellite orbiting a central body. And they go under a number of names: frame-dragging, Lense-Thirring Effect, Geodetic Effect, de Sitter Precession, Schiff Precession, and so on. Depending on whether the satellite is in orbit or stationary and whether the central body is rotating. GR effects are all very small, depending on the motion of nearby masses (falling off as 1/r³), not large effects depending on distant ones as Mach would have it.

Even the original Lense-Thirring example of the rotating spherical shell, which is often quoted as support for Mach's Principle, actually helps to disprove it, since the inertial frame inside the shell differs from the inertial frame at infinity, and the rotation rate inside fails to match the rotation rate of the shell itself.

Interesting. How much is the difference and against what exactly was the local rotation compared?

Wikipedia gives a long list of experiments designed to measure precessional effects, with varying degrees of success. The best result seems to be Gravity Probe B:

Principal investigators at Stanford University reported on May 4, 2011, that they had accurately measured the framing effect relative to the distant star IM Pegasi, and the calculations proved to be in line with the prediction of Einstein's theory. The results, published in Physical Review Letters measured the geodetic effect with an error of about 0.2 percent. The results reported the frame dragging effect (caused by the Earth's rotation) added up to 37 milliarcseconds with an error of about 19 percent.

 

[A.T.]

Wikipedia gives a long list of experiments designed to measure precessional effects, with varying degrees of success. The best result seems to be Gravity Probe B:

The precession here is caused by the nearby Earth. But if there was no rotating mass nearby the probe wouldn't precess relative to the distant stars, correct? The question in post #18 referred to the later scenario.

GR effects are all very small, depending on the motion of nearby masses (falling off as 1/r3), not large effects depending on distant ones as Mach would have it.

Difficult to judge without a quantitative Machian theory.

Even the original Lense-Thirring example of the rotating spherical shell, which is often quoted as support for Mach's Principle, actually helps to disprove it, since the inertial frame inside the shell differs from the inertial frame at infinity, and the rotation rate inside fails to match the rotation rate of the shell itself.

As far I understand it, the Machian idea doesn't exclude the effect of near masses. I interpret the term "distant stars" as "all the other matter in the universe, including nearby matter". The question is:

Absent of known local effects, why does the local rotation measurement match the rotation relative to the distant matter?

 

[PAllen]

Absent of known local effects, why does the local rotation measurement match the rotation relative to the distant matter?

Since GR admits solutions in which this is false, it does not contain any explanation of this fact.

 

[Jonathan Scott]

In GR, frame dragging due to rotation has been confirmed (e.g. by Gravity Probe B), and it is also predicted that linear acceleration will cause a linear form of frame dragging too, although so far there is no known experimental way to confirm this.

What is really interesting is that if you calculate the frame-dragging effect of incorrectly assuming that you are at rest when you are actually rotating or accelerating relative to other masses in the universe (so that relative to you, the rest of the universe is rotating or accelerating), then the result is definitely of the right order of magnitude to explain inertia. This seems a tremendous coincidence when the values involved are so huge, and suggests that the match should have some physical significance, and perhaps be exact. This is mentioned in MTW "Gravitation" for example in the section on the "Sum for Inertia", and is a constant theme in "Gravitation and Inertia".

The odd thing is that according to GR, the strength of gravity is a universal constant, which cannot vary according to the distribution of matter both locally and within the universe as a whole, so it therefore apparently cannot adjust in any way to make this astonishing match anything other than a temporary coincidence. As Machian theories typically imply a variable G, the current experimental evidence (for example from Lunar Laser Ranging) that G does not vary detectably with time and location rules out most such theories directly.

I've personally found this coincidence sufficiently compelling that I've been trying to see if it's possible to find a Machian theory which is compatible with experimental observations of constant G.

It is quite trivial to find a type of Machian theory which gives the same results as standard GR in the vicinity of a central mass, in that it is possible to match the GR equations for the Schwarzschild solution exactly with a Machian equivalent in which G is an abbreviation for the effectively constant gravitational effect of everything in the universe except the local central mass. In this theory, the Einstein field equations are an approximation which is only very accurate for a case with a single central mass, where the effect of everything else in the universe can be assumed to be constant or negligible.

However, it is much more difficult to find a Machian explanation for why the Lunar Laser Ranging experiment suggests that any current rate of variation in G is less than 1.5% within the age of the universe, and without such an explanation, Machian theories do not currently appear to be viable.

 

[A.T.]

Since GR admits solutions in which this is false, it does not contain any explanation of this fact.

But in the Gravity Probe B experiment the precession was measured relative to a distant star, under the assumption that without the Earth nearby there would be no precession relative to that distant star. How is this assumption justified?

 

[RiccardoVen]

This is mentioned in MTW "Gravitation" for example in the section on the "Sum for Inertia", and is a constant theme in "Gravitation and Inertia".

Thanks JS, this spoiler is what I was more or less searching for justifing a bit more my interest in reading Gravitation and Inertia. The more I read, the more I feel this could be a parallel way to learn GR ( I mean not diving into Machian theories, but understanding really in deep inertial forces and inertia within GR context ).

 

[Bill_K]

The question is:

Absent of known local effects, why does the local rotation measurement match the rotation relative to the distant matter?

More generally, why is the particular universe we live in appear to be homogeneous, isotropic, nonrotating and flat. Of all the possible cosmologies that have been studied, ours is the simplest and the most boring. This is a question that cosmic inflation needs to answer.

 

[PAllen]

But in the Gravity Probe B experiment the precession was measured relative to a distant star, under the assumption that without the Earth nearby there would be no precession relative to that distant star. How is this assumption justified?

W assume our universe is an FLRW solution. There is no reason for this to be true within GR as a theory. We pick boundary specific conditions such that the solution with those conditions matches observation. Unless the theory forces those boundary conditions, it provides no explanation of what is behind them. Einstein strongly hoped GR would not need arbitrary boundary conditions, and considered it major defect that it does. It is up to some future theory to provide an explanatory framework.

 

[WannabeNewton]

Is it just a coincidence that the local measurement of rotation (based on inertia, Sagnac) matches the global measurement of rotation (based on light from distant stars) ?

See post #3. To repeat, all of those fail to agree in Kerr space-time, which is a particular instance of what Bill elucidated.

We had a really long thread on this in the past, see: https://www.physicsforums.com/showthread.php?t=729416

EDIT: Also just as an aside, the Sagnac effect is not a local measure of rotation. If it was then it would always agree with the Fermi-Walker definition of rotation but it doesn't as mentioned in post #3. It is a quasi-local measure of rotation since it isn't as global as rotation with respect to spatial infinity but still not entirely local since it requires knowledge of the axial Killing field along the entire closed circuit and on the symmetry axis of the space-time.