General Relativity

[Raze]

That's kind of the point of Occam. If you have two apparently arbitrary ways to describe the same data, pick the one with the fewest entities.

But how do you know which of the infinite possible reference frames is the simplest?

 

[Perpetual Student]

Sure, but that doesn't make everything equivalent. You can call a road race of Phobos equivalent to a road race on earth. But once you introduce cosmology, well, you really do have something unique and special. There may be an absurd number of possible road races in the universe, each with its simplest reference frame, but there's only one cosmos. And there's only one co-moving reference frame for that one cosmos. On a certain level, yes, it's not any more valid than any other reference frame. And Occam's razor is ultimately about convenience, not truth. But nonetheless, there still remains one reference frame which is unique for everyone, everywhere. I don't think you can construct any other reference frame which is similarly unique for everyone.

Isn't that the point here? For millennia mankind used the surface of earth as a special reference frame for describing the universe (turtles all the way down and all). Over the years we learned that there is nothing special about the surface of the earth as a reference.
The CMB rest frame is perhaps the only currently available universal frame of reference for best describing the universe, at least that part of it that we can observe. I don't know how we can escape the conclusion that that unique frame of reference tells us what the universe is actually doing. GR is simply incomplete in that respect.

 

[Raze]

Isn't that the point here? For millennia mankind used the surface of earth as a special reference frame for describing the universe (turtles all the way down and all). Over the years we learned that there is nothing special about the surface of the earth as a reference.
The CMB rest frame is perhaps the only currently available universal frame of reference for best describing the universe, at least that part of it that we can observe. I don't know how we can escape the conclusion that that unique frame of reference tells us what the universe is actually doing. GR is simply incomplete in that respect.

Why? Is this an argument that the CMB rest frame would result in a simpler mathematical description (the Occam's Razor argument).

Another question: does GR already include the possibility of using the CMB as a rest frame? If so how is GR incomplete?

 

[sol invictus]

That's kind of the point of Occam. If you have two apparently arbitrary ways to describe the same data, pick the one with the fewest entities.

What's an "entity"?

Sure, but that doesn't make everything equivalent. You can call a road race of Phobos equivalent to a road race on earth. But once you introduce cosmology, well, you really do have something unique and special. There may be an absurd number of possible road races in the universe, each with its simplest reference frame, but there's only one cosmos.

How do you know? All we can see is a part of it, a part which we can be nearly certain is a very small part.

And there's only one co-moving reference frame for that one cosmos. On a certain level, yes, it's not any more valid than any other reference frame. And Occam's razor is ultimately about convenience, not truth. But nonetheless, there still remains one reference frame which is unique for everyone, everywhere. I don't think you can construct any other reference frame which is similarly unique for everyone.

There's one set of reference frames - it's certainly not unique (for example if the universe is spatially flat, rotations and translations preserve all the properties you're talking about, as do time reparametrizations).

That set is in any case defined only up to perturbations, which - while present - are admittedly small on the scales we can observe. There's absolutely no reason to think they remain small on larger scales, however.

Another thing to note that this frame is obviously not inertial, because it defines a rest frame. Frames in constant motion with respect to that rest frame will see something different (in particular, they will see a dipole temperature anisotropy in the CMB).

But again, I'm not arguing with the assertion that that class of frames are the most convenient to describe cosmology on large scales. They are - but there is nothing fundamental about that, any more than there is about the statement that the road race on Phobos is most conveniently described using Phobocentric coordinates.

 

[sol invictus]

I don't know how we can escape the conclusion that that unique frame of reference tells us what the universe is actually doing.

Very, very easily.

First, it's not unique even if we ignore the fact that there are perturbations (see my reply to Zig above).

Second, there are perturbations which destroy the whole concept of a precise unique frame. The universe is not homogeneous and isotropic, it's just approximately so, and even that only on a particular range of scales.

Third, we can only see part of the universe, and have no evidence at all that the perturbations remain small on larger scales (i.e. no reason to think it remains approximately homogeneous and isotropic on larger than horizon scales).

Fourth, even if we ignore all of the above, your objection would only make sense if there was only one possibility - if the universe must always have that unique frame. But we know for certain that isn't true (because there are perturbations, and because everything we know about physics says it isn't).

Fifth, even if we ignore all of the above, the ability of GR to describe more than one frame is a feature without which it would simply be incomplete. Asking that it only be able to describe physics in one special frame is like asking for a theory of linguistics that only works on one language, or requiring that map-making theory only admit one projection, or that arithmetic only work in Roman numerals... it's utter nonsense.

The whole point of GR - the stroke of genius that led to it - was Einstein's realization that physics cannot possibly depend on the coordinates we humans choose to describe it with, and that following that fact through actually carries profound and mathematically powerful consequences.

 

[sol invictus]

Let me give another example. Suppose you're presented with an arrangement of two concentric metal pipes, one inside the other with the space in between filled with an insulator. The pipes are fairly long and straight, and you're asked to calculate their capacitance. What do you do?

Well, if you've taken any physics you know exactly how to start. You measure the radii of the pipes, then write down the equations for capacitance - and you use cylindrical coordinates, because that's the approximate symmetry of the problem (approximate because no real pipe is perfectly round, and on microscopic scales they're not even remotely close to it). Then you do a simple integral and get the capacitance - approximately. I certainly don't dispute that that is the quickest and easiest way to go.

Now, suppose someone comes along and insists on doing the calculation using spherical coordinates. If she does everything correctly, of course she will get the same, correct answer. She'll have to work harder, but the physics is exactly the same - obviously! She's simply labelled points differently, and how could that possibly matter?

Is there anything more correct about the first method than the second? Are the pipes "in" cylindrical coordinates, but not "in" spherical? (That's a question for you, PS.) Is it a deficiency of the theory that it cannot tell us which coordinates "really" describe those pipes? Can we conclude anything about the world from the fact that the section of pipe we can see is roughly cylindrically symmetric?

 

[Ziggurat]

How do you know? All we can see is a part of it, a part which we can be nearly certain is a very small part.

In a sense, yes, we don't know. We assume large-scale homogeneity. With that assumption, the conclusion becomes inevitable. Without that assumption, we can't conclude much of anything about cosmology. So I'll stick with it for now.

There's one set of reference frames - it's certainly not unique (for example if the universe is spatially flat, rotations and translations preserve all the properties you're talking about, as do time reparametrizations).

Close enough for the current discussion.

Another thing to note that this frame is obviously not inertial

I never said it was. I said it was unique.

But again, I'm not arguing with the assertion that that class of frames are the most convenient to describe cosmology on large scales. They are - but there is nothing fundamental about that, any more than there is about the statement that the road race on Phobos is most conveniently described using Phobocentric coordinates.

I didn't use the term "fundamental", but rather "unique". The Phobos frame is not unique. The co-moving cosmological frame (or if you insist, set of frames) is unique. And as far as I am aware, it is the only reference frame which is unique. Whether you choose to consider that to be indicative of something "fundamental" seems to be a semantic issue of what you consider "fundamental", and that question holds little interest to me.

 

[sol invictus]

In a sense, yes, we don't know. We assume large-scale homogeneity. With that assumption, the conclusion becomes inevitable. Without that assumption, we can't conclude much of anything about cosmology. So I'll stick with it for now.

Actually we don't need that assumption for much of anything - most of cosmology doesn't depend on what's outside our current horizon (even the asymptotic future in most realistic scenarios).

Close enough for the current discussion.

I don't agree at all. For one thing the existence of perturbations destroys the entire argument (such as it was), because it means that there isn't even a unique class of frames. How can there be something fundamentally important about an approximation?

I never said it was. I said it was unique.

But it isn't unique even if we ignore perturbations.

I didn't use the term "fundamental", but rather "unique". The Phobos frame is not unique.

We were originally discussing the frame in which Phobos isn't rotating. That's not unique, but it's not much less unique (in any sense I can think of) than the cosmic "rest frame".

The co-moving cosmological frame (or if you insist, set of frames) is unique. And as far as I am aware, it is the only reference frame which is unique.

But again, it's not.

The part of the CMB we can see is approximately rotationally invariant, and (probably) approximately translation invariant. To the extent those are good approximations, we can use those symmetries to define a special class of reference frames. Those are convenient because the approximate symmetries are manifest in them, and hence they are relatively simple to use.

But that's as far as it goes - there is no sense in which those reference frames tell us what the universe is "really" doing (the idea is absurd). Please see my example above with the pipes, it might help.

 

[Perpetual Student]

Let me give another example. Suppose you're presented with an arrangement of two concentric metal pipes, one inside the other with the space in between filled with an insulator. The pipes are fairly long and straight, and you're asked to calculate their capacitance. What do you do?

Well, if you've taken any physics you know exactly how to start. You measure the radii of the pipes, then write down the equations for capacitance - and you use cylindrical coordinates, because that's the approximate symmetry of the problem (approximate because no real pipe is perfectly round, and on microscopic scales they're not even remotely close to it). Then you do a simple integral and get the capacitance - approximately. I certainly don't dispute that that is the quickest and easiest way to go.

Now, suppose someone comes along and insists on doing the calculation using spherical coordinates. If she does everything correctly, of course she will get the same, correct answer. She'll have to work harder, but the physics is exactly the same - obviously! She's simply labelled points differently, and how could that possibly matter?

Is there anything more correct about the first method than the second? Are the pipes "in" cylindrical coordinates, but not "in" spherical? (That's a question for you, PS.) Is it a deficiency of the theory that it cannot tell us which coordinates "really" describe those pipes? Can we conclude anything about the world from the fact that the section of pipe we can see is roughly cylindrically symmetric?

Thanks for taking the time to indulge me in this question. First, I will admit that my resolve in this matter has been weakened by your arguments, but my intuition will not yield.
I do see the point you are making above. There is nothing more "correct" about the two approaches. Both will yield the same capacitance. Neither method is really describing the capacitance of the pipes in a more "real" way.
But this example is only dealing with an electrical property of that particular geometry for a capacitor. It is not designed to tell us anything about the actual geometry of the configuration. Because the spherical coordinates lead to the same answer, can we conclude that the capacitor consists of two concentric spheres? Are two concentric spheres the same as two concentric cylinders? We know that is not the case from simple geometry and you defined the problem as consisting of two concentric cylinders, in the first place.
So, even though neither approach better describes the device, one of them does use the "real" geometry and is consequently simpler, as you said. Could we not say the same about GR, as we use it to describe the whole universe. A spot on the surface of Phobos will not yield a better answer than the rest frame of the CMB, but the latter is simpler and perhaps more accurately describes the real universe.
I also understand that Einstein's great realization was "that physics cannot possibly depend on the coordinates we humans choose to describe it with, and that following that fact through actually carries profound and mathematically powerful consequences." But GR is after all only a model. Do you not agree it is incomplete if we cannot use it to conclude that the whole universe is not revolving around a spot on the surface of Phobos?

 

[Ziggurat]

I don't agree at all. For one thing the existence of perturbations destroys the entire argument (such as it was), because it means that there isn't even a unique class of frames. How can there be something fundamentally important about an approximation?

The "close enough" was in reference to the fact that the co-moving frame is really a set of frames, not a single frame. They are related to each other in rather trivial ways, unlike the set of all reference frames available, or even the various Phobos-like frames for racing in.

We were originally discussing the frame in which Phobos isn't rotating. That's not unique, but it's not much less unique (in any sense I can think of) than the cosmic "rest frame".

Sure it is. You know it is. You're arguing otherwise because you want to point out that mathematically all these frames get handled the same way, which is true. Which is true, but doesn't change my point.

But that's as far as it goes - there is no sense in which those reference frames tell us what the universe is "really" doing (the idea is absurd). Please see my example above with the pipes, it might help.

I saw your pipes example already. And the thing about your pipe example is that the simplicity of solving the problem in cylindrical coordinates does in fact indicate something about the pipe: namely, the pipe has cylindrical symmetry. The pipe is really being cylindrical. The co-moving reference frame likewise does indeed indicate something real about the universe: the average motion of mass in the universe. It really is doing that. The validity of alternative reference frames or coordinates doesn't change that.

 

[sol invictus]

Because the spherical coordinates lead to the same answer, can we conclude that the capacitor consists of two concentric spheres?

No, of course not. Any coordinate choice would give the same answer.

Are two concentric spheres the same as two concentric cylinders?

No, they aren't. There is a real physical difference that real, measurable quantities depend on. And indeed, it is the case that the part of the CMB we can see is approximately symmetric under a certain group of symmetry transformations, and we know that because we measured it.

Could we not say the same about GR, as we use it to describe the whole universe. A spot on the surface of Phobos will not yield a better answer than the rest frame of the CMB, but the latter is simpler and perhaps more accurately describes the real universe.

If you want to describe the part of the CMB we can observe, it's simplest to use coordinates in which its approximate symmetries are manifest, yes. That's all that can be said.

I also understand that Einstein's great realization was "that physics cannot possibly depend on the coordinates we humans choose to describe it with, and that following that fact through actually carries profound and mathematically powerful consequences." But GR is after all only a model. Do you not agree it is incomplete if we cannot use it to conclude that the whole universe is not revolving around a spot on the surface of Phobos?

Do you not agree that Maxwell's equations are incomplete if they cannot tell us that we must use cylindrical coordinates to describe those pipes?

We can use Maxwell's equations plus measurements of the capacitance to determine that the pipes are roughly cylindrically symmetric. Similarly, we can use GR plus cosmological measurements to determine that the part of the CMB we can see is roughly homogeneous and isotropic.

And we can say is that no physical quantities of relevance to cosmology depend in any particularly simple way on the distance from the axis of rotation of Phobos. But that's it, that's all we can say.

What's this obsession with the CMB, anyway? Sure, it's the largest scale thing we can observe right now. But 50 years ago we hadn't observed it, and 50 years from now we might be observing something even larger. And even the CMB is only approximately homogeneous and isotropic.

 

[sol invictus]

Sure it is. You know it is.

Actually I'm honestly not sure it is in any well-defined sense. But if you think so, go ahead and try to argue for it. (You could try to define your frame at every point by requiring that the CMB have zero dipole, but then the large scale structure distribution won't have zero dipole due to perturbations and peculiar motions. On the other side I can choose my Phobocentric coordinates so Phobos has zero rotation, and then try to extend those coordinates in the simplest analogue to a rigidly rotating frame there is in an FRW cosmology, which I think is unique if we ignore those perturbations.)

I saw your pipes example already. And the thing about your pipe example is that the simplicity of solving the problem in cylindrical coordinates does in fact indicate something about the pipe: namely, the pipe has cylindrical symmetry. The pipe is really being cylindrical. The co-moving reference frame likewise does indeed indicate something real about the universe: the average motion of mass in the universe. It really is doing that. The validity of alternative reference frames or coordinates doesn't change that.

I'm not saying there isn't a real fact about the universe that accounts for the simplicity of certain classes of frames for describing certain observables. I'm saying three other things:

1) The fact some physical situation may have a certain set of symmetries does not force us to use a set of coordinates in which those symmetries are manifest, nor is that lack of being forced a deficiency in our theories. Quite the contrary, it's a requirement of any complete theory that it make sense in any coordinates.

2) The part of the CMB we can see is only approximately symmetric, and we know we're only seeing a small part of the whole thing.

3) The CMB is just one set of observations. Other sets of observations have different approximate symmetries, and the CMB isn't particularly unique or special except in that now, in 2010, it's the biggest thing we can observe.

 

[Raze]

Because the spherical coordinates lead to the same answer, can we conclude that the capacitor consists of two concentric spheres?

How do you get this conclusion? Spherical coordinates just means you define the location of a point based upon the radial distance from the origin, an inclination angle from a location directly above the point (ETA-from a fixed zenith direction), and an azimuth angle from a reference plane perpendicular to the inclination angle that passes through the origin.

It's just a way to define the location of the point.

With cylindrical coordinates you have a radius, an angle and a height. That's the difference.

 

[Perpetual Student]

Do you not agree that Maxwell's equations are incomplete if they cannot tell us that we must use cylindrical coordinates to describe those pipes?

We can use Maxwell's equations plus measurements of the capacitance to determine that the pipes are roughly cylindrically symmetric. Similarly, we can use GR plus cosmological measurements to determine that the part of the CMB we can see is roughly homogeneous and isotropic.

Well, I would venture to say that Maxwell's equations were never intended for that purpose and it would be unfair to characterize them as incomplete. So, perhaps the same thing should be said about GR in the context of this discussion. Are not all models limited to a particular context and purpose?
That leads me to the position that we should not expect GR to decide whether a spot on Phobos or the CMB is an more accurate description of the workings of the universe. GR doesn't care and was not constructed to care.
Consequently, we should use something like common sense, Occam's razor or the simplicity of the mathematics to tell us what is really happening.
I will go to my grave (or ashes) absolutely firm in the belief that the universe does not revolve around a spot on Phobos and I have not spent my life on the back of a turtle.

 

[Perpetual Student]

How do you get this conclusion? Spherical coordinates just means you define the location of a point based upon the radial distance from the origin, an inclination angle from a location directly above the point (ETA-from a fixed zenith direction), and an azimuth angle from a reference plane perpendicular to the inclination angle that passes through the origin.

It's just a way to define the location of the point.

With cylindrical coordinates you have a radius, an angle and a height. That's the difference.

Exactly! So?

 

[Tim Thompson]

Model & Reality

I have never heard of a theory involving a rotating universe. Is there such a theory?

Yes. See the Gödel metric & Gödel, 1949. Gödel's original paper is not freely available, but many of the 436 citations are.

... but that does not change the reality of the nature of the universe.

I don't think it is possible under any circumstances to know what is the reality of the universe. I think the best we can do is know the relationship of consistency between our model of the universe and our observations of the universe. When we find the two in high accord we tend to treat the model as if it is reality, but we must keep in mind that is it always only a model.

 

[schrodingasdawg]

Several months ago on a thread about relativity, there was a discussion concerning the concept that all frames of reference are equally valid under general relativity. The physicists who participated asserted that (as an extreme example) it would be equally valid to view the whole universe as revolving around Phobos (one of the moons of Mars) compared to any other perspective (the CMBR, for example). The mathematics, of course would be vastly more complicated, but that would not invalidate that particular consequence of GR.

This is correct. One can devise a coordinate system in which Phobos is at rest, and in which everything else (Mars, the planets, starts, galaxies, etc.) is revolving around it. A complicated metric will be needed to describe the universe using such a coordinate system, and this metric will describe a rather contrived looking spacetime with all the right gravitational forces needed to keep the universe in motion.

At that time, I argued that we all really know that the whole universe is not really revolving around Phobos, even though GR allows that perspective for anyone who might be inclined to use it. The professionals told me I was dead wrong! -- All frames of reference are equally valid!

The professionals are right. To say that any particular frame of reference (or any class of reference frames) describes 'reality' is a metaphysical statement.

I like sol's post on the matter:

Frames are just human labeling conventions . They have no more connection to reality or intrinsic meaning than the sequence of symbols C-A-T does to the animal it describes in English.

I think this sums up GR's attitude towards coordinates quite concisely.

 

[Raze]

Because the spherical coordinates lead to the same answer, can we conclude that the capacitor consists of two concentric spheres?

How do you get this conclusion? Spherical coordinates just means you define the location of a point based upon the radial distance from the origin, an inclination angle from a location directly above the point (ETA-from a fixed zenith direction), and an azimuth angle from a reference plane perpendicular to the inclination angle that passes through the origin.

It's just a way to define the location of the point.

With cylindrical coordinates you have a radius, an angle and a height. That's the difference.

Exactly! So?

Well, I failed to realize that your question was rhetorical. My apologies.

 

[Perpetual Student]

Is it really essential to accept that reality is so utterly relative because GR makes it possible to see it that way? 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?
Doesn't this tell us that we need something more than GR to understand the universe?

 

[Vorpal]

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

Of course it is. Or rather, it's as essential as your earlier appeal to Occam's razor. A theory that treats all frames equally is simpler than one that manufactures special classes of (say) "fundamental frame" and "fictitious frame". If "entities" cuts across anything meaningful, it's that, because we're ascribing objective reality to some frames but not others. The presence of so-called fictitious forces is not at all comparable, because they are just a coordinate effect, and coordinates are neither objectively real nor claimed to be.

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?

Yes. There is no observational reason to treat either as "more real".

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

If you insist on this, I think you better abandon Occam in the interests of self-consistency, because what you actually want is a theory which makes a metaphysical, unempirical distinction.