General Relativity

[Farsight]

Is it really essential to accept that reality is so utterly relative because GR makes it possible to see it that way?

No, not at all. IMHO the people who advocate this don't actually understand general relativity. When you read the original at The Foundation of the General Theory of Relativity (3.6Mbytes) Einstein is talking about the equations of motion. Yes, motion is relative, your motion does affect your measurements, and motion through space is affected by a concentration of energy tied up in the mass of a planet. But at the same time the CMBR really is a de-facto absolute reference frame.

Is it really just as valid to see all the resulting complex and convoluted motions and the required fictitious forces to explain all the motions of the universe from the perspective of a spot on the surface of Phobos as it is to view the universe from, say, a non-rotating point in intergalactic space?

No. It's mathematical science fiction.

Doesn't this tell us that we need something more than GR to understand the universe?

General relativity doesn't cover everything, so yes. But you shouldn't think it ought to be scrapped. The real problem you're experiencing is that people say they understand it when actually, they don't.

 

[Vorpal]

No, not at all. IMHO the people who advocate this don't actually understand general relativity. When you read the original at The Foundation of the General Theory of Relativity (3.6Mbytes) Einstein is talking about the equations of motion.

You don't understand what "equations of motions" means in physics. But that's a minor issue. I have read Einstein's work, and directing attention to §13 of your source, Einstein states it very explicitly that he considers the connection coefficients to be the components of the gravitational field. But this is manifestly coordinate-dependent; e.g., in flat spacetime, take cylindrical coordinates (r,θ,z). Then the only nonzero coefficients are:
[1] Γ^r_θθ = -r, Γ^θ_rθ = Γ^θ_θr = 1/r,
and so under this interpretation, in cylindrical coordinates there is a nonvanishing gravitational field. But the corresponding geodesic equations are:
[2] r" - r(θ')² = 0, θ" + (2/r)r'θ' = 0, z" = 0.
This is very obviously what Newtonian mechanics calls the centrifugal force, or explicitly if we make θ rotate with a constant angular velocity. If you wanted Einstein's authority to support PS's programme of cutting out fictitious forces, then you did not succeed, since it does the exact opposite: the fictitious forces that PS dislikes, are, according to Einstein, the very components of the gravitational field!

Yes, motion is relative, your motion does affect your measurements, and motion through space is affected by a concentration of energy tied up in the mass of a planet. But at the same time the CMBR really is a de-facto absolute reference frame.

You don't understand what "absolute reference frame" means in physics either, and that's a much more important deficiency. The maximal-isotropy frame is special in some senses, but not in terms of the laws of physics, so it is not "absolute" in the sense actually meant when discussing relativity.

The real problem you're experiencing is that people say they understand it when actually, they don't.

There is certainly a lot of that going on, but here it's mostly in the singular.

 

[Farsight]

You don't understand what "equations of motions" means in physics. But that's a minor issue.

Oh yes I do. Einstein talked about curvilinear motion through space, and about clocks clocking up motion. And he repeats "equations of motion" throughout The Foundation of the General Theory of Relativity.

I have read Einstein's work, and directing attention to §13 of your source, Einstein states it very explicitly that he considers the connection coefficients to be the components of the gravitational field.

And at the foot of page 178 he says "the law of motion" which he repeats at the top of page 179.

But this is manifestly coordinate-dependent; e.g., in flat spacetime, take cylindrical coordinates (r,θ,z). Then the only nonzero coefficients are: [1] Γ^r_θθ = -r, Γ^θ_rθ = Γ^θ_θr = 1/r,
and so under this interpretation, in cylindrical coordinates there is a nonvanishing gravitational field. But the corresponding geodesic equations are:
[2] r" - r(θ')² = 0, θ" + (2/r)r'θ' = 0, z" = 0. This is very obviously what Newtonian mechanics calls the centrifugal force, or explicitly if we make θ rotate with a constant angular velocity. If you wanted Einstein's authority to support PS's programme of cutting out fictitious forces, then you did not succeed, since it does the exact opposite: the fictitious forces that PS dislikes, are, according to Einstein, the very components of the gravitational field!

I didn't want that. But your example is not in line with §13. The geodetic straight line is defined independently of a system of reference. Your cylindrical system of coordinates is an aspect of your motion that shapes your measurement and creates those fictitious forces. A real gravitational field is different. It's there because g_μv varies. All observers will agree on this, it isn't relative, it isn't the result of your selected coordinates and your motion, it is in no way fictitious.

You don't understand what "absolute reference frame" means in physics either, and that's a much more important deficiency.

I said de-facto. And I do understand about laws of physics being "special" in that absolute reference frame. But I also understand that the laws of physics do not in reality exist. They are man-made rules drawn up from observations and measurements, all of which are affected by our motion and by the space we move through. I also understand that a reference frame does not actually exist. You cannot look up to the sky and point one out. It is an artefact of measurement. You can use the CMBR to absolutely determine your motion through the universe. GR is all about motion, and you can't get any more absolute than that.

The maximal-isotropy frame is special in some senses, but not in terms of the laws of physics, so it is not "absolute" in the sense actually meant when discussing relativity.

It's the rest frame of the universe. Physics is about the universe, and we can be 100% confident that it isn't rotating round Phobos. Absolutely.

 

[sol invictus]

I didn't want that. But your example is not in line with §13. The geodetic straight line is defined independently of a system of reference.

That could be interpreted in a way that makes it correct, but I doubt it's what you meant. A geodesic in flat space will be "straight" written in Cartesian coordinates - i.e. its trajectory will be linear in the coordinates - but that same geodesic will not look straight in curvilinear coordinates on the same space.

Your cylindrical system of coordinates is an aspect of your motion

Nonsense. Coordinate choices need have nothing whatsoever to do with the motion of any particular observer.

A real gravitational field is different.

Not according to Einstein - and not according to me, for that matter. There is no true distinction (or if there is, I don't know what it is).

It's there because g_μv varies.

g_μv varies in cylindrical coordinates on flat Minkowski space. Fail.

All observers will agree on this, it isn't relative

All observers will agree on the results of every physical experiment - which of course does not include the constancy of the metric.

Farsight, it's clear to everyone that you don't have any idea what you're talking about. Please stick to your own crank threads. This is an educational forum.

 

[schrodingasdawg]

Is it really essential to accept that reality is so utterly relative because GR makes it possible to see it that way?

Yes. Look at special relativity for an analogue. In SR, all inertial frames of reference are equally valid. It's possible to, say, suggest that there is one special reference frame that is at absolute rest, i.e. the frame at which the ether is at rest, and to suppose that the ether (and thus the EM fields and all) transform in such a way that the ether rest frame and all other inertial frames cannot be distinguished experimentally. I believe this is the gist of the Lorentz Ether Theory: SR + some special, but metaphysical, rest frame. Obviously SR is the better theory because it has fewer entities (no ether, no absolute rest frame) and makes the same predictions.

Now imagine we had some analogue of LET for GR: we have some theory that makes all the exact same predictions of GR, but in addition suggests there is some undetectable entity that is at rest, and is non-rotating, in some particular coordinate system: therefore, this coordinate system is 'special' and describes a universe at rest. GR is the better theory (by Occam's razor), since it suggests the existence of fewer entities and makes the same predictions.

 

[Perpetual Student]

Yes. Look at special relativity for an analogue. In SR, all inertial frames of reference are equally valid. It's possible to, say, suggest that there is one special reference frame that is at absolute rest, i.e. the frame at which the ether is at rest, and to suppose that the ether (and thus the EM fields and all) transform in such a way that the ether rest frame and all other inertial frames cannot be distinguished experimentally. I believe this is the gist of the Lorentz Ether Theory: SR + some special, but metaphysical, rest frame. Obviously SR is the better theory because it has fewer entities (no ether, no absolute rest frame) and makes the same predictions.

Now imagine we had some analogue of LET for GR: we have some theory that makes all the exact same predictions of GR, but in addition suggests there is some undetectable entity that is at rest, and is non-rotating, in some particular coordinate system: therefore, this coordinate system is 'special' and describes a universe at rest. GR is the better theory (by Occam's razor), since it suggests the existence of fewer entities and makes the same predictions.

Isn't that analogy a bit off, since there is no experiment one can conduct in an inertial frame to distinguish it in any way as special, but one can conduct an experiment to distinguish rotation from non-rotation?

 

[sol invictus]

Isn't that analogy a bit off, since there is no experiment one can conduct in an inertial frame to distinguish it in any way as special, but one can conduct an experiment to distinguish rotation from non-rotation?

No! Again, using rotating coordinates cannot possibly change physical results, any more than using spherical coordinates for the pipes will give a different capacitance.

Note that one can from the inside distinguish a rotating lab in empty (flat) space from a non-rotating lab in empty (flat) space, but that's not what we're talking about. Those are two different physical situations. We're talking about using two different coordinate systems to describe the same physical situation.

You made a very similar statement earlier:

No, we know that all inertial frames provide the same physics. That is not true of accelerating frames.

Sure it is. They predict exactly the same physics, just like inertial frames. Anything else would be nonsense.

There's a very useful analogy (which in fact is so precise it's hardly an analogy). Consider a flat map of the surface of the earth. Since the earth's surface is curved, the map must be distorted. That means that when you use it to determine the distance between two locations, say Tokyo and Capetown, you can't just measure the distance on the map using a straightedge and multiply by some scale. Instead, the distance will be a more complicated function that depends on the locations of the two cities and the map projection. But given that function, you can use the map to compute the actual distance.

If you had a whole set of maps, each equipped with its corresponding function, you could compute the distance between Tokyo and Capetown (or any other two locations) using each map and you'd always get the same, correct answer. If one map+function didn't give you the correct answer, you'd know that either the map or the function is wrong, or the map is correct be isn't describing the same planet.

It's exactly the same in GR. If two frames predict different physics, either you made a mistake or they're describing two physically different spacetimes. But given one spacetime, we can use any frame we want and we will always get consistent and correct answers, just as we can make any projection we want and we will always get a useable map. It's just that some frames/projections are more simple than others for certain purposes.

 

[Vorpal]

And at the foot of page 178 he says "the law of motion" which he repeats at the top of page 179.

As usual you're paying more attention to the word choice rather than

I didn't want that. But your example is not in line with §13.

It is perfectly in line with it, what I've said in other threads on this topic, and also very explictly confirms what sol has been saying longer than anyone else.

A real gravitational field is different.

Maybe you should pay attention to the last sentence of §13. The connection coefficients Γ^α_μν are explicitly dependent on the coordinate components of g_μv and fail to form a tensor. That's about as coordinate-dependent as it can possibly get.

The deep irony is that just about the only thing you've been right about (in another thread) is that Einstein's view that Γ^α_μν are the components of the gravitational field has been historically rejected by many physicists precisely because it is coordinate-dependent. This is actually a pretty minor intepretational issue, since it's possible to do all the observervational aspects of GTR without ever mentioning a "gravitational field", but it shows that you can't even be bothered to be self-consistent.

It's there because g_μv varies. All observers will agree on this, it isn't relative, it isn't the result of your selected coordinates and your motion, it is in no way fictitious.

... that's actually quite remarkable. It was always obvious that you don't understand what you're trying to talk about, but it is now clear that you haven't even tried expressing g_μv in any curvilinear coordinates, ever. And since the procedure in this case can be carried out by any decent high-school calculus student, involving nothing more complicated than differentiation and substitution, that's very strange.

Cartesian: g_μv = diag(-1,1,1,1)
Cylindrical: g_μv = diag(-1,1,r²,1)

This is a bit like talking about Newtonian physics without having any idea what "slope" is, much less "derivative" or "differential equation."

It's the rest frame of the universe. Physics is about the universe, and we can be 100% confident that it isn't rotating round Phobos. Absolutely.

Probably, yes. I think we might test for this in the approximation of the galaxies as a fluid (which is pretty standard in cosmological models) by seeing if the vorticity vanishes. If it does, then this statement is coordinate-independent and should ensure that, on the large-scale, there is no rotation about anything, Phobos or otherwise. But note that has nothing to do with picking out an extra-special frame of reference.

 

[Perpetual Student]

No! Again, using rotating coordinates cannot possibly change physical results, any more than using spherical coordinates for the pipes will give a different capacitance.

OK, I understand that.

Note that one can from the inside distinguish a rotating lab in empty (flat) space from a non-rotating lab in empty (flat) space, but that's not what we're talking about. Those are two different physical situations. We're talking about using two different coordinate systems to describe the same physical situation.

Isn't the fact that we can make that distinction of any significance? I understand that we can use any coordinate system (including a rotating one) to describe the same physical situation. If I were out in empty space in my lab and saw that I was rotating, would it not be unreasonable for me to reject the notion that I was rotating, but that the whole universe is revolving around me? I understand that the mathematics of GR allow me to see it either way, but would you not say that, in reality, the universe is not revolving around me? My rotation can easily be achieved by a simple application of energy. Getting the whole universe to revolve around me would be inconceivable. Is that not of any significance in deciding what is really happening?

You made a very similar statement earlier:

I am well aware of that. I get it that GR allows one to look at it either way, but I find it difficult to simply accept that is as far as our knowledge can take us because of GR and that we have no other tools to use to make these kind of distinctions.
Here I am in intergalactic space observing the universe around me. So, I press a button and activate some rockets to get my ship to rotate. I understand that by creating fictitious forces and convoluted physics I can also conclude my little rockets set the whole universe in motion. Do we really want to leave it at that and not find a way to determine that the former perspective is a superior reflection of reality, and not merely more convenient mathematically?

Originally Posted by sol invictus
Sure it is. They predict exactly the same physics, just like inertial frames. Anything else would be nonsense.

There's a very useful analogy (which in fact is so precise it's hardly an analogy). Consider a flat map of the surface of the earth. Since the earth's surface is curved, the map must be distorted. That means that when you use it to determine the distance between two locations, say Tokyo and Capetown, you can't just measure the distance on the map using a straightedge and multiply by some scale. Instead, the distance will be a more complicated function that depends on the locations of the two cities and the map projection. But given that function, you can use the map to compute the actual distance.

If you had a whole set of maps, each equipped with its corresponding function, you could compute the distance between Tokyo and Capetown (or any other two locations) using each map and you'd always get the same, correct answer. If one map+function didn't give you the correct answer, you'd know that either the map or the function is wrong, or the map is correct be isn't describing the same planet.

It's exactly the same in GR. If two frames predict different physics, either you made a mistake or they're describing two physically different spacetimes. But given one spacetime, we can use any frame we want and we will always get consistent and correct answers, just as we can make any projection we want and we will always get a useable map. It's just that some frames/projections are more simple than others for certain purposes.

OK, using the above analogy, each map and correct function gives us the same correct answer. That's great, but now let's ask which map is the correct one? You would say they are all equally valid, and that my question makes no sense.
Not true! The earth actually has one actual shape and size. These other models have value and are great tools but they do not tell us what the earth actually is. To understand what the earth really is we must look at the actual earth. So, GR gives us an infinite number of ways to examine the universe. But just like the earth itself is the only "real" map of itself, there is only one real universe, and it does not revolve around my little pathetic ship!

 

[Vorpal]

Isn't the fact that we can make that distinction of any significance? I understand that we can use any coordinate system (including a rotating one) to describe the same physical situation. If I were out in empty space in my lab and saw that I was rotating, would it not be unreasonable for me to reject the notion that I was rotating, but that the whole universe is revolving around me?

Yes, but not for reasons of any particular frame being special. Besides the stress on your ship walls (which is by itself sufficient to distinguish the two cases), the case where the universe rotates about an axis that happens to go through you has a nonvanishing vorticity tensor, whereas the case of you rotating does not. You keep replacing the argument about the same physical situation in different coordinate systems with different physical situation in different coordinate systems.

OK, using the above analogy, each map and correct function gives us the same correct answer. That's great, but now let's ask which map is the correct one? You would say they are all equally valid, and that my question makes no sense.
Not true! The earth actually has one actual shape and size. These other models have value and are great tools but they do not tell us what the earth actually is.

They are, and they do. They're all diffeomorphic to the Earth (at least almost everywhere), so they all have the same geometry as any other map and the Earth.

___
Side question: is the von Stockum dust a provably unique solution under the conditions of axisymmetry and rigid rotation of a fluid?

 

[schrodingasdawg]

If I were out in empty space in my lab and saw that I was rotating, would it not be unreasonable for me to reject the notion that I was rotating, but that the whole universe is revolving around me? I understand that the mathematics of GR allow me to see it either way, but would you not say that, in reality, the universe is not revolving around me? My rotation can easily be achieved by a simple application of energy. Getting the whole universe to revolve around me would be inconceivable. Is that not of any significance in deciding what is really happening?

Just as getting the whole universe to revolve around you would be inconceivable, so would getting the whole universe to stop if it already were revolving around you. You can apply energy to yourself so that you begin rotating, or stop rotating, in some particular coordinate system. For instance, suppose you are stationary with respect to one of the comoving frames: you can apply energy and rotate, so that you feel your arms being pulled to your sides (of course, this is just their inertia in this CS). Suppose you were describing the same phenomenon in a CS which describes the entire universe as revolving around you with the same angular velocity that you were spinning with respect to the comoving frame (but in the opposite direction): then you would have begun spinning, and applied energy to yourself so that you were no longer spinning; and when you ceased spinning, the feeling of your arms being pulled outwards would be due to the gravitational field of the revolving universe.

Okay, that was wordy, and perhaps poorly stated. The point is that the measurable effects of you spinning v. you not spinning (with respect to either coordinate system), such as the feeling of your arms being tugged at, is an issue of something physically going on. The issue of which coordinate system to use is just a labeling issue. Where as in the coming frame you went from not spinning to spinning, in the revolving frame you went from spinning to not spinning. The question "was I really spinning" can only be answered with "in these particular coordinates . . . "

. . . I can also conclude my little rockets set the whole universe in motion.

I think this may be a point of confusion for you. To go from describing some portion of the universe as 'stationary' with respect to some comoving frame to describing it as revolving around a point is a change of coordinates. Using your rockets to change the angular velocity of your spacecraft doesn't set the universe into motion any more than turning around rotates the universe. Regardless of whether you use a comoving or revolving frame, the angular velocity of your ship is changed by the rockets: it's your ship that was set into motion, not the universe.

(Of course, to be fair, one can always construct a reference frame that has changing angular velocity [with respect to a comoving frame], such as one that starts off stationary and accelerates up to some angular velocity. The reference frame could even be constructed so that your ship is not rotating at any point, prior to, during, or after being boosted. But the rockets were still having a physical effect on your ship, not the universe: in that coordinate system, the universe's motion would've changed as it did regardless of whether your ship was accounted for; the rockets merely counteracted the gravitational forces [which exist in this angularly accelerating CS] that would've spun the ship had they not been counteracted.)

 

[Uncayimmy]

Note that one can from the inside distinguish a rotating lab in empty (flat) space from a non-rotating lab in empty (flat) space, but that's not what we're talking about. Those are two different physical situations. We're talking about using two different coordinate systems to describe the same physical situation.

I think this one paragraph encapsulates the problem in communication. I hold no expertise on the physics end of things, but I think I might have an angle on the perception thing. I keep reading threads like this and wondering if I'm not thinking hard enough or if I actually get what's going on.

While I have not done the math, it is my understanding that relativity allows me to pick any set of coordinates as a frame of reference (not using the official term reference frame in an attempt to keep it in layman's terms) and describe the universe. In other words, no matter how you look at it, the "rules" (equations) are going to allow you make predictions that will be born out by experiment.

Where Newton's laws "fail" is that they are inadequate for describing the universe under some frames of reference or at least some conditions (may not be using the correct terminology). While those "laws" of motion are perfectly suitable for everyday life, they are not useful for all conditions. For my own lack of a better term, Newton's laws "prefer" that you hang out here on earth for the most part and look at things without getting into badass speeds.

Where I think the confusion lies is trying to reconcile relativity with what we mere mortals understand based on our perceptions and knowledge. For example, I'm cool with believing that the earth rotates on its axis while orbiting the sun along with the other planets. Relativity says (I think) that you can describe all of this with the earth being "motionless" in a frame of reference, and all the laws of physics will still apply.

This leads to the inevitable, "Yeh, but the earth is really orbiting the sun, right?" The response seems to be, "It doesn't make any difference. The laws of physics are the same."

"Yeh, but the earth is really orbiting the sun, right? It's not really stationary with everything else flying about in paths that just don't seem realistic."

"But it doesn't matter. When you pull your car into your garage, are you really going 3MPH? It's convenient to look at it that way, but from the vantage point of another galaxy, the earth is rotating while orbiting the sun and the whole solar system is orbiting the galaxy. Is that any less real than your 3MPH?"

That exchange seems to be the gist of what I'm reading here. If I'm wrong, please correct me. Assuming I've got that part right, here's the question I'd like the experts to address. If reasonable, please address it from your expert level as well as that of the Average Joe.

When I pull my car into my garage, is there any possible way for me to know that I pulled in at 3MPH rather (say) than the entire frigging planet moved towards my stationary car? If there is, that makes me "comfortable" that there exists a "reality" of some sorts. If there is, however, that doesn't mean there's a preferred reference frame because relativity, I believe, could explain the actions from any reference frame you could imagine and do it accurately.

For me personally, this is simply curiosity. I'm not trying to overturn the world of physics or solve some metaphysical conundrum.

Thanks for any time devoted to this. It is appreciated.

 

[dasmiller]

This leads to the inevitable, "Yeh, but the earth is really orbiting the sun, right?" The response seems to be, "It doesn't make any difference. The laws of physics are the same."

"Yeh, but the earth is really orbiting the sun, right? It's not really stationary with everything else flying about in paths that just don't seem realistic."

"But it doesn't matter.

And I suspect that the word "really" is a big part of the problem, because it implies that there's some underlying "real" reference frame and we just haven't figured out what it is (but in our hearts, we think we know).

If we go back to Sol's map analogy -yes, there are a number of ways to make 2D maps of a 3D globe, but which one is the "real" one? Why, none of them, of course. It's not that we haven't figured out the "real" one yet - there isn't one. Mercator projections have their advantages, as do Behrman, Miller, etc. But none will give accurate results for all cases if you're simply doing 2D measurements on them. And if you're doing the math required to get accurate results, then you can get accurate results from any of them, and then they're all "real."

 

[schrodingasdawg]

When I pull my car into my garage, is there any possible way for me to know that I pulled in at 3MPH rather (say) than the entire frigging planet moved towards my stationary car?

When you pull your car into the garage at 3 mph, there's no way to know that it was your car moving rather than the Earth moving, so both descriptions are equally valid. One is just more convenient for our everyday purposes.

When you drive a car at 80 mph on the freeway, it's equally valid to describe both the car as moving with respect to the stationary freeway and the freeway as moving with respect to the stationary car. We can model either case using a coordinate system, which is just a way of assigning four real numbers to every event in space and time. The case is described by some coordinate system where the spatial parts are Cartesian and the origin is some point on the asphalt, not moving with respect to the street; the latter case is also a simple coordinate system, but one where the car can be described as stationary.

Both coordinate systems are valid, but something to keep in mind is that in the coordinate system where your car is stationary, you need to keep your foot on the accelerator to remain stationary: for the same reason that someone on a treadmill needs to keep running to stay in place.

When you accelerate a car from 0 mph to 80 mph at a rate of 20 mph/s on a northbound lane, the speed of your car is changing in either of the above described coordinate systems. In the CS where the road is stationary, your car is accelerating from 0 to 80 mph and is traveling north; in the CS that an 80 mph car would be stationary in, your car is de-accelerating from going 80 mph southbound to a complete stop. Either description is valid.

A third coordinate system can be constructed, in which the accelerating car is stationary. In this system, the road begins stationary, but begins to move to the south and accelerates to moving at 80 mph. As the road starts moving, you and your car experience a constant force pushing you to the south with a magnitude of 20 mph/s times your mass; and you need to apply your foot to the gas pedal to counteract this constant force.

The thing to keep in mind when using this third coordinate system is that you used the gas pedal to keep your car stationary against the southward constant force: using the gas pedal didn't cause the force.

(Another thing to keep in mind is that while Newton's mechanics usually describes the southward force as 'fictitious'---the mathematical artifact of a non-inertial frame of reference---general relativity describes this as a very real [at least in this coordinate system] gravitational field. But I suppose the relativist philosophy can be extended to Newton's mechanics anyway.)

So, none of that seems like an immediate answer to your question, and even seems to go off-topic. But I think the examples can be instructive. I basically want to say that while motion is relative---and thus, whether you pulled into your garage or the garage moved around your car isn't knowable---the fact of the matter is that you're controlling the motion of your car, not the motion of the planet, when you drive; and this is true regardless of what coordinate system you're using. (Of course, to be pedantic, there's always that bit about Newton's third law and you're actually applying a negligible force to the Earth with your car... but whatever.)

 

[Uncayimmy]

When you pull your car into the garage at 3 mph, there's no way to know that it was your car moving rather than the Earth moving, so both descriptions are equally valid. One is just more convenient for our everyday purposes.

<snipped excellent explanation>

So, none of that seems like an immediate answer to your question, and even seems to go off-topic. But I think the examples can be instructive. I basically want to say that while motion is relative---and thus, whether you pulled into your garage or the garage moved around your car isn't knowable---the fact of the matter is that you're controlling the motion of your car, not the motion of the planet, when you drive; and this is true regardless of what coordinate system you're using. (Of course, to be pedantic, there's always that bit about Newton's third law and you're actually applying a negligible force to the Earth with your car... but whatever.)

I suppose that if I took a physics course in college I would have late night gabfests about this stuff, so it's probably old hat for many of you. You're definitely addressing what I think is the core issue. Here's where I begin to stumble.

Suppose in your highway example I look at the car next to me. It would seem that your descriptions would have to apply to both of us. It would seem that it can't be the earth moving relative to my stationary car and the other car moving on a stationary earth at the same time. It would seem it's one or the other. So the natural inclination in my puny mind is to think that one of them is "real" while the other is merely a description. It would also seem that given enough blood flow to the brain I could figure out a way to determine which one of many is "real."

If you could take that ball and run with it, I would be appreciative. I'm not disputing anything you've said. I'm just trying to wrap my head around it.

 

[sol invictus]

Isn't the fact that we can make that distinction of any significance?

Sure. It says two different physical situations are "really" different, and two different metric describing the same physical space (i.e. related by a coordinate transformation) are "really" the same.

The infinity of metrics that describe our universe includes a set in which it rotates around Phobos.

If I were out in empty space in my lab and saw that I was rotating, would it not be unreasonable for me to reject the notion that I was rotating, but that the whole universe is revolving around me?

That depends on what you mean precisely. By "saw that I was rotating" I will assume you mean you're in a sealed lab, and you measured forces consistent with the lab rotating around its axis in a locally flat space.

In that situation it would not be reasonable to reject the idea that the universe is rigidly rotating under the influence of "fictitious forces" (or an axi-symmetric gravity field, take your pick) around your (stationary) lab's axis at exactly the rate corresponding to the forces you measure, because that's a completely equivalent description to the one you prefer.

I understand that the mathematics of GR allow me to see it either way, but would you not say that, in reality, the universe is not revolving around me? My rotation can easily be achieved by a simple application of energy. Getting the whole universe to revolve around me would be inconceivable.

Changing your velocity can easily be achieved by walking. Getting the entire earth to move would be inconceivable.

Is that not of any significance in deciding what is really happening?

Well, is the earth "really" at rest?

OK, using the above analogy, each map and correct function gives us the same correct answer. That's great, but now let's ask which map is the correct one? You would say they are all equally valid, and that my question makes no sense.
Not true! The earth actually has one actual shape and size. These other models have value and are great tools but they do not tell us what the earth actually is. To understand what the earth really is we must look at the actual earth. So, GR gives us an infinite number of ways to examine the universe. But just like the earth itself is the only "real" map of itself, there is only one real universe, and it does not revolve around my little pathetic ship!

I don't understand what you're saying. The earth itself is not a map at all, so how can it be the only "real" map?

Is that city "really" Genoa or "really" Genova?

"The universe is rotating around Phobos under the influence of a certain set of fictitious forces" is a sentence that stands for a set of physically observable predictions. If those predictions are verified, it's just as good a description as any other.

 

[sol invictus]

Suppose in your highway example I look at the car next to me. It would seem that your descriptions would have to apply to both of us. It would seem that it can't be the earth moving relative to my stationary car and the other car moving on a stationary earth at the same time. It would seem it's one or the other. So the natural inclination in my puny mind is to think that one of them is "real" while the other is merely a description. It would also seem that given enough blood flow to the brain I could figure out a way to determine which one of many is "real."

You seem to have jumped from saying "both descriptions can't be valid at the same time" to saying "therefore one or the other is real".

Try that on the map analogy. Two maps, two projections (the function that tells you how "stretched" the map is, and therefore how to convert map distances into real distances). The projections are different, so they can't both be valid for both maps. You should pick one map and one projection and use that for everything, or carefully switch from one to the other. But obviously neither is more real than the other.

 

[Perpetual Student]

I don't understand what you're saying. The earth itself is not a map at all, so how can it be the only "real" map?

Sorry for the poor choice of words. The point I attempted to make is that even though we know that all of our maps and accompanying functions give the same correct results, we do know these results only have meaning (or purpose) because they tell us something about the "real" earth. The real earth has the simplest of all accompanying functions. Similarly, even though GR gives us an infinite set of maps and functions, are they not all describing some real universe, which also has some simplest of all accompanying functions? Would this not tell us that Occam's razor guides us to a best description of the real universe?

 

[Perpetual Student]

I think this may be a point of confusion for you. To go from describing some portion of the universe as 'stationary' with respect to some comoving frame to describing it as revolving around a point is a change of coordinates. Using your rockets to change the angular velocity of your spacecraft doesn't set the universe into motion any more than turning around rotates the universe. Regardless of whether you use a comoving or revolving frame, the angular velocity of your ship is changed by the rockets: it's your ship that was set into motion, not the universe.

(Of course, to be fair, one can always construct a reference frame that has changing angular velocity [with respect to a comoving frame], such as one that starts off stationary and accelerates up to some angular velocity. The reference frame could even be constructed so that your ship is not rotating at any point, prior to, during, or after being boosted. But the rockets were still having a physical effect on your ship, not the universe: in that coordinate system, the universe's motion would've changed as it did regardless of whether your ship was accounted for; the rockets merely counteracted the gravitational forces [which exist in this angularly accelerating CS] that would've spun the ship had they not been counteracted.)

I don't have any problem with the above, but it seems to me to contradict sol's position.

 

[sol invictus]

Sorry for the poor choice of words. The point I attempted to make is that even though we know that all of our maps and accompanying functions give the same correct results, we do know these results only have meaning (or purpose) because they tell us something about the "real" earth. The real earth has the simplest of all accompanying functions.

That doesn't make any more sense than what you said before. What's the "function" you claim the "real earth" "has"?

The minute you write down any such function, you're no longer talking about the "real earth", you're talking about a map (or metric) of it. But then we're back where we started.

Similarly, even though GR gives us an infinite set of maps and functions, are they not all describing some real universe, which also has some simplest of all accompanying functions? Would this not tell us that Occam's razor guides us to a best description of the real universe?

That doesn't make sense for the same reason as above.

I don't have any problem with the above, but it seems to me to contradict sol's position.

How?