# Mach's Principle

1. Mar 27, 2005

### moving finger

The core of "Mach's Principle" is something like this: the inertia of a body is determined in relation to all other bodies in the universe (in short, "matter there governs inertia here").

Newton was aware of something akin to Mach's principle, and it is said the principle played a central role in Einstein's development of GR.

Imagine a simple gedanken experiment.

We are aboard a space-station in free space, with no significant gravitational masses in the vicinity of the space-station. We can induce a kind of artificial gravity aboard the space station if we set it spinning (recall "2001 - A Space Odyssey" for a perfect example).

Question 1 : What is the space station spinning relative to?

Obviously it is not spinning relative to itself (it is stationary in it's own reference frame). And since we believe there is no "aether" or absolute space, it is not spinning relative to that either.

Mach's principle would suggest it is the fact it is spinning relative to the background stars that allows us to know that it is spinning (and so also must the masses inside the space station "know" it is spinning relative to the background stars, otherwise they would not experience the artificial gravity).

Question 2 : What happens if we could simultaneously remove all of the background stars (remember, this IS a gedanken experiment)? Would the space-station still be spinning? and if "yes", what is it now spinning relative to?

Thanks

MF

2. Mar 27, 2005

### Garth

These are good, deep questions. As well as a space station, we might also want to consider the whole universe. At times it is asked whether the whole universe is spinning, but as you ask: "What is it spinning relative to?" What is it that determines the inertial compass against which such a spin might be measured?

Mach's Principle suggests that the inertial compass is locked into the distribution of matter and energy in the universe, however such a position is not fully concordant with the principles of General Relativity. Mach's Principle suggests that there is a preferred frame of reference, the Centre of Momentum of the whole universe, which can be identified with the co-moving frame of reference of the Cosmic Microwave Background, in which the CMB is globally isotropic, in contradiction to the Equivalence Principle.

The Brans Dicke theory modified GR in order to fully include Mach's Principle, however that theory does not seem to be concordant with observation. However a modified version of the Brans Dicke theory "A New Self Creation Cosmology" not only includes Mach's Principle but is also concordant with present observational and cosmological constraints.

Garth

3. Mar 27, 2005

### pervect

Staff Emeritus

For whatever it's worth, while GR was inspired by Mach's principle, GR is not in and of itself Machian, in the sense that rotation in GR is absolute, not relative.

Thus things (in GR) don't need anything to rotate "relative to" - whether or not they are rotating is an absolute.

Thus it's possible in GR to have a universe with a non-zero total angular momentum, this is not forbidden by any laws.

4. Mar 27, 2005

### yogi

I have posed this several times before - its worth rereading more than once. From Einsteins 1920 address:

“...to deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics to not harmonize with this view. For the mechanical behavior of a corporal system hovering freely in empty space not only depends upon relative positions (distances) and relative velocities, but also on its state of rotation, which physically may be taken as a characteristic not appertaining to the system itself. In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space. Since he classes his absolute space together with real things, for him rotation relative to absolute space is also something real. Newton might no less well have called his absolute space “Ether”; what is essential is merely that beside observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.

It is true that Mach tried to avoid having to accept as real something which is not observable by endeavoring to substitute in mechanics a mean acceleration with reference to the totality of the masses of the universe in place of an acceleration with reference to absolute space. But inertial resistance opposed to relative acceleration of distant masses presupposes action at a distance; and as the modern physicist does not believe that he may accept this action at a distance, he comes back once more, if he follows Mack, to the ether, which has to serve as medium for the effects of inertia. But this conception of the ether to which we are lead by Mack’s way of thinking differs essentially from the ether conceived by Newton, by Fresnel and by Lorentz. Mack’s ether not only conditions the behavior of inert masses, but is also conditioned in its state by them.

5. Mar 27, 2005

### yogi

“..every attempt to deny the physical reality of space collapses in the face of the law of inertia. For if acceleration is to be taken as real, then space must also be real within which bodies are conceived as accelerated. Newton saw this with perfect clarity and consequently he called space ‘absolute” ..the forces between particles were regarded as unconditionally associated with the particles themselves. ...Mere empty space was not admitted as a carrier for physical changes and processes. It was only ..the stage on which the drama of material happenings was played.”

“...The ether was invented, penetrating everything, filling the whole of space, and admitted as a new kind of matter. ... it was overlooked that by this procedure, space itself had been brought to life...It (the ether) was thus to some degree identical with space itself.... In this way the field theory was born as a illegitimate child of Newtonian physics.”

“To become fully conscious of this change in outlook was a task for a highly original mind whose insight could go straight to essentials, a mind that never got stuck in formulas. Faraday was this favored spirit. His instinct revolted at the idea of forces acting directly at a distance which seemed contrary to every elementary observation.” If one electrified body attracts or repels a second body, this was for him brought about not by a direct action from the first body to the second, but through an intermediary action. The first body brings the space immediately around it into a certain condition which spreads itself into more distant parts of space according to a certain spatiotemporal law of propagation. This condition of space was called ‘the electric field.’ The second body experiences a force because it lies in the field of the first, and vice versa. The ‘field’ thus provided a conceptual apparatus which rendered unnecessary the idea of action at a distance. Faraday also had the bold idea that under appropriate circumstances fields might detach themselves from the bodies producing them and speed away through space as free fields; this was his interpretation of light.”

Einstein --- as I read his words - does not appear to rule out the notion that local space is conditioned by all the other matter in the universe - in other words, the distant stars act in total upon space - and we measure that affect when we accelerate local masses relative thereto - a two step process??? Is Einstein saying Mach's principle is still the root cause of inertia -but not by direct action, but nonetheless the primary source?

Last edited: Mar 27, 2005
6. Mar 27, 2005

### moving finger

Lol – you pre-empted me, Garth. I was going to lead up to this. One solution to Einstein's field equations was found in 1949 by Austrian logician Kurt Gödel, and it involves a rotating universe (Gödel K., 1949, 'An example of a new type of cosmological solutions of Einstein's field equations of gravitation', Rev. Mod. Phys. 21, 447). It was shown that certain motions within such a rotating universe would lead to time travel.

Einstein concluded that Mach's principle had to be founded on the existence some kind of ether, because he rejected the idea of action-at-a-distance (he referred to this as "spooky action-at-a-distance" in the EPR paradox). But with the benefit of our recent understanding of non-local effects in QM (ie that there CAN be superluminal "action at a distance"), we now know that Einstein's naive intuition about the absurdity of spooky action-at-a-distance was wrong. Does Mach's principle need to be re-examined in light of this?

Seems strange to me that we dispense with the idea of absolute space on the one hand (for positions and velocities in SR), but then we have to re-introduce absolute space on the other (for accelerations and gravity in GR). Could it be that Einstein's intuition was in fact wrong, that there is no absolute space at all, and GR needs to be re-examined in this context? Has anyone tried to do this?

MF

ps : A form of GR that did not assume absolute space would then rule out the time-travel Godel universe (which would be intellectually satisfying for some!).

7. Mar 28, 2005

### yogi

I think it is bit early to rule out Einstein's intuition - quantum entanglement is not understood - it may not mean that fields propagate faster tha c. But is it really necessary in the case of inertia that the effect of far away masses travel faster than c - the influence of the distant stars exists as more or less an unvarying field - and the field of each star conditions local space - so when inertia is experienced we are not depending upon a new signal from each and every mass every second - but a continuous (ongoing) property of space.

Here is a thought experiment that, if it could be performed, might lead to some understanding of inertia if it is due to distant matter - assume a small test mass near a large massive planet - there is a large G force - but this force can be canceled by placing an identical planet at the same distance from the test mass so that their G fields disappear at the center of the test mass (the test mass is midway between the two large high density planets). The question posed is whether these two large planets with their cancelling G fields will affect the inertia of the inbetween test mass ever so slightly.

8. Mar 28, 2005

### moving finger

Yes, I agree with this, but then why did Einstein seem to have such a problem with such a matter-defined field? I quote his words in the text you submitted :
"inertial resistance opposed to relative acceleration of distant masses presupposes action at a distance; and as the modern physicist does not believe that he may accept this action at a distance, he comes back once more, if he follows Mack, to the ether".
What was Einstein's problem with accepting that distant masses provide a background field against which we measure rotation, and why did he reject that and suggest the only alternative was an ether (which in turn seems to have led him to his axiom that GR must be founded on a concept of absolute space)?

I'm not sure we would expect it to, whether rotation is "absolute" or relative to some background field?

The more I think about it, the more problem I have with the idea of "absolute rotation". As a concept, this seems to me as ridiculous as the concept of absolute (linear unaccelerated) motion. If rotation is in some way "absolute", what is it exactly (in the absence of the background stars) that defines the preferred rotational rest-frame (ie the frame in which the body is at rest and is not rotating)? It seems to me that you cannot escape the logic that all motion, whether it is linear, accelerated, or rotational, must be relative to "something".

MF

9. Mar 28, 2005

### joshuaw

A good book to read on this: "Foundations of Space Time Theories" by Michael Friedman.
He steps one through the reason's for changing to a 4 dimensional consideration which in turn makes rotation and acceleration absolute(if I read it correctly :) ). The introduction is really helpful to this. The rest of the book expands a little more on the topics.

10. Mar 28, 2005

### pervect

Staff Emeritus
There is a simple test to deterimne whether or not one is in a rotating coordinate system or not. If ene has a ball of cofee grounds arranged in a sphere at rest relative to the coordinate system, the volume of the ball of this sphere of coffee grounds will expand (due to the centrifugal "generalized forces"). In a non-rotating system, the volume will stay the same.

I put the simple conclusion first, now I'll talk a little bit about how one can arrive at it. (This may be scary, that's why I put the simple part first :-)).

In empty space, you can think of the Riemann as having six components, three of which represent "stretching" tidal forces - just like the tidal forces the moon exerts on the Earth. They are measured in "acceleration/unit length".

There are also three magnetic-like components of the Reimann that we don't have to get into at the moment.

That's a total of six components, but there are only 5 degrees of freedom for the "gravitational field" at a point in empty space. (There are some more degrees of freedom if the matter density isn't zero, but in empty space you have and need only five independent components). The reason there are only five is that there is a relationship among the components. Specifically, there is an identity which says that the sum of the three components of the stretching tidal forces are zero in a local orthonormal coordinate system, (i.e. the sum of the x,y, and z components of the forces, otherwise known as the trace, is zero).

This is a highly mathematical statement, but it has a physical interpretation, as "Baez's coffee grounds".

http://math.ucr.edu/home/baez/gr/outline2.html
(or see his longer paper on the same topic)

Baez's inteprretation of this identity states that the tidal forces (in empty space, and a non-rotating coordinate system) do not change the volume of a spherical ball off coffee grounds, just the shape.

Using this "coffee grounds" description of this identity, it becomes very clear how to conduct a test of whether or not one is rotating, or at rest.

11. Mar 28, 2005

### Garth

The question is: "What would happen if we performed the experiment in an otherwise empty universe?"
In other words, how do the "coffee grounds" 'know' whether they are rotating or not; what does their inertial compass 'lock onto'?
Garth

Last edited: Mar 28, 2005
12. Mar 28, 2005

### yogi

Pervect - With regard to Baez proposition that rotation will cause the central sphere of coffee grounds to distort - this would be true if space behaved as a physical fluid - in which case there can only be one plane (axis of rotation). But space may not be so limited - it may be that it moves within itself w/o interfereing with its own motion - e.g., as 3 dimensional vortex wherein the effective spatial rotation is isotropic. Perhaps there is something similar happening on the quantum scale - recall Feynman's frustration in trying to pin down the electron spin axis

Now if the space we call the cosmos behaves as a 3 dimensional vortex, all matter contained therein will be acted upon by the centripetal component of the spatial rotation - ergo such particles will experience (v^2)/r radial forces - so - you can see where I am headed (most likely wrong headed) but ideas are cheap.

13. Mar 28, 2005

### moving finger

Garth and I don't often see eye to eye but on this one I agree.

This is the central question, and I don't see an answer in Baez's paper. Its fine to throw in lots of maths and 6-dimensional spaces and Riemann and Ricci tensors and all the rest of it..... but at the end of the day it still comes back to the same question : A rotating body somehow "knows" that it is rotating, therefore there must be a reference frame. What is it that determines that reference frame in empty space?

If the answer is that there is an absolute space (ie an ether) then this absolute space must provide a reference frame for all kinds of motion, not just for rotation......

14. Mar 28, 2005

### Chronos

I don't think that is the case. In empty space, a rotating body would have no clue it was rotating. There is nothing for it to rotate with respect to. The same argument applies to a rotating universe - how would it know?

15. Mar 28, 2005

### moving finger

I agree.

OK, Mach, Garth, Chronos and I seem to be of similar minds in this respect (please correct me if I am wrong) - that rotation must be with respect to a frame of reference, and in empty space (devoid of all matter and energy) there is no frame of reference, hence rotation is meaningless. For the same reason, it is meaningless to talk of the entire universe rotating.

Pervect and joshuaw on the other hand maintain that Einstein & GR assumes an absolute space for rotation.

yogi seems to be on the fence? (but leaning towards Mach?)

So far these are subjective opinions.

How can we move forward on this?

Has anyone tried formulating a GTR based on rotation not being absolute, but being instead relative to the background stars (perhaps relative the background gravitational field)?

MF

16. Mar 29, 2005

### Garth

Yes, Brans, C.H. and Dicke, R.H.: ("Mach’s Principle and a Relativistic Theory of Gravitation"1961, Phys. Rev. 124, 925), and amongst others, myself: A New Self Creation Cosmology.

Garth

17. Mar 29, 2005

### pervect

Staff Emeritus
That question is a bit on the philosophical side, I'm afraid. I suppose one answer would be the famous quote "Matter tells space how to curve, and space tells matter how to move." So the (philosophical) answer using this approach would be "space".

18. Mar 29, 2005

### pervect

Staff Emeritus
As I said once before, according to standard relativity, it doesn't really make sense to talk about "space moving" at all. This follows directly from the assumption (which seems to be true as far as we can determine experimentally) that all experimental results will be the same regardless of one's velocity.

Seemingly paradoxically, it can make sense to talk about whether an extended region of space is rotating, even though there is no way to determine its velocity.

A good example of this is the rotating Earth. Aside from the "coffee grounds" test (which is a bit hard to apply on earth , as one has to put the coffee grounds into "free fall", another test for rotating frames is the fact that one cannot synchronize all clocks in a rotating frame in an Einsteinian manner. If one starts synchronizing clocks East-West, starting at the east and moving west, synchronizing two clocks by the midpoint method (a light signal emitted at the midpoint must reach both clocks at the same time), one finds that one one has returned to the starting point, the last pair of clocks is not synchronized. The magnitude of the synchornization error is on the order of 4*A*w/c^2, A being the area of the loop, and w being the angular velocity of rotation.

19. Mar 29, 2005

### moving finger

I disagree. The question is not simply a philosophical question (any more than the principle of relativity in SR is a philosophical question).

When we talk of "how do the coffee grounds know whether they are rotating or not?" what we mean is : do the coffee grounds rotate (a) relative to themselves, (b) relative to something else, or (c) relative to some absolute reference frame?

It seems we discount (a) as being meaningless, which leaves us with (b) or (c).

If the answer is that coffee grounds rotate relative to "space" then that begs additional questions such as (d) what determines the space metric (is it mass/energy, in which case the background stars DO in fact determine the reference frame?) and (e) if space acts as an absolute frame for rotation, why does it not also do so for position and linear velocity?

Sorry, but (strictly logically) this result simply means that the experiments in question are not dependent on absolute velocity - it does not necessarily mean that there is no absolute velocity, and it does not mean that there is no sense to talk about "space moving".

But how do you know that you are measuring absolute rotation of the earth, as opposed to rotation of the earth relative to some background reference frame such as the "fixed stars"? I would humbly suggest that you cannot (in this experiment you describe) distinguish between these two alternative hypotheses, and both hypotheses fit the facts you have presented.

Does anyone know if there have been any attempts to experimentally measure absolute rotation, to distinguish it from rotation relative to some background reference frame (a la Michelson-Morley)?

MF

"And if you take one from three hundred and sixty-five what remains?"
"Three hundred and sixty-four, of course."
Humpty Dumpty looked doubtful, "I'd rather see that done on paper," he said.

20. Mar 29, 2005

### Stingray

The space station is spinning with respect to someone standing at its center :tongue:.

To be more a bit more serious, let's say we have an astronaut on this space station who measures the metric in his immediate vicinity (after fixing some coordinate system). He will of course know his (normalized) 4-velocity with respect to this frame. From this, he can find his 4-acceleration. Using the metric, he can then compute the magnitude of his acceleration. This number will be independent of the coordinates chosen.

Now most people would call the magnitude of the 4-acceleration an 'acceleration.' Since the definition doesn't require us to define a reference frame, there's not necessarily any meaning to ask what our astronaut is accelerating with respect to. If you really want to force it, then you could say that his worldline is accelerating with respect to instantaneously tangent geodesics. This establishes absolute acceleration in any theory using the standard notions of classical spacetime (the specific field equations are irrelevant to everything I've said).

Talking about absolute rotation is more difficult. It is most naturally done by constructing more invariants (called path curvatures) that depend only on the astronaut's local measurements. In most nontrivial circumstances, we can even construct a unique set of 4 orthonormal vectors at the astronaut for each instant of his proper time (a tetrad frame). This is called a Frenet-Serret frame, by the way.

Anyway, as I remember it, you can naturally identify one of the invariants with "circular motion" in a particular sense. Basically, the astronaut's worldline will infinitesimally "rotate" with respect to a local congruence of geodesics defined in some natural way. The point in all of this, though, is that you can come up with reasonable definitions that are completely local and coordinate independent.

By the way, if you're interested in Mach's principle, D. Sciama did some interesting work trying to show how it works (or doesn't work) in GR.

21. Mar 30, 2005

### Garth

We don't have a conveniently empty universe to do the experiment in! However the Gravity Probe B experiment might be thought of as such an experiment, as the Machian theory of SCC predicts only 5/6 the GR geodetic precession, and that is being measured at this moment. We shall know the result in early 2006.

Garth

22. Mar 30, 2005

### moving finger

OK, it will take me some time to digest the rest of your post, but I think we can dismiss this opening comment? Anyone standing at the centre of the space station is presumably co-rotating with the space station. If they are not, then yes I agree the station is rotating relative to them..... or maybe they are rotating relative to the station..... or maybe they are both rotating relative to some external reference frame.....that's the problem.

MF

23. Mar 30, 2005

### Stingray

Yes, you can.

24. Mar 30, 2005

### pervect

Staff Emeritus
The distinction I draw between philosophy and science is that scientific questions are testable with an experiment.

Emptying out all the matter in the universe is simple something that can't be done. And if it could be done, how would one observe the results?

There is some small amount of science in the quesion, because it does talk about an experimental result, which is why I said

I meant that quite literally - it's not totally philosophical, but it's close.

The problem is that the experiment is not one that can ever be performed. So one can argue about the question endlessly (that's the problem with philosophical questions), because question can't be settled experimentally.

I would like to re-emphasize the point that I am talking about standard relativity here. I even explicitly said I was talking about standard relativity.

In the context of standard relativity, there is NO EXPERIMENT that can determine one's velocity. The concept doesn't really make any sense in standard theory, because there is no way to measure it, even in principle.

Possibly "doesn't make sense" isn't the best word choice. I'm not sure how to say it more literally - perhaps "is not productive", or "does not have and can never have any empirical basis" might have been more clear.

In some other non-standard theory, the concept could make sense. (Yogi seems to have his own theories, for instance).

Basically, with a ring laser gyro (making use of the Sagnac effect, which is a consequence of the synchronization issues in a roating frame I mentioned in one of my posts) one can measure one's rotation.

One does not need to refer to the rest of the universe to make the measurement, it can be done in a small closed room.

This is what makes rotation absolute (and what makes any other measurement of acceleration absolute, for that matter). It's because the measurement of rotation can be performed in a small closed room, without any reference to other objects, that it is an absolute measurement.

Contrast this with velocity - one can measure one's velocity relative to another object, but one cannot determine one's velocity from experiments performed in a small closed room. This is what makes velocity relative and not absolute.

25. Mar 30, 2005

### yogi

It has been suggested that space rotates about black holes - I am not sure how good the evidence is for this proposition - or even what it means - but there is definitely a dichotomy between the notion of space as some sort of dynamic - and space as inert. Modern physics asserts that things are one way or the other - but different phenomena seem to require shifting foundations.