Exploring Newton's Bucket Paradox

[Dale]

I don't know much about Mach's principle, but discussions about it always seem to turn into these rather silly "otherwise empty universe" discussions, which makes me question the value of Mach's principle.

Does Mach's principle have any concrete testable predictions? If not, what is its value?

 

[Garth]

Does Mach's principle have any concrete testable predictions? If not, what is its value?

In one version of the principle it suggests the Newtonian Gravitational constant is not actually constant but varies from place to place.

The Brans Dicke theory, which fully incorporates Mach's principle into GR, made observational predictions that do not seem to be consistent with observation.

Garth

 

[aaj]

Hi. First post here. I have no formal math or physics training, but read popular books on physics and am pretty well read as far as that goes. Now for the question.

I'm fascinated by the Newton's Bucket problem and fortunately for me it's cleared my head of the 2 brothers paradox (one on earth, one in ship, ship ages) with regard to which one is considered moving and which is stationary.

For a description of Newton's Bucket, here's a good one:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Newton_bucket.html

I've never liked the traditional idea that the brother that is considered moving (and therefore aging) is the one that is accelerating away because once acceleration stops and the ship continues at near light speed, the aging process continues yet the ship is only moving relative to the Earth and not accelerating away from it.

Newton's Bucket solves that problem by inferring that the ship is moving near light speed relative to either the stars or some universal fabric that is static or almost static relative to the stars.

Newton's bucket implies that if the universe were empty (I suppose this would include dark matter and energy) except for the bucket and a single observer, the bucket would seemingly have to behave strangely. For example, if the observer were spinning around the bucket (and the bucket around the observer) but both in the same direction as far as the two axis of rotation are concerned, the bucket could not be said to be spinning and therefore would not exhibit inertial forces or the resultant concave water. If the observer and bucket were spinning opposite to each other, then what? Would the water then become concave relative to the velocity of the observer? Or is a greater mass (or something else altogether) required such as massive galaxies? And if either or both are causing the water to become concave, then what exactly is causing it. I realize the simple answer is inertia, but this paradox implies that inertia would cease to exist in an empty universe and with the observer and bucket moving in the same direction or possibly in different directions as well.

Inertia would have to cease to exist in an empty universe that contained only a bucket of water and a single observer moving in the same direction around it as there would be absolutely no frame of reference with regard to acceleration. With no inertia, one could not feel any effects of acceleration so if the bucket exploded, or the observer sneezed, which would move relative to the other, and which one would age when applied to the two brother paradox.

Glad to have found this forum.

The way I interpret 'Newton's bucket' experiment is that it does indeed show that there is some absolute reference and that relative motions are not all that matter. Now I don't think, it automatically implies the existence of absolute space, as Newton and most contemporaries thought. Indeed as Einstein showed, the absolute reference turned out to be space-time rather than space.

In space-time, only relative velocities matter but accelerations are still absolute. Hence, that should explain Newton's bucket experiment. There is no ambiguity about whether water in a bucket is rotating or not because rotation is an accelerated motion. Hence, it has nothing to do with gravitational pull of the rest of the universe and so a concave water shape should result in a rotated bucket even if its the only object in the universe.

 

[yuiop]

The way I interpret 'Newton's bucket' experiment is that it does indeed show that there is some absolute reference and that relative motions are not all that matter. Now I don't think, it automatically implies the existence of absolute space, as Newton and most contemporaries thought. Indeed as Einstein showed, the absolute reference turned out to be space-time rather than space.

In space-time, only relative velocities matter but accelerations are still absolute. Hence, that should explain Newton's bucket experiment. There is no ambiguity about whether water in a bucket is rotating or not because rotation is an accelerated motion. Hence, it has nothing to do with gravitational pull of the rest of the universe and so a concave water shape should result in a rotated bucket even if its the only object in the universe.

Can you be sure acceleration is absolute?

Calculations using general relativity have shown that a massive rotating shell would induce a force that causes the surface of stationary water in a stationary bucket at its centre to curve exactly as if the if the water was rotating. The relativistic principle suggests there is no measurement that can destinguish a rotating bucket in a universe of of stationary stars from a stationary bucket in a universe of rotating stars. The mass of the rotating stars will drag spacetime as per the Lense Therring effect causing the water in the stationary bucket to climb up the sides of the bucket as if it was rotating. The stars will not be thrown outward by centripetal forces because the spacetime is co-moving with the stars.

Now imagine a universe with a one stationary bucket and one atom at the edge of the universe visible from the bucket. The atom is rotating around the bucket at very high speed but there is no way that the mass of a single atom at such a great distance can induce any significant gravitational field or curvature in the surface of the water in the bucket. By invoking the principle of relativity, rotating the bucket and water relative to the distant stationary atom will not induce any curvature in the surface of the water. The single atom is an aproximation of an "otherwise empty universe"

The same can be said for linear acceleration. There is no difference between a rocket accelerating in a stationary universe and a stationary rocket in an accelerating universe. The rocket engine is simply resisting the gravitational field that is drawing the rest of the universe into an event horizon behind the rocket.

Further evidence that gravitational fields might be the source of inertia is this. A perfectly elastic ball is placed in a box and set bouncing from side to side horizontally. The box is far out in space. As the box is gradually lowered towards a large massive body we would expect that if the inertial mass is increasing with increasing proximity to a massive body and if momentum is conserved, that the ball would slow down. That is exactly what we do observe (from a distance). When we bring the box back up we note the ball has speeded up again. Similar experiments accelerating objects horizontally would show they behave as if they have greater inertial mass lower down nearer the massive body, suggesting Mach's principle of inertia being a function of the total gravity of the all the surrounding mass is not far off the mark. Now when we take the box infinitely far away from the massive body (or a long way away, anyway) that the ball still has inertia seeming to contradict Mach's principle. The solution is that in our universe, you can not get infinitely far from any massive body without getting closer to other massive bodies. There is always a "zero point" gravitational field wherever you are and although we might find it mathematically convenient to call this residual gravitational potential zero, it is not in fact zero and this could account for baryon particles having a non zero inertial mass, when seemingly at a zero gravitational potential. Even in the largest of voids, the surrounding mass ensures the gravitational potential is never zero and so the inertial mass is never zero.

So my argument is that if we take a take an informal description of Mach's principle as "Inertia of a body is a property of its motion relative to the fixed stars" and restate it as "Inertia of a body is a property of its motion relative to the spacetime determined by the distribution and motion of matter in space" then Mach's principle is pretty compatible with relativity. The important notion is that inertia is not an intrinsic property of mass, independent of its surroundings.

 

[aaj]

The important notion is that inertia is not an intrinsic property of mass, independent of its surroundings.

This is interesting. Just as a curiousity, has there been any experiment conducted to verify this notion? I haven't thought over this much but should it not be possible to simply test this out by performing an experiment on an object, once with no heavy object close by and once more with many heavy objects in its immediate surrounding? By observing whether the object responds differently to the same force, it might be possible to test out the hypothesis that inertia is not an intrinsic property of mass. Ofcourse, I can quite guess that technical limitations might be a big reason why we cannot achieve the sensitivity required for the above kind of experiment.

On another line of thought, if the hypothesis of inertia not being an intrinsic property of mass is true, how come we have never quite observed this effect through astronomical observations? I mean galaxies also move through space. Has it ever been observed that the inertia of an entire cluster of stars has changed simply because of their changed position in the universe? If it hasn't and they have changed position, it would imply that the mass density of the universe is pretty even in all directions.

 

[Mentz114]

The idea that inertia is a reaction between mass and some field is explored in this paper, which is published in Physics Letters A and on the arXiv

http://arxiv.org/abs/physics/9802031

The authors ascribe inertia to the EM ZPF, but actually it would work with any ZPF that interacted with baryons. here's a brief extract -

If correct, this concept would substitute for Mach’s principle and imply that no further mass-giving Higgs-type fields may be required to explain the inertia of material objects, although extensions to include the zero-point fields of the other fundamental interactions may be necessary for a complete theory of inertia.

Which sounds like Kev's proposal.

 

[aaj]

The important notion is that inertia is not an intrinsic property of mass, independent of its surroundings.

Its also intetresting to note that the above hypothesis seeks to make a clear distinction between mass and inertia. Most common definitions of mass itself are in terms of inertia. For instance, 1kg of mass may be defined as that mass which accelerates at 1m/s^2 in response to a force of 1N. Now if we delink mass and inertia as per the quoted hypothesis, how then do we define mass?

I am guessing it would be in terms of the ability to curve spacetime. So depending on the surroundings, an object's inertia may be different but are we saying that its ability to curve spacetime around itself would be unaffected?

So would a universe consisting of only one atom still be curved in the vicinity of the atom?

It seems we would then we forced to have to have two masses for each object. a) The Inertial mass which would be a measure of the inertia of the object and which the hypothesis says depends on its surroundings and b) the curvature mass which would be a measure of the ability of the object to curve spacetime.

But then, gravity is a manifestation of curvature so we are harking back tothe times when we had the concept of inertial and gravitational masses. And so many experiments have showed that these two masses have always been found to be the same with ever increasing accuracy. If this is the case, we are left with two conclusions:

a) The two masses seem to be equal because inertia is indeed and intrinsic property of the body and is determined by the same quantity that curves spacetime and is unaffected by its surroundings

b) Inertia may be determined by surroundings but we have never noticed any fluctuation because the universe is very even in all directions to an astonishing degree.

But then point b still begs the question why it is that the quantity which determines inertia is so so nearly equal to the quantity which is responsible for curving spactime in the vicinity of the object?

 

[Mentz114]

aaj

But then point b still begs the question why it is that the quantity which determines inertia is so so nearly equal to the quantity which is responsible for curving spactime in the vicinity of the object?

If the ZPF hypothesis is true, then both inertia and gravity result from the very same cancellation effect - and so must be identical. This is one of the best points about this hypothesis, unification of gravitational mass and inertial mass.

 

[Ken G]

What bothers me about this idea is that it seems to require that the presence of gravity alters the physics of a system in a way other than due its tidal effects, which seems to violate the principle of equivalence. In other words, if you put a box around a system, then the effect of gravity on the internal workings of that system should only come in via its tidal influences. But if you put a force on a point particle in that box, and claim that gravity from external sources are responsible for the way it accelerates, then you cannot have the equivalence principle. Note there is not a problem with kev's thought experiment about a ball bouncing back and forth in a box, because tidal stresses across that box must be responsible for the behavior observed, but inertia itself is a property of a point particle.

 

[Mentz114]

I agree that it would be a major flaw if the gravitational mass and inertial mass changed according to some local field strength. I think the authors of the cited paper assume an absolute vacuum, one that looks the same to all inertial observers and is in fact the source of inertial and gravitational effects when interacting with matter.

I'm keeping an open mind about this. No one else has attempted to 'explain' F=ma and it is an ingenious idea that maybe could give rise to a decent theory.

 

[Ken G]

I see something of a "Catch 22" here. If it responds to a local field strength, inertia seems to refute relativity, but at least you have a falsifiable theory. If it does not, then how will you ever establish the connection? It sounds a lot like the claim "the total distribution of mass in the universe is why the speed of light is what it is"-- how could anyone falsify that claim? I see Mach's principle as a way to break one's mind out of a box that might limit you to missing a theory like general relativity, but having the theory of GR, I'm not sure where we need Mach's principle. It's true that GR is a differential theory, so needs the external application of some kind of boundary conditions (does it not?), and one might then say we use Mach to inform the boundary conditions. But even that would be backwards logic-- we always apply whatever boundary conditions that seem to work, so if Mach hadn't worked we would use a different boundary condition. It doesn't establish that Mach informs our boundary condition-- getting results that agree with experiment do that. This is the fundamental problem of mixing philosophical principles into physics-- science just isn't done that way, except in the "inspiration" phase.

 

[Mentz114]

..., but at least you have a falsifiable theory.

We aren't talking about fully-fledged theory but an hypothesis. When I said a 'decent' theory, I mean it must be falsifiable. I don't consider what Haisch et al have presented to anything like correct.

This is the fundamental problem of mixing philosophical principles into physics-- science just isn't done that way, except in the "inspiration" phase.

If you're talking about Mach's conjecture, I agree. I've never seen any use for it.

 

[Buckethead]

Can you be sure acceleration is absolute?

Calculations using general relativity have shown that a massive rotating shell would induce a force that causes the surface of stationary water in a stationary bucket at its centre to curve exactly as if the if the water was rotating. The relativistic principle suggests there is no measurement that can destinguish a rotating bucket in a universe of of stationary stars from a stationary bucket in a universe of rotating stars. The mass of the rotating stars will drag spacetime as per the Lense Therring effect causing the water in the stationary bucket to climb up the sides of the bucket as if it was rotating. The stars will not be thrown outward by centripetal forces because the spacetime is co-moving with the stars.

This last sentance is astonishing to me, a real eye opener. If spacetime moves with the stars, then doesn't this automatically imply that the spatial location of the stars (by stars I do mean all matter in the universe) and spacetime itself are one in the same thing? Doesn't this mean that Mach's principle and the idea of absolute spacetime are the same thing?

Now imagine a universe with a one stationary bucket and one atom at the edge of the universe visible from the bucket. The atom is rotating around the bucket at very high speed but there is no way that the mass of a single atom at such a great distance can induce any significant gravitational field or curvature in the surface of the water in the bucket. By invoking the principle of relativity, rotating the bucket and water relative to the distant stationary atom will not induce any curvature in the surface of the water. The single atom is an aproximation of an "otherwise empty universe"

I'm not convinced that there is a relationship between gravity and absolute spacetime/Mach's principle as wouldn't we see a change in inertia if we were far out in space away from strong forces of gravity? Whatever spacetime is "made of" it has to be fairly uniform across the universe. It must be influenced by either virtual particles, dark energy, dark matter, or some strange type of pervasive field.

It seems about half of the posts here have argued that a bucket can be said to spin if and only if the water is climbing it's side independant of whether or not there is other matter in the universe and I'm assuming also independant of any absolute spacetime (if it can exist without matter). But in the very minimum, an absolute spacetime must exist or the idea of rotation reduces to simply a "seemingly stationary bucket with water in a concave shape". I could not conclude that anything was spinning from this observation. This would lead me to believe that some outside force such as gravity or otherwise was surrounding the bucket and forcing the water into this shape. This is the reason I do not think that a bucket in an empty universe (or in a universe without an absolute spacetime) can have a concave shape. This implies a lack of inertia (no more Newton's laws).

This lack of inertia does indeed bring up some additional strange observations. For example, what would happen if you shone a laser beam? If nothing unusual happened (it's light propagated out in a straight line) then we have a good argument as to why the water should go concave. If a straight line were definable by a laser, then certainly a bucket could be said to spin (and could be observed going concave) as it's water molocules tried to follow the path of the laser light. But without the presence of absolute spacetime, light could not propagate in a line or in any definable fashion.

It seems to me there can be nothing logical happening to light, a spinning bucket, or a linearly accelerating object unless these things are happening in a frame of reference and in the very least this frame of reference must be a grid of spacetime and at most could be the relative position of all matter in the universe.

To get back to Kev's comment about the stars not being affected by centrifugal forces: If this were indeed true, then spacetime's rotational velocity is defined by the location of the matter in the universe and follows it exactly. This would imply that Mach's principle is true. If Kev's comment were not true, then this would imply that spacetime and the matter in the universe were rotating relative to each other and centrifugal forces would be acting on stars in strange ways and it would also imply that spacetime must be made up of some form of matter or field that defined a grid by which inertia is subject. I'm not sure I can buy that, but I suppose it's possible.

I wish I had time to respond to more of the comments in this thread, but I am very much enjoying all that I am reading and I appreciate that this thread is being kept alive.

 

[newTonn]

Hi. First post here. I have no formal math or physics training, but read popular books on physics and am pretty well read as far as that goes. Now for the question.

I'm fascinated by the Newton's Bucket problem and fortunately for me it's cleared my head of the 2 brothers paradox (one on earth, one in ship, ship ages) with regard to which one is considered moving and which is stationary.

For a description of Newton's Bucket, here's a good one:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Newton_bucket.html

I've never liked the traditional idea that the brother that is considered moving (and therefore aging) is the one that is accelerating away because once acceleration stops and the ship continues at near light speed, the aging process continues yet the ship is only moving relative to the Earth and not accelerating away from it.

Newton's Bucket solves that problem by inferring that the ship is moving near light speed relative to either the stars or some universal fabric that is static or almost static relative to the stars.

Newton's bucket implies that if the universe were empty (I suppose this would include dark matter and energy) except for the bucket and a single observer, the bucket would seemingly have to behave strangely. For example, if the observer were spinning around the bucket (and the bucket around the observer) but both in the same direction as far as the two axis of rotation are concerned, the bucket could not be said to be spinning and therefore would not exhibit inertial forces or the resultant concave water. If the observer and bucket were spinning opposite to each other, then what? Would the water then become concave relative to the velocity of the observer? Or is a greater mass (or something else altogether) required such as massive galaxies? And if either or both are causing the water to become concave, then what exactly is causing it. I realize the simple answer is inertia, but this paradox implies that inertia would cease to exist in an empty universe and with the observer and bucket moving in the same direction or possibly in different directions as well.

Inertia would have to cease to exist in an empty universe that contained only a bucket of water and a single observer moving in the same direction around it as there would be absolutely no frame of reference with regard to acceleration. With no inertia, one could not feel any effects of acceleration so if the bucket exploded, or the observer sneezed, which would move relative to the other, and which one would age when applied to the two brother paradox.

Glad to have found this forum.

Dear all
let us view it in another angle.a liquid(water here)will exert presurre in all direction to the walls of the container(radially).when the bucket starts rotating,the extreme end molecule of water which is pressed against the wall will be moved together with the wall,because it is pressed against the wall,this in turn will be transferred to the next molecule and so on...upto the centre.
when a molecule near to centre starts to move in a circular path,it will exert more pressure tangentially and in fact ,it will be tranfered to the next layer of molecules(which is already pushing the other layer molecule due to circular motion) and forces added so on... and that force is not enough to break the wall of the bucket,but enough to raise the external molecule to a small height against the atmospheric pressure and gravity which is pulling it down.
When this external molecule is elevated,the penultimate will occuppy its space and so on.ultimately the surface will become concave..
Now when even the bucket stops rotating,the water will continue spinning because the water molecules still possesses kinetic energy and the pressure exerted to the walls of bucket is tangetial instead of normal as in the begining.gravity and atmospheric pressure of course will take some time to act against this spinning ,ultimately to halt it
Please correct me if i am wrong in understanding actual problem

 

[Buckethead]

Dear all
let us view it in another angle.a liquid(water here)will exert presurre in all direction to the walls of the container(radially).when the bucket starts rotating,the extreme end molecule of water which is pressed against the wall will be moved together with the wall,because it is pressed against the wall,this in turn will be transferred to the next molecule and so on...upto the centre.

This is the reason that water flows up the side of the bucket, but this is not the problem. The problem is determining how the water knows that it is moving in the first place and hence moving up the side of the bucket.

 

[Buckethead]

But that's why I asked if anyone really believed you could not get a dip in a bucket in an otherwise empty universe. I certainly don't believe it. So if you could, then you have to use the bucket to tell you whether or not it's rotating-- the effort to invert that logic is the source of the problem (that's where philosophy enters and muddies the science).
General relativity predicts the result of that experiment. Why do we need Mach? Don't get me wrong, I realize that asking the questions Mach did helped Einstein think "outside the box". That is generally what I view philosophy is for-- to free our thinking to see what the possibilities are. But we tend to cling to it long after it has ceased its usefulness, and mistake it for part of the theory.

From what I understand Einstein first bonded with Mach's idea, but then fell back on just accepting that there is an absolute universe. When Einstein talks about curvature of spacetime due to gravity in GR, what exactly is he referring to? (not mathamatically, but philosophically). One couldn't say it's a gravitational field because the warping of space is a result of gravitation. Is he referring to the virtual frame of reference that is created when all the mass and it's revolving/positional properties are taken into account (which would be Mach's principle)? I cannot accept that he is simply referring to a mathamatical virtual spacetime that has no physical basis in the real world. This would be nonsense.

So please, without a physical spacetime with which to say a water molocule is moving relative to, how can you still conclude that the water in the bucket will take a concave shape? The answer to this question is in my opinion very important to this discussion.

 

[Buckethead]

I don't know much about Mach's principle, but discussions about it always seem to turn into these rather silly "otherwise empty universe" discussions, which makes me question the value of Mach's principle.

Does Mach's principle have any concrete testable predictions? If not, what is its value?

I think it may. For example, if it can be shown that massive objects (or some other form of matter in the universe) and it's relative rotation/speed to an object determines the inertia (or mass) of the object, then this would indicate that counter structures could be built to alter the mass of objects. Having control over the mass/inertia of on object can open up all sorts of doors. More efficient ways to propel ships through space and countering gravity being a couple.

 

[Mentz114]

So please, without a physical spacetime with which to say a water molocule is moving relative to, how can you still conclude that the water in the bucket will take a concave shape? The answer to this question is in my opinion very important to this discussion.

Rotation can only be meaningfully defined for an extended object ( the water). Parts of the water are moving relative to other parts. There is no need to invoke physical space-time.

 

[Buckethead]

Rotation can only be meaningfully defined for an extended object ( the water). Parts of the water are moving relative to other parts. There is no need to invoke physical space-time.

I don't believe it's true when you say parts of the water are moving relative to other parts. They are all aligned and moving together. Let me restate this Newton's bucket in another way to eliminate the rotation.

Imagine a long straight stiff rod lying in empty space. Near one end of the rod and with it's nozzel parallel to it is a canon with a tennis ball inside. Just behind the tennis ball inside the canon is a laser pointing in the same direction as the nozzle of the canon.

The canon is fired, and shortly after that the laser is also fired. Two questions. Is the trajectory of the tennis ball parallel to the rod and secondly does the laser beam strike the rear of the tennis ball?

The seemingly obvious answers are yes in both cases, but I'm not so sure these are the correct answers. The reason being there is no physical relationship between the rod and the tennis ball or between the tennis ball and the laser beam.

The tennis ball can be thought of as representing water molocules in the bucket, and if it cannot be determined that the tennis ball will move parallel to the rod, then it cannot be determined that the molocules of water in the bucket will try and move tangent to the rotation of the water.

 

[newTonn]

This is the reason that water flows up the side of the bucket, but this is not the problem. The problem is determining how the water knows that it is moving in the first place and hence moving up the side of the bucket.

Can you please explain it further?water is moving because it is pressed against a moving object(bucket).since it cannot break the walls of bucket ,it is taking the next easiest path,that is tangentially with a small horizontal angle upwards.
I cannot understant why physical space time is necessary to explain this experiment.

 

[Mentz114]

Buckethead:

I appreciate that you're trying to make a subtle point, but your example fails in the first sentence.

Imagine a long straight stiff rod lying in empty space...

How do you define 'straight' ? You need some kind of reference to compare the rod with ( line of sight ?).
I don't see how this relates to the rotating water, sorry.

It is the inertial mass of the water that makes it climb the sides of the bucket, so I think it all boils down to this - will a solo object possesses inertia ?

In my opinion it will. It is simpler to believe that inertia is a local thing, either an intrinsic property of matter, or the result of a local interaction.

 

[yuiop]

What bothers me about this idea is that it seems to require that the presence of gravity alters the physics of a system in a way other than due its tidal effects, which seems to violate the principle of equivalence. In other words, if you put a box around a system, then the effect of gravity on the internal workings of that system should only come in via its tidal influences. But if you put a force on a point particle in that box, and claim that gravity from external sources are responsible for the way it accelerates, then you cannot have the equivalence principle. Note there is not a problem with kev's thought experiment about a ball bouncing back and forth in a box, because tidal stresses across that box must be responsible for the behavior observed, but inertia itself is a property of a point particle.

I would just like to point out I was talking about a ball bouncing back and forth horizontally in the box so tidal stresses do not come into that thought experiment, because tidal stresses are vertical.

Having given some more thought to the subject it seems that GR is not fully compatible with Mach's principle as pointed out by Garth. (By the way, you have to admire Garth's honesty and intellectual integrity in his handling of his own theory with respect to the GPB experiment). It would seem that inertial mass is influenced by it environment (the distribution of mass around it) but there is also a residual intrinsic inertial mass independent of its surroundings, just as there is a residual invariant rest mass/ energy. It could be thought of as Mach's principle is partly incorporated into GR but Einstein did not go the whole way. In other words, in GR a rotating bucket in a universe of "fixed stars" is not exactly the same thing as a stationary bucket in a universe of stars rotating around it, and GR can tell us if the universe is rotating or not. Mach's principle on the other hand can not tell a rotating universe from a static universe. That is my interpretation anyway, but welcome the input of the experts here ;)

 

[Buckethead]

Buckethead:

I appreciate that you're trying to make a subtle point, but your example fails in the first sentence.
How do you define 'straight' ? You need some kind of reference to compare the rod with ( line of sight ?)

The rod was manufactured just before the universe suddenly vanished. It was determined to be straight using a laser which had a straight beam when the universe existed (I'm actually not being sarcastic here even though it might sound that way)

I don't see how this relates to the rotating water, sorry.

There are three factors involved in the water rising in a bucket. 1) The water is in motion relative to something. 2) The water has mass. 3) Anything in motion with mass wants to move in a straight line. If it can be shown that inertia (mass) vanishes, or the ability to move in a "straight" line vanishes or relative motion vanishes, then the water in the bucket will have a problem. In a spinning bucket, the issue of relative motion can be answered because the water is being forced to deviate from a "straight" line, so I have removed this as being a factor and instead am just focusing on the definition of a "straight" line in my new example. The issue of inertia is also a factor, but my example is just focusing on the definition of a straight line in empty space since if a straight line becomes undefined in an empty universe this is enough of a reason for the water to be confused.

It is the inertial mass of the water that makes it climb the sides of the bucket, so I think it all boils down to this - will a solo object possesses inertia ?

In my opinion it will. It is simpler to believe that inertia is a local thing, either an intrinsic property of matter, or the result of a local interaction.

I think that inertia for objects with "mass" or more properly objects that are made of matter is a long standing assumption. I think objects are given the property of inertia because of something acting on those objects, not because inertia is an inherent property of matter.

 

[Mentz114]

JesseM:

In other words, in GR a rotating bucket in a universe of "fixed stars" is not exactly the same thing as a stationary bucket in a universe of stars rotating around it, and GR can tell us if the universe is rotating or not. Mach's principle on the other hand can not tell a rotating universe from a static universe.

Well put. That's what motivates me to call it 'Mach's conjecture'. Why is it a 'principle' if our best theory of gravity clearly disagrees ?

Buckethead:

I think objects are given the property of inertia because of something acting on those objects, not because inertia is an inherent property of matter.

Let's hope some future experiment can decide this, and lay Mach's thing to rest.

 

[Ken G]

I would just like to point out I was talking about a ball bouncing back and forth horizontally in the box so tidal stresses do not come into that thought experiment, because tidal stresses are vertical.

No, tidal stresses in a central gravitational field are not vertical on a box. This is why the Moon makes tides on the Earth-- it stretches the Earth along the Moon-Earth line, and squashes it in directions perpendicular to that line. Both effects are about equally important in making tides. In other words, I predict the effect you describe would not happen in the constant gravitational field of a huge plane of mass. If I'm right, that invalidates the argument. (And I think I am, or else your effect would occur in a reference frame in constant acceleration relative to the box, and that doesn't come out of my Lorentz transformation.)

 

[Buckethead]

It occurred to me that a Machian universe is a sort of democracy of mass. The mass of the "fixed stars" of mach represent the majority vote and define a sort of absolute reference frame. I think it is this implication of an absolute inertial reference frame that caused Einstein to ultimately reject the Machian viewpoint and declare it is incompatible with general relativity.

To see this on a smaller scale imagine a universe that comprises just the Earth and the Moon. Now the Earth seen from the Moon has a slightly bulged shape. Since the Earth represents the majority of mass in our reduced universe then it is declared stationary in the machian viewpoint. The bulged shape of the Earth is caused by a rotating or spiralling gravity "field". Einstein required that gravity (space curvature) is shaped by mass.Since the only objects of any significant size in this universe are the Earth and the Moon and since the Earth is considered stationary (by Mach) then the gravity "field" that is causing the stationary Earth to bulge at the equator can only be generated by the orbiting moon. The mass and motion of the Moon is insufficient to fully account for the bulge of the Earth and I imagine it this sort of reasoning that makes the Mach's principle incompatible with general relativity.

Now if we find a reference frame in which the total angular momentum of our reduced universe is zero then (I'm assuming) the gravitational curvature and the paths of the gravitational bodies can be all be accounted for by the combined gravitational effects of all the masses.


The subtle difference between the viewpoints of Mach and Einstein is that while the inertia of the water in the bucket is defined by the fixed stars in Mach's view, it is defined by the combined masses and motions of the stars and the bucket in Einstein's view.

It's hard to believe a year has passed since this thread started, but life beckoned and I had to abandon this for awhile. Still my enthusiasm for this subject seems to beckon as well. I just re-read this entire thread and I am totally blown away by all of the thoughtful posts discussing this topic. This is an amazing topic and I hope all of the previous posters and others will continue to chime in. Since posting over a year ago I have of course learned some new things (I haven't stopped reading) and some posts I wasn't completely able to comprehend fully back then came out in a new light which made re-reading that much more exciting.

I wanted to reply to so many posts, but chose this one as Kev seems to be thinking in parallel with what I am trying to pursue and represents some of the deepest parts of this thread so chose this one to start. I hope to get to others as well.

It occurred to me that a Machian universe is a sort of democracy of mass. The mass of the "fixed stars" of mach represent the majority vote and define a sort of absolute reference frame..

I believe very very strongly in this. The "concrete ring" phenonmenon shown to be true in GR is one of the reasons, but there is more to this as I will explain in my next post which will address the single atom around the bucket.

To see this on a smaller scale imagine a universe that comprises just the Earth and the Moon. Now the Earth seen from the Moon has a slightly bulged shape. Since the Earth represents the majority of mass in our reduced universe then it is declared stationary in the machian viewpoint. The bulged shape of the Earth is caused by a rotating or spiralling gravity "field". Einstein required that gravity (space curvature) is shaped by mass.Since the only objects of any significant size in this universe are the Earth and the Moon and since the Earth is considered stationary (by Mach) then the gravity "field" that is causing the stationary Earth to bulge at the equator can only be generated by the orbiting moon. The mass and motion of the Moon is insufficient to fully account for the bulge of the Earth and I imagine it this sort of reasoning that makes the Mach's principle incompatible with general relativity...

OK, now we are getting to a very philosophical crossroads that I think is very important. I think that in this reduced universe, the Earth will NOT bulge at the equator. If the universe is as it is today, then suddenly everything but the Earth and Moon were to disappear, one cannot assume that the Machian static frame of reference that was defined by the universe will remain in the state it was in. After all, this reference was by definition defined by the position of all the matter in the universe. Now that it's gone, the frame is subject to change. If indeed your "democracy" holds the answer, then it would be the Earth itself, (having the majority of the mass in this new universe) and it's rotation that would define the new Machian framework. The frame would rotate with the Earth, thereby rendering the Earth as "not rotating" and the bulge would cease except for any limited gravitational influence of the moon.

Actually, the Earth would not be completely at a state of rest as the moon would also influence the frame due to it's mass, but would be a very small "vote" and can be mostly disregarded.

The subtle difference between the viewpoints of Mach and Einstein is that while the inertia of the water in the bucket is defined by the fixed stars in Mach's view, it is defined by the combined masses and motions of the stars and the bucket in Einstein's view.

I do not understand the difference between these two viewpoints. They seem to me to be one in the same. Can you explain further?

 

[Buckethead]

Can you be sure acceleration is absolute?

Calculations using general relativity have shown that a massive rotating shell would induce a force that causes the surface of stationary water in a stationary bucket at its centre to curve exactly as if the if the water was rotating. The relativistic principle suggests there is no measurement that can destinguish a rotating bucket in a universe of of stationary stars from a stationary bucket in a universe of rotating stars. The mass of the rotating stars will drag spacetime as per the Lense Therring effect causing the water in the stationary bucket to climb up the sides of the bucket as if it was rotating. The stars will not be thrown outward by centripetal forces because the spacetime is co-moving with the stars.

Now imagine a universe with a one stationary bucket and one atom at the edge of the universe visible from the bucket. The atom is rotating around the bucket at very high speed but there is no way that the mass of a single atom at such a great distance can induce any significant gravitational field or curvature in the surface of the water in the bucket. By invoking the principle of relativity, rotating the bucket and water relative to the distant stationary atom will not induce any curvature in the surface of the water. The single atom is an approximation of an "otherwise empty universe" .

The atom it seems to me will indeed have no effect on the bucket. BUT, the lack of gravity I do not think is the reason. I again think it goes back to the "democracy of mass" (what a great phrase). The bucket in this universe defines the Machian frame by which inertia is defined because of it's relative mass compared to the atom. The atom is under this influence and if it is indeed revolving around the bucket it is experiencing acceleration due to it's change in vector.

Now, if we change the atom into the Sun, we have a different scenario on our hands. The Sun now has virtually all the mass in the universe so it defines the Machian frame and if it's relationship (and vectors) have not changed, then what we have is the entire Machian frame revolving around the bucket, which is to say the Sun is now stationary and the Bucket is revolving around the Sun. It will be the bucket that will be experiencing acceleration now, instead of the Sun. And it will be the bucket that will have to use it's side jets to stay in orbit (since gravity is too weak).

So I guess what I'm getting at is that gravity does not play a part in this at all, only (relative) mass, in other words "democracy of mass". So what I would like to know is, in the "ring of concrete" where the rotation of the ring has an influence on the water in the bucket, GR shows that this is due to gravity, but if the ring where the size of the universe and it's mass nothing more than that of a planet, AND (very important) it and the bucket were the only things in the universe, would the ring strongly influence the bucket according to GR? I think the ring would influence the bucket, because it defines the Machian frame, But what does GR have to say about this?

So my argument is that if we take a take an informal description of Mach's principle as "Inertia of a body is a property of its motion relative to the fixed stars" and restate it as "Inertia of a body is a property of its motion relative to the spacetime determined by the distribution and motion of matter in space" then Mach's principle is pretty compatible with relativity. The important notion is that inertia is not an intrinsic property of mass, independent of its surroundings.

I like this, and I think I'll use it as my standard definition of Mach's principle.

 

[Buckethead]

Buckethead:
Your logic is wrong.

Observers on the bricks could determine that the distance between the bricks remains constant over time. Therefore something must be keeping them apart. In the absence of any other candidate, centripetal force is deduced.

See above. You just keep ignoring the extended object argument. Why ?


Again - your universe is not empty - there are two bricks in it, and observers can detect the rotation without reference to any outside frame.

I agree with your determination of rotation by observation of the rope, but ONLY in a non-empty universe. In a universe filled with stars, one can observe the bricks moving relative to the stars and more importantly, so can the rope. In other words there is a reason the rope is going taught, because the bricks are rotating relative to the stars. In an empty universe however, this luxury does not exist. Since relativity says that an object can only be in relative and not absolute motion, there is no way to determine in an empty universe if two bricks are rotating around each other by simple virtue of the fact that their distance over time does not change. You state that a taught rope can determine this, but what I am suggesting is that you are putting the cart before the horse. Imagine you are the rope, and it is your job to determine if you should go taught or not (or you will be beheaded). The way you would determine this is by observing if the two bricks are trying to go past each other in the same way that two ships are trying to pass each other. If they are, then you (as a rope) are responsible for preventing this from happening and the result is a taught rope. But in an empty universe, with two bricks tied to you how are you going to determine if the two bricks are trying to pass each other? Their relative distance is not changing with time, so you cannot use this as a determining factor. In fact, if you released yourself from one of the bricks, would it suddenly take off? Why? Was it moving in the first place? You cannot say it was moving, because it was not moving relative to the other brick (it's distance did not change). In fact, it seems since it is not moving relative to the other brick, it must be stationary. Again, in a universe of stars this problem does not exist because it is easy to see the bricks are rotating relative to the stars, but in an empty universe and because SR says you can only determine motion by relative motion, if the bricks are not moving relative to each other, then you cannot (as a rope) say you must go taught, in fact you must go loose as there is no relative motion.

 

[Ken G]

Another important point to make is that for any discussion on Mach's principle, we must first stipulate that we simply do not know what would happen in an "otherwise empty universe", expressly because we have no such universe to do experiments in. If we instead start with the presumption that we do know what would happen to water in a bucket in an "otherwise empty universe", then we cannot discuss Mach's principle, as we have to have already incorporated it or outlawed it by fiat when we specify what happens to that bucket.

 

[Buckethead]

Another important point to make is that for any discussion on Mach's principle, we must first stipulate that we simply do not know what would happen in an "otherwise empty universe", expressly because we have no such universe to do experiments in. If we instead start with the presumption that we do know what would happen to water in a bucket in an "otherwise empty universe", then we cannot discuss Mach's principle, as we have to have already incorporated it or outlawed it by fiat when we specify what happens to that bucket.

Wouldn't what you are saying only hold true in a completely empty universe? By virtue of the fact that the bucket and atom (or Sun) exist and have mass, we have a starting point for incorporating Mach's principle. If we invision Mach's principle to be a effect generated by the mass in the universe, and their relative motions, then this would apply to any amount of mass.

 

[Ken G]

I don't think so, to apply Mach's principle to the bucket of water, you need more than just the bucket and the water. You basically need a boundary condition for your spacetime, at infinity or at least embedded in something substantial that you can consider to be stationary. The mass invoked by Mach's principle must be effectively infinite, in other words. If you just use the bucket itself, then you are asking a different question, about water sloshing inside a bucket, rather than water and bucket moving together. If the water is moving relative to the bucket, you'll have frictional forces that will be much more important than anything that looks like a gravity, and if the water is not moving relative to the bucket, then there's no gravity from the bucket that can make the water bulge. Mach's principle comes from a huge distant mass distribution that can have a significant enough gravity to anchor the concept of being absolutely stationary.

Put differently, I would say that Mach's principle basically asks the question, do we characterize motion by the presence of various effects we attribute to motion (say, ficticious forces), or does the motion and those various other effects both originate as results of some deeper phenomenon. The latter is Mach's claim-- that there is some deeper influence that an effectively infinite mass distribution has, which simply would not be there if that mass were not there. The former is the situation without Mach's principle. So, in an otherwise empty universe, if you spin a bucket and you need Mach's principle to get the ficticious forces, then the water would simply not bulge in that bucket-- there would never be any way to tell whether it was the bucket or the water that was originally spinning, and also no way to tell which one ultimately adopts the other's speed (i.e., which one has the greater inertia). The equilibrium would always be a stationary bucket and stationary water in it, in effect whichever object was taken to be the stationary one is the one that would have all the inertia.

Alternatively, in a universe where motion is as motion does and no more, then we could still have a bucket and water that were all stationary in their own frame, yet still showed ficticious forces creating a bulge, even in an otherwise empty universe. That's the universe with no Mach's principle. Which universe are we in? How could we ever tell? And what do we do with seemingly physically based questions that actually come with no way to answer? I would simply restate Mach's principle as the general observation that any elements of our universe that are unavoidable and inescapable could play an implicit role in all our physical theories in ways that we can never test or understand.

 

[Buckethead]

I don't think so, to apply Mach's principle to the bucket of water, you need more than just the bucket and the water. You basically need a boundary condition for your spacetime, at infinity or at least embedded in something substantial that you can consider to be stationary. The mass invoked by Mach's principle must be effectively infinite, in other words. If you just use the bucket itself, then you are asking a different question, about water sloshing inside a bucket, rather than water and bucket moving together. If the water is moving relative to the bucket, you'll have frictional forces that will be much more important than anything that looks like a gravity, and if the water is not moving relative to the bucket, then there's no gravity from the bucket that can make the water bulge. Mach's principle comes from a huge distant mass distribution that can have a significant enough gravity to anchor the concept of being absolutely stationary.

Put differently, I would say that Mach's principle basically asks the question, do we characterize motion by the presence of various effects we attribute to motion (say, ficticious forces), or does the motion and those various other effects both originate as results of some deeper phenomenon. The latter is Mach's claim-- that there is some deeper influence that an effectively infinite mass distribution has, which simply would not be there if that mass were not there. The former is the situation without Mach's principle. So, in an otherwise empty universe, if you spin a bucket and you need Mach's principle to get the ficticious forces, then the water would simply not bulge in that bucket-- there would never be any way to tell whether it was the bucket or the water that was originally spinning, and also no way to tell which one ultimately adopts the other's speed (i.e., which one has the greater inertia). The equilibrium would always be a stationary bucket and stationary water in it, in effect whichever object was taken to be the stationary one is the one that would have all the inertia.

Alternatively, in a universe where motion is as motion does and no more, then we could still have a bucket and water that were all stationary in their own frame, yet still showed ficticious forces creating a bulge, even in an otherwise empty universe. That's the universe with no Mach's principle. Which universe are we in? How could we ever tell? And what do we do with seemingly physically based questions that actually come with no way to answer? I would simply restate Mach's principle as the general observation that any elements of our universe that are unavoidable and inescapable could play an implicit role in all our physical theories in ways that we can never test or understand.

I think we have to accept that even a minimal universe (a universe with any amount of mass) will have to allow us to measure relative linear and relative rotational motion. If we cannot allow even that then all logic goes out the window. With this in mind I think we have to allow either of the 2 scenarios you suggest, either a universe "where motion is as motion does" or a universe where Mach's principle holds, again, regardless of how much mass is in the universe. If we accept the first, then if I read you correctly you are accepting an absolute frame of reference. Otherwise "motion is as motion does" does not mean anything. If a single bucket spins and it is showing concaveness, then (by definition) the bucket is spinning. And if it is spinning then it must be spinning relative to something even if that something is nothing we can define. I do not favor this as it implies that the absolute frame of reference is moving relative to the bucket and there is no logic behind this. This implies that the frame somehow as an independant nature relative to the bucket. As if it could be somehow "fixed" allowing the bucket to spin relative to it. But again, as you have explained, unless you can tell if it's the frame spinning and not the bucket or the bucket spinning and not the frame, then we go in circles. It then becomes your POV (the spinning frame or the spinning bucket) that determines if the water should be concave or flat.

On the other hand if we allow Mach's principle to be described as something real formed by the motion and rotation of an object and if this "frame" is influenced by a democracy of mass then clear concise predictions about the water in the bucket can be made. For example, if there is only a bucket of water and nothing else in the universe, then the bucket can never become concave. You can try and spin it, and it will remain flat, in other words it can never spin. The Machian frame will spin with it. If you try and move it in a linear direction, then it will show no movement of the water to one side of the bucket, again it will remain flat because the Machian frame gets pushed right along with it.

If we now introduce a smaller and distant object (a pebble, millions of light years away) into our universe, according to democracy of mass if we try and spin the bucket, the water in the bucket will still not rise, but what we will observe is that suddenly and seemingly inexplicably, the distant pebble will appear to begin revolving around the bucket! In addition if we try and move the bucket from it's current location, the water still does not slosh up one side, because it is not moving relative to the frame, but instead the distant pebble now moves closer or further from the bucket since it is moving relative to the frame.

If we now introduce more and more massive and relatively stationary objects to the universe, the democracy of mass continues to define the relative position and relative rotation of the Machian frame. If (most) all of the mass in this new universe are stationary with respect to each other then the Machian frame will encompass their position and their "so called" rotation (which would not actually exist since rotation would be defined by relative rotation with respect to the Machian frame and there would be none). In such a universe (such as our own) smaller objects such as galaxies can then be defined as rotating, or moving relative to this machian frame and their inertial forces would be measured directly relative to this frame as well.

 

[Ken G]

I think we have to accept that even a minimal universe (a universe with any amount of mass) will have to allow us to measure relative linear and relative rotational motion. If we cannot allow even that then all logic goes out the window. With this in mind I think we have to allow either of the 2 scenarios you suggest, either a universe "where motion is as motion does" or a universe where Mach's principle holds, again, regardless of how much mass is in the universe.

There's still another possibility-- if we have a universe with a kind of "minimal" mass in it, we could simply have weaker ficticious forces than in our universe, for the same acceleration. So yes, we could have relative rotation, but just a weaker centrifugal force. This of course would have to mean that inertia works differently than in Newton's laws, but that's exactly what we don't know about such "minimal" universes. It might be fun to imagine the various possible forms of Newton's laws that reduce to the familiar one in a "maximal" Machian universe, but they would be impossible to test.

If we accept the first, then if I read you correctly you are accepting an absolute frame of reference. Otherwise "motion is as motion does" does not mean anything.

It requires absolute frames only in the same way that special relativity treats inertial frames in a special way-- the frames that have no ficticious forces. I should have said "acceleration is as acceleration does", and by that I mean, the appearance of ficticious forces. In this picture, we don't say we have ficticious forces because we have an absolute acceleration, but rather we say that the presence of ficticious forces provide the definition of absolute acceleration (that's essentially how an accelerometer works).

If a single bucket spins and it is showing concaveness, then (by definition) the bucket is spinning. And if it is spinning then it must be spinning relative to something even if that something is nothing we can define.

Right, that's the "acceleration is as acceleration does" non-Machian approach.

I do not favor this as it implies that the absolute frame of reference is moving relative to the bucket and there is no logic behind this.

Mach didn't like it much either, but it's probably the picture that has best survived general relativity, though I believe that issue is still debated among real GR experts (of which I am not one).

On the other hand if we allow Mach's principle to be described as something real formed by the motion and rotation of an object and if this "frame" is influenced by a democracy of mass then clear concise predictions about the water in the bucket can be made.

Yes, the whole approach to the "center of mass" of a system is very much a kind of "vote", as you say. It still has strange properties though-- as you say, if we have a spinning bucket with 99% of the mass of the universe, and an outside observer with 1% of the mass, the spinning bucket could "vote" that the observer is actually in orbit and the bucket is not spinning at all, and we conclude the bucket is 1% spinning and the observer is 99% orbiting. Hence we only expect a 1% bulge in the water in the bucket. Now in a universe where the observer had a million times more mass, the bulge is back to its usual scale. But the problem is, this would hold no matter how small those masses actually are, so the gravitational constant G would have to be "renormalized" based on the mass in the universe, otherwise the influences would be too small with our current G to do anything. I prefer to think of G as a fundamental constant, and only the nature of spacetime is influenced by the mass. That's why I think you need the rest of the universe to have essentially infinite mass for Mach's principle to seem reasonable, because then the gravitational influence is not negligible, it "anchors" the spacetime. Nevertheless, I could not argue that your way of renormalizing G to whatever is the total mass is impossible or wrong.

For example, if there is only a bucket of water and nothing else in the universe, then the bucket can never become concave. You can try and spin it, and it will remain flat, in other words it can never spin.

Yes, that's the fundamental question-- can a bucket spin if it is the whole universe? Mach says no, "motion is as motion does" says yes. It would be an issue of what is possible in the "initial conditions" of such a universe. Now, how would we ever know which holds true in our universe? It seems a matter of personal philosophy, as we can never do experiments in such a universe, and the testable distinctions in GR are debated even among the experts.

 

[yuiop]

Hi,

It is has been a while since I have given this subject any serious thought so the following comments are just some casual thoughts to resume the conversation and see where we are at.

First of all Mach's principle is hard to prove or refute because it it is not clear what exactly is meant by that principle and there are no "Mach's principle equations" to calculate exactly what it predicts or exactly how it differs from General Relativity.

Reading between the lines I get the impression the principle that Mach was trying to establish was a fully relativistic notion of acceleration that only has meaning relative to other objects. Einstein of course was drawn to Mach's idea because of its relativistic nature but ultimately he rejected that notion in formulating his final version of GR. So here is the surprise. General Relativity is not fully relativistic. Here is an example. Say you are a universe like ours but it contains only you and a glowing particle many light years away. The particle appears to circumnavigate a large circle once every minute. Now is it you rotating at 1 rpm or are you stationary and the particle is orbiting you? Mach's principle would seem to indicate that either view point of view is equally valid (the fully relativistic idea). However if we assume for a moment that it is you that is stationary (after all by the democracy of mass your mass is orders of magnitude larger) then the particle would be orbiting at velocity much greater than the speed of light. This is why I think Einstien rejected the fully Machian universe. Much as Einstein liked the idea of everything being fully relativistic, he really hated the idea of anything exceeding the speed of light, so he settled for a not fully relativistic description of the universe which gives an absolute nature to accelerating motion which includes rotation.

Here are some other points to consider. The Schwarzschild metric describes the spacetime around a non-rotating body in an "otherwise empty universe" and the Kerr metric describes the spacetime around a rotating body in an "otherwise empty universe". (Both metrics assume an uncharged body). Whether or not the body is rotating or not, is relative to the spacetime it is embedded in and is not relative to any other bodies. The Schwarzschild or Kerr body curves and shapes the spacetime around it. The vacuum outside of the body is not entirely nothing. After all you cannot curve and shape nothing.

In modern cosmology it is known that distant galaxies are receding at velocities that greatly exceed the speed of light. However, this is not considered a violation of General Relativity because the distant receding galaxies are stationary with respect to the expanding spacetime that they are embedded in. Again, what looks like a vacuum is not entirely nothing because a pure vacuum that is entirely nothing can not expand or do anything else for that matter. This sort of relates to the ZPE field that Mentz referred to. It is also generally accepted that if a body accelerates sufficiently quickly that it will see virtual particles popping out of the vacuum. This is the "Unruh effect" and again it only requires that a body is accelerating relative to the vacuum or spacetime and is not relative to any other bodies. Again the vacuum should not be thought of as entirely nothing.

Finally a little thought experiment. Imagine an Earth sized body in an "otherwise empty universe" that is rotating so fast that it oceans would be flung into space by centripetal forces but from the Machian viewpoint it is "unaware" that it rotating and retains its oceans and perfectly spherical shape. Now imagine a single particle popping up anywhere in this otherwise empty universe due to some quantum fluctuation. Would the Earth like body suddenly lose its oceans as a result of the appearance of this single tiny particle? That seems unlikely.

My intuition is that unlike Special Relativity which is fully relativistic and where motion only has meaning relative to other bodies, General Relativity has an absolute nature relative to spacetime as far as rotation and linear acceleration are concerned. It would seem to me that in General Relativity a body has an existence relative to the spacetime around it, even in an otherwise "apparently empty" universe.

 

[Buckethead]

There's still another possibility-- if we have a universe with a kind of "minimal" mass in it, we could simply have weaker ficticious forces than in our universe, for the same acceleration. So yes, we could have relative rotation, but just a weaker centrifugal force. This of course would have to mean that inertia works differently than in Newton's laws, but that's exactly what we don't know about such "minimal" universes. It might be fun to imagine the various possible forms of Newton's laws that reduce to the familiar one in a "maximal" Machian universe, but they would be impossible to test..

I don't feel easy about this theory. One has to wonder about the nature of these fictitious forces and why they would suddenly appear when a bucket starts to spin in the presence of other matter.

Yes, the whole approach to the "center of mass" of a system is very much a kind of "vote", as you say. It still has strange properties though-- as you say, if we have a spinning bucket with 99% of the mass of the universe, and an outside observer with 1% of the mass, the spinning bucket could "vote" that the observer is actually in orbit and the bucket is not spinning at all, and we conclude the bucket is 1% spinning and the observer is 99% orbiting. Hence we only expect a 1% bulge in the water in the bucket. Now in a universe where the observer had a million times more mass, the bulge is back to its usual scale. But the problem is, this would hold no matter how small those masses actually are, so the gravitational constant G would have to be "renormalized" based on the mass in the universe, otherwise the influences would be too small with our current G to do anything. I prefer to think of G as a fundamental constant, and only the nature of spacetime is influenced by the mass. That's why I think you need the rest of the universe to have essentially infinite mass for Mach's principle to seem reasonable, because then the gravitational influence is not negligible, it "anchors" the spacetime. Nevertheless, I could not argue that your way of renormalizing G to whatever is the total mass is impossible or wrong.

You bring up some really interesting questions here. I too would prefer to see G remain constant and if we chose to use this as a given, then it can help define the nature of the Machian frame. For example, we can now say that the sum total influence of the Machian frame generated by individual masses are a ratio of the masses and (very important) the relative motions between the masses (both linear and rotational). It is not necessary to assign an absolute value to the strength of the frame (and indeed we cannot) if G is to remain constant. In other words we can say the inertia of an object in any size universe remains constant if the mass and velocity of the object in question remains small compared to the overall mass and overall relative velocity of the rest of the universe. Even a bucket with a ratio of 1:100 will approach the inertia of a bucket with a ratio vastly larger than that.

All of this of course implies that the Machian frame is not related directly to gravity. This begs the question of what exactly is it. It may be that it's influence extends in all directions indefinitely and does not decrease in relative influence with distance. This would make sense in a minimal universe. Also it is strickingly different from gravity in that it's influence is directly related to it's relative velocity and rotation.

Fortunately some predictions can be made if this Machian frame is described this way. For example the Machian frame would not be static throughout the universe as Mach originally invisioned. Take 2 galactic clusters spaced 100 million ly apart. If one were to rotate clockwise and the other counterclockwise, the mid point between them would have a Machian space that moved in a linear direction perpendicular to the line drawn between the two clusters. Light traveling though this space against the flow of the movement would slow down (relative to an outside observer) whereas light traveling the other way would speed up. Light traveling near one of the rotating clusters would be deflected and so on. By closely examining galactic lensing effects we could predict rotations and relative movements of clusters.