[varsigma (OP)]

If Isaac Newton could do it, why not you?

The story goes, Newton was living in the countryside, avoiding the outbreak of the London plague, and was sitting in an orchard.

Nowadays, a high school student can "do the math" that predicts what an apple will do when it falls from a tree.
Once it is free to fall in a gravitational field, it accelerates at g, this has a local value and the planet has an average g, which differs (is differential) over the surface.

According to Mach's principle (who came along a bit later than the 17th century), the local value of g depends on all the interactions between an inert mass, and all the mass along any line from the COM of the massive object.

Which means, although  you have the "sense" of gravity having a pull in one direction, all the mass around you, in any mountains etc, is pulling at the same strength (given by Newton's constant which is very small).

You feel, therefore, a product of all this, centred in the middle of the earth and say "gravity is a central force". This force is an idea, it's an analytical idea. The force is also a product, over all the small chunks of matter in your vicinity (i.e. the planet earth, mostly), which you find the centre of, it's all directed vertically, but if you went lower into the well, any analytical "force lines" would get closer together because of the geometry: gravity is geometrical.

In fact Einstein showed that gravity is the geometry of spacetime. Newton showed that you can take a flat section of spacetime and draw a curve that follows "acceleration" for a falling object. Einstein showed that it isn't really flat, except locally where apples fall in straight lines, and see the same "sum over little bits of matter", distributed around them.

Forces are first order approximations (analytically) connected to the curvature in four dimensions; an approximation of the geometrical distribution (of a matter-field).

Now, I have a friend who, quite some years ago was enrolled in a graduate degree in mathematical physics. I went to the same uni, but wasn't studying math except for calculus, I wanted to study electronics and communications theory.

Anyhoo, he still hasn't finished the thesis so still isn't a postgrad. But his thesis is a review of Mach's principle, and I honestly can't tell you what it's about because it's all tensor calculus. But I sorta got something about a spinning bucket filled with water, and all the mass in the universe around it; i.e. in its local frame. So if the bucket is in space, in a "field-free" free region, what happens? I might be contracting a few things there (yuk yuk).

[Kryptid]

But I sorta got something about a spinning bucket filled with water, and all the mass in the universe around it; i.e. in its local frame. So if the bucket is in space, in a "field-free" free region, what happens?

How can it be in a field-free region when "all the mass in the universe" is around it?

Also, you should change the title of this thread to be in the form of a question.

[varsigma (OP)]

How can it be in a field-free region when "all the mass in the universe" is around it?

Also, you should change the title of this thread to be in the form of a question.

How can a region of spacetime be flat, when it has mass, the mass in the universe, all around it? According to Mach's principle, all of spacetime is curved. So the question is as I just asked: how can we determine if a bucket of water, spinning in space, is 'climbing' the sides because of the inertia of liquids, or because there's a universe filled with mass around it?

Say you're an observer on the rim, the universe is rotating around you, take it from there.

[varsigma (OP)]

Some of you may know that gravity waves propagate at the speed of light.

This throws Mach's principle into the restriction that no interaction can occur faster than this, and the bucket can only interact with the mass around it at this speed. The effect seen because water has inertial mass, is a local effect, disconnected from the visible universe by the restriction on signal transmission; of gravitational, electromagnetic, or any other kind.

But EInstein wrote that the idea was helpful in formulating his general theory. I guess my friend is trying to see if he missed something.

[Kryptid]

Say you're an observer on the rim, the universe is rotating around you, take it from there.

You would know that you are rotating because rotation is a form of acceleration. Acceleration isn't relative, it's absolute.

[Eternal Student]

Hi.

  I have incredibly little idea what it is you're asking or hoping to discuss.

According to Mach's principle (who came along a bit later than the 17th century), the local value of g depends on all the interactions between an inert mass, and all the mass along any line from the COM of the massive object.

   That is not a common version of any statement that is usually considered to be Mach's principle (to the best of my knowledge).   However, it doesn't seem that Mach's principle has been completely standardised and presented as a single scientific principle.   The origins of the idea are from books and articles that concern Philosophy as much as Science.

But I sorta got something about a spinning bucket filled with water, and all the mass in the universe around it; i.e. in its local frame. So if the bucket is in space, in a "field-free" free region, what happens? I might be contracting a few things there (yuk yuk).

     Spinning buckets and Mach's principle get a mention in Wikipedia: https://en.wikipedia.org/wiki/Mach%27s_principle.
    Sadly the article in Wikipedia isn't all that clear  (i.m.o.) but anyone reading this post might like to see what the article says.  It gives some background that I'm not going to repeat.

There are some YT videos on the topic.    This one seems reasonable and lasts 8 minutes:
"Mach's principle" by Sixty Symbols.

So the question is as I just asked: how can we determine if a bucket of water, spinning in space, is 'climbing' the sides because of the inertia of liquids, or because there's a universe filled with mass around it?

   Your sentence seems to have been rushed.  The bucket doesn't climb the sides,  the water climbs the sides of the bucket (after it starts to follow the same rottation as the bucket that it was in).
    The issue is partially resolved by the phenomena of "frame dragging" in General relativity.   If you didn't spin the bucket and the water in it but just left them alone  and instead started to spin the rest of the universe around it, then the water would soon start to climb the sides of the bucket.   I haven't done the calculations but it could be much the same situation that we're more familiar with in everyday life - it could be indistinguishable from the situation where the bucket had been spun and the water (through the viscosity of the liquid and it's interaction with the walls of the bucket) started to spin with it.

You would know that you are rotating because rotation is a form of acceleration. Acceleration isn't relative, it's absolute.

    Yes, that's the standard theory.   I don't know what @varsigma  was really trying to discuss but it could be that they were considering something like this:
     If you apply acceleration to every mass (or source of gravitation) in the rest of the universe that matches the acceleration you were (previously assumed to be) experiencing then the acceleration that you had might stop being something that is absolute, you may not be able to detect it in any way what-so-ever.   For example, an accelerometer in your possesion might not show or record any acceleration existed.

That's enough for one post,  I'm really not sure if it is what you (@varsigma) wanted to discuss anyway.

Best Wishes.

[varsigma (OP)]

Your sentence seems to have been rushed.  The bucket doesn't climb the sides,  the water climbs the sides of the bucket (after it starts to follow the same rottation as the bucket that it was in).

If I have a bucket with water in it, how much water is in the bucket? It's a "bucket" of water, right? Everyone knows it starts to climb the sides if you rotate it in place, if you stop the bucket the water keeps moving, right?

My otherwise rushed version of Mach's principle isn't too far off either. This is how they describe it in Wikipedia, from the link you posted:

The proposition is that the existence of absolute rotation (the distinction of local inertial frames vs. rotating reference frames) is determined by the large-scale distribution of matter ...

.

[Eternal Student]

Hi again.

   I can only apologise if my reply seemed hostile, @varsigma .    It wasn't intended to be.   I simply don't know what it was you wanted to discuss.

Everyone knows it starts to climb the sides if you rotate it in place, if you stop the bucket the water keeps moving, right?

    I'm quite stupid sometimes and I don't mind too much if that is known by people reading this forum.   When I first read your original post I wondered how the bucket was being spun.   It could have been swung by someones arms so that it passed over their heads but the water stayed in the bucket due to the speed of that motion.
    What I'm saying is that it wasn't clear to me what was going on and what you wanted to discuss.   I'm still not certain what you wanted to discuss or what the question is.

Best Wishes.

[Eternal Student]

Hi again.

Some of you may know that gravity waves propagate at the speed of light.

   You may need to clarify that.  Do you really mean gravity waves or gravitational waves?   Gravitational waves are the things they detect at LIGO.  Gravity waves are just waves where the returning force is gravity, like ocean waves and they don't usually travel at light speed.

This throws Mach's principle into the restriction that no interaction can occur faster than this

     I don't think there is a well established speed of gravity despite what might be written in Popular Science articles.   Yes, gravitational waves do seem to propagate at the speed of light but that is only one type of phenomena:   They are the consequence of one type of interaction that creates distortions in the metric which spread through space at a certain speed.

    1.   Particles that mediate the force of gravity, gravitons, have not been identified.  So there isn't a well established speed for the graviton.

    2.   The Riemann curvature tensor, R_abcd,  or what we might call "gravity" is dynamic, it changes with time.  However, most of those changes don't produce gravitational waves that you can detect.  Let's try to be clear about this:  Gravitational waves are observed when massive bodies undergo an acceleration, if the massive body was just following uniform motion then no gravitational wave would be detected.   LIGO detect gravitational waves from black hole mergers where they orbit around each other before merging.   It is the orbiting that is important because anything in orbit is experiencing an acceleration and is not in uniform motion.

    The Einstein Field Equations are basically  G_μν = k. T_μν.    The thing on the left describes the curvature of spacetime.  The thing on the right describes the mass and energy content of the universe and is called the stress-energy tensor.   The stress-energy tensor has all the usual information you would expect in the older Newtonian theory of gravity,  for example the distribution of mass (or the location of massive objects) is some information that is encapsulated in that tensor.  However, the stress energy tensor also contains information about the distribution and flow of momentum (i.e. information about the movement of those massive objects).
    If GR is right then  momentum contributes to the stress-energy tensor just as much as the location of some mass.   So if you change the momentum of one (or all of the) massive object(s) in space then the curvature tensor throughout the whole universe should respond immediately.   It doesn't propagate outward at the speed of light or anything like that, the EFE is an equation and if you change the right hand side then the left hand side changes immediately.  In this respect the speed of gravity is infinite, changes must occur in the curvature of the whole universe instantly if you change the stress energy tensor.
    The situation is complicated because you can't, in reality, just change the momentum of one object spontaneously and without any other effects.   If you applied a force to one massive object then it must actually be coming from somewhere and whatever provided that force will itself be accelerated in the other direction.  In this respect you can't just create a +ve flow of momentum in one location without creating an equivalent flow of momentum in the other direction (so that there is no net change).

Best Wishes.

LATE EDITING:  minor adjustment in the use of the word "net".

[varsigma (OP)]

Yes, sorry, that actually was a bit rushed and I should have said gravitational waves; an ocean wave is called a gravity wave because gravity is involved, whereas a capillary surface wave is dependent on surface tension, not gravity. This can be observed because an ocean wave can have smaller capillary waves on its surface. Anyway, those are examples of waves in a medium.

Whereas fields like gravity and electromagnetism propagate through all of space. If you consider two electrons anywhere in the universe, the electric field between them must have extended through all of space in the early universe. SInce being close together at that time, the two electrons have moved around relative to each other, experienced local changes in position, etc, but the initial field can't go anywhere, ergo the field has existed since electric fields between charged particles first appeared.

But the speed of light and the speed of gravitational waves are fixed by the special theory; what that theory says is strongly connected to information transfer. An observer, far from a source of charge can accelerate independently of the field, so sees a field changing because they have a secular motion relative to the fixed field. The approach then is, what information does the observer obtain "from the field" by accelerating?

I think part of the problem, with the discussion is the human tendency to focus on local changes and ignore the background. It's just what we do.

p.s. apologies for not quoting your response; I started with one but something borked and I lost it.

[Eternal Student]

Hi again.

But the speed of light and the speed of gravitational waves are fixed by the special theory

   The speed of light is fixed by special relativity.   However, the speed of gravitational waves is not fixed by special relativity, you can't even have gravitational waves without General Relativity.   
    It is considered good fortune and not at all co-incidence that gravitational waves should also travel at the speed of light, if that's what you meant.  It seems to imply something very fundamental.   However, I don't think it was manifestly obvious that gravitational waves would travel at the speed of light just because special relativity limits the speed of most things to the speed of light. 
    I mean, I wouldn't have guessed that some galaxies show recession exceeding the speed of light - but they do.  Special relativity imposes no limit on the speed of expansion of space.    What makes you so certain that special relativity would restrict the "speed of gravity" (which has no precise meaning) or even just the speed of gravitational waves (which at least does have a sensible definition)?

The approach then is, what information does the observer obtain "from the field" by accelerating?

   Are we drifting off topic or is this to be related to Mach's principle?
Anyway, the transmission of information should have a speed restriction, that much I can agree with.   

I think part of the problem, with the discussion is the human tendency to focus on local changes and ignore the background. It's just what we do.

   "the discussion".... do you mean this post in this forum  or  the general discussion about Mach's principle in the wider community of physicists,   or the general discussion that is science?   I still don't really know what it was you were hoping to discuss and, as @evan_au  pointed out earlier, if there was any sort of question here.
    None the less, you have a very good point there and the original entry topic, Mach's principle, does serve as a suitable context to illustrate that point.

Best Wishes.

[Halc]

   The speed of light is fixed by special relativity.   However, the speed of gravitational waves is not fixed by special relativity, you can't even have gravitational waves without General Relativity.   
    It is considered good fortune and not at all co-incidence that gravitational waves should also travel at the speed of light, if that's what you meant.  It seems to imply something very fundamental.   However, I don't think it was manifestly obvious that gravitational waves would travel at the speed of light just because special relativity limits the speed of most things to the speed of light.

Quite so. Aside from SR not dealing with gravity, as you've pointed out above, gravitational wave speed is a function of the geometry of spacetime, and thus would sink even special relativity (violating the Galiean principle of relativity premise) if they traveled at any other speed. 

I mean, I wouldn't have guessed that some galaxies show recession exceeding the speed of light

Recession speeds of specific galaxies are technically a celerity, not a velocity. Special relativity has no problem with celerity faster than c, which is why I could get to Betelgeuse before I die if I had a fast enough ship.

What makes you so certain that special relativity would restrict the "speed of gravity" (which has no precise meaning)

Indeed. 'Speed of gravity' makes it sound like massive objects emit some sort of tractor beam at things that travels at some speed and is responsible for the attraction. I agree, 'speed of gravity' is an incoherent concept. The field is already here and it can't not be everywhere, so it doesn't travel. Only changes to the field travel, and those changes are the gravitational waves. Gravitational waves carry information. Gravity does not.

[varsigma (OP)]

   The speed of light is fixed by special relativity.   However, the speed of gravitational waves is not fixed by special relativity, you can't even have gravitational waves without General Relativity.

I think Einstein's special theory showed that anything that has energy can't travel faster than the speed of light.

And if gravity is the curvature, and the gravitational waves are massless, what you need now is to show how their propagation speed isn't because they're massless and have energy, like light. Light propagates in space much like gravitational waves.

[Eternal Student]

Hi again.

I think Einstein's special theory showed that anything that has energy can't travel faster than the speed of light.

   You may need to provide a reference for that, or write down the proof if it's short.   I'm not familiar with it  (of course, there's a lot of things I don't know).
   If you replaced the word "anything" with  "any particle" then you might be correct and I wouldn't bother challenging that too much. 

And if gravity is the curvature, and the gravitational waves are massless, what you need now is to show how their propagation speed isn't because they're massless and have energy, like light. Light propagates in space much like gravitational waves.

    Gravitational waves aren't particles.   They are just changes in the metric and those changes seem to propagate through space.  Special Relativity places restrictions on the speed of movement of particles not on the movement of energy (which is just abstract), or changes in some field.   
    However, in most of physics there is some particle associated with an energy transfer from one place to another, so then there is some overlap - that transfer is limited in speed.   Similarly, most fields are associated with a particle and then you have a limit to the speed at which changes can propagate through that field.

   You also suggested gravitational waves are waves like light but I'm not sure that's relevant:  Special relativity never showed that light would travel at the speed c  just because it's an electromagnetic wave.  The speed of light was determined as a consequence of its wave mechanics but that was in a different theory (Maxwell's) and not from SR.   

    Anyway, as discussed in an earlier post, since gravitational waves do seem to propagate at the speed c, we now have a strong suggestion that a massless particle (let's call it a graviton) is involved with gravity.   It's just you couldn't go at this the other way round:  Since we haven't found a graviton, there was no guarnatee that the speed of gravitational waves would be c just from Special relativity.

Best Wishes.

[varsigma (OP)]

It's just you couldn't go at this the other way round:  Since we haven't found a graviton, there was no guarnatee that the speed of gravitational waves would be c just from Special relativity.

I don't know that I agree with that. The special theory demonstrated that nothing with energy can have a speed > c.
Masslessness means the speed is c; gravitational waves are massless.

I think that would have been obvious in the first half of the 20th century. Unless the thinking was the waves couldn't exist, say. Or they do have mass. I don't think either of those arguments stands up to scrutiny for the gravitational waves, even unobserved ones.

[Eternal Student]

Hi again.

I don't know that I agree with that. The special theory demonstrated that nothing with energy can have a speed > c.

   You have said something like that before.

   I asked for a reference or proof:

You may need to provide a reference for that, or write down the proof if it's short..... If you replaced the word "anything" with  "any particle" then you might be correct and I wouldn't bother challenging that....

You don't have to provide a reference or proof - I don't run this forum.   This is just a friendly discussion on a forum but I can't go along with that assertion.

    It was not guaranteed that gravitational waves would propagate at the speed of light, they are not particles just changes in the metric.
    See, for example, the common textbook "Gravitation", Misner, Thorne and Wheeler, p. 957   where it is shown that High-amplitude gravitational waves propagating on a background of curved spacetime develop a trailing edge that propagates at speeds less than c.
     This doesn't establish that they could travel at speeds greater than c but it does show that gravitational waves are not like massless particles, they are not bound to one unique speed of propagation like a massless particle would be.

Best Wishes.   

[varsigma (OP)]

This doesn't establish that they could travel at speeds greater than c but it does show that gravitational waves are not like massless particles, they are not bound to one unique speed of propagation like a massless particle would be.

ok, well, there is this:

In 1936, Einstein and Nathan Rosen submitted a paper to Physical Review in which they claimed gravitational waves could not exist in the full general theory of relativity because any such solution of the field equations would have a singularity. The journal sent their manuscript to be reviewed by Howard P. Robertson, who anonymously reported that the singularities in question were simply the harmless coordinate singularities of the employed cylindrical coordinates. Einstein, who was unfamiliar with the concept of peer review, angrily withdrew the manuscript, never to publish in Physical Review again. Nonetheless, his assistant Leopold Infeld, who had been in contact with Robertson, convinced Einstein that the criticism was correct, and the paper was rewritten with the opposite conclusion and published elsewhere.[23][24]: 79ff  In 1956, Felix Pirani remedied the confusion caused by the use of various coordinate systems by rephrasing the gravitational waves in terms of the manifestly observable Riemann curvature tensor.[25]

At the time, Pirani's work was overshadowed by the community's focus on a different question: whether gravitational waves could transmit energy. This matter was settled by a thought experiment proposed by Richard Feynman during the first "GR" conference at Chapel Hill in 1957. In short, his argument known as the "sticky bead argument" notes that if one takes a rod with beads then the effect of a passing gravitational wave would be to move the beads along the rod; friction would then produce heat, implying that the passing wave had done work. Shortly after, Hermann Bondi, published a detailed version of the "sticky bead argument".[23] This later lead to a series of articles (1959 to 1989) by Bondi and Pirani that established the existence of plane wave solutions for gravitational waves.[26]

https://en.wikipedia.org/wiki/Gravitational_wave#History

[Eternal Student]

Hi again.

I'm not sure what that extract was intended to show and I'm still uncertain what it was you wanted to discuss.  I may be less than helpful or just plainly getting in the way, so I'll exit the conversation.

Best Wishes.

[varsigma (OP)]

I'm not sure what that extract was intended to show and I'm still uncertain what it was you wanted to discuss.

I agree. I'm not sure now either. A hint: I think I have mild Asperger's.

[Eternal Student]

mild Asperger's.

   That's common among scientists, so I shouldn't worry about it too much.  Use the forum again if you find something you do want to discuss.

Best Wishes.