General Relativity

[Roboramma]

It seems to me that the solutions to quadratic equations are simply ways in which the equation can work out to equality. If you are looking for a solution to a specific equation because the equation models some particular system that you're looking at, you know that at least one of those ways must match your actual system. In that case, it makes sense to look at the other properties of the system, that are not modeled by the equation, to see if they are consistent or not with each of the possible solutions. If they aren't consistent with a particular solution, then that solution can be discarded.

I don't see how this applies to coordinate transforms in GR.

 

[Perpetual Student]

It seems to me that the solutions to quadratic equations are simply ways in which the equation can work out to equality. If you are looking for a solution to a specific equation because the equation models some particular system that you're looking at, you know that at least one of those ways must match your actual system. In that case, it makes sense to look at the other properties of the system, that are not modeled by the equation, to see if they are consistent or not with each of the possible solutions. If they aren't consistent with a particular solution, then that solution can be discarded.

I don't see how this applies to coordinate transforms in GR.

It was intended as an analogy. Because quadratics can give us useless answers, we do not abandon quadratics and we do not accept all the useless answers.

 

[sol invictus]

It was intended as an analogy. Because quadratics can give us useless answers, we do not abandon quadratics and we do not accept all the useless answers.

But the coordinate system you mention is far from useless, especially if you happen to be in Princeton, NJ. When was the last time you took the velocity of the solar system in its orbit around the center of the Milky Way into account in calculating the time it will take you to drive to the grocery store?

 

[corbin]

I think the short answer to your conundrum, PS, is that you've run into a place where science ends and philosophy begins. There really IS no scientific reason to believe that one explanation for motion is better than any other.

I've recently become very interested in this subject myself... still need to digest the Wiki articles on it.

 

[Perpetual Student]

But the coordinate system you mention is far from useless, especially if you happen to be in Princeton, NJ. When was the last time you took the velocity of the solar system in its orbit around the center of the Milky Way into account in calculating the time it will take you to drive to the grocery store?

Consider this:
Just as √-1 is very meaningful in many contexts but useless in others, the Princeton, NJ coordinate system is useful in the context you mention but meaningless when doing cosmology. It's when we do cosmology and ask questions about the nature of the universe that we reject Princeton, NJ as being a meaningful basis for a coordinate system.
So, does it not seem that the Princeton, NJ system in the context of the whole universe is analogous to an "imaginary" answer -- as √-1 would be when we get $i100.00 as an answer for a financial question?

 

[Perpetual Student]

...
A universe that comes with its own coordinates seems to me to be quite bizarre, but YMMV.

Why?

 

[Roboramma]

Consider this:
Just as √-1 is very meaningful in many contexts but useless in others, the Princeton, NJ coordinate system is useful in the context you mention but meaningless when doing cosmology. It's when we do cosmology and ask questions about the nature of the universe that we reject Princeton, NJ as being a meaningful basis for a coordinate system.
So, does it not seem that the Princeton, NJ system in the context of the whole universe is analogous to an "imaginary" answer -- as √-1 would be when we get $i100.00 as an answer for a financial question?

The issue that I have with your analogy is that when we discard √-1 as an answer to a financial question there are valid reasons to do so. What reason do you have to discard a coordinate system in which Princeton, NJ is put at rest?

 

[theprestige]

It's when we do cosmology and ask questions about the nature of the universe that we reject Princeton, NJ as being a meaningful basis for a coordinate system.

Isn't it rather the whole point that all coordinate systems are equally meaningful? I think the real reason cosmologists don't use PNJ Coordinates is that they're not particularly convenient. But "convenient" and "meaningful" are two entirely different things.

 

[W.D.Clinger]

So, my question is, could it not be that even though GR renders all coordinate systems valid, the one where everything revolves around Princeton NJ, for example, would be rejected as a real description of the entire universe even though it might have some specific utility?

Yes, but you may not have expected me to highlight the critical word in your question. See below.

I am anticipating the response that there can be no preferred frame under GR -- end of discussion.

Although that response is correct, it should be the beginning of discussion.

In this context, "coordinate system" is synonymous with a coordinate patch or chart in the sense of differential geometry. Any such chart is just one of many possible homeomorphisms between an open subset of the spacetime manifold (or manifold with boundary) and an open subset of 4-dimensional Euclidean space (or space with boundary), regarded as Minkowski space.

The entire spacetime manifold is covered by a full atlas, which is a collection of such charts subject to a condition that says they play nicely together. (Their compositions of the form f(g^-1(x)) are diffeomorphisms, and the higher order derivatives exist also.) In general, it takes more than one chart to cover a manifold. The 2-sphere, for example, cannot be covered by a single chart.

So far as we know, a chart that's approximately at rest with respect to the cosmic microwave background radiation covers as much of the known universe as any other chart can cover.

For all I know, a chart that says the residents of Princeton are being accelerated directly upward at 9.8 m/s² may not be able to cover so much of the universe.

Locally preferred charts may run into coordinate singularities or other pathologies when you try to extend them to cover large sections of the universe. That's why, for all I know, a chart "where everything revolves around Princeton NJ" might have to "be rejected as a real description of the entire universe".

We've seen an example of that in both the Black holes and mathematics of black hole denialism threads. Schwarzschild coordinates work just fine as a static description of spacetime around an isolated star or black hole, but they run into a coordinate singularity at the event horizon of a black hole. To obtain a chart that includes the event horizon, you have to give up the illusion of staticity and use different coordinates, such as Painlevé-Gullstrand or Lemaître or Eddington-Finkelstein or Kruskal-Szekeres coordinates. Of the coordinate systems just mentioned, it is my understanding that Kruskal-Szekeres coordinates are the only ones that can describe the largest possible spacetime manifold that contains an isolated black hole and satisfies Einstein's field equations.

That gives you an example of why some coordinate systems must be "rejected as a real description of the entire universe", even though those coordinate systems make perfect sense (and may even be preferred!) when describing some small piece of the universe.

 

[Perpetual Student]

Thanks for all the thoughtful responses. Here's another angle:

Earlier in this thread there was a lot of discussion about using the rest frame of the CMB as a basis for establishing the "real" reference frame of the universe. That idea was roundly rejected because it would violate GR. Now that's an interesting expression: "violating" GR.
Would we say that because we reject $i100 as an answer to a financial question that we are "violating" quadratic equations? Why is the one example a "violation" and the other simply an obvious and practical decision? The CMB is the largest and most pervasive thing we know of in the universe. The CMB has a crucial historical significance for the universe. Why would we be "violating" some equation if we make the practical and obvious decision that the CMB tells us what the actual frame of reference of the whole universe really is, while Princeton, NJ has some value only in some very local and very specific analysis?

 

[sol invictus]

Consider this:
Just as √-1 is very meaningful in many contexts but useless in others, the Princeton, NJ coordinate system is useful in the context you mention but meaningless when doing cosmology. It's when we do cosmology and ask questions about the nature of the universe that we reject Princeton, NJ as being a meaningful basis for a coordinate system.
So, does it not seem that the Princeton, NJ system in the context of the whole universe is analogous to an "imaginary" answer -- as √-1 would be when we get $i100.00 as an answer for a financial question?

There are many problems with that. The biggest one is that an imaginary or complex amount of money may simply not make sense at all. But the coordinate system based on Princeton does make sense cosmologically - it's just extremely inconvenient. There's no bright line you can draw with coordinate systems - they are related to each other by continuous transformations, so how do you decide where to put the boundary between "meaningful for cosmology" and "not meaningful for cosmology"?

Why?

My answer is that coordinates are a human convention. A universe endowed with its own special coordinates would have to one with some sort of underlying "grid" built into its fundamental structure, a grid with a particular shape (Cartesian versus polar, for example, or square versus hexagonal versus...).

Such a thing is possible, but it's very bizarre. It stinks of the worst kind of "looking under the lamp-post" logic - it's like believing that English is the language of the universe.

Thanks for all the thoughtful responses. Here's another angle:

Earlier in this thread there was a lot of discussion about using the rest frame of the CMB as a basis for establishing the "real" reference frame of the universe. That idea was roundly rejected because it would violate GR. Now that's an interesting expression: "violating" GR.
Would we say that because we reject $i100 as an answer to a financial question that we are "violating" quadratic equations? Why is the one example a "violation" and the other simply an obvious and practical decision? The CMB is the largest and most pervasive thing we know of in the universe. The CMB has a crucial historical significance for the universe. Why would we be "violating" some equation if we make the practical and obvious decision that the CMB tells us what the actual frame of reference of the whole universe really is, while Princeton, NJ has some value only in some very local and very specific analysis?

It's because coordinate invariance is the single most basic and fundamental ingredient of general relativity. By itself it defines the theory almost uniquely. Understanding the constraints coordinate invariance imposes is what allowed Einstein to formulate the theory in the first place. It's analogous to what is sometimes called "gauge invariance", if that means anything to you - the freedom to change coordinate reflects a redundancy in our preferred description of the system, not a fact about physics or reality. Choosing to call that round red object "apple" versus "pomme" doesn't change anything about the object.

To abandon that is an absolutely radical change in the basic structure. Such a change to a very successful theory would require compelling justification.

 

[Perpetual Student]

... coordinate systems ... are related to each other by continuous transformations...

That is, perhaps, the most persuasive point of all.
However, for this layman, just as we have a grounding for time (with the universe some 13.75 years old), I would like to think of space as also having some grounding. So looking at the infinity of coordinate systems that are related by continuous transformations, I can argue that the one with the simplest description of the universe is the preferred one. It is likely that it would be one, or close to one, with the CMB at rest. It may not currently have much of a scientific basis, but it makes good common sense.
Here's another consideration: I'm not sure what this means, but I have seen the universe described as either infinite and bounded or finite. In either case, there should be no difficulty in concluding it has a center, unless some geometric rationale prohibits it -- like being on the surface of the analog of a sphere in four dimensions. Does GR necessarily give us such a geometry?

 

[theprestige]

That is, perhaps, the most persuasive point of all.

But you remain unpersuaded, right?

However, for this layman, just as we have a grounding for time (with the universe some 13.75 years old), I would like to think of space as also having some grounding.

As a layman looking for spatial grounding, you could do a lot worse than a coordinate system centered on your town square (i.e., a localized version of your PNJ coordinates).

For this layman, I'm having a hard time understanding why cosmic space needs "some grounding", or why it would matter to me from a layman's perspective if it didn't have it.

So looking at the infinity of coordinate systems that are related by continuous transformations, I can argue that the one with the simplest description of the universe is the preferred one.

And if there isn't a single coordinate system that gives a complete "description of the universe" (whatever that means)?

What if the idea of "simplest description" depends on what part of the universe you're looking at, and which questions you're trying to answer?

If the whole point is that cosmologists can freely switch from one coordinate system to another, whenever it is convenient to do so, why would they bother trying to define one as "preferred"? Wouldn't it make more sense to simply develop an appreciation for the capabilities of each one, and simply prefer whichever one is most convenient for the question at hand?

It is likely that it would be one, or close to one, with the CMB at rest.

Even if a coordinate system at rest with regard to the CMB were somehow "preferred", what would that actually mean to cosmologists? We already know that GR provides a description of the universe that is independent of any particular coordinate system. So "preferring" CMB coordinates wouldn't really be helpful. Indeed, such a preference would end up being an unnecessary entity that would have to promptly be discarded as soon as it was acknowledged.

And cosmologists would be discarding it anyway, whenever some other system was more convenient to their inquiries.

So from the perspective of GR it can't truly be "preferred", and from the perspective of practical applications it pretty much won't be preferred.

It may not currently have much of a scientific basis, but it makes good common sense.

It's been this layman's experience that appeals to "good common sense" is counter-productive to establishing a good scientific basis for anything. Besides, to the extent that "common sense" is relevant, it suggests that cosmologists have no need of a preferred coordinate system, and would ignore one half the time anyway.

 

[drgsrinivas]

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?

If you believe in commonsense and logic, you will immediately throw away your Relativity.

If you believe in Relativity, then don't bother about logic and rationality.

When you don't have to bother about logic, then you don't have to bother about the experimental evidence also- You can (mis)interpret any data/ observation as strongly supportive of your belief system because you don't have to be logical while interpreting.

For example, if someone believes that ants are the biggest enemies to mankind, then every movement of every ant may be interpreted as part of an organised coup against the humans.

What is important is your belief- Do you believe in Logic or Relativity? Identify your belief and stick to the same. Two opposite religions can't go together.

You can't expect things to be logical at one time and ignore the same logic at another time as per your convenience.

www.debunkingrelativity.wordpress.com

 

[sol invictus]

Here's another consideration: I'm not sure what this means, but I have seen the universe described as either infinite and bounded or finite. In either case, there should be no difficulty in concluding it has a center, unless some geometric rationale prohibits it -- like being on the surface of the analog of a sphere in four dimensions. Does GR necessarily give us such a geometry?

All the evidence we have is consistent with the universe not having a center. None of the standard cosmologies have either a boundary or a center; instead, they are homogeneous (every point is identical to every other point, which precludes both centers and boundaries) and isotropic (all directions are equivalent).

 

[Perpetual Student]

If you believe in commonsense and logic, you will immediately throw away your Relativity.

If you believe in Relativity, then don't bother about logic and rationality.

When you don't have to bother about logic, then you don't have to bother about the experimental evidence also- You can (mis)interpret any data/ observation as strongly supportive of your belief system because you don't have to be logical while interpreting.

For example, if someone believes that ants are the biggest enemies to mankind, then every movement of every ant may be interpreted as part of an organised coup against the humans.

What is important is your belief- Do you believe in Logic or Relativity? Identify your belief and stick to the same. Two opposite religions can't go together.

You can't expect things to be logical at one time and ignore the same logic at another time as per your convenience.

www.debunkingrelativity.wordpress.com

Nonsense! All science is based on logic and logic is what resulted in the discovery of relativity and continues to drive its development. Many aspects of relativity and quantum theory are counter-intuitive but that does not mean that they defy logic.

 

[Ziggurat]

If you believe in commonsense and logic, you will immediately throw away your Relativity.

If you believe in Relativity, then don't bother about logic and rationality.

Edited by kmortis: 

removed personal comment

 

[edd]

http://abstrusegoose.com/449

 

[Perpetual Student]

So, it seems, based on the physicists who have respond here (thanks to all), that I've traveled on a road leading to another dead end. We spend all our lives (at least, I have) with an encompassing feeling that we are in some "place." This "place" leads me to think in terms of my day-to-day coordinate system. I am here and stationary; my wife is in the office; my grandchildren are 120 miles to the west, etc. But I know that is a narrow and unrealistic perspective. I am also rotating, revolving and spiraling in some complex dance, when compared to the CMB.
Nevertheless, my "intellectual" big picture perspective would like the universe to be a well defined "place" in the same way. The vast structures we see shaping the observable universe give me a sense of place in the universe, but GR and the impossibility of a preferred coordinate system seems to take some of that away -- and I find that disturbing. If I did not have such a strong dedication to science and the scientific method, it would be tempting for me to give in to the dark side and take up some crackpot anti-relativity belief system.

 

[Ziggurat]

Nevertheless, my "intellectual" big picture perspective would like the universe to be a well defined "place" in the same way. The vast structures we see shaping the observable universe give me a sense of place in the universe, but GR and the impossibility of a preferred coordinate system seems to take some of that away -- and I find that disturbing. If I did not have such a strong dedication to science and the scientific method, it would be tempting for me to give in to the dark side and take up some crackpot anti-relativity belief system.

If you relax what you mean by "preferred", then GR actually does provide something that may suit your purposes, and you actually alluded to it. And that's the co-moving reference frame of the universe, which we can observe by watching the CMB. This reference frame isn't preferred in the sense that the laws of physics are any different in this frame from any other frame. But it is still a unique reference frame in terms of a number of observable details of the universe, such as the CMB being essentially isotropic. In more tangible terms than the CMB, the co-moving reference frame is the reference frame in which mater is (on average) stationary within the universe. Local measurements can't distinguish this reference frame from any other reference frame (so again, the laws of physics are no different), but we're not confined to local measurements, and large-scale measurements (like the CMB) are sensitive to it. So if you want a reference frame on which you can hang a sense of place without everything becoming seemingly completely arbitrary, well, the co-moving reference frame can serve that purpose perfectly well.