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

[jcsd]

Upon further reflection - It does appear that Sagnac type experiments where the area enclosed by the path(s) is finite, will disclose absolute rotation - at least with respect to local space - consider my last two posts scratched

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).

 

[pervect]

But would they work in an empty universe? What would they 'lock onto'?

Although you may think that this question is also 'rather philosophical' it falls into the venerable tradition of 'gedanken' or 'thought' experiments that help to develop the principles upon which physical theory may be built.

Garth

Standard General Relativity says that ring laser gyroscopes will work in an empty universe. To analyze the problem, first has to set up the metric of the empty universe. I'm assuming that it's flat (Lorentzian) - I'd have to think a bit to see if this followed directly from Einstein's equations or not. There certainly should be a locally Lorentzin frame, the question is whether the universe is cosmologically Lorentzian. The empty "Milne" universe we were talking about a while back does admit a Lorentzian coordinate system as I recall.

Assuming that the metric is Lorentzian, or that the ring laser gyroscope assembly is small relative to the cosmological dimensions of the universe so that the metric is locally Lorentzian (either assumption will do), there wouldn't be any difference in the experimental results between carrying the experiment out in intergalactic space, or in a totally empty universe, according to GR. You'd have to make sure that the mass of the ring-laser gyroscope was small, as well, so that the ring laser gyroscope assembly did not perturb the metric of the universe much. This would put some constraints on the gravitatioanl constant G.

 

[pmb_phy]

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").

. Yup. I agree. However I recall that there are many forms of this law.

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

The "distant stars."

re - "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"

Its more than a referrence. The mass of the distant stars actually exerts forces. E.g. If you created a hollow sphere and placed a ball inside and started rotating the sphere then you'd see the ball dragged around and that frame which is attached to the sphere would be an inertial frame of reference.

re - "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?"

I don't think there is a well agreed upon answer to this question.

Thanks

MF [/QUOTE]

 

[Garth]

Yes pervect

Standard General Relativity says that ring laser gyroscopes will work in an empty universe.

I agree, however the question is: "Is GR correct about this?"

Pete - some mechanism is required to transmit the rotation of the hollow sphere to the ball. The Lense-Thirring effect of GR may provide that which is necessary, however the Post-Newtonian Approximation breaks down when the hollow sphere approaches the thickness required to resolve the matter. The Brans-Dicke and SCC theories require a scalar field that determines the inertial mass of fundamental particles and that field affects the outcome of such experiments as the Gravity Probe B. So eyes down to watch for the results of that when it is published!

Garth

 

[pervect]

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?".

 

[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