Is Mach's Principle incorporated into General Relativity?

[WCOLtd]

In this mock-up universe there exists two observers on a large disc of great mass, and there is a third observer a great distance away. In this scenario we take the observation of the spinning disc from the distant (third) observer.

I was thinking about Special Relativity and Mach's Principle and I thought about an enormous spinning rigid disk, one that spun such that it rotated at such that the edge of the disc approached .999999c or something like that.

I was thinking about the implications of Mach's principle when applied to for this large rigid, isolated disc.

According to whatever observers are on the disc, they would not observe each other to be moving at all, they are at rest to all other observers on the disc.

Yet if a distant observer were to look at this disc from a great distance. The observer would see that the edge of the disc is not rigid, it would appear to warping. - because the edge of the disc could not travel faster than the speed of light. Which it would if the speed of the outside part of the disc is 2pi times the rate of rotation.

This would appear to me to be some kind of contradiction, because the outside observer would be influenced by the inside observer as if the disc were not warped and the observer observing the thing on the outer part of the disc would see both the warpage, and observe the acted influence as if the disc were not warped.

I am pretty confident so far, but then I started to think of something that I am a bit less sure of.

I thought that maybe that wasn't possible, I thought that the centripetal force would be so great that the outer observers would detach themselves from the disc before the contradiction ever occurred - because it would simply have to.

Then I thought that because of Mach's principle the centripetal force of the distant observer should be reflexive of that centripetal force. - I think this is a stretch - but it's as if the distant observer is also spinning around the center - because the true motion of rotation is relative - so that distant observer would experience a centripetal force as well.

So then I thought that maybe the interaction of galaxies could be like that - that expansion may be correlated with the sum distribution of all centrifuges.

 

[Dmitry67]

Contrary to the classical mechanics, there are no absolutely rigid bodies in the relativity - even theoretically.

 

[WCOLtd]

The rigidity of the body is a supposition take a third observer a great distance away who sees the disc as rotating at great speed. The disc would appear to be warped or non rigid, you see, I don't make the assertion that the body is "absolutely rigid" - but I define "rigidity" by the inferred at rest observational and inertial references fixed at any point along the disc. That means that one observer will see any other as being fixed, while distant outside observers may see the disc as rotating.

 

[Mentz114]

because the true motion of rotation is relative - so that distant observer would experience a centripetal force as well.

I don't think this is correct. Rotation is absolute, and everyone on the disc will be able to detect ( i.e. feel ) the centripetal acceleration. In other words, there is no frame of reference in which the acceleration is zero.

Mach's 'principle' is based on this misunderstanding.

 

[Jonathan Scott]

I don't think this is correct. Rotation is absolute, and everyone on the disc will be able to detect ( i.e. feel ) the centripetal acceleration. In other words, there is no frame of reference in which the acceleration is zero.

Mach's 'principle' is based on this misunderstanding.

When Mach's principle is applied to rotations, it effectively says that centrifugal and coriolis forces experienced in a rotating frame of reference can alternatively be described as the gravitational effects of the rest of the universe effectively rotating in the other direction. This means that unless the rotating system consisted of most of the universe, observers on the disk would still be experiencing a rotating frame of reference and all its side-effects.

This works reasonably well even within GR (as an alternative way of viewing frame-dragging effects), although it doesn't extend exactly to the whole universe in the GR case.

 

[Mentz114]

When Mach's principle is applied to rotations, it effectively says that centrifugal and coriolis forces experienced in a rotating frame of reference can alternatively be described as the gravitational effects of the rest of the universe effectively rotating in the other direction.

I've never seen a convincing demonstration of this. If I start rotating my bucket of water how can it be influenced immediately by galaxies millions of light years away ?

 

[Jonathan Scott]

I've never seen a convincing demonstration of this. If I start rotating my bucket of water how can it be influenced immediately by galaxies millions of light years away ?

That's no different from any other "action at a distance" effect, especially other gravitational effects, and would normally be described in terms of a field or in terms of the effect on space-time geometry. GR already says that a nearby moving or rotating source will cause rotational frame-dragging effects of the appropriate order of magnitude. (Some people claim, especially in support of the extremely implausible "Woodward effect", that Mach's principle frame-dragging effects cannot be described in terms of a field and are really due to a form of action at a distance, but I feel I can safely ignore them).

 

[Mentz114]

Does this 'action at a distance' propagate instantaneously like Wheeler and Feynman's advanced and retarded effects ?

If the only objects in the universe were a spinning disc and some inhabitants living on it,
I'm pretty confident they would be aware they were spinning, even in the absence of other matter. Without any astronomical reference points, they'd have to invent the spinning top before coming up with a correct cosmology !

 

[WCOLtd]

I don't think it's like that at all, it has to do with gravitational fields.
How would they know they were spinning? What frame of reference to they measure that motion?

Think of it like this.
Try to imagine a universe in which there exists two bodies, and you the observer.

In one frame of reference, you stare down at the two bodies and you see 1 smaller body rotating around a larger body. According to Newton, the centrifugal force is what keeps the smaller body from falling into the larger one.

Now imagine that you rotate yourself in such a way staring down at the larger mass, that the smaller body appears to be in a fixed position relative to you. The larger mass appears to be spinning faster.

According to this perspective, the smaller body is no longer rotating around the larger one, so it should, according to Newton, fall into the larger body as if there was no centripetal force.

Ernst Mach proposed that the distribution and the character of the motion of mass is what determines inertial reference, whether something experiences a centripetal force or not.

For instance In a Newtonian sense, many large galaxies and star systems would not be able to hold - because the force of centrifuge would be too great.

check out my blog
"Gravitational Kinematics: Inertial definitions"

 

[Jonathan Scott]

Does this 'action at a distance' propagate instantaneously like Wheeler and Feynman's advanced and retarded effects ?

If the only objects in the universe were a spinning disc and some inhabitants living on it,
I'm pretty confident they would be aware they were spinning, even in the absence of other matter. Without any astronomical reference points, they'd have to invent the spinning top before coming up with a correct cosmology !

In GR, what they would notice depends on the mass and radius of the disk. If the mass were large and close enough, the frame-dragging effect could almost nullify the effects of the rotation. GR doesn't naturally rescale this effect, so it depends on the total mass and its distribution in this "universe", but in general Mach's principle (at least in the form demonstrated by Sciama) would expect to work for a universe with any mass and size, rescaling automatically as necessary.

I think that in theory GR is necessarily limited to a maximum rotational frame-dragging effect of slightly less than 1 to 1 (that is, the dragged frame cannot quite experience the same angular velocity as the source mass), for example near a maximally rotating Kerr central mass solution, so it cannot totally reproduce Mach's principle.

 

[Mentz114]

WCOLtd

How would they know they were spinning? What frame of reference to they measure that motion?

Speaking of the observers on the disc, they will know they are spinning because they can measure the centripetal force. The effective force is directed away from the centre
of the disc, which provides a frame of reference.

I don't see the relevance of your analogy here.

According to this perspective, the smaller body is no longer rotating around the larger one, so it should, according to Newton, fall into the larger body as if there was no centripetal force.

Are you saying that if I align my frame with the rotating body it will fall into the other body ?

In fact, I am in the frame of a smaller body rotating around a larger one and I'm not noticeably falling into the sun.

For instance In a Newtonian sense, many large galaxies and star systems would not be able to hold - because the force of centrifuge would be too great.

Are sure about that ?

 

[WCOLtd]

Ok, I can see now that my answers were rather unsatisfactory. This is my favorite subject in all of science, because I thought of it independently of Einstein - and made a huge fool of myself in the process.

To answer your first question about centripetal force. I am going to need you to pay close attention, and to use your imagination to picture these scenarios.

Ok. Think about things inertially, if you were to jump out of an airplane - neglecting air resistance, you would feel as though you were weightless.

What does that experience weightlessness mean anyway? well it means you're not being acted upon by a force, you do not experience a push. Remember this.

Now imagine you have a rocket pushing you upwards, the accelerating rocket enacts a force on your feet, and if you put a scale on the floor of the rocket you can measure that force.

Much like a rocket ship, the ground enacts a force on the observer, so if you put a scale on the ground of the Earth you can measure the force. If you didn't have eyes and were pretty clever you might guess that the ground of the Earth was much like an expanding surface in all directions.

Now it isn't really expanding at an accelerated rate in all directions, but it feels just like it. Inertially the two cases are analogous.

But the more important thing, is that when you jump out of the aircraft, it is as though for the first time, you are not experiencing a force, that case is analogous to a person moving at constant velocity in space. So while the person is falling at an accelerated rate - we take that acceleration at that radius to be considered the "at rest" reference frame.

The person is accelerating and experiencing no force. How is this possible? F=ma, surely he would. Now in fact it may be more correct to say that the person is not accelerating, that it is the ground rushing up to meet him.

Now we could bicker about how we interpret the situation you may say "Surely things are being pulled down by the Earth's gravitational pull" and I may say "No, the things falling to Earth is just a misconception - the Earth is expanding at an accelerated rate things just appear to be falling at -9.8m/s^2 relative to us, we have taken the point of reference of the expanding ground."
You might contend that it would be ridiculous if the Earth were expanding because then the Earth would expand - rather quickly into the sun. And I might say that it's ridiculous to say that the force pulls objects a function of their weight because a bowling ball and a feather fall in the same rate - why is that so. Wouldn't it be much easier to explain that phenomenon as if the feather and a bowling ball were at some fixed position and a rocket ship were accelerating by it. (I would probably have the more difficult position to explain things) but you get the idea, it goes back and forth.
Yet we can both agree on one thing, observation, and we both know that the gravitational force - whether it be an expansion or a pull, is proportional to the product of the masses and the inverse of the distance. And this is all Newtoners need to agree to show how their theory, in the end destroys itself.
We also know, that things do not experience a force and do experience a relative acceleration when in free fall.
From these observations we can infer that an object that is not accelerating towards the ground is thus experiencing a force.
Now take the case that the Earth is rotating
So whether or not something experiences a force or not depends on the acceleration and the distribution of the masses.

if you split the Earth up into a bunch of little mass intervals, you will notice that the intervals near the outside, where the relative speed is greatest, the linear component of acceleration both towards and away from the falling object is at it's greatest.

Because the force experienced by the falling ball is equal to the relative acceleration with the ground.

Because on one side the intervals are accelerating toward the object faster than the it is accelerating, the object will experience a force, pushing it back and slightly to the side.
On the other side, the intervals of mass are accelerating away from the object faster than it's accelerating, so it experiences a pull on that side. This causes a perpendicular component of force to the free fall. If it spins fast enough, the perpendicular component of force might equal that of free fall, and it orbits around.

I hope that's sufficient I could give you equations, and show you that rotating bodies influence a rotation, it's not a profound effect, but it's there.

The second answer is sort of, I am not saying that if you align your frame with the rotating body you will fall into the other body; I am saying if you align all frames with the rotating body then you most certainly will fall.

As a kid I always wondered how satellites could stay at a fixed position in the sky, because I used to only think in those terms, that rotation is completely relative, that since the satellite is fixed to the rotation of the earth, it should fall to the earth, but it is nowhere near enough to only take into account the rotation of the earth, you must also take into account all the trillions upon trillions of other reference frames, dust particles, planets in the solar system, and distant stars many billions of light years away.

And to answer your third question come to think about it, I am not sure of galaxies, I think I was only told that - I haven't seen it for myself. To be honest though I am very unsure of many things, but I am confident that this idea is true, because I thought about it, and I think it makes sense.

 

[yuiop]

For instance In a Newtonian sense, many large galaxies and star systems would not be able to hold - because the force of centrifuge would be too great.

Although this observation is true, it is assumed that there is additional matter in the form of 'dark matter' that is not easily observed that keeps the stars in stable orbit despite their apparent excess orbital velocities. That is a separate issue to the main theme of this thread.

According to this perspective, the smaller body is no longer rotating around the larger one, so it should, according to Newton, fall into the larger body as if there was no centripetal force.

As a kid I always wondered how satellites could stay at a fixed position in the sky, because I used to only think in those terms, that rotation is completely relative, that since the satellite is fixed to the rotation of the earth, it should fall to the earth, but it is nowhere near enough to only take into account the rotation of the earth, you must also take into account all the trillions upon trillions of other reference frames, dust particles, planets in the solar system, and distant stars many billions of light years away.

Let's focus on your example of the geostationary satellite. It appears to be motionless from the point of view of an observer on the ground and that observer can imagine that the Earth is stationary and that all the galaxies of the universe are orbiting arounfd the Earth, but we understand that the satellite is in orbit and that the Earth is also rotating.

Mach's principle implies that we can consider the Earth and the geostationary satellite to be stationary (not rotating) and in the reference frame of the observer on the ground, the combined effect of all the stars and galaxies orbiting around the Earth causes an antigravitational effect that suspends the stationary satellite above the stationary Earth.
At first Einstein was attracted to Mach's when he was developing General Relativity, because it has the relativistic aspect of Special Relativity, but ultimately Einstein rejected Mach's principle and it is not incorporated into General Relativity for the simple reason that the universe does not work like that. The often quoted catch phrase that "everything is relative" is misleading and simply not true in General Relativity. Rotation is absolute and not relative.