I am the first in the world

[Vorpal]

A coordinate system is a way to get coordinates.

Not so sure about that. Out of GTR and semi-Riemannian geometry Hartle's, Wald's, and O'Neill's books disagree. Out of the mathematicians' differential geometry books, Spivak's, do Carmo's, and Chern's disagree. I haven't bothered looking for more, but all of those either explicitly define coordinate charts and coordinate systems to be synonymous (e.g., Wald) or define a coordinate system as a chart that's a member of the atlas of charts, which just puts an extra differential compatibility condition that has nothing to do with measurement of lengths or angles. In other words, nothing at all like what you're doing.

If you wanted some notion of "coordinate system" distinct from charts, you really should have made your meaning explicitly clear beforehand.

I thought you might bring up coordinate charts, I was thinking I might want to head you off at the pass in one of the last posts, but decided not to. I kind of tried to by bringing up the "within a given region" phrase (basically the same idea as coordinate charts), as well as the "modulo some frame" bit so that you would not be able to bring in relativistic effects either as a point of contention. Since I was referring to only one "region", I thought you would pick up on the meaning of why geordinates could cause problems in said region, but alas, such is life.

You brought up charts, because coordinate systems are charts. If you actually wanted to talk about frame fields or tetrads, simply do so. And if that's not what you want, define your own concept.

Stop thinking in highlevel math for a minute and consider how one measures things like coordinates. Here is a metric:

ds^2 = dx^2 + x^2 dy^2.

I bet you thought I meant that x and y were Cartesian huh?

Um, no. Why would I think that if you've given me that metric?

If you take the metric I wrote above and interpret the x and y as 2-D cartesian then you would say that the space was warped, ...

Why would I say that? It's very obviously flat. The nonzero connection coefficients are Γ^x_xy = Γ^x_yx = 1/r and Γ^y_xx = -r. The curvature is identically zero. Look, I've been saying that coordinates are arbitrary labels and you expect that I would be fooled by a label? How much credit are you giving me here, exactly?

The above statement for ds is in coordinate-free language.

No it isn't. In coordinate-free language, the metric is a symmetric nondegenerate (0,2)-tensor field of constant index. That's it. In contrast, what you just said is so manifestly coordinate-dependent that I'm beginning to wonder whether you're trolling.

If you can make the metric you gave above be coordinate-free as you claim, do it, because I have no idea what you are referring to exactly. Maybe I would learn something, my guess though is that your concepts are confused and trying to put the metric you gave in coordinate-free language would show that.

Ok. Since you apparently accept that coordinate-based characterization, let's also slightly generalize and say that a spherically symmetric spacetime is one where the metric can be put in the form
[latex]$ds^2 = -e^{2\nu}dt^2 + e^{2\lambda}dr^2 + r^2(d\theta^2+\sin^2\theta d\phi^2)$[/latex]
where ν,λ are functions of t,r only. Therefore, a spacetime is spherically symmetric if, and only if, it is a product manifold of two-dimensional manifolds, Lorentzian with metric g₁ Riemannian with metric g₂ and unit Gaussian curvature, such that [latex]$g = g_1 + e^{\lambda}g_2$[/latex] where λ is an arbitrary smooth function satisfying [latex]$g_2(d\lambda) = 0$[/latex]. Easy.

I suppose if one wanted to be completely correct, one would implement the appropriate projections explicitly, but that's trivial. In any case, you asked for a coordinate-free characterization of spherical symmetry, and there it is.

I care because I love precision when it comes to thought. There was no claim by whom? When I first learned about them in a class in school the instructor literally said that r is the radius. In another class I took the claim was that r was the reduced circumference (the book "Spacetime Physics" was used).

There was no claim by anyone at least minimally competent in GTR or differential geometry that the radial coordinate necessarily represents the sort of distance that you would get if you crawled along a radial path with ruler (which is what you're measuring). Some call the Schwarzschild radial coordinate a radius, sure. So what?

(just a stupid point, it is "different kinds of radii", I suck at English some times so do not be afraid in pointing out any syntax or spelling errors on my part either)

D'oh. Well, I knew that, but English is not my native language.

... will have a GRC of r (assuming we want the GRC of that spatial slice and r is not a function of the angles), and was certainly well known before Crothers. The point was that he noticed this about the Schw. Sol. and was the first I could tell of to be annoyingly vocal about it, unless you can show me someone else who did the same.

Well, I've never heard anyone being annoyingly vocal about it. Most folks find this completely obvious after familiarizing themselves with how the line elements works. It's like decrying the phrase "square of a vector" because it really should be "norm-squared of a vector".

Look, let's take some perspective. You came into this thread not just to underline something you see as abuse of terminology, but paint it as a genuine misuse that leads to horrible errors in physics, to the effect that everyone might be mistaken in believing that there are such things as black holes. So please forgive me for feeling underwhelmed; I'm expecting some kind of genuine error that your concept fixes, rather than what amounts to a criticism of word usage.

Ughghg, by now I hope you see that coordinates in coordinate systems are not just labels. Think of them as lables when you want to state things in a coordinate-free way, such as laws of nature, generalized formulas to find length, area etc. But if coordinates are just labels, go tell an experimentalist to find what p is -- oh, and it is just a label by the way. Is that momentum, pressure, context please!

Coordinates are just labels. The experimentalist typically measures things in an ONB (orthonormal basis), which is a particular type of tetrad, and would therefore give the tetrad components of the stress-energy tensor rather than the coordinate components.

Sorry if that sounds smug, but I told you people in the field forget that coordinates in a coordinate system have an operational meaning when a specific solution is rendered, and you went and proved as much by your reply.

You really need to understand the difference between coordinates, which are arbitrary labels, and vectors, which are genuine geometric objects.

 

[tensordyne]

knowing is half the battle!

Not so sure about that. Out of GTR and semi-Riemannian geometry Hartle's, Wald's, and O'Neill's books disagree. Out of the mathematicians' differential geometry books, Spivak's, do Carmo's, and Chern's disagree. I haven't bothered looking for more, but all of those either explicitly define coordinate charts and coordinate systems to be synonymous (e.g., Wald) or define a coordinate system as a chart that's a member of the atlas of charts, which just puts an extra differential compatibility condition that has nothing to do with measurement of lengths or angles. In other words, nothing at all like what you're doing.

I know that the following quote from wikipedia is not from as august of sources as the ones you mentioned, but here is how a coordinate system is described there at least:

"In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element."

That is how I intuitively think about coordinate systems as well. A way to uniquely "lable" points (I guess we should say events if talking about GR)within a given region. What I am trying to point out is that for a given coordinate system (or chart if you care about applicability within a region), when you say a variable like x is a coordinate, the x name is not also just a lable, it has more meaning then that in certain cases. For instance, in 2D Cartesian, (x,y) means that if you go x to the right and y up you will get the point (x,y). In some cases variable names that go with coordinates have no further meaning besides allowing bijectiveness of points to "lables".

There are problems of course with just using a coordinate system in the context of GR or (psuedo-)Riemannian geometry for the whole of some space. You have to measure locally (in a region of space where the metric does not change too much, and if it does change too much, then use a smaller region where it does not).

If you wanted some notion of "coordinate system" distinct from charts, you really should have made your meaning explicitly clear beforehand.

Sorry, I was trying to keep things simple. My conception of coordinate systems is as far as I can tell, what the concensus view is, so I do not see why there is a problem.

You brought up charts, because coordinate systems are charts. If you actually wanted to talk about frame fields or tetrads, simply do so. And if that's not what you want, define your own concept.

I did not bring up charts, you did. The first post with the word chart in it was yours, not mine. I wanted to side-step the issue if I could. Alas.

Um, no. Why would I think that if you've given me that metric?

Because variable names used in conjunction with coordinates often have a specific meaning (but not always). If I give you a metric in (x,y) coordinates don't tell me that your mind does not think I am more then likely referring to Cartesian coordinates then not. Who knows though, maybe your some kind of purist or something.

Why would I say that? It's very obviously flat. The nonzero connection coefficients are Γ^x_xy = Γ^x_yx = 1/r and Γ^y_xx = -r. The curvature is identically zero. Look, I've been saying that coordinates are arbitrary labels and you expect that I would be fooled by a label? How much credit are you giving me here, exactly?

Hmm, well, I messed up and did not give a metric that was not flat. I am not sure if it is "very obviously" so, but whatever. The geometry is still 'warped' in comparison to flat 2D Euclidean for the example I gave. For the example replace the metric with one that is not flat and my argument should then work.

Coordinates do give "arbitrary" lables to points (not arbitrary with respect to a given coordinate system though, because then, what would be the point in that?), never disagreed with that. The point is that if I give you a variable, say x, and say it is part of a coordinate, then I am saying that it has a way of measuring it, at least in principle.

No it isn't. In coordinate-free language, the metric is a symmetric nondegenerate (0,2)-tensor field of constant index. That's it. In contrast, what you just said is so manifestly coordinate-dependent that I'm beginning to wonder whether you're trolling.

Well, I can assure you I am not trolling (I do not do that). In general, to do coordinate-free you have to talk about vectors and so on. Let's put it this way, the expression I gave was coordinate system independent, you can use any coordinate system you want and the form of the line element will remain the same. For a given metric in a given coordinate system, replace the metric expression for what it is in a new coordinate system and do the same for the coordinates in the differentials and the result will be identical to the original. It is invariant.

Ok. Since you apparently accept that coordinate-based characterization, let's also slightly generalize and say that a spherically symmetric spacetime is one where the metric can be put in the form
[latex]$ds^2 = -e^{2\nu}dt^2 + e^{2\lambda}dr^2 + r^2(d\theta^2+\sin^2\theta d\phi^2)$[/latex]
where ν,λ are functions of t,r only. Therefore, a spacetime is spherically symmetric if, and only if, it is a product manifold of two-dimensional manifolds, Lorentzian with metric g₁ Riemannian with metric g₂ and unit Gaussian curvature, such that [latex]$g = g_1 + e^{\lambda}g_2$[/latex] where λ is an arbitrary smooth function satisfying [latex]$g_2(d\lambda) = 0$[/latex]. Easy.

The metric you first give in the part above is in a specific coordinate system (Schw. Coordinates). The functions nu and lambda are unknown (at least up to that part of the derivation). At that point in the derivation, r as a coordinate in the coordinate system is also usually not specified.

wikipedia again:

"A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its ideas without reference to any particular coordinate system."

Since the metric you gave is in terms of a given coordinate system, how can it be coordinate-free? Also, the expression for the line element I gave works in all coordinate systems. Unless you can show me a definition somewhere that meets your criteria, I have to say it is a bust for now.

I suppose if one wanted to be completely correct, one would implement the appropriate projections explicitly, but that's trivial. In any case, you asked for a coordinate-free characterization of spherical symmetry, and there it is.

OK, see above.

There was no claim by anyone at least minimally competent in GTR or differential geometry that the radial coordinate necessarily represents the sort of distance that you would get if you crawled along a radial path with ruler (which is what you're measuring). Some call the Schwarzschild radial coordinate a radius, sure. So what?

Never said anyone competent in GR made any such claim. I said that a radius as it is conventionally understood involves the crawling you refer to above.
I said that trying to use such a coordinate in the Schw. Universe case would not be possible because it would blow up due to the origin. I said that even if you try and use a kind of offset radius you would see that the relationship between the offset radius and the r in Schw. solution is not an identity up to some constant but is of a different form then that. I said that interpreting the r in the Schw. solution as "reduced circumference", "aerial radius", the surface example you gave, are all fine, but that the best interpretation is as the GRC because the r in the Schw. sol. depends on the geometry, and those other definitions do not. That is what I said and that is 'so what?'.

I know it might be a set of subtle points, but I think they are worth making all the same.

Well, I've never heard anyone being annoyingly vocal about it. Most folks find this completely obvious after familiarizing themselves with how the line elements works. It's like decrying the phrase "square of a vector" because it really should be "norm-squared of a vector".

I guess two hear-says does not make things cancel out to be a say. I for one can tell you that I did not know that the GRC of the Schw. solution when you set t, r constant was r. Maybe that is just me though.

Look, let's take some perspective. You came into this thread not just to underline something you see as abuse of terminology, but paint it as a genuine misuse that leads to horrible errors in physics, to the effect that everyone might be mistaken in believing that there are such things as black holes. So please forgive me for feeling underwhelmed; I'm expecting some kind of genuine error that your concept fixes, rather than what amounts to a criticism of word usage.

Actually, I came on the thread because it looked like it was dead. I wanted to throw out some crazy ideas and see if that would cause the discussion to become better then what it was up until that point (which to be honest, was pretty dreadful). I had a feeling no one would respond. Oh well, you did.

As for the physics of blackholes, all I can do is be honest. I found a solution that makes me question some aspects of the current concensus. I wanted to get out there the information about this whole radii thing first because I thought it would be less contentious (and I would be able to defend it better) as a topic to start out with and you might enjoy the ideas surrounding GRC / geordinates, etc. Word usage is extremely important but concepts are even more important. The current concept about what r is in the Schw. sol is incomplete, as far as I am concerned. I have as yet not covered some of my thoughts on blackholes and why I doubt their existence.

Coordinates are just labels. The experimentalist typically measures things in an ONB (orthonormal basis), which is a particular type of tetrad, and would therefore give the tetrad components of the stress-energy tensor rather than the coordinate components.

For ****'s sake. Yes, a coordinate is just a lable. No, you should not have to make an experimenter measure coordinates in an ONB (although I am sure you are not claiming this per se). You should be able to express the stress-energy tensor in any coordinate system you want (as long as it is a reasonable coordinate system, that is also part of the whole point of GR) even if that leads to nonortho covariant and contravariant vectors. Do we really have to get into the whole topic of tensors and invariants? Covariant versus contravariant, etc. etc.

When referring to a specific coordinate in a given coordinate system there is a specific way to measure said coordinate. Variables representing such coordinates therefore have more meaning then just as variable names. (x,y) in cartesian for instance: the x is not just a variable name, it has associated with it an idea of how to find various x's. If the way you find something like a coordinate is geometry dependent, then I am calling it a geordinate and find the idea interesting. I hope that clears some things up.

You really need to understand the difference between coordinates, which are arbitrary labels, and vectors, which are genuine geometric objects.

You really need to stop telling me what I really need to understand. Unless you are a mind-reader, how do you know for sure that I conceptually do not understand something? Oh well, just found that bit kind of rude is all.

I know about Tangent Vector Spaces in case you want to bring in another topic that is off-point. I know that coordinate systems in the topic of GR can cause problems if one does not stay local and that is why we use patches or charts or tangent vectors.

I also know that vectors can be a problem if you extend them too far because they can get bent by the geometry, so to speak (that is why we use tinsy tiny vectors dx^{\mu} or Tetrads ala differential geometry or the other things mentioned thus far). I know about non-holonomic versus holonomic coordinates. I know about Killing Vectors. etc. etc. Is there something else you would like me to know about?

All the best to you all.

 

[tensordyne]

product manifold?



"Ok. Since you apparently accept that coordinate-based characterization, let's also slightly generalize and say that a spherically symmetric spacetime is one where the metric can be put in the form
[latex]$ds^2 = -e^{2\nu}dt^2 + e^{2\lambda}dr^2 + r^2(d\theta^2+\sin^2\theta d\phi^2)$[/latex]
where ν,λ are functions of t,r only. Therefore, a spacetime is spherically symmetric if, and only if, it is a product manifold of two-dimensional manifolds, Lorentzian with metric g₁ Riemannian with metric g₂ and unit Gaussian curvature, such that [latex]$g = g_1 + e^{\lambda}g_2$[/latex] where λ is an arbitrary smooth function satisfying [latex]$g_2(d\lambda) = 0$[/latex]. Easy."

I did not talk about the g part of the above and wanted to go over that. At first I did not know what you were getting at, now I do.

First off, product manifold? That is a weird way to put it. So both g_1 and
g_2 are metrics I take it, and each is (locally?) Lorentzian with unit Gaussian Curvature (why unit? constant over some surface sounds more natural). Why not just say that is the requirement for g and be done with it?

What concerns me though is the product manifold statement. I think the term is being used incorrectly. If one has manifolds X and Y, the product manifold, as I have always seen it, is (X,Y), the ordered tuple of the cartesian product of points in each (there are other considerations I suppose). The overall metric becomes

[latex]
ds^2 = g^X_{ab} dX^a dX^b + g^Y_{\mu\nu} dY^{\mu} dY^{\nu}
[/latex]

in component form and where the dX's and dY's are from the coordinates in each manifold (I hope the g's are pretty clear in this regard as well). Unless you are doing some kind of fancy math formulation or something. It all just seems very odd. I like to keep things in component form if possible, because it causes less confusion generally.

This post is just a side note. Pay it no heed if you want to just go into about coordinates being lables and all that.

 

[tensordyne]

While I am on the subject, I asked for the metric you gave to be in coordinate-free (component-free) form. You gave (from what I would guess in a correct form) a coordinate-free classification of spherically symmetric, locally lorentz metrics. That is not the same thing.

Plus, how does one interpret g_2(d\lambda)? I have seen that kind of notation used before in a book by Frankel that I think was called "The Geometry of Physics". I think you did a differential pullback or something of that nature. Oh well, if you want to make that more clear I am all ears.

 

[Reality Check]

I wanted to get out there the information about this whole radii thing first because I thought it would be less contentious (and I would be able to defend it better) as a topic to start out with and you might enjoy the ideas surrounding GRC / geordinates, etc. Word usage is extremely important but concepts are even more important. The current concept about what r is in the Schw. sol is incomplete, as far as I am concerned.

What information about the whole radii thing?
Everyone who has ever learned about the Schwarzschild metric has also learned that r is the radial coordinate, i.e. a label for a surface.
Sometimes people are lazy and when talking about the Schwarzschild metric just call r the radius.
That is quite trivial.

I have as yet not covered some of my thoughts on blackholes and why I doubt their existence.

I hope that you know about the physical evidence for black holes. For example see the On the Physical Reality of Black Holes thread.

 

[tensordyne]

Thanks for the link RC. I have not as yet put down all my thoughts on the matter so I hope that you will not hold it against me if I am reserved about that for now.

I learned the same thing as you point out. I disagree that it is trivial because the concept of a radius is different then the concept of a surface. Beyond that though, the point I have been trying to make is that the r in Schw. Sol. is geometry dependent (which is odd, because coordinates are not supposed to depend on geometry). That the r is in fact the Gaussian radius of curvature (proveably so in fact) of the surface when you set t and r constant. I have argued that this gives a better insight into the Schwarzschild Geometry itself. That is about it. Now you are up to speed I hope.

 

[Reality Check]

I learned the same thing as you point out. I disagree that it is trivial because the concept of a radius is different then the concept of a surface.

It is trivial beacuse the definition of a radial coordinate in the Schw. Sol. is that it labels the surfaces.
That has nothing to do with that the concept of a radius is different from the concept of a surface.

Beyond that though, the point I have been trying to make is that the r in Schw. Sol. is geometry dependent (which is odd, because coordinates are not supposed to depend on geometry).

I am not sure what you mean.
The Schw. Sol. has one geometry as far as I know given by the metric. Thus the r in Schw. Sol. is part of that one geometry.
If you change the geometry then you have a different solution where r may be defined differently.

ETA: Perhaps you can list a couple of the different geometries for the Schw. Sol. and how r changes between them?

 

[tensordyne]

Good, this gives me an opportunity to get some thoughts together. The triviality question is somewhat annoying but I will leave that alone since the other questions are very good.

RC, I am not sure what level of math and physics you have, which could be a problem. I will try and keep things highlevel.

A.
Lets say we have a system of coordinates that work by specifying time, two angles (polar and the other one) and a radial-like coordinate (divide the circumference by 2pi, the usual in other words). If we use this coordinate system in a Schw. universe and things are properly centered you will get that the metric is the Schw. sol. as you would expect.

B.
Now imagine we use a system of "coordinates" where time and the angles are the same as before but instead of finding r in the conventional way we define it to be the Gaussian radius of curvature for spheres centered properly on the origin. This will also work because the Gaussian radius of curvature (GRC) is the same on the whole of a sphere in Schwarzschild's solution.

Now, what is the deal then? I say B. is a better way to look at the situation as per geometry. I say this for a couple of reasons. First off, in A. we did some non-local (in the sense of having to set up rings of reduced circumferii or something) measurements to get the reduced circumference, the GRC can be measured locally, and you are supposed to measure things locally in GR if you can help it. I also say B. is better for understanding geometry, because as you noted, it comes from the derivation. You could use A. in any space because it truly is a coordinate system. B. is not a coordinate system in the way A. is, because B. works only in spherically symmetric geometries (in which case you could say A. is better for specifying events in all geometries).

B. is better in the sense of understanding the geometry of Schw. because there is a rich set of math associated with Guassian curvature. In A. you have just a set of surfaces. You can still get whatever you want by using A., but B. is richer for getting into the geometry. For me at least it provides a very illuminating picture.

Also, physicists and mathematicians have a way of thinking that the coordinate system used in the Schw. Sol. is unrelated to the geometry, but it is, since part of the derivation assumes spherical symmetry and you get a coordinate out of it that can be interpreted in a geometric instead of merely a coordinate-based way. It is not always possible to interpret coordinates in a geometric way for various geometries.

The geordinate idea: if one of your ordinates depends on what the geometry is then that ordinate is a geordinate. The whole set of ordinates would then be a geordinate system.

As for examples:

Say, for instance, you have a space that is highly warped and you use the B. geordinate system. For a given t and angles you may have several points with the same GRC. Even on the same sphere there could be points with differing Gaussian radii of curvatures.

That is about it. I am not trying to say anything out of the ordinary, just note what should be known to most people who know about Schwarschild's Solution.

 

[The Man]

Now you can calculate the minimum speed for an UFO so the Aliens will not see people on Earth only buildings!

Perhaps what they want to do is to see the people in buildings. Having no affirmative direct contact so far, one is led to the inescapable conclusion that if there are Aliens visiting they must be simply voyeuristic (and perhaps into a little S&M with the probing). Alien fetishes may not be as alien as we might expect.


ETA:

Sorry, I now return you to your previously scheduled black hole space time manifolds.

 

[Reality Check]

A.
Lets say we have a system of coordinates ...

B.
Now imagine we use a system of "coordinates"
....

That does not answer my question. All you have done is change the coordinate system which does not change the geometry AFAIK.

The question was related to my understanding (maybe wrong) that the geometry for a manifold is specified by adding a metric to the manifold. For example to get a Riemannian manifold, you add a differentiable structure to a manifold and then a metric to define lengths and angles.

You implied that the Schw. Sol. has multiple (infinite?) geometries. So I asked:
Perhaps you can list a couple of the different geometries for the Schw. Sol. and how r changes between them?

 

[Vorpal]

I know that the following quote from wikipedia is not from as august of sources as the ones you mentioned, but here is how a coordinate system is described there at least: "In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element."

That is how I intuitively think about coordinate systems as well. A way to uniquely "lable" points (I guess we should say events if talking about GR)within a given region. What I am trying to point out is that for a given coordinate system (or chart if you care about applicability within a region), when you say a variable like x is a coordinate, the x name is not also just a lable, it has more meaning then that in certain cases. For instance, in 2D Cartesian, (x,y) means that if you go x to the right and y up you will get the point (x,y). In some cases variable names that go with coordinates have no further meaning besides allowing bijectiveness of points to "lables".

That's literally true for every coordinate chart onto a convex set; e.g., in polar (r,θ), starting from (0,0) and going R along r-direction and then Θ along θ-direction, with the operations understood to keep the other coordinate fixed, just as you have in Cartesian. If you mean that "going" means a certain length, rather than in this trivial parametrization manner, then that meaning is inhereted entirely from the Euclidean metric. It's not part of what Cartesian coordinate mean; one can sensibly talk about the Schwarzschild metric in Cartesian coordinates:
[latex]$g_{00} = 1-\frac{2M}{r},\;g_{ab} = -\delta_{ab}-\frac{2M/r}{1-2M/r}\frac{x^ax^b}{r^2}$[/latex],
where a,b>0, [latex]$r^2 = (x^1)^2 + \ldots$[/latex] in the usual way and the rest of the components are zero. It can be found from the standard transformation between spherical and Cartesian coordinate.

And it's not just GTR or differential geometry. For example, Rⁿ has Cartesian coordinates as a vector space regardless of whether it is further equipped with a metric or inner product to make it properly Euclidean. Normally, this would a kind of overly pedantic point, but you're someone who repeatedly insisted on valuing exacting precision in the usage of these terms, so I'm asking for a little consistency here.

I did not bring up charts, you did. The first post with the word chart in it was yours, not mine. I wanted to side-step the issue if I could. Alas.

If you use terminology in a nonstandard way while insisting on precision, don't be too surprised at that.

Because variable names used in conjunction with coordinates often have a specific meaning (but not always). If I give you a metric in (x,y) coordinates don't tell me that your mind does not think I am more then likely referring to Cartesian coordinates then not. Who knows though, maybe your some kind of purist or something.

Sorry, but I'm telling you that my mind did not do this for the situation you gave me. If you simply say you have coordinates called x,y, then yes, I'd either assume you were talking about Cartesian coordinates, or at the very least wonder whether you were (depending on context of the discussion). But you don't tell me the names beforehand and the metric is so simple as to be seen whole at a glance, so it didn't and I don't see why it should. Yes, I realize people have notational expectations, and so do I. But it is those very expectation on that prevented me from thinking "Cartesian", simply because the form is so different from what I would expect to see. Actually, I was momentarily undecided between "polar" and "hyperbolic" (change the sign of one the terms, and it'll look like the Rindler chart of Minkowski spacetime), because those are the two closest in form, if you care to dissect thought processes further.

Coordinates do give "arbitrary" lables to points (not arbitrary with respect to a given coordinate system though, because then, what would be the point in that?), never disagreed with that. The point is that if I give you a variable, say x, and say it is part of a coordinate, then I am saying that it has a way of measuring it, at least in principle.

Ok, but that's not conforming to standard modern terminology, and having a frame field or tetrad does something analogous anyway. I don't understand why you feel the need to put the task to coordinates.

Well, I can assure you I am not trolling (I do not do that). In general, to do coordinate-free you have to talk about vectors and so on. Let's put it this way, the expression I gave was coordinate system independent, you can use any coordinate system you want and the form of the line element will remain the same. For a given metric in a given coordinate system, replace the metric expression for what it is in a new coordinate system and do the same for the coordinates in the differentials and the result will be identical to the original. It is invariant.

I said "coordinate-free language", an expression I assumed you understood because you proceeded to use it yourself several times after that. It means exactly what it says. A valid tensorial equation is of course always independent of which coordinates are used, but your equation still picks a coordinate system, even if that choice is arbitrary.

The original point of this was that you seemed to support your position by characterizing spherically symmetric geometries in terms of some generalization of spherical coordinates. I was simply pointing out that such geometrical concepts don't need coordinates at all in order to be meaningful, but rather the metric itself. You disagreed, and it balooned from there.

Since the metric you gave is in terms of a given coordinate system, how can it be coordinate-free?

Whut? No it isn't. It's in terms of of the metrics of two other manifolds, and the concept of "metric" itself is not dependent on any coordinate system. See the section you quoted just above the one you're responding to for that. The initial coordinate part was simply motivation for that characterization for spherical symmetry, because it's actually easy to see how they're related.

Also, the expression for the line element I gave works in all coordinate systems. Unless you can show me a definition somewhere that meets your criteria, I have to say it is a bust for now.

In post 51, you were talking about the "the most general metric". I gave you a coordinate-free definition in post 55 because you asked for it; you even quoted it. Well, fine. A line element is a smooth section of the square of the cotangent bundle satisfying certain properties, which you can translate yourself from the definition of metric. I won't bother because you said you weren't interested in that stuff, and frankly at this point it feels like you're just moving the goalposts.

You really need to stop telling me what I really need to understand. Unless you are a mind-reader, how do you know for sure that I conceptually do not understand something? Oh well, just found that bit kind of rude is all.

Well, the claim that ds² = g_μνdx^μdx^ν is expressed in a coordinate-independent language seemed just so completely bizarre that you either being a troll or genuinely misunderstanding the notion of "coordinate" was a reasonable conclusion.

 

[Vorpal]

First off, product manifold? That is a weird way to put it. So both g_1 and
g_2 are metrics I take it, and each is (locally?) Lorentzian with unit Gaussian Curvature (why unit? constant over some surface sounds more natural).

One if Lorentzian, the other Riemannian. They have to be fixed, so it must have the same Gaussian curvature because otherwise the product wouldn't make sense; having it be a unit is an arbitrary choice.

What's odd about? It's a sensible definition. I'm sure there are probably a dozen ways to characterize spherical symmetry without coordinates. A physicist might prefer say that it has an isometry group that contains SO(3), or something. If you like that better, feel free to consider that instead. I just used a product manifold because it has direct correspondence to the 'standard form' of a spherically symmetric metric, which is what we started with.

Why not just say that is the requirement for g and be done with it?

I'm not sure what you're referring to. I did give a requirement for g.

What concerns me though is the product manifold statement. I think the term is being used incorrectly. If one has manifolds X and Y, the product manifold, as I have always seen it, is (X,Y), the ordered tuple of the cartesian product of points in each (there are other considerations I suppose). The overall metric becomes
[latex]ds^2 = g^X_{ab} dX^a dX^b + g^Y_{\mu\nu} dY^{\mu} dY^{\nu}[/latex]
in component form and where the dX's and dY's are from the coordinates in each manifold (I hope the g's are pretty clear in this regard as well). Unless you are doing some kind of fancy math formulation or something.

I don't see the problem. Without any components, one would just say the metric is the sum of the projection-pullbacks of the two metrics, the latter scaled by that factor. Pretty standard example of a warped product.

It all just seems very odd. I like to keep things in component form if possible, because it causes less confusion generally.

Hey, you asked me to do put it in a coordinate-free language.

 

[Vorpal]

Now imagine we use a system of "coordinates" where time and the angles are the same as before but instead of finding r in the conventional way we define it to be the Gaussian radius of curvature for spheres centered properly on the origin. ...
B. is better in the sense of understanding the geometry of Schw. because there is a rich set of math associated with Guassian curvature. ...
The geordinate idea: if one of your ordinates depends on what the geometry is then that ordinate is a geordinate. The whole set of ordinates would then be a geordinate system.

Wait wait... so you're saying that using coordinates that are adapted to geometrical aspects of your manifold in some way ('geordinates') gives you an advantage in understanding the geometrical aspects of your manifold...

I'm sorry, but what you were saying about this insight not being trivial...? I think you need to go back to that part.

 

[tensordyne]

That does not answer my question. All you have done is change the coordinate system which does not change the geometry AFAIK.

The question was related to my understanding (maybe wrong) that the geometry for a manifold is specified by adding a metric to the manifold. For example to get a Riemannian manifold, you add a differentiable structure to a manifold and then a metric to define lengths and angles.

You implied that the Schw. Sol. has multiple (infinite?) geometries. So I asked:
Perhaps you can list a couple of the different geometries for the Schw. Sol. and how r changes between them?

Sorry for not answering your question.

The Schw. Sol. has only one geometry. I am not trying at all to imply it has more then one, only one. I am just trying to explain a feature about that manifold as it pertains to the Schw. Sol. That is it.

Start out with a manifold, it is in essence a shape. If you put on a coordinate system covering a portion of it the manifold and figure out how lengths work, that gives a metric. Alternatively you could just have a metric in a given coordinate system and that gives you at least part of some manifold.

The whole thing about the other manifolds was to show how a coordinate system is different then a geordinate system. That is it.

 

[Reality Check]

The Schw. Sol. has only one geometry. I am not trying at all to imply it has more then one, only one. I am just trying to explain a feature about that manifold as it pertains to the Schw. Sol. That is it.

So I conclude that your statement that

Originally Posted by tensordyne
Beyond that though, the point I have been trying to make is that the r in Schw. Sol. is geometry dependent (which is odd, because coordinates are not supposed to depend on geometry).

was incorrect since there is only one geometry in the Schw. Sol. and so its radial coordinate is r.
Of course we can look at other solutions to the EFE which can have different geometries (metrics), these metrics may have a coordinate called r and that r will in general be different from the r in the Schw. Sol. .
This is no surprise.

The whole thing about the other manifolds was to show how a coordinate system is different then a geordinate system. That is it.

Your 'geordinate system' also seems trivial. It is just the coordinate system that someone likes, i.e. a subjective decision where you like Gaussian curvature but another person might like surfaces.

 

[tensordyne]

Wait wait... so you're saying that using coordinates that are adapted to geometrical aspects of your manifold in some way ('geordinates') gives you an advantage in understanding the geometrical aspects of your manifold...

I'm sorry, but what you were saying about this insight not being trivial...? I think you need to go back to that part.

That is too bad you don't like the idea, but at least it is explained now. It helps me quite a bit in imagining Schw. Geometry, but, c'est la vie. By the way, RC said the coordinate system transforms are trivial, if I understand him correctly. Understanding aspects of geometry is always a good thing. Unless you define trivial, it is a subjective term.

Q: Is it possible to have a coordinate system that has coordinates for a specific geometry that are based on the geometry and yet still serve perfectly well as coordinates.

A (before I learned about this stuff): Hmm, not sure.

A (now): Yes.

Whatever. Your coordinate-free description is still odd to me. I asked you to give me a coordinate-free form of

[latex]
(1) ds^2 = -e^{2\nu} dt^2 + e^{2\lambda} dr^2 + r^2 d\Omega^2.
[/latex]

That is what I asked. You gave

[latex]
(2) g = g_1 + e^{\lambda} g_2
[/latex]

Along with (hopefully my memory is right on the following)

[latex]
(3) g_2(d\lambda) = 0.
[/latex]

Where is the square of the length differential in either above. I grant you, it has g's, but that is not what I asked for. I asked for (1) in coordinate-free form. That IS what I asked for, and I asked it for you because I knew it was not possible. So if you say, "oh, it is implied", wrong answer, (1) in coordinate-free form.

Now tell me, how does (2) and (3) specifically give (1)? Specifically, not generalities, specifically. My point is that if I give a specific metric, a coordinate-free anything does not make sense.

Manifold + Coordinate System = Specific Metric,

at least for how it is normally done.

(2) I interpret as a matrix equation possibly but the plus is odd and makes me doubt this interpretation, (3) I am still not sure how to interpret (sorry, you finally got me on the math on this one).

The whole point with the

[latex]
ds^2 = dx^2 + x^2 dy^2
[/latex]

thing was to show that variables (coordinate variables in this case), are not enough with just their name to understand what a metric is saying. The x,y above could be cartesian, polar, other. Without knowing how the coordinates are measured you are just doing math and not physics. I think you have said as much so I will stop on that point.

I think you might be saying that if you have a unit gaussian sphere called say

[latex]
S^2
[/latex]

and a plane

[latex]
R^2
[/latex]

then topologically at least, the space that is the result is

[latex]S^2 \times R^2[/latex]?

Or somewhere there abouts (have to change R^2 to a half-plane maybe but I think that is not correct since the origin is not compact or something).

The lambda that you use in expression (2) is a function of r which is a coordinate. Oh well, no big deal, I will let that slide.

On the

[latex]
(4) ds^2 = g_{\mu\nu} dx^{\mu} dx^{\nu}
[/latex]

what is your problem with that as far as coordinate-free goes? It meets the definition I found on wikipedia, the x coordinates can be whatever you want. It is not like I am specifying that they have to be any particular kind of coordinate system in (4) above. Heck, the above description does not even say what the dimension has to be, it should work as an expression in a whole slew of geometries (Riemannian and pseudo-Riemannian to be exact). It has components (indices). Is that the issue?

The rest of what you have written since then I see no apperent problems with or do not really care to address.

 

[tensordyne]

triviolata

So I conclude that your statement that

...

was incorrect since there is only one geometry in the Schw. Sol. and so its radial coordinate is r.

I hate to beat a dead horse, but for any given mass the Schw. geometry is uniquely defined. In this geometry, a specification for the metric has a variable in it called r that is in what are called Schwarzschild Coordinates. There are a lot of ways of interpreting said coordinate r, one of which is of a geometric nature and depends on the geometry being of a specific type (spherically symmetric) for it to work.

Of course we can look at other solutions to the EFE which can have different geometries (metrics), these metrics may have a coordinate called r and that r will in general be different from the r in the Schw. Sol. .
This is no surprise.

I am not surprised by that either, I was kind of surprised though that a coordinate could be defined by its geometry in certain cases.

Your 'geordinate system' also seems trivial. It is just the coordinate system that someone likes, i.e. a subjective decision where you like Gaussian curvature but another person might like surfaces.

That makes it arbitrary. I do not think it was necessarily trivial to realize that a geometry dependent variable can be used as a coordinate, but I will let you decide on that one for yourself.

I can say it is a trivial matter to make statements about how "trivial" something is on someone elses realization (which is a true realization by the way, note Vorpal has not complained about it).

That, truly is, trivially easy to do.

Why don't you try doing something less trivial, like, I don't know, checking whether the coordinate description I gave works. Oh yeah, that would take work and thought, and it is much easier to go on about trivialities.

Here, let me help you some, the Gaussian Curvature K is

[latex]
K = R_{1212} / g
[/latex]

where the 1 and 2 are for theta and phi in the Schwarzschild Solution (that is just how it is conventionally written based on Gauss's original work). The R is the Riemann tensor and g is the determinant of the metric in Schw. Sol. You should be able to look up everything online if you do not know what the expressions mean and do it as long as you know how to do partial derivatives and some algabra. If I am correct then

[latex]
K = 1 / r^2
[/latex]

Eh, whatever. It's trivial after all.

 

[sol invictus]

I'm sure there are probably a dozen ways to characterize spherical symmetry without coordinates. A physicist might prefer say that it has an isometry group that contains SO(3), or something.

That's how I would do it.

I don't see the problem. Without any components, one would just say the metric is the sum of the projection-pullbacks of the two metrics, the latter scaled by that factor. Pretty standard example of a warped product.

Sometimes, at least in physics, "product manifold" is used as a kind of synonym-by-default for direct product manifold, i.e. excludes warped products. But it's not precise language.

 

[sol invictus]

On the

[latex]
(4) ds^2 = g_{\mu\nu} dx^{\mu} dx^{\nu}
[/latex]

what is your problem with that as far as coordinate-free goes?

[latex]x^{\mu}[/latex] is a coordinate.

I notice you haven't responded to my observation that your "solution" isn't a solution (given the information you provided about it).

 

[Vorpal]

Where is the square of the length differential in either above. I grant you, it has g's, but that is not what I asked for.

If it has g's, you're basically done. Think of a given curve as an immersion of another one-dimensional manifold, and equip it with the pullback metric. The curve is now a manifold equipped with a metric that gives square of its length along it. Voila: the square of the length differential in coordinate-free language.

I'm not sure why you're so resistant about the fact that all of the core concepts of differential geometry are definable in a coordinate-free language. If you're not interested in that kind of thing, ok; if you don't find it useful, ok; but this insistence that the concepts need coordinates in order to be meaningful just doesn't mesh with the facts.

Now tell me, how does (2) and (3) specifically give (1)? Specifically, not generalities, specifically.

This is nuts. It's a very general statement in the first place, so of course it requires a "generality". What do you want me to do? Characterize a specific geometry in coordinate-free language? You haven't given me one in the first place! Although for any exact solution of GTR, I don't see why that wouldn't be possible in principle, whether or not it's worth doing (and sometimes requires stepping into algebraic geometry). I've seen it done for several nontrivial geometries, including Schwarzschild.

... thing was to show that variables (coordinate variables in this case), are not enough with just their name to understand what a metric is saying.

Why would I you believe I thought this? I've been saying the complete opposite for a long time now. Names are completely and utterly irrelevant mathematically or physically. They have practical value in that having common notational expectations is convenient for communication, but while that's an important concern, it's not a mathematical or physical one.

This began when I made the statement that conventionally, one would characterize spherical symmetry in terms of the existence of a coordinate chart in which the metric takes a certain form, and further said one can put this criterion into coordinate-independent language. Later, I repeated this so there wouldn't be misunderstanding:

Since you apparently accept that coordinate-based characterization [of spherical symmetry for Riemannian manifolds], let's also slightly generalize and say that a spherically symmetric spacetime is one where the metric can be put in the form
[...]
... Therefore, a spacetime is spherically symmetric if, and only if, it is a product manifold ... [(2),(3)].

The condition characterizes spherically symmetric spacetimes. However, what I did mess up on was that (1)'s λ does not correspond to (2)'s λ, and should have made it more general for clarity. Mea culpa; I didn't pay enough attention to the metric because the point was simply that it's possible to characterize spherical symmetry without any reference to coordinates, and as I said before, (1) was simply motivation as to why it would be the case.

One can fix this oversight by altering (1) to read (1') [latex]$ds^2 = Fdt^2+2Gdtdr+Hdr^2 + e^{2\lambda}(d\theta^2 + \sin^2d\phi^2)$[/latex] with every coefficient a smooth function of t,r only, as before. It's strictly more general than (1), so everything satisfying (1) automatically satisfies this criterion as well.

The lambda that you use in expression (2) is a function of r which is a coordinate. Oh well, no big deal, I will let that slide.

No, it isn't, because there is no r in (2) at all.

what is your problem with that as far as coordinate-free goes? It meets the definition I found on wikipedia, the x coordinates can be whatever you want.

It's not coordinate-free. It's coordinate-independent in just the sense you say here (the coordinates can be whatever), but not coordinate-free (because they're still there). Those meanings are distinct, and represent different approaches to geometry. If you got this idea from wikipedia, then it is simply wrong.