Is Inertia Tied to the Stars According to Mach's Principle?

[Torog]

Mach, Newton and others observed that centrifugal forces appear in a object when it rotates in relation to the stars. Einstein was convinced by this and tried, unsuccessfully as far as I understand, to incorporate what he called Mach’s Principle into General Relativity.

From Wikipedia -”Mach’s principle”

“Einstein was convinced that a valid theory of gravity would necessarily have to include the relativity of inertia:

So strongly did Einstein believe at that time in the relativity of inertia that in 1918 he stated as being on an equal footing three principles on which a satisfactory theory of gravitation should rest:

1. The principle of relativity as expressed by general covariance.
2. The principle of equivalence.
3. Mach's principle (the first time this term entered the literature): … that the gµν are completely determined by the mass of bodies, more generally by Tµν.
   

In 1922, Einstein noted that others were satisfied to proceed without this [third] criterion and added, "This contentedness will appear incomprehensible to a later generation however."

It must be said that, as far as I can see, to this day Mach's principle has not brought physics decisively farther. It must also be said that the origin of inertia is and remains the most obscure subject in the theory of particles and fields. Mach's principle may therefore have a future – but not without the quantum theory.

 Abraham Pais, in Subtle is the Lord: the Science and the Life of Albert Einstein (Oxford University Press, 2005), pp. 287–288.”

To illustrate it from my point of view:

A little thought experiment: A scientist is in a closed space capsule with no windows. The capsule is set spinning. The scientist has a gyroscope and can use his thrusters to stop the spacecraft from spinning. Another closed spacecraft at a distance (the other side of the solar system or the other side of the galaxy) is doing the same thing. Both scientists do their work and declare themselves not rotating, open the door, lean out and wave to each other. They notice that the capsules are not rotating in relation to each other. How did they do this? They used no external clues, all they used was the gyroscope to determine their state of rotation. They both locked onto the same thing to stop the rotation - inertia. They also notice that the stars aren’t moving.

So what do you have: Inertial forces are locked to the stars. This means that inertial forces cannot reside within an object but must be an interaction between mass and the distant stars.

In fact it is not possible to rotate an object in relation to the stars without developing inertial of centrifugal forces. Also a spinning gyroscope’s axis will always point to a fixed place in the sky unless there is pressure on the axis to make it precess

Michaelson and Morley, and others proved that there is no such thing as absolute movement.

Acceleration, however is absolute. This is why the axis of the gyroscope points to a fixed point in the firmament. Linear acceleration is also absolute. If an object develops inertial forces it will be found that it is accelerating in relation to the stars and the forces developed are not in relation to local objects or a reference frame that we have arbitrarily defined. (let’s leave out acceleration due to gravity for now)

As was stated in the quote at the top “It must be said that, as far as I can see, to this day Mach's principle has not brought physics decisively farther.” This may be correct but it doesn’t mean that Mach’s principle isn’t true. Inertial forces are tied to the stars, hence rotation is absolute and we must figure this out. It is not enough to ignore facts because they don’t fit our favourite theory. I even saw somebody say that Mach’s principle was passè. How can a fact of physics be passè.

 

[PeterDonis]

unsuccessfully as far as I understand

That depends on what your definition is of Mach's Principle. Some physicists seem to think GR incorporates it just fine; for example, John Wheeler, one of the key figures in GR (the "W" in MTW, the classic GR textbook), co-authored a textbook called "Gravitation and Inertia", which is almost entirely about how GR incorporates Mach's Principle.

A scientist is in a closed space capsule with no windows. The capsule is set spinning. The scientist has a gyroscope and can use his thrusters to stop the spacecraft from spinning. Another closed spacecraft at a distance (the other side of the solar system or the other side of the galaxy) is doing the same thing. Both scientists do their work and declare themselves not rotating, open the door, lean out and wave to each other. They notice that the capsules are not rotating in relation to each other.

You left out one key assumption: that spacetime itself is flat. In a curved spacetime this experiment might not actually produce the result you describe: two scientists in closed capsules, widely separated in space, who both use local gyroscopes to determine that their capsules are not rotating, and then open a door (I would say a window since we don't want air to leak out ) and look at each other, will not necessarily find that they are not rotating relative to each other. In other words, in a curved spacetime, "not rotating relative to local gyroscopes" is not necessarily the same as "not rotating relative to the distant stars".

In fact this kind of thought experiment is one of the reasons why Wheeler says GR does incorporate Mach's Principle--because Mach's Principle, in GR terms, says that the geometry of spacetime, which is what determines how gyroscopes behave, is not fixed but dynamically depends on the distribution of matter and energy in the universe. The distant stars contribute to that, but so do other things. For example, if your gyroscopes are orbiting the Earth, the Earth's mass and spin will affect their behavior (look up "de Sitter precession" and "Lense-Thirring precession"--the latter is what was tested for and confirmed by Gravity Probe B) so that they precess relative to the distant stars--in other words, if you are riding along with the gyroscopes in orbit about the Earth, and use the gyroscopes to make sure your spaceship is not rotating, and then look out at the distant stars, you will find that you are rotating relative to them--not just in the sense that you are orbiting the Earth, but in addition to that: when you have completed exactly one orbit around the Earth, the distant stars will not have returned to the same positions in your sky as before.

Inertial forces are locked to the stars

Not necessarily. See above. What they are "locked" to is the overall distribution of matter and energy; but, as above, the distant stars are not the only things that contribute to that.

a spinning gyroscope’s axis will always point to a fixed place in the sky

Not necessarily. See above. Note also that even in flat spacetime, if we take a gyroscope and put it in circular motion, applying force only at its center of mass (zero torque), the gyroscope will precess relative to the distant stars (i.e., after exactly one orbit around the circle, the stars will not be in the same direction relative to the gyroscope). This is called Thomas precession.

Acceleration, however is absolute

More precisely, proper acceleration--what an accelerometer measures--is absolute. But proper acceleration and rotation are different things. See below.

This is why the axis of the gyroscope points to a fixed point in the firmament

I'm not sure I understand your reasoning here (also, the statement as you make it is incorrect, see above). It might be helpful to draw a careful distinction between two kinds of "rotation"--the rotation of the gyroscope itself, as the thing that keeps its axis "fixed" and makes it a good reference to use for a "direction in space", and the "rotation" of an object moving in a circular path about something else. The latter might or might not be associated with any proper acceleration; in the case of Thomas precession of a gyroscope moving in a circle in flat spacetime, it is, but in the cases of de Sitter precession and Lense-Thirring precession of a gyroscope in a free-fall orbit around a planet, it isn't (the orbit is free-fall, with zero proper acceleration). But the gyroscope's axis serves as a reference just as well in both cases.

Linear acceleration is also absolute.

Yes, this is true, but not for the reason you give. Linear acceleration is proper acceleration, and proper acceleration is absolute because, as above, it is a direct local observable--just use an accelerometer.

If an object develops inertial forces it will be found that it is accelerating in relation to the stars

This is not necessarily true either. A rocket ship that is "hovering" above a planet like the Earth, at a fixed altitude and fixed spatial position, might not be moving at all relative to the distant stars (that depends on how the planet itself is moving, and we can certainly imagine a planet that is at rest relative to the distant stars). But it will still have nonzero proper acceleration. In fact, by the equivalence principle, the crew has no way to tell, just from measurements made inside the rocket, whether they are in fact hovering motionless above a planet or accelerating linearly in free space with no gravity present. The only way they can tell is to look out the window at distant objects.

the forces developed are not in relation to local objects or a reference frame that we have arbitrarily defined

This is not correct. See above.

let’s leave out acceleration due to gravity for now

This is a good idea, because "acceleration due to gravity" is not proper acceleration--objects moving solely under gravity are in free fall. So it's best to ignore any kind of "acceleration" other than proper acceleration.

Inertial forces are tied to the stars, hence rotation is absolute

This is not correct as you state it. There are a lot of issues involved here and you have overlooked some and appear to be mistaken about others. See above.

 

[Torog]

Thank you Peter, a lot to think about here. I am really impressed by the breadth and speed of your answer.

I'll get back with further thoughts soon.

Regards,

 

[Chronos]

You are correct in that GR is not fully consistent with Mach's principle regarding inertia. Einstein himself acknowledged this fact. The seminal paper; On the Origin of Inertia, by Sciama: http://articles.adsabs.harvard.edu/...=2&data_type=GIF&type=SCREEN_VIEW&classic=YES, has a nice summar. You may also find this discussion of interest; A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke, https://arxiv.org/abs/0708.3518

 

[Torog]

Thank you Chronos for the links. Very illuminating (the bits in English I could understand) and encouraging.

Regards,

 

[spacejunkie]

Under Barbour's precise definitions of Mach's Principle, linked below, GR is "weakly" Machian (definition 2) in that it is slightly less predictive than required by the strong version. The definition of Mach's Principle
Found. Phys. 40 1263-1284 (2010)
https://arxiv.org/abs/1007.3368

 

[Torog]

That depends on what your definition is of Mach's Principle. Some physicists seem to think GR incorporates it just fine; for example, John Wheeler, one of the key figures in GR (the "W" in MTW, the classic GR textbook), co-authored a textbook called "Gravitation and Inertia", which is almost entirely about how GR incorporates Mach's Principle.,

Peter

Thanks I understand most of your criticism and the reason why the gyroscope might not be perfectly stable. This comes from the twists and the dents in the underlying space – or spacetime if you prefer.

Let’s just say that the preponderance of inertial forces come from distant matter. Otherwise you would have to say that there is no connection between the fact that centrifugal forces appear as objects rotate in relation to the stars. Or just say that any of the observations and effects that make up Mach’s Principle are fictitious and not observable.

I also understand the principle of equivalence although I don’t like it very much. The fields created by linear acceleration and by a gravitational object are drastically different as you know. The fact that it may not be possible to differentiate in a small lab is not very significant – increase the sensitivity of the testing equipment and you could, in theory, always tell which kind of field you were in. I do understand that both forces – or both examples of free fall you mentioned – come from your Spacetime – in one case a linear acceleration and in the other a local dent.

You also say that some physicists believe that Mach’s principle is included “just fine”. Again from Wikipedia on Mach’s principle:
“There have been other attempts to formulate a theory which is more fully Machian, such as the Brans–Dicke theory and the Hoyle–Narlikar theory of gravity, but most physicists argue that none have been fully successful. At an exit poll of experts, held in Tübingen in 1993, when asked the question, 'Is general relativity perfectly Machian?', 3 respondents replied 'yes' and 22 replied 'no'. To the question, 'Is general relativity with appropriate boundary conditions of closure of some kind very Machian?' the result was 14 'yes' and 7 'no'.”

Why after 100 years of GR is there still so much discussion and disagreement on its application?

The most difficult thing I found was trying to use “quotes” as you did in your response to me. I managed to get your response up as a quote but for the life of me I couldn’t figure out way to separate my response from the quote.

All the best

 

[PeterDonis]

I managed to get your response up as a quote but for the life of me I couldn’t figure out way to separate my response from the quote.

Just make sure that your response starts after the end of quote tag. I used magic moderator powers to edit your post to fix that.

 

[PeterDonis]

This comes from the twists and the dents in the underlying space – or spacetime if you prefer.

I would not quite put it this way. First of all, Thomas precession happens in flat spacetime, which has no "twists and dents". Second, the relationship between the way the gyroscope is pointing and the direction of some distant object like a star depends on the gyroscope's path through spacetime, not just on the geometry of spacetime. But the geometry of spacetime certainly is a factor, yes.

The fields created by linear acceleration and by a gravitational object are drastically different as you know.

Only over a large enough extent of spacetime. Over a small extent of spacetime they are only a little different--and the difference gets smaller as the extent of spacetime over which you are looking gets smaller. So it's not always "drastic".

Why after 100 years of GR is there still so much discussion and disagreement on its application?

There isn't any disagreement on the "application" of GR. Everyone agrees on how to use the theory to make predictions that can be compared with experiment. For example, everyone agrees that the distribution of matter and energy in a spacetime determines its geometry via the Einstein Field Equation, and that in turn determines what "inertial forces" are present at a given point in spacetime. If you hand a bunch of relativity physicists with different opinions on Mach's Principle the same distribution of matter and energy and ask them to predict what the inertial forces will be in the resulting spacetime, they will all give the same answer.

The question about Mach's Principle is not an "application", it's more of a "philosophical" question about whether the theory has a certain property, the definition of which not everyone agrees on. For example, not everyone agrees on whether the fact I described just now, that the distribution of matter and energy in a spacetime determines its geometry and hence the inertial forces, means that GR fully incorporates Mach's Principle--even though, as above, they all give the same answer about the actual experimental prediction, what the inertial forces are.

 

[Torog]

Just make sure that your response starts after the end of quote tag. I used magic moderator powers to edit your post to fix that.

Got it, Thanks

 

[Torog]

Under Barbour's precise definitions of Mach's Principle, linked below, GR is "weakly" Machian (definition 2) in that it is slightly less predictive than required by the strong version.The definition of Mach's Principle
Found. Phys. 40 1263-1284 (2010)
https://arxiv.org/abs/1007.3368

Spacejunkie,

I found this definition of Mach's Principle very difficult to follow. Nowhere near as clear in writing as the paper linked by Chronos (http://articles.adsabs.harvard.edu/...=2&data_type=GIF&type=SCREEN_VIEW&classic=YES) by Sciama.

Thanks all the same

 

[timmdeeg]

Mach
As was stated in the quote at the top “It must be said that, as far as I can see, to this day Mach's principle has not brought physics decisively farther.” This may be correct but it doesn’t mean that Mach’s principle isn’t true. Inertial forces are tied to the stars, hence rotation is absolute and we must figure this out. It is not enough to ignore facts because they don’t fit our favourite theory. I even saw somebody say that Mach’s principle was passè. How can a fact of physics be passè.

Interesting discussion. Perhaps this is of interest:

Max Jammer in "Concepts of Mass", 2000, page 150:

It could be shown that a particle in an otherwise empty universe can possesses inertia or that the first Machian effect is not at all a truly physical effect but can be eliminated by an appropriate choice of a coordinate system. Einstein's confidence in the principle gradually waned, so much that eventually, a year before his death, he declared that "one should no longer speak at all of Mach's principle."

 

[Dale]

Why after 100 years of GR is there still so much discussion and disagreement on its application?

What you cited was not disagreement on GR, it was disagreement on what "Machian" means. Mach's principle is very appealing in broad terms, but very difficult to express in a testable form. The best attempt to date, Brans Dickie gravity, seems to indicate that the universe is no more Machian than GR.

 

[PAllen]

@Peter, the fact that topology and even global geometry may depend on boundary conditions is a limitation on how Machian GR is, true? As an extreme, eternal BH solutions and Minkowski space all have no matter or energy at all in them, but they have completely different inertial structures.

 

[PeterDonis]

the fact that topology and even global geometry may depend on boundary conditions is a limitation on how Machian GR is, true?

For some (perhaps most) meanings of "Machian", yes.

eternal BH solutions and Minkowski space all have no matter or energy at all in them, but they have completely different inertial structures

Yes, this is a good example of how the Einstein Field Equation alone is not sufficient to determine a global geometry, mathematically speaking.

In terms of physically realistic solutions, though, this issue doesn't really arise. Any physically realistic solution for, e.g., a black hole will have a region of nonzero stress-energy somewhere, because some object with stress-energy in it will have collapsed to form the hole. Even the asymptotically flat boundary condition can be replaced by a smooth merging of the black hole solution into something like an FRW background spacetime describing the universe in which the hole exists. And the FRW spacetime describing the universe as a whole doesn't need a boundary condition--unless you call the fact that our current best-fit solution is spatially infinite a "boundary condition". (Which might be a tenable position--IIRC Einstein said that for a GR solution describing the universe to be truly Machian, the universe would have to be spatially closed.)

 

[stevendaryl]

That depends on what your definition is of Mach's Principle. Some physicists seem to think GR incorporates it just fine; for example, John Wheeler, one of the key figures in GR (the "W" in MTW, the classic GR textbook), co-authored a textbook called "Gravitation and Inertia", which is almost entirely about how GR incorporates Mach's Principle.

What definition of Mach's principle makes GR Machian? I thought that Mach's principle was the claim that acceleration is relative to other matter, so there could be no difference between (1) a bucket of water rotating clockwise and (2) the rest of the universe rotating counterclockwise. But in GR, even in an empty universe, there is a notion of geodesics, so acceleration doesn't have to be relative to anything.

 

[PeterDonis]

What definition of Mach's principle makes GR Machian?

The one that, roughly speaking, says that the spacetime geometry, and therefore the inertial properties of worldlines (which ones are geodesics, and what proper acceleration is associated with non-geodesic worldlines), are dynamical, i.e., determined by the distribution of matter and energy in the universe. (There are some possible caveats even for this definition, which PAllen and I have had an exchange about in this thread.)

 

[Alfredo Tifi]

So, if Wheeler is right, the result of another thought experiment: an unfolding Newton's bucket in a completely flat universe, is that water in it does not rise. Rotation, even rotation relatively to the unfolding rope, should be equivalent to rest. Inertia would be completely absent. Am I right?

does[/I] incorporate Mach's Principle--because Mach's Principle, in GR terms, says that the geometry of spacetime, which is what determines how gyroscopes behave, is not fixed but dynamically depends on the distribution of matter and energy in the universe. The distant stars contribute to that, but so do other things.

 

[timmdeeg]

Rotation, even rotation relatively to the unfolding rope, should be equivalent to rest. Inertia would be completely absent. Am I right?

I don't think so. As far as I can tell today most physicists would agree that proper acceleration is a local phenomenon, thus the water in the bucket would rise. See also the statement of Max Jammer in post #12.

 

[Alfredo Tifi]

I don't think so. As far as I can tell today most physicists would agree that proper acceleration is a local phenomenon, thus the water in the bucket would rise. See also the statement of Max Jammer in post #12.

Since Max Jammer and late Einstein I get the idea that absolute rotation in an empty space (that is relative to nothing) does make sense. Am I still wrong?

 

[timmdeeg]

I would agree to that but am not sure if "relative to nothing" is the wording a physicist would use. Perhaps it's better to say proper acceleration is Lorentz-invariant, so it's not observer dependent. A counterexample is the velocity of a body relative to an observer.

 

[Torog]

A different angle…. (Assume flat Spacetime)

I am mystified by Kinetic energy & Inertia. It is assumed that inertia has no limit and experimental evidence seems to bear this out. It is by far the most powerful force in nature. It would theoretically be possible to use any amount of energy to accelerate an object - for example, the energy of an atomic bomb to accelerate a bullet. But where is the energy once the bullet is accelerated? If a laboratory was set up near the accelerated bullet and you looked for the energy, there would be no observable physical change in the projectile. We know that it is possible to quantify kinetic energy knowing the mass and the speed and with this it is possible to know exactly how much energy we will recover when the bullet hits something. This is a convenience but it doesn't tell us where the energy was in the interim.

It is also theoretically possible (as I understand it) to put more kinetic energy into a mass than the atomic energy you could derive from the same mass with fission or fusion.

Say a bullet is traveling in an inertial frame and has a certain calculated amount of kinetic energy. If it then moves into another inertial frame and has an apparently different speed it will have a different amount of kinetic energy. So can we change the amount of energy in the bullet by defining the reference frame? It seems absurd that we can decide at a distance how much kinetic energy the bullet holds.

Here is another little thought experiment. Fire a bullet at a brick and then fire a brick at a bullet. Examine the the evidence of the collisions and it will not be possible to tell in either case if the brick or the bullet was moving and which had the energy. So where does the energy go in the acceleration? In an absurd example you could fire a bullet at a wall and say that the kinetic energy moved to the wall and waited there to be released when the bullet arrived. So in effect we do not know where the energy (kinetic) is when we accelerate something.

If a bullet is accelerated by its charge it will travel in a straight line on what could be called inertial rails. The definition of the straight line involves the inertial frame of reference and the whole Universe. If the inertia of the bullet was residing in the bullet (a common assumption) it might go in any direction once accelerated. The fact that it goes in a straight line (in the frame of reference of the gun) shows that the inertial frame or field is controlling the path of the bullet. Its inertia is defined by an external field – that's what keeps the bullet on its straight path.

 

[timmdeeg]

Say a bullet is traveling in an inertial frame and has a certain calculated amount of kinetic energy. If it then moves into another inertial frame and has an apparently different speed it will have a different amount of kinetic energy. So can we change the amount of energy in the bullet by defining the reference frame? It seems absurd that we can decide at a distance how much kinetic energy the bullet holds.

Not really. Kinetic energy is indeed frame dependent. E.g. in the frame in which you and the bullet are at rest (relative speed between you and the bullet is zero) the kinetic energy of the bullet is zero.

In an absurd example you could fire a bullet at a wall and say that the kinetic energy moved to the wall and waited there to be released when the bullet arrived. So in effect we do not know where the energy (kinetic) is when we accelerate something.

In the frame in which the wall is at rest the bullet has a certain amount of kinetic energy. After it hits the wall, this amount is transformed into deformation of the bullet and the wall and into heat. So the energy is conserved before and after the bullet hits the wall.

 

[Torog]

In the frame in which the wall is at rest the bullet has a certain amount of kinetic energy. After it hits the wall, this amount is transformed into deformation of the bullet and the wall and into heat. So the energy is conserved before and after the bullet hits the wall.

I'm not worried about the conservation of energy. I would just like to know where it is if is use my heat energy to accelerate something.

 

[Dale]

I'm not worried about the conservation of energy. I would just like to know where it is if is use my heat energy to accelerate something.

Energy is conserved in many cases, and also localized in many cases. What it is not is frame invariant. So you can perfectly well localize the heat energy used to accelerate something, but it will depend on the reference frame.

 

[timmdeeg]

I would just like to know where it is if is use my heat energy to accelerate something.

Then it corresponds to the kinetic energy of the bullet in the frame in which you are at rest.

 

[PeterDonis]

if Wheeler is right, the result of another thought experiment: an unfolding Newton's bucket in a completely flat universe, is that water in it does not rise. Rotation, even rotation relatively to the unfolding rope, should be equivalent to rest. Inertia would be completely absent. Am I right?

No. Flat spacetime has a geometry, and determines inertial properties such as those that make the water rise in Newton's bucket. You can have regions of spacetime that are flat in a spacetime that is not flat everywhere--for example, the interior of a perfectly spherically symmetric shell of matter. You could do the Newton's bucket in the vacuum region inside the shell and the water would rise.

 

[Ken G]

I think one way to think about what Mach's principle does, most generally, is it places Newton's first law at the level of an emergent phenomenon, rather than a law of nature in its own right. The first law,in modern terms, essentially says that objects with no force on them follow geodesics, but Newton and Galileo simply assumed the geodesics could be known in advance-- straight lines. It seems to me the main thing Mach is saying is, why should we know what the geodesics are in advance? They should have a dynamical history as well. So it's not so much the question "what are the geodesics in an empty universe," but even more it is, "what determines the geodesics in an empty universe in the first place." So an empty universe could simply be a singular situation where there is nohing (but arbitrary boundary conditions) to say what the geodesics are, that wouldn't violate Mach's principle because that principle doesn't need to determine the conditions in a pathologically empty universes, any more than Laplace's equation does for a static electric field in an infinite universe empty of charge.

 

[timmdeeg]

I think one way to think about what Mach's principle does, most generally, is it places Newton's first law at the level of an emergent phenomenon, rather than a law of nature in its own right. The first law,in modern terms, essentially says that objects with no force on them follow geodesics, but Newton and Galileo simply assumed the geodesics could be known in advance-- straight lines. It seems to me the main thing Mach is saying is, why should we know what the geodesics are in advance?

Could you elaborate a little more? To my understanding Newton's first law refers to inertial reference frames while Mach's principle refers to proper acceleration and thus not to geodesics. What am I missing? I don't see the connection.

 

[Ken G]

Newton's first law is essentially the rule for the behavior of objects that have zero proper acceleration as seen by observers that have zero proper acceleration. General relativity generalizes that rule by saying the objects will follow geodesics. However, GR does not assume the geodesics must be straight lines, instead it allows them to evolve dynamically based on the history of the interaction between mass and spacetime. As put by Wheeler, mass tells spacetime how to curve and spacetime tells mass how to move (in the absence of forces). I'm saying that the fundamental break from Newton and Galileo that this statement represents is the admission that we must not assume we know what the geodesics are by fiat, the geodesics are emergent properties not fundamental laws. This is the key advance of GR--- it is always an advance when a theory takes what was simply assumed as true (like Kepler's statement that orbits are ellipses), and provides a law for demonstrating the emergence of that property, thus determining why that is the case (and when it is not the case, as with galaxies and dark matter),

But such advances come at a price-- we need additional information to understand the emergent property. We need to know the mass distribution to know if orbits will be ellipses or something else, for example. So we need boundary conditions, not only in space but also time, which Galileo and Newton did not need and did not put in their version of the law of inertia. Given that need, we can see that an empty universe that has always been empty is an inherently problematic example, as it lacks any information of the nature of mass telling space how to curve, and therefore it may accept multiple possible solutions for the geodesics. Instead, we would have to supply that information manually, a problem that cannot be blamed on Mach's principle because Mach's principle is only the statement that, for example, a rotating bucket will have the water rise because the water is not following the geodesics that are dictated by the dynamical history of the distribution of stars and galaxies and so on. Hence, one cannot claim that Mach's principle is inadequate by citing the fact that it doesn't resolve the ambiguities of an empty universe-- it is the universe itself that is inadequate in such a case.