context is sometimes important!
Coordinates are independent of geometry.
Agreed.
Your algorithm analogy is mistaken: "a ruler here or a compass there" requires a metric. Coordinates are blind to this. With some very minimal assumptions about the manifold, they do reflect the local topology. That's about all they can do beyond being labels for points.
Disagree.
A coordinate system is a way to get coordinates. If one locally measures with a ruler, the metric does not matter (and one must locally measure in GR). Coordinates are blind to what exactly? The way in which one measures them? That does not make a lot of sense. Something else perhaps?
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? Nope, I meant for the x to act completely like r and y to act completely like [latex]\theta[/latex] in polar coordinates. Coordinates do not have any intrinsic meaning, coordinates though in a coordinate system have operational meaning. There is a fixed way in which to determine what a coordinate is for a given point for a given coordinate system and it does follow something like an algorithm (just a spatially based algorith: put your ruler here, measure that time when the light occurs, etc). Even if you just have a look-up table for points, for example, you would still have to look them up, which is an algorithm (finite number of steps, has an end, steps are definite in nature, etc. etc.).
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, but since you know that the (x,y) are really like [latex](r, \theta)[/latex] as they are conventionally understood, you know that the space is flat. That is kind of what I am getting at when it concerns Schw. Sol. My critiques are from an operational point of view. What is being measured and how is it being measured are the questions.
To clearly understand what a metric is telling you, you have to know both how the coordinates work (as per what the coordinate system is) as well as what the metric is in terms of those coordinates, when dealing with a specific case. (mentally though you could just "look" at a particular manifold to see topology and shape without caring about coordinates, but that is another story).
You're treating coordinates as if they had intrinsic meaning. If I tell you that I have a two-dimensional manifold and a coordinate chart with x,y satisfying 0<x,y<1, you would have no idea whether the manifold is infinite like a plane, or finite like a square, or anything else. Talking about size or shape doesn't even make sense with that information alone. The only thing you know is my choice of labels.
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.
In the above you did not give a coordinate system, which is the general topic I have been addressing, so while I have to agree with your conclusions in the post above, it does not change anything, or even address any issues I brought up.
Conventionally, one would say that the metric has the form [latex]$ds^2 = A(r)dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)$[/latex] in that coordinate chart. It's possible to translate that into intrinsic, coordinate-free language, but it would still depend on the metric.
The metric you listed is already specialized. I guess we could bring up spaces such as metric spaces ala topology that are even more generalized then what is used in GR, but I think we should just stick to Riemannian Geometry for now. In Riemannian Geometry of course the most general metric is (as I am sure you know already)
[latex]ds^2 = g_{\mu\nu} dx^{\mu} dx^{\nu}[/latex].
The above statement for ds is in coordinate-free language. What in the world are you referring to then? Oh yeah, in coordinate-free language the most general form of the metric is always used (as well as perhaps any tensors in coordinate-free or covariant form). Let's not get into Differential Forms either, that would just confuse things (although when I hear "coordinate-free", differential forms immediately comes to mind for some reason).
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. Sounds like fun to me either way.
A coordinate chart is just a way of mapping some points to some n-tuples of real numbers in a nice, bicontinuous way (jargon: a homeomorphism between an open set of the manifold an an open set of Rn). If you're going to a different manifold, then you are also going to have a different coordinate chart--by definition!
Yes, a coordinate chart is a region of some space with a coordinate system (or mapping, which can be thought of as a kind of coordinate system if one wants to) and some rules about how regions that share between the charts have some bijective properties in terms of various point-set operations (you are not the only one who knows some math jargon, or math for that matter). GR uses Riemannian Geometry, so the coordinate charts, beyond just being coordinate charts, should also be diffeomorphic (usually one assumes [latex]C^\infty[/latex]) or whatever. But this is getting off-topic.
You misunderstand coordinates, though that doesn't necessarily mean that "geordinates" are doomed. In order to make sense of having "the same" coordinates for different manifolds, one needs some sort of device to identify either points between those manifolds in a suitable way, or their coordinate charts directly. (A trivial example of the former kind of device would be a diffeomorphism, but that's not very interesting here because diffeomorphic spacetimes are also physically equivalent.)
The point is not to try to save geordinate systems (something that acts like a coordinate system but has one or more ordinates that depend on the geometry) as a concept since that needs no saving, but to point out that -
a. It is possible to have ordinates in the kind of charts I was relaying to you that depend on the geometry of the space and yet still give a unique set of designators for points in said chart (I only had one chart for the whole space in the example I gave, but the same principle applies).
b. That if one is not careful (uses charts, makes sure the geometry is OK for the geordinates in question, etc.), then the possibility exists that points may not be uniquely defined by said geordinates + coordinates for a given geometry.
Points a. and b. are the only ones I care to defend. I do not misunderstand coordinates, I just have an understanding I think you have probably not considered too much yet. It is worth considering because I think it will give you a better insight into GR and Differential Geometry, but, whatever. I can only show a horse the water and all that.
That's correct, and obvious from the dr² term of the Schwarzschild metric. I don't really understand why consider it to be a problem, though. The Schwarzschild geometry is spherically symmetric, and the Schwarzschild coordinate r is just a parametrization of the nested spheres (in t=const) such that they have the geometry Euclidean spheres of radius r. There was no claim that it represents distance in the first place.
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). Really, it is best understood as being the GRC as far as I am concerned.
It's far from the only such definition, even within the same Schwarzschild geometry. For example, the isotropic radius is characterized by round light-cones. The way 'radius' is used is having nested spheres satisfying such-and-such criterion. Different criteria give different kinds of radius.
(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)
The metric in Schw. Sol. is
[latex]ds^2 = (1-2M/r) dt^2 - (1-2M/r)^{-1} dr^2 - r^2 d\Omega^2[/latex]
(I prefer (1, -1, -1, -1) for my signature)
with
[latex]d\Omega^2 = d\theta^2 + \sin^2 \theta d\phi^2[/latex]
(square of solid angle line element)
If you look at the derivation for the Schw. Sol. given in books on the subject you will see that they start off with [latex](t, r, \theta, \phi)[/latex] and they really mean it as the coordinate system of 3D spherical coordinates with time added (for whatever kind of criterion you want for r as long as it meets the normal monotonically increasing condition, but I get the impression it is as one would normally imagine r with equal rulings and so on). Then they say they want spherical symmetry and so forth for the solution that is sought, which specializes the metric to
[latex]ds^2 = A(r) dt^2 - B(r) dr^2 - C(r) d\Omega^2[/latex].
At this point r still operationally means the same as the r from 3D spherical coordinate systems. Then they go, oh well, just make
[latex]C(r) = \rho^2[/latex],
but we don't like writing [latex]\rho[/latex], so lets just write [latex]\rho[/latex] as r instead, it doesn't matter, we are GR people and don't care about coordinate systems after all. That is a logical error. It implicitly says that
[latex]r = \rho[/latex], which is not the case.
How do I know? First off, note that no coordinate changes were made to t or the angles, so the argument that I should also be concerned about those coordinates when it comes to their operational meaning is invalid, as far as I can tell.
Let me just write down the Schw. Sol. without the implicit error (you shouldn't have a problem with me doing this, I am just keeping the original lable for the coordinate transform after all, and it is "just" a lable!).
[latex]
ds^2 = (1-2M/\rho) dt^2 - (1-2M/\rho)^{-1} d\rho^2 - r^2 d\Omega^2
[/latex]
Now, how does [latex]\rho[/latex] behave as a function of r? First off, there are two problems, we know that in the above metric that if t and the angles are set to constant and [latex]\rho[/latex] is integrated from 0 to whatever then the integral will diverge from below. Also, is it even the case that when r = 0 that [latex]\rho = 0[/latex] as well? That would need to be shown, or assumed, or something for goodness sakes.
For now, the best one can do is just integrate from some constant
[latex]\rho[/latex] to say [latex]\rho = \rho[/latex] and see what happens.
Please don't make me do the integral, but I can assure you that the result is something like
[latex]r = f(\rho) - C[/latex]
where f is not the identity function and C a constant of integration. This is important because
[latex]
\int dr = \int^{\rho = \rho}_{const} d\rho (1-2M/\rho)^{-1/2}
[/latex]
is the most sensible way to define the relation between r and
[latex]\rho[/latex], given the geometry and everything else. But then, I get that by understanding good old fashioned Analytic Geometry, so maybe you will have to think about that one for a bit because your head is so high in the clouds of Differential Geometry. Differential Geometry is just another tool, don't forget about your other math tools.
Actually, isotropic coordinates are good examples: if you object to using "radius" in "Schwarzschild radius", for consistency's sake, you should object to things like "angle" for the Schwarzschild φ and θ, because they do not faithfully represent angular measurements (but isotropic ones do). And what about "Schwarzschild time", since there are so many ways of measuring time?
I have heard of isotropic coordinates for Schw. Sol. before. Not sure what to say beyond general doubts. Would have to look into specifics before being able to comment.
That's not thanks to Crothers; that's thanks to Schwarzschild. Some books also define the surface of constant r (and t=const, again) as being a Euclidean sphere of area 4πr², which is so completely and obviously equivalent that Crothers didn't need to discover it.
The r as conventionally used is really [latex]\rho[/latex] and operationally means the same as the GRC, or reduced circumference, or the sphere example you gave, or the aerial whatsits, or who knows what else. The difference is that by knowing that the r that is conventionally used (my rho) is the GRC, one gains a much greater appreciation for what the geometry actually is (and isn't that the point of doing GR?). Unique shells just does not tell you as much, even if they obey some sane set of rules.
Oh yeah, any metric with
[latex]ds^2 = ... + r^2 d\Omega^2[/latex]
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.
They are just labels. If you want them to be more, you'll have to define your own concept.
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!
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.
Or is it some kind of common fallacy, or myth. Not everything in physics is coordinate-free! Sometimes you need to know how a coordinate is measured to understand a problem when a specific solution is found!
Plus, I would rather be getting into the whole blackhole thing (this is much more contentious and my thoughts on the matter are still evolving, plus, you have made some remarks I find interesting). This should have been an easy one for you to agree with, given some thought. So before you reply, please think more carefully about what I am saying. I am trying to make a point and so far you have not gotten the point. It is a basic point really, in the grand scheme of things.
OK, best of luck.
