General Relativity

[Reality Check]

I wouldn't say I've mastered it, but it isn't rocket science, you start with vectors then move up to matrices. A tensor is a matrix.

Science courses start with vectors, move up to matrices and introduce tensor notation as a neat way to represent matrices. A tensor is not a matrix. A matrix is a representation of a tensor in a Cartesian coordinate system.

 

[W.D.Clinger]

I wouldn't say I've mastered it, but it isn't rocket science, you start with vectors then move up to matrices. A tensor is a matrix.

Science courses start with vectors, move up to matrices and introduce tensor notation as a neat way to represent matrices. A tensor is not a matrix. A matrix is a representation of a tensor in a Cartesian coordinate system.

Tensors can be represented by matrices even in non-Cartesian coordinate systems, but Farsight probably ought to acquaint himself with differential calculus before he tries to explain the relationship between tensor fields and tangent bundles.

 

[Perpetual Student]

This is my original post:

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.
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! To my dissatisfaction, that’s where the discussion ended.
After several months of further reflection, it still seems to me that if that is the case, if we cannot use Occam’s razor (or some similar concept), to conclude that the universe is not really revolving around Phobos, it is a fundamental flaw of GR. It simply contradicts common sense, intuition and rationality to view things otherwise. And, as far as I can tell, there is no utility in viewing the universe in such an absurd manner. Any comments?

Subsequently, an even stronger assertion was made (I believe by sol invictus) that, if one could demonstrate that some preferred reference frame actually existed, that would invalidate GR. In other words, GR will not permit a preferred frame of reference.
I am providing the following In order to put this question in the context of my level of understanding. I have made some effort to better understand the mathematical basis of GR in order to grasp this mysterious consequence of GR, and so far I have succeeded in gaining a general understanding of the tensor equation of GR and I have studied some specifics like the Schwarzschild metric, the Minkowski metric and I have a fair handle on why tensors are coordinate independent and the importance of that in GR.
But, as I continue my quest, I’m not sure what I am looking for. This tensor equation does allow one to examine the universe from any inertial or accelerated coordinate system, but I see no violation of the equation by discovering that there might be a best (or preferred) coordinate system for the whole universe, like the that of the CMB. Certainly, such a preferred reference frame would not be provided by the Einstein tensor equation but some other consideration or theory outside of GR. Further help with this question would be very much appreciated.

By the way, I have enjoyed this effort so much that I intend to continue for the foreseeable future as far as this old brain will allow. The Bernhard Wuensch (MIT) lectures and the Leonard Susskind (Stanford University) lectures on tensors, SR, GR and other topics are quite accessible and very entertaining and there exist so many other lectures, papers and reference sources on these topics. It's a big help that I did study some mathematics years ago. And -- what a great time to be alive!

 

[epepke]

Science courses start with vectors, move up to matrices and introduce tensor notation as a neat way to represent matrices. A tensor is not a matrix. A matrix is a representation of a tensor in a Cartesian coordinate system.

A matrix does not necessarily represent a tensor (most don't), just as an array does not necessarily represent a vector.

 

[theprestige]

This tensor equation does allow one to examine the universe from any inertial or accelerated coordinate system, but I see no violation of the equation by discovering that there might be a best (or preferred) coordinate system for the whole universe, like the that of the CMB. Certainly, such a preferred reference frame would not be provided by the Einstein tensor equation but some other consideration or theory outside of GR. Further help with this question would be very much appreciated.

I think the thing about GR and preferred frames is kind of tautological: GR is a way of describing observed reality that depends on there not being a preferred frame. GR is very good at describing observed reality, but the description only makes sense if there's no preferred frame.

So naturally, any description of reality that includes a preferred frame violates GR, because GR is a description that excludes preferred frames.

Notably, every explanation of observed reality that includes a preferred frame, that has been attempted so far, is demonstrably less explanatory than GR.

So the challenge is to find an explanation of observed reality, that explains everything that GR explains, but does so in the context of a preferred frame.

 

[Perpetual Student]

I think the thing about GR and preferred frames is kind of tautological: GR is a way of describing observed reality that depends on there not being a preferred frame. GR is very good at describing observed reality, but the description only makes sense if there's no preferred frame.

Thanks for your response.
It seems that you are saying that for GR to be a functional model, the concept of a preferred frame must be abandoned. But that is quite different from saying that the demonstrated validity of GR somehow "proves" there is no preferred frame. We can suspend many aspects of reality (and often do) for some mathematical model to give us good information.

So naturally, any description of reality that includes a preferred frame violates GR, because GR is a description that excludes preferred frames.

For one model of our universe to rule out some other model it seems to me that there must be some mathematically demonstrated contradiction, otherwise both models would have a standing.

Notably, every explanation of observed reality that includes a preferred frame, that has been attempted so far, is demonstrably less explanatory than GR

That may be true. I really don't know. It seems that the whole universe can be viewed as an object which has a time and a place, which would imply that there is some frame in which it is, on average, stationary. The CMB seems to provide some information about that frame.

So the challenge is to find an explanation of observed reality, that explains everything that GR explains, but does so in the context of a preferred frame.

I still have not seen a mathematical demonstration that would prohibit us from regarding the frame I described above as the universal frame of our reality while continuing to use GR as our most productive model.

 

[Vorpal]

This tensor equation does allow one to examine the universe from any inertial or accelerated coordinate system, but I see no violation of the equation by discovering that there might be a best (or preferred) coordinate system for the whole universe, like the that of the CMB. Certainly, such a preferred reference frame would not be provided by the Einstein tensor equation but some other consideration or theory outside of GR. Further help with this question would be very much appreciated.

Define "best". At least in the way you've characterized previously in this thread, it seems that those other considerations are human convenience. GTR would not in any way be falsified by that sort of "best". For example, if I have a spherically symmetric spacetime, almost always it's simpler for me personally if I do calculations in coordinates that reflect that symmetry.

But so what? The mere fact that there special frame picked out by the presense of a symmetric state that makes things 'simple' is not relevant. What's important is whether this frame actually influences the dynamics. And you can't put in a preferred frame into the dynamics (the matter Lagrangian) without breaking stuff. If the frame is held fixed, then the stress-energy tensor won't have vanishing covariant derivative, but it must by the EFE since the Einstein tensor does--which is a completely geometric Bianchi identity. If it isn't, then you'll probably be getting some completely non-local physics instead.

There is a very large difference between 'the frame is tailored for some symmetry of the physical state' and 'this frame is physically important'. All the argument toward the CMB you've had so far are basically variations of the former, and none address the latter. Basically, you're conflating
1) 'Preferred' in the sense of 'picked out by actual matter configuration'
2) 'Preferred' in the sense of contributing to physical dynamics.
Does the Lagrangian for the CMB have explicit dependence on some spacetime coordinates?

 

[Perpetual Student]

Define "best". At least in the way you've characterized previously in this thread, it seems that those other considerations are human convenience. GTR would not in any way be falsified by that sort of "best". For example, if I have a spherically symmetric spacetime, almost always it's simpler for me personally if I do calculations in coordinates that reflect that symmetry.

Perhaps you are correct that "best" in the context I have in mind might be characterized as "human convenience," but I do believe it goes beyond that. Yes, it does seem clear that GR will offer no help in making any such claim. But I do not believe that's where the story necessarily ends, since the universe is a definable real object and has a time, a place and a global reality. I don't think it's any more a human convenience to give the whole universe a preferred frame than it is to say the earth is spherical. The concept of a sphere is a human construction but I see spheres as also having a reality in a platonic sense. I know many will disagree with that.

But so what? The mere fact that there special frame picked out by the presense of a symmetric state that makes things 'simple' is not relevant. What's important is whether this frame actually influences the dynamics. And you can't put in a preferred frame into the dynamics (the matter Lagrangian) without breaking stuff. If the frame is held fixed, then the stress-energy tensor won't have vanishing covariant derivative, but it must by the EFE since the Einstein tensor does--which is a completely geometric Bianchi identity. If it isn't, then you'll probably be getting some completely non-local physics instead.

There is a very large difference between 'the frame is tailored for some symmetry of the physical state' and 'this frame is physically important'. All the argument toward the CMB you've had so far are basically variations of the former, and none address the latter. Basically, you're conflating
1) 'Preferred' in the sense of 'picked out by actual matter configuration'
2) 'Preferred' in the sense of contributing to physical dynamics.
Does the Lagrangian for the CMB have explicit dependence on some spacetime coordinates?

Could you elaborate on the distinction between 1 and 2 above? So far my studies seem to confirm that my definition of "best" as described above will not break anything in GR and I see no justification of the claim that GR will not allow a preferred frame in the context I have defined. But these are provisional thoughts since I am far from done in my quest in this area. If anyone can offer any ideas or specific aspects of GR that may shed more light of this question, please let me know.

 

[theprestige]

For one model of our universe to rule out some other model it seems to me that there must be some mathematically demonstrated contradiction, otherwise both models would have a standing.

I'll let our actual physicists discuss the mathematical contradictions between frame-dependent models and GR's frame-independent model

Am I even characterizing it correctly? I certainly find this particular question interesting, and I look forward to learning more about it.

It seems that the whole universe can be viewed as an object which has a time and a place...

It does? I'm pretty sure that the conclusion we're led to, by our best and most successfully predictive explanations of observed reality, is that it seems the whole universe cannot be viewed in this way.

Would you be willing to share the details of your observations/models/experimental results that lead you to say it seems the way you say it does?

.. which would imply that there is some frame in which it is, on average, stationary. The CMB seems to provide some information about that frame.

Again, it does? Again, would you be willing to share the details of the observations/models/experiments that lead you to say so?

 

[Perpetual Student]

I'll let our actual physicists discuss the mathematical contradictions between frame-dependent models and GR's frame-independent model
Am I even characterizing it correctly? I certainly find this particular question interesting, and I look forward to learning more about it.

So far, it does not appear that there exists any such mathematical contradiction, but my knowledge is limited and I'm open to the possibility.

It does? I'm pretty sure that the conclusion we're led to, by our best and most successfully predictive explanations of observed reality, is that it seems the whole universe cannot be viewed in this way.

Would you be willing to share the details of your observations/models/experimental results that lead you to say it seems the way you say it does?


Again, it does? Again, would you be willing to share the details of the observations/models/experiments that lead you to say so?

I have no "observations/models/experimental results." I have what I believe to be a logical argument. Any particle or collection of particles can be viewed as an object. Physicists do it all the time. Is there anything about the whole universe that would make it an exception? All the theories concerning the CMB make it a universal object; why would it not give us information about the reference frame of the whole universe? I understand that such a frame could not be preferred under GR. My question is why could we not regard it as a universal frame and -- in this sense -- preferred?
I have yet to see how this would violate GR. As an analogy, Leonardo da Vinci's Mona Lisa could be deformed through an infinite number of continuous mappings -- all such alternate images would contain the same information and in this sense would be equally valid, but we naturally would prefer the original -- as a preferred image.

 

[H'ethetheth]

So far, it does not appear that there exists any such mathematical contradiction, but my knowledge is limited and I'm open to the possibility.



I have no "observations/models/experimental results." I have what I believe to be a logical argument. Any particle or collection of particles can be viewed as an object. Physicists do it all the time. Is there anything about the whole universe that would make it an exception? All the theories concerning the CMB make it a universal object; why would it not give us information about the reference frame of the whole universe? I understand that such a frame could not be preferred under GR. My question is why could we not regard it as a universal frame and -- in this sense -- preferred?
I have yet to see how this would violate GR. As an analogy, Leonardo da Vinci's Mona Lisa could be deformed through an infinite number of continuous mappings -- all such alternate images would contain the same information and in this sense would be equally valid, but we naturally would prefer the original -- as a preferred image.

I don't think there is a contradiction there. My question is: What would this accomplish? It's certainly not true that this CMB frame would produce the most elegant equations of motions in every case, nor would it be remotely practical in any case not involving intergalactic motions.
If you combine this fact with the fact that the choice of frame is fundamentally irrelevant under GR anyway, I don't see what preferring this frame is good for.

 

[Farsight]

It might be relevant to quantum mechanics and unifying it with relativity. See J.S. Bell's Concept of Local Causality by Travis Norsen. Have a look at pages 2 and 3:

"Indeed, Bell even went so far as to suggest, in response to his theorem and the relevant experimental data,17,18 the rejection of “fundamental relativity” and the return to a Lorentzian view in which there is a dynamically privileged (though probably empirically undetectable) reference frame:..."

“...I would say that the cheapest resolution is something like going back to relativity as it was before Einstein, when people like Lorentz and Poincar´e thought that there was an aether – a preferred frame of reference – but that our measuring instruments were distorted by motion in such a way that we could not detect motion through the aether."

I was really surprised when I read that.

 

[sol invictus]

PS - I've tried this analogy before without much success, but perhaps it will make more sense now that you've studied tensors a bit. Coordinates are nothing more or less than labels we use to identify points in spacetime. In a similar way, words are labels we use to identify objects and concepts. Different coordinate systems (or frames, same thing) assign different labels to the same spacetime point; different languages assign different words to the same physical object.

Is there a "best" language?

 

[dafydd]

PS - I've tried this analogy before without much success, but perhaps it will make more sense now that you've studied tensors a bit. Coordinates are nothing more or less than labels we use to identify points in spacetime. In a similar way, words are labels we use to identify objects and concepts. Different coordinate systems (or frames, same thing) assign different labels to the same spacetime point; different languages assign different words to the same physical object.

Is there a "best" language?

Welsh.

 

[theprestige]

Welsh.

Therefore it follows that the "best" reference frame is the rest frame of of Wales.

 

[Mashuna]

Therefore it follows that the "best" reference frame is the rest frame of of Wales.

It's also why if you want to translate English to German, you have to translate English to Welsh first, then Welsh to German.

 

[Perpetual Student]

I don't think there is a contradiction there. My question is: What would this accomplish? It's certainly not true that this CMB frame would produce the most elegant equations of motions in every case, nor would it be remotely practical in any case not involving intergalactic motions.
If you combine this fact with the fact that the choice of frame is fundamentally irrelevant under GR anyway, I don't see what preferring this frame is good for.

Potential technological implications aside, one can ask that question of any knowledge. What does it accomplish to know that the universe is approximately 13.75 years old? If the CMB tells us something about the rest frame of the universe, it contributes to our understanding of reality.

 

[Perpetual Student]

PS - I've tried this analogy before without much success, but perhaps it will make more sense now that you've studied tensors a bit. Coordinates are nothing more or less than labels we use to identify points in spacetime. In a similar way, words are labels we use to identify objects and concepts. Different coordinate systems (or frames, same thing) assign different labels to the same spacetime point; different languages assign different words to the same physical object.

Is there a "best" language?

Yes, that analogy is quite clear given my newly found understanding of the significance of the way tensors transform under continuous coordinate changes. I see that the EFE do provide a clear and an elegant perspective of the universe. And the experimental confirmation of GR is impressive.
My continued discomfort with this is that the EFE also seem to leave a kind of residual void in the sense that somehow the universe is without foundation. This is obviously not very scientific, but an emotional reaction -- perhaps having some philosophic implications that I don't like.
I find it very difficult to accept that the planets do not, in reality, revolve around the center of mass of the solar system, which then revolves around the center of mass of the galaxy, which revolves around some galactic center of mass, etc. etc., with the whole mass of the universe being stationary (on average) in some universal rest frame, which may be demonstrated by the rest frame of the CMB. Does that picture not resonate at all with you?
I would be interested to learn how you might respond to my analogy of the Mona Lisa, described above?

 

[Fredrik]

I would be interested to learn how you might respond to my analogy of the Mona Lisa, described above?

You say that we should prefer the original over the copies, but what if all we have are copies? (To use the language analogy: Think of reality as a chair, and the GR descriptions of it as the words for chair in many different languages. No matter what language you use, you're still just dealing with a representation of the chair, not with the actual chair). If the best theory we have doesn't say that one of copies is special, then why should we? Because our intuition tells us that one of them will be more "real" than the others? This would be a bad reason, since science has proved our intuition wrong many times.

 

[theprestige]

I find it very difficult to accept that the planets do not, in reality, revolve around the center of mass of the solar system, which then revolves around the center of mass of the galaxy, which revolves around some galactic center of mass, etc. etc., with the whole mass of the universe being stationary (on average) in some universal rest frame, which may be demonstrated by the rest frame of the CMB.

But in reality, the only time the planets actually revolve around the center of mass of the solar system is when they are considered from the rest frame of the center of mass of the solar system.

In any other frame--including the frame of the CMB, the planets are doing something quite different.