Mach's Principle: Inertia, Newton, Einstein & Gedanken Experiments

[Chronos]

I'm not very sympathetic towards 'empty universe' arguments. This universe is not empty and is not constrained by extrapolations based on an 'empty' universe. It's a good starting point, but you can only carry it so far before it becomes unphysical.

 

[Garth]

GR has a good track record of being right to date, but demanding that it be applicable to the condtions of an empty or almost empty universe seems to me to be pushing the theory pretty hard. I don't really see how the question can actually be answered experimentally, either.

So make my answer a definite maybe, with a side order of "how could we tell, anyway?".

Gravity Probe B will be able to tell, the predictions for General Relativity, the Brans Dicke theory and Self Creation Cosmology are:
Gravity Probe B:
GR prediction: Geodetic effect 6.6144 arcseconds/yr
Gravitomagnetic effect 40.9 millarcseconds/yr

BD prediction: Geodetic effect {(3w+4)/3w+6)}6.6144 arcseconds/yr
Gravitomagnetic effect {(2w+3)/(2w+4)}40.9 millarcseconds/yr

SCC prediction:Geodetic effect 5.5120 arcseconds/yr
Gravitomagnetic effect 40.9 millarcseconds/yr

Wait and see!

Garth

 

[Garth]

I'm not very sympathetic towards 'empty universe' arguments. This universe is not empty and is not constrained by extrapolations based on an 'empty' universe. It's a good starting point, but you can only carry it so far before it becomes unphysical.

That depends on how valuable you consider gedanken experiments to be.

Garth

 

[Ich]

perfect,

when you say that gr predicts an absolute rotation even for an empty universe, doesn´t that mean that spacetime has some properties in itself and can thus not be regarded as "empty" in the narrowest sense?
And what in those properites allows for an absolute angular velocity but not rotation, position, and velocity?

 

[pervect]

Space-time can be described by its metric, g_ab, which you can think of as a matrix. The metric tells you how to compute the Lorentz inteval, which is invariant for all observers, i.e.

ds^2 = dt^2 - dx^2 - dy^2 - dz^2

represents a metric which is diagional, with the 4 diagional elements being (1,-1,-1,-1)
This is the usual Lorentzian metric of flat space-time.

So space-time is described by a set of numbers - 10 numbers in general. Even empty space-time needs some set of numbers to describe it. For clarity, I would reserve the phrase "empty space-time" to it's usual meaning to talk about the case where there is no matter density. Philosophically, it is correct to note that the metric coefficients always exist, and that they determine the properties of space-time.

A rotating space-time can never have a diagonal metric - specifically it will have components like g_01, g_02, g_03. A space-time with a diagonal metric can never be rotating. Non-orthogonal coordiante systems are a complicating factor - if one insists that the vectors x,y,z at any point are all orthogonal, determining whether or not a coordinate system is rotating becomes as easy as inspecting the metric to see whehter or not it's diagional.

With a bit of algebra + calculus, you can find the formula for the above metric if you assume that coordinate system is rotating with an angular velocity w around the z axis. (i.e. you assume that x' = x*cos(wt) - y*sin(wt), y' = y*cos(wt) - x*sin(wt)). However, this is a rather long and tedious calculation, so I won't go into it. If you cary it out, though, you'll confirm that the metric coefficients of the rotating coordinate system are not diagonal and hopefully gain some insight as to why the metric of a rotating coordinate system cannot be diagonal.

It's also possible to approach the description of space-time in terms of the Riemann curvature tensor rather than the metric coefficients. See the post where I talked about the "ball of coffee grounds" earlier for more details on this approach.

In either case, the metric coefficients or the Riemann curvature tensor completely describes the properties of space-time - and one can determine whether one is rotating or not by inspection of either the metric or the Riemann curvature tensor.

I should probably add for clarity that tidal forces are a subset of the components of the Riemann curvature. It turns out that knowing the tidal forces at a point in space-time gives a sufficient amount of information about the Riemann curvature tensor there to determine whether or not the coordinate system is rotating.

 

[Chronos]

Ok, I'm going to take one more stab at this. In a universe that solely consisted of a space station, spacetime itself would cease to exist at the perimeter of the space station. The space station would effectively wrap around upon itself and the center of mass would exist at all points within the space station. Objects inside the space station would, however, be free to move, rotate, and experiencing inertial effects - but strictly relative to each other. The space station, as a whole, would not be a valid reference frame.

 

[Garth]

Chronos Why would space-time cease to exist at the perimeter of the space station?

pervect You have erected a coordinate system: x, y, z, t in your universe.
If it were an empty universe how would you know that the coordinate system is not rotating?
If there is matter in the universe and rotation can be determined by the presence of centrifugal and coriolis inertial forces, what is the mechanism by which that coordinate system is locked onto the inertial centre of momentum of the matter? (Foucault's pendulum - what is it that tells the pendulum how to precess so that its determination of the rotation of the Earth coincides with that determined by observing the distant fixed stars)?

Garth

 

[Creator]

Ok, I'm going to take one more stab at this. In a universe that solely consisted of a space station, spacetime itself would cease to exist at the perimeter of the space station. The space station would effectively wrap around upon itself and the center of mass would exist at all points within the space station.

Err... is your space station rotating? he, he

Creator

 

[pervect]

pervect You have erected a coordinate system: x, y, z, t in your universe.
If it were an empty universe how would you know that the coordinate system is not rotating?
If there is matter in the universe and rotation can be determined by the presence of centrifugal and coriolis inertial forces, what is the mechanism by which that coordinate system is locked onto the inertial centre of momentum of the matter? (Foucault's pendulum - what is it that tells the pendulum how to precess so that its determination of the rotation of the Earth coincides with that determined by observing the distant fixed stars)?

Garth

I thought I went into "how",, though I skipped some of the detail. If you ensure that x,y,and z are all at right angles, you can tell from the metric whether or not your coorinate system is rotating by seeing if the metric is diagonal. Perhaps a simpler way equivalent way to describe this - you look to see if light follows a straight line path.

You can also determine if you're rotating from measurement of the Riemann curvature tensor, in particular the subset of the Riemannc curvature tensor which are the tidal forces - or by looking at the volume of the "ball of coffee grounds".

And of course there is the ring laser gyroscope, which relies on the sagnac effect (which is just a property of light in rotating frames).

So there are a lot of ways to tell if you are rotating.

As far as "why" or "what", I don't think there's any further answer, except that it might be useful to say that it's logically impossible to imagine a universe where the second derivative of position can be determined and rotation cannot be determined. Since we can determine acceleration, we can determine rotation, there is no way to have acceleration be absolute and not have rotation be absolute that is logically consistent.

[add]
It's also logically incosistent to imagine a universe in which rotation is relative and in which 'c' is constant. Consider the apparent velocity of an object 1 light year away in a frame rotating at 1 reveolution/second, for instance - it is much greater than the speed of light.

 

[Garth]

pervect The problem is that our experience is only of a universe with matter in it, and our inertial compasses are aligned with the average distribution of matter in motion in that universe. We find it hard to imagine otherwise. Now it may well be true that there is an absolute non-rotating frame, irrespective of the matter in the universe, however other possibilities exist that are being explored at the moment by GPB. That is why that experiment is so important!

Garth

 

[Creator]

Ruyomg Wang tries to explain the Saganc effect result of roatation of the appartus rather than being in a roataing refernce frame using his fibre optic conveyor as an example (of course the result of the experiment is explqaijned very easily within the context of relavity by simply making sure you are looking at the total path of a beam of light).

I've heard of it. Can you supply some of the details.

It is interesting that there are many different physical explanations that result in the same Sagnac eqn.
In light of the discussion that an active ring laser gyro reveals absolute rotation, I find it very intriguing that an exactly equivalent Sagnac phase shift equation can be derived from ( velocity) doppler shift eqns. alone. This seems to imply to me that a non-inertial frame effect can be derivable from a simple (special relativistic) effect? , without all the full blown GR?


Creator

 

[Garth]

What frame of reference is established in which the Saganc effect equations are framed? How do you know whether that frame of reference is itself rotating or not?

Garth

 

[Ich]

as i understand pervect´s post, spacetime in GR is far more than a relation between massiive bodies. It has measurable properties and can even contain energy. So if i dont´t mistake somthing, it makes perfect sense to talk about rotation wrt spacetime, with no extra masses needed as a reference.
Since GR seems to work, i tend to share this view of things. Peronally, I also have no conceptual problems with GR not being completely machian. I rather add spacetime to my list of "physically existing entities" to get a complete description of the universe as we know it.
Of course, if further tests show that GR should be superseded by, for example, Garth´s theory, i´d give it another thought. But until then, i´m comfortable with it.

 

[Garth]

as i understand pervect´s post, spacetime in GR is far more than a relation between massiive bodies. It has measurable properties and can even contain energy. So if i dont´t mistake somthing, it makes perfect sense to talk about rotation wrt spacetime, with no extra masses needed as a reference.
Since GR seems to work, i tend to share this view of things. Peronally, I also have no conceptual problems with GR not being completely machian. I rather add spacetime to my list of "physically existing entities" to get a complete description of the universe as we know it.

However there is the question of consistency within GR. The Einstein Equivalence Principle subsumes the 'No preferred frames' principle of SR, and that would suggest that there should not be an 'absolute non-rotating inertial frame' defined in isolation of any matter in the universe.

Garth

 

[Ich]

There is no "No preferred frames" principle in SR. To the contrary, SR includes Newton´s mechanics and therefore makes a difference between rotating and non-rotating frames.
What pervect pointed out to me is that even when "you can´t apply the concept of motion" (linear motion meant by Einstein) to spacetime in GR, you obviously still have the concept of absolute rotation.
Ok, that´s strange. But then, there are many even stranger things in physics.

 

[Garth]

There is no "No preferred frames" principle in SR. .

I disagree

Garth

 

[Ich]

SR subsumes Galiei´s relativity principle from Newtonian mechanics and extends it to electrodynamics. It states that physics is the same in all inertial systems. Neither Newton nor Maxwell nor Einstein said that rotation had no physical effects.
From my limited knowledge of GR i can´t tell how it handles rotating frames. Maybe as some kind of acceleration effects, but still rotation is absolute. What´s new in GR is that rotation of spacetime is partly influenced (and not defined) by rotating masses.

 

[Garth]

But the question is: "What defines an inertial (either non-accelerating or non-rotating) frame of reference?" Non-accelerating or non-rotating with respect to what?

As you have gathered, I am Machian in that I believe that inertial frames have to be defined with respect to the distribution of the rest of the matter in motion in the universe.

However, one can further ask: "What was it that decided that frame of reference (probably identified now by the globally isotropic CMB frame) in the first place?"

Garth

 

[Ich]

Non-accelerating with respect to spacetime. As I pointed out, I have no problems with spacetime being an entity in itself.
And for CMB, it is just light with some special energy distribution. Maybe it really shows us how the universe expanded, but I don´t expect physics to be bifferent in this frame.

 

[Stingray]

But the question is: "What defines an inertial (either non-accelerating or non-rotating) frame of reference?" Non-accelerating or non-rotating with respect to what?

As I stated before, you can define it with respect to local geodesics. If you overinterpret the principle of relativity, you'll be left with nothing resembling reality...

 

[Garth]

As I stated before, you can define it with respect to local geodesics. If you overinterpret the principle of relativity, you'll be left with nothing resembling reality...

To be precise the point I was making was that SR is built on the principle of "no preferred inertial frames of reference".

However Einstein himself asked the same question as Bishop Berkeley and Ernst Mach about what was it that decided which frames were to be inertial or not? The situation is particularly acute in the gedanken of a test particle in an otherwise empty universe.

Einstein was somewhat satisfied that his GR theory partially included Mach's Principle, but it is generally understood by him and most other researchers in the field that GR does not fully include it. Hence the point of such work as that of Brans Dicke and others including myself.

Merely erecting a coordinate system, and a metric to go with it, is only doing mathematics and not physics. You have to physically define how the mathematical symbols relate to physical realities. This means basically repeatedly asking the question, when writing a mathematical representation of a physical quantity, "How do you measure it?"

Rather that "over interpreting PR and being left with nothing resembling reality", the opposite seems to be the case.

Garth

 

[Stingray]

To be precise the point I was making was that SR is built on the principle of "no preferred inertial frames of reference".

Definitions of "inertial" can be found in better textbooks. It is not mysterious, especially when thinking in terms of Minkowski's formulation of SR. There, you can measure the metric, and all coordinate systems which diagonalize it are called inertial.

Merely erecting a coordinate system, and a metric to go with it, is only doing mathematics and not physics. You have to physically define how the mathematical symbols relate to physical realities. This means basically repeatedly asking the question, when writing a mathematical representation of a physical quantity, "How do you measure it?"

The physical relevance of the metric is well-known. It is not conceptually difficult to measure it in a small region around some worldline (e.g. the lone particle). Again, look in textbooks.

Lastly, Ich said

It [SR] states that physics is the same in all inertial systems.

It is actually the same in all frames. Although SR wasn't originally formulated this way, it usually is now. The problem is that the physics is the "same" only for a properly defined sense of "same" . It is IMO more a requirement for mathematical statements than of physical principle.

For example, you can write down Newtonian gravity so that it looks the same in all frames as well. It's unfortunate that this formulation is rarely discussed in a standard physics education. It is really very elegant, and shows a surprising amount of similarity between Einstein's and Newton's theories.

 

[Ich]

It is actually the same in all frames. Although SR wasn't originally formulated this way, it usually is now. The problem is that the physics is the "same" only for a properly defined sense of "same" . It is IMO more a requirement for mathematical statements than of physical principle.

For example, you can write down Newtonian gravity so that it looks the same in all frames as well. It's unfortunate that this formulation is rarely discussed in a standard physics education. It is really very elegant, and shows a surprising amount of similarity between Einstein's and Newton's theories.

That´s beyond my scope; I guess what you mean has something to do with covariance and this sort of stuff.
What I meant is that there is no way to tell whether you are inertially moving or not, but there are thousands of experiments to tell whether your apparatus is rotating or not.

 

[moving finger]

To be precise the point I was making was that SR is built on the principle of "no preferred inertial frames of reference".

However Einstein himself asked the same question as Bishop Berkeley and Ernst Mach about what was it that decided which frames were to be inertial or not? The situation is particularly acute in the gedanken of a test particle in an otherwise empty universe.

Einstein was somewhat satisfied that his GR theory partially included Mach's Principle, but it is generally understood by him and most other researchers in the field that GR does not fully include it. Hence the point of such work as that of Brans Dicke and others including myself.

Merely erecting a coordinate system, and a metric to go with it, is only doing mathematics and not physics. You have to physically define how the mathematical symbols relate to physical realities. This means basically repeatedly asking the question, when writing a mathematical representation of a physical quantity, "How do you measure it?"

Rather that "over interpreting PR and being left with nothing resembling reality", the opposite seems to be the case.

Garth

Garth - I agree 100%

MF

 

[moving finger]

What I meant is that there is no way to tell whether you are inertially moving or not, but there are thousands of experiments to tell whether your apparatus is rotating or not.

But the real question is whether that "rotation" you measure is an absolute rotation (with respect to some mystical absolute non-rotating reference frame), or whether it is a relative rotation (with respect to, perhaps, the rest of mass-energy in the universe). The former is perhaps assumed in GR, the latter is Machian. These two are very different concepts.

MF

 

[Ich]

But the real question is whether that "rotation" you measure is an absolute rotation (with respect to some mystical absolute non-rotating reference frame), or whether it is a relative rotation (with respect to, perhaps, the rest of mass-energy in the universe).
MF

Let me rephrase that: the real question is whether that "rotation" you measure is a rotation with respect to local spacetime, or whether it is a rotation with respect to the rest of mass-energy in the universe, of which your probe knows via some mystical distance effect.
I´m not the one to decide which is true, but: Einstein, himself a "Machian", came up ith a working theory which passed every experimental test so far. And this theory works strictly local via the mediation of a spacetime with some physical properties in itself.
Even if you like more the strictly machian view, be sure you would need a whole bunch of "New Physics" to implement it. Einstein tried, but failed.
Maybe Garth has the solution, but until there is some evidence for his theory, i stick to GR.

 

[Garth]

Let me rephrase that: the real question is whether that "rotation" you measure is a rotation with respect to local spacetime, or whether it is a rotation with respect to the rest of mass-energy in the universe, of which your probe knows via some mystical distance effect.
I´m not the one to decide which is true, but: Einstein, himself a "Machian", came up ith a working theory which passed every experimental test so far. And this theory works strictly local via the mediation of a spacetime with some physical properties in itself.
Even if you like more the strictly machian view, be sure you would need a whole bunch of "New Physics" to implement it. Einstein tried, but failed.
Maybe Garth has the solution, but until there is some evidence for his theory, i stick to GR.

Although Einstein was a Machian at heart he worried that GR did not fully include that Principle.

I would argue that Mach's Principle is actually inconsistent with the principles of GR as it can identify a preferred inertial frame of reference, that is the one co-moving with the centre of mass/momentum of the entire universe. At any event in space-time I would identify that with the globally isotropic CMB frame at that event.

SCC does have 'evidence' to support it: viz: It is concordant with cosmological constraints without the need to invoke Inflation, non-baryonic Dark Matter or Dark Energy. It predicts the Pioneer anomaly as a clock drift between atomic and ephemeris time, and it predicts a secular speeding up of the Earth's rotation at a rate equal to Hubble’s constant that is indeed observed. These are described in the preprint Self Creation Cosmology - An Alternative Gravitational Theory .

Garth

 

[Ich]

I downloaded your paper and will read it when I have time. But don´t expect too much of enlightenment for me, I even did not understand GR.

 

[moving finger]

Let me rephrase that: the real question is whether that "rotation" you measure is a rotation with respect to local spacetime,

To me, THIS is the mystical concept. What meaning does "local spacetime" have in the absence of mass/energy?

or whether it is a rotation with respect to the rest of mass-energy in the universe, of which your probe knows via some mystical distance effect.

why mystical? gravitational fields propagate (as far as we know) without any distance limit, and all mass/energy produces a gravitational field - therefore every atom in your body is experiencing right now the combined gravitational fields of all of the rest of the mass/energy in the universe. What is mystical about that?

I´m not the one to decide which is true, but: Einstein, himself a "Machian", came up ith a working theory which passed every experimental test so far. And this theory works strictly local via the mediation of a spacetime with some physical properties in itself.

Simply because nobody so far has been able to carry out a test which distinguishes between rotation with respect to absolute space, or with respect to local spacetime (whatever that means) or with respect to the rest of the cosmos.

Even if you like more the strictly machian view, be sure you would need a whole bunch of "New Physics" to implement it.

That is the whole point of this thread

MF

 

[Ich]

Simply because nobody so far has been able to carry out a test which distinguishes between rotation with respect to absolute space, or with respect to local spacetime (whatever that means) or with respect to the rest of the cosmos.

AFAIK they will do next year. Should we bet whether GR will win again?

 

[moving finger]

AFAIK they will do next year. Should we bet whether GR will win again?

what do you think the results will prove? That there is an absolute rotational rest-frame that is NOT the same as the Machian rest-frame?

MF

 

[Ich]

They will not prove anything, but they may show evidence that frame-dragging is as predicted by GR. Any alternative theory then would have to reproduce this result. Or better, predict this result before.
Or even better, predict a different result and be shown to be more exact than GR. I think Garth´s predicts 5/6 ? (Sorry still had no time to read the paper)

 

[moving finger]

They will not prove anything, but they may show evidence that frame-dragging is as predicted by GR. Any alternative theory then would have to reproduce this result. Or better, predict this result before.
Or even better, predict a different result and be shown to be more exact than GR. I think Garth´s predicts 5/6 ? (Sorry still had no time to read the paper)

I don't think the expectation is as clear-cut as you suggest. Yes, Garth suggests that his SCC theory can be distinguished from GR by Gravity Probe B, but SCC is not the only Machian theory in town. Frame dragging is apparently a direct consequence of the equations of GR, but it is also a Machian effect.

I'm not an expert on GR, so it is not clear to me how one can distinguish in this way between the alleged GR assumption of an "absolute rotational reference frame", and a Machian "relative rotational reference frame" (relative to the background stars)? Can anyone enlighten me?

Thanks

MF

 

[Garth]

The difference is to be found in the gedanken experiment of a spinning spherical mass in an otherwise empty universe. In GR the frame dragging, Lense-Thirring effect, will rotate space-time at a small fraction of the rate of rotation of the mass. In a fully Machian theory space-time will be dragged completely with the mass, so that relative to the space-time metric the mass will not be rotating, for there is nothing to compare its rotation to, and inertial centrifugal and coriolis forces will disappear.

B.T.W. In SCC the geodetic precession is 5/6 GR whereas the Lense-Thirring effect is the same as GR. The universe not being otherwise empty! In SCC inertial mass is also determined by the distribution of mass in motion in the rest of the universe.

Garth

 

[moving finger]

The difference is to be found in the gedanken experiment of a spinning spherical mass in an otherwise empty universe. In GR the frame dragging, Lense-Thirring effect, will rotate space-time at a small fraction of the rate of rotation of the mass. In a fully Machian theory space-time will be dragged completely with the mass, so that relative to the space-time metric the mass will not be rotating, for there is nothing to compare its rotation to, and inertial centrifugal and coriolis forces will disappear.

Thanks, Garth!

Is it expected that Gravity Probe B will be able to distinguish between the two above, given that the Earth is NOT spinning in an empty universe? In other words, the frame dragging caused by the Earth should rotate space-time at a fraction of the rotation rate of the Earth both in GR and in a Machian view, because in the latter the background stars will have an effect?

Are there any straightforward publications you are aware of which compare and contrast the expected magnitude of the frame dragging that Gravity Probe B will measure (a) assuming GR is corrrect and (b) assuming the universe is Machian?

(I know you like to promote the SCC theory, but I'm looking for a generic "Machian" prediction rather than the SCC prediction, if that's possible )

Cheers

MF