Black holes

[Reality Check]

This is embarrassing, RC.

This is embarrassingly ignorant, Farsight
Here's the deal:
Alice and Bob see that a rock chucked into a black hole never gets to the event horizon. That is what GR states happens to any object falling toward an event horizon for any external observer.

But that is not wht I said:

Originally Posted by Reality Check
And one more time - you are wrong.
Alice is not a popsicle - she is an observer.
Bob is not a popsicle - he is an observer.
I will make this post even clearer:
Let there be an observer Alice who is using Schwarzschild coordinates.
Let there be an observer Bob who is using Kruskal–Szekeres coordinates
Neither observer is falling into the black hole. They are both observing the black hole from outside.
Alice measures that there is a singularity at the event horizon.
Bob measures that nothing special happens at the event horizon.

i.e. Bob measures that there is no singularity at the event horizon.

Also:
Alice measures the coordinate speed of light will be zero at the event horizon.
Bob measures the coordinate speed of light will be c at the event horizon.

The existence of a singularity at the event horizon is an artifact of the coordinate system selected. That is why it is called ... a coordinate singularity !

 

[Reality Check]

No. I'm saying a reference frame is an artefact of measurement.

If you want do measurements then an artifact of doing the measurements is the selection of a coordinate system to do those measurement in.
Duh!
You do not have to keep stating the obvious.

Then show it to me.

Ok: I select Kruskal–Szekeres coordinates as my frame of reference .

Unless you have the delusion that a frame of reference is a physical thing like a lump of rock. Then I cannot cater to that ignorance and "show" it to you.

I have chosen to use Kruskal–Szekeres coordinates. I observe that "the metric is perfectly well defined and non-singular at the event horizon". I observe that the coordinate speed of light is c at the event horizon.

Of course I could have chosen Schwarzschild coordinates instead. Then I would have observed that the metric is singular at the event horizon and that the coordinate speed of light is 0 there.

 

[sol invictus]

What, pray tell, do you mean by identifications on x or y?

Whether they run from -infinity to +infinity, and if instead they run over a finite range, whether some points are identical to others or not, or if some other boundary condition gets imposed. Zig gave a good example.

In case you do not, since you claim you can tell what the geometry is just given the form of a metric, here is one.

ds² = u² du² + v² dv²
So what is the geometry like in this case?

It's locally the flat Euclidean plane, as one can see immediately from a trivial coordinate transformation. What it is globally depends on coordinate ranges, identifications, boundary conditions, etc. (information which can be considered part of the metric, since it affects distances and geodesics), which you didn't specify.

 

[Farsight]

Tensordyne: your contribution is appreciated.

I'm coming late to this thread, and to all threads in which Farsight's ideas are proposed and discussed. Apologies for going over ground that has, undoubtedly, been gone over before.
What is this "hard scientific evidence"?

Essentially optical clocks losing synchronisation when separated by a vertical elevation of only a foot. I've also mentioned the GPS clock adjustment and the Shapiro delay, but they're essentially the same thing. The optical clock uses aluminium rather than caesium, and a UV frequency rather than a microwave frequency, but it works along the same lines, and employs electromagnetic phenomena. When these move at a lower rate, the clock runs slower. We use the idealised parallel-mirror light clock extensively in relativity, see for example this instance, and we know that parallel-mirror light clocks would keep time with optical clocks at different elevations. So we know that this scenario applies:

|----------------|
|----------------|

Think of the two light beams as racehorses.

...So making an allowance for sloppy language ("using the motion of light" is not quite the same as "the length of the path travelled by light in vacuum", but close enough), Farsight is right about the meter.

Thank you.

But I can't see how he can possibly be right re the second (the unit of time), even allowing for sloppy language.

I am right about it. Think it through. Frequency is defined in Hertz, which is cycles per second. You cannot say that light has any particular frequency before you've defined the second. You count how many waves come past you to define the second. When they're moving slower your second is bigger.

In any case, his stated conclusion (we "always deem the local speed of light to be 299,792,458 m/s") is correct if not well expressed).

This is why people say the speed of light is constant. What they really mean by this is the locally measured speed of light is constant. What you don't hear so much is that the coordinate speed of light varies in a non-inertial reference frame, like the room you're in. There's nothign wrong with this per se, but it misses the trick that the true speed of light varies in that room too. If it didn't, optical clocks at different elevations would stay synchronised.

that clocks clock up regular cyclic motion rather than "the flow of time", and by reading the original Einstein to understand that that gravitational time dilation is the result of a reduced rate of motion caused in turn by a concentration of energy "conditioning" the surrounding space.

I have no idea what this is supposed to mean; does anyone (even Farsight)?

Yes of course. Open up a clock. In a mechanical clock you see cogs whirring. In a quartz wristwatch you see a crystal vibrating. Pick any clock and it's the same story. You don't see time flowing through it. As for the Einstein thing, see his 1920 Leyden Address

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration".

He says matter here, but he refers to energy elsewhere, such as in the Foundation of the General Theory of Relativity where he said:

"the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy"

And what's the relevance of "the original Einstein"? I mean, he came up with GR, and published a paper or two on it. But he's not a god; his word is not inerrant.

No, but we're talking GR here, and that hard scientific evidence is backed up by what Einstein said, see this post of mine on a previous thread.

Is it just me (that does not understand what Farsight is trying to say), or has he displayed a rather gross misunderstanding of relativity?

No. GR as taught today is no longer in line with Einstein's GR.

If nothing else, Farsight seems to be trying to have his GR cake and eat it.

Relativity is the sleeping beauty of physics. I'm doing my bit to hack through the thicket. It's a save the planet thing.

gravitational time dilation" is an effect you can derive from GR, and it is unambiguous. You can do experiments to test this GR prediction, and as far as I know, every such test has produced results consistent with GR (to the experimental uncertainties).

Yep. GR is a really well-tested theory, see Clifford Will's paper. But you don't see time flowing, and you don't see time flowing slower. Have a look at A World Without Time: The Forgotten Legacy of Godel and Einstein. Then ask yourself what you can see going slower.

To understand what GR predicts concerning the observed behavior of light near black holes, you need to first understand GR (duh!). While the answers may be somewhat tricky to work out, and there will certainly be some subtleties, not least because the mathematical framework that GR is expressed in is not intuitive), they will nonetheless be unambiguous.

Yep.

[But Farsight seems to be introducing his own ideas - beyond GR - and mixing them in, without making any attempt to distinguish the two.

Nope. I'm not some "my theory" guy.

What am I missing?

The hard scientific evidence.

...And if nothing else, then where are the experimental and observational results showing inconsistency with GR?

They aren't coming from me. I'm rooting for GR.

And please note that I've mentioned vacuum impedance before now.

What is Farsight referring to?

The impedance of free space, usually written as Z₀ = √(μ₀/ε₀). It's described as a constant, but it isn't actually constant. Remember those light beams? The speed of light c = √(1/ε₀μ₀). Do your own research. Think for yourself.

 

[Farsight]

Farsight has no such results, but he's using a rhetorical tactic made famous by Michael Mozina: He's saying our mathematics may be right, but we don't understand the physics.

I'm presenting evidence for the original frozen star black hole interpretation, backed up by Einstein references and a solid argument. Clinger has no counter argument, and is trying blind you with maths and diss me instead. Anyhow, have a read of this article for a bit of history:

"Robert Oppenheimer predicted in 1939 that a supermassive star could collapse, thus forming a "frozen star" in nature, rather than just in mathematics. The collapse would seem to slow down, actually freezing in time at the point it crosses rs. The light from the star would experience a heavy redshift at rs.

Unfortunately, many physicists considered this to only be a feature of the highly symmetrical nature of the Schwartzchild metric, believing that in nature such a collapse would not actually take place due to asymmetries.

It wasn't until 1967 - nearly 50 years after the discovery of rs - that physicists Stephen Hawking and Roger Penrose showed that not only were black holes a direct result of general relativity, but also that there was no way of halting such a collapse. The discovery of pulsars supported this theory and, shortly thereafter, physicist John Wheeler coined the term "black hole" for the phenomenon in a December 29, 1967 lecture".

Note though that Ann Ewing wrote black hole down at a meeting three years before Wheeler is said to have coined the term. And note that the bottom line about what I'm saying is that Hawking was wrong. Heresy!

Gotta go. Sleep tight all. Sweet dreams.

 

[tensordyne]

Whether they run from -infinity to +infinity, and if instead they run over a finite range, whether some points are identical to others or not, or if some other boundary condition gets imposed. Zig gave a good example.

Now it seems the metric is gaining requirements for greater specificity that it did not originally have.

The claim that one can understand what the geometry of a space is like when the metric is only given abstractly is false on its face.

More assertions without evidence. I can tell you exactly what geometry the metric you posted describes.

So I gave an abstract metric. I did not describe how the coordinate variables were to be determined or thought about in any way. So here is the response to the abstract metric I gave.

It's locally the flat Euclidean plane, as one can see immediately from a trivial coordinate transformation. What it is globally depends on coordinate ranges, identifications, boundary conditions, etc. (information which can be considered part of the metric, since it affects distances and geodesics), which you didn't specify.

It is flat Euclidean eh?

I did not specify such information as listed above on purpose, as it was in keeping with my point. Look back to the counterclaim from post #324. Your response to my statement that if I gave a metric, but only abstractly gave its form, was that that is not a problem as per determining its geometry.

As per that claim and the metric I gave, the geometry for such a metric could be just about anything I want (some restrictions I suppose), as I did not specify how to go about determining what u and v are in terms of anything.

Locally about some (u,v) specified point I could make the geometry pseudo-euclidean, highly curved Euclidean, flat Euclidean, etc. etc. Hell, u and v could be complex valued (octonians anyone?), as I did not specify that either.

I hope this makes my point.

 

[DeiRenDopa]

Thanks for the answers, Farsight.

This will be a several-part response.

Tensordyne: your contribution is appreciated.

I'm coming late to this thread, and to all threads in which Farsight's ideas are proposed and discussed. Apologies for going over ground that has, undoubtedly, been gone over before.
What is this "hard scientific evidence"?

Essentially optical clocks losing synchronisation when separated by a vertical elevation of only a foot.

Let me see if I understand you correctly: the extent to which these "optical clocks" "lose synchronisation when separated by a vertical elevation of only a foot" is an effect predicted by GR, and the quantitative extent to which they (the clocks) do so (lose synchronization) is the same as predicted by GR (to within the experimental uncertainties). Is that an accurate summary?

I've also mentioned the GPS clock adjustment and the Shapiro delay, but they're essentially the same thing. The optical clock uses aluminium rather than caesium, and a UV frequency rather than a microwave frequency, but it works along the same lines, and employs electromagnetic phenomena. When these move at a lower rate,

Slow down!

When what moves? The optical clocks? The electromagnetic phenomena? A UV frequency?

And what does "at a lower rate" mean?

the clock runs slower.

I kinda wondered if this gross misunderstanding was at the root of your assertion ("It's based on the hard scientific evidence that the speed of light varies with gravitational potential" - the "it" is not important here), now you've confirmed my suspicion, thanks.

I think you need to go read a good textbook on relativity, Farsight. The only way you can tell if a 'clock runs slower' is by comparing it with another clock in the same reference frame!

We use the idealised parallel-mirror light clock extensively in relativity, see for example this instance, and we know that parallel-mirror light clocks would keep time with optical clocks at different elevations.

No doubt you are clear, in your own mind, what you intend to mean by this; sadly, this reader is confused.

A parallel-mirror light clock ('first clock') will keep time with an optical clock ('second clock') if the two are in the same reference frame. If the first and second clocks are in different reference frames, they will not, necessarily, 'keep the same time'.

But your sentence - "parallel-mirror light clocks would keep time with optical clocks at different elevations" - are you talking about two (or more?) of the first kind of clock? Two (or more?) of the second kind of clock? And how many "elevations" do you have, two? three? more??

So we know that this scenario applies:

|----------------|
|----------------|

Think of the two light beams as racehorses.

Huh?

I have no idea what you wrote is supposed to mean.

Care to try again?

But I can't see how he can possibly be right re the second (the unit of time), even allowing for sloppy language.

I am right about it. Think it through. Frequency is defined in Hertz, which is cycles per second. You cannot say that light has any particular frequency before you've defined the second. You count how many waves come past you to define the second. When they're moving slower your second is bigger.

Weird.

I could have sworn you were speaking English.

The definition of the unit of time, the second, says nothing about "light [having] any particular frequency"; it is about the frequency at which a tunable microwave cavity produces a maximum fluorescence signal.

What part of the definition do you not understand?

This is why people say the speed of light is constant. What they really mean by this is the locally measured speed of light is constant. What you don't hear so much is that the coordinate speed of light varies [...]

You know what I'm going to say next, right?

What do you mean by "the coordinate speed of light"?

There's nothign wrong with this per se, but it misses the trick that the true speed of light varies in that room too.

The "true speed of light", eh? "True".

What is this?

(to be continued)

 

[Vorpal]

Now it seems the metric is gaining requirements for greater specificity that it did not originally have.

If we're going to get all technical about it, then it most certainly did. The metric is a rank-2 tensor field, and so its domain is part of its definition just in the same way as f(x) = x² over (0,1), over the real line, and over the complex plane, are all different functions despite having the same functional form. The further specifications on the metric are no more illegitimate than asking to specify the domain/codomain information for functions.

So I gave an abstract metric.

Well, actually, you didn't. What you gave earlier was:

ds² = u² du² + v² dv²

which is not properly speaking a metric. Though it carries a lot of information nevertheless, and the first thing one would normally do is take u²(du⊗du) + v²(dv⊗dv), etc.

 

[Reality Check]

What do you mean by "the coordinate speed of light"?

I doubt that Farsight can answer this given that he thinks that coordinates have no effect on the measurement of the speed of light (he asserts that the speed of light at an event horizon is always zero).

The coordinate speed of light is the speed of light as measured in a given coordinate system. It is what an observer would deduce from measuring a coordinate distance and coordinate time and dividing the two.
For example:

• in Schwarzschild coordinates, the coordinate speed of light is zero at the event horizon of a black hole.
• in Kruskal–Szekeres coordinates, the coordinate speed of light is c at the event horizon of a black hole.

 

[tensordyne]

If we're going to get all technical about it, then it most certainly did. The metric is a rank-2 tensor field, and so its domain is part of its definition just in the same way as f(x) = x² over (0,1), over the real line, and over the complex plane, are all different functions despite having the same functional form. The further specifications on the metric are no more illegitimate than asking to specify the domain/codomain information for functions.

Fair enough, but it still does not change the point that even if I specify limits for u and v in the metric I gave but not specify how to interpret u and v, then how can one say what the geometry associated with the metric is that I gave?

If we are being specific, then both the interpretation of the coordinate variables (how such coordinates are measured), as well as their limits and possible periodicities, should be given. If such information is not given, I would argue, one really has not specified a metric in a way that allows one to think about the geometry associated with such a metric, which is the main point I have been trying to make all along (I did not discuss ranges on coordinates, but it is a good point, so I include it now [although one could say that such information is also part of how one determines a coordinate variable interpretation, among other things]).

Notice that three posters made claims about what they thought the metric I gave would imply in terms of concrete geometry (is it flat, is it Euclidean, etc), even though I did not specify in any way how to interpret the coordinates in the metric. What if u = x and v = (-1)^(1/4) y, with x and y being determined as per Cartesian coordinates, then the metric I gave would not be Euclidean, it would be like a 1 + 1 space-time. Just as equally, one could specify u and v in terms of many other possibilities, each giving a different geometry in its own right.

 

[Ziggurat]

Fair enough, but it still does not change the point that even if I specify limits for u and v in the metric I gave but not specify how to interpret u and v, then how can one say what the geometry associated with the metric is that I gave?

If we are being specific, then both the interpretation of the coordinate variables (how such coordinates are measured), as well as their limits and possible periodicities, should be given. If such information is not given, I would argue, one really has not specified a metric in a way that allows one to think about the geometry associated with such a metric, which is the main point I have been trying to make all along (I did not discuss ranges on coordinates, but it is a good point, so I include it now [although one could say that such information is also part of how one determines a coordinate variable interpretation, among other things]).

But you're wrong. Specifying the metric, the range, and any boundary conditions (like theta = 0 is the same as theta = 2pi) is in fact all that's required, because only one geometry will be possible once you do that. Additional descriptors are certainly useful, because extracting the geometry from the metric alone can sometimes be a pain in the ass, but it's not strictly necessary.

Notice that three posters made claims about what they thought the metric I gave would imply in terms of concrete geometry (is it flat, is it Euclidean, etc), even though I did not specify in any way how to interpret the coordinates in the metric. What if u = x and v = (-1)^(1/4) y, with x and y being determined as per Cartesian coordinates, then the metric I gave would not be Euclidean, it would be like a 1 + 1 space-time.

Then v is complex, and our assumptions that it was real were wrong. But then, we needed to know the range of v to begin with (that's been a requirement Sol gave from the start), and you didn't provide that. Had you provided that information, then no guidance would have been needed.

Just as equally, one could specify u and v in terms of many other possibilities, each giving a different geometry in its own right.

They can only be different geometries if u and/or v have different ranges or boundary mappings.

 

[tensordyne]

I am curious what you mean by Geometry then. As per usual in such cases I find Wikipedia most enlightening. Here is there definition of Geometry:

Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metria "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

Before I go into too much logic I would like to see if you agree with this definition in whole or in part. Specifically, if one knows a given geometry, then one knows how "questions of shape, size, relative position of figures, and the properties of space" are worked out.

I am not particularly trying to trap you or anyone else on this subject, it is just I see a bias in the thinking of people who know about GR and related subjects, a kind of abstraction bias. There is nothing wrong with abstraction (I love it myself), but visualizing concrete situations requires a more visually geometric way of thinking. That mode of thought seems to be verboten or not even consciously known about by some posters here, or, at least that is how it seems to me.

Either way, let me know Ziggurat if you find the definition of geometry good enough or not. I ask this because I do not want to play semantic games.

 

[W.D.Clinger]

I'm afraid I have to disagree with Ziggurat and with sol invictus.

They can only be different geometries if u and/or v have different ranges or boundary mappings.

Maybe, but neither of you have given a sufficiently precise definition of what you mean by boundary mappings.

Consider the cylindrical manifold I described above, in which charts f and g both use -5 < y < +5. Now reverse g's y coordinates from the mapping I described, without changing the metric form. (If I'm not mistaken, that gives you an infinitely wide Möbius strip instead of a cylinder.) Do you want that to count as a change of boundary mappings, even though (by the definition of chart) neither chart includes a boundary? (That's a real question. Your answer will give me a better idea of what you might mean by boundary conditions, and it might give you a greater appreciation for how hard it is to get these definitions right.) Now reverse g's x coordinates as well. We're back to a cylinder.

The ranges of the coordinates are arbitrary, and may be completely different in different charts that have exactly the same domain. Consider once more the original charts f and g I described above for a cylindrical manifold. Both of those charts used -5 < y < +5, but you'd get exactly the same manifold and geometry if f were to use -5 < y < +5 while g uses 230 < y < 240.

For that particular example, changing g to use 230 < y < 240 would leave the metric form unchanged. In general, however, changing the coordinate range may have to be accompanied by changes in the metric form. Different charts don't have to use the same metric form.

Both you and sol invictus appear to be assuming that all charts use the same coordinate system and metric form. That assumption is often seen in the research literature on relativity, but its ubiquity doesn't mean it's the right way to think about manifolds. When the metric form is stated in coordinate form, that metric form is chart-dependent, which is exactly the same thing as saying it's coordinate-dependent. Physicists tend to assume there's always just one set of coordinates that covers the entire manifold, but that's just as wrong as assuming the entire manifold is covered by a single chart.

tensordyne has drawn our attention to the importance of implicit assumptions and notational conventions. The moral of the story, as I see it, is that the basic definitions are important, and the informality with which these matters are often discussed can lead to ambiguity and misunderstanding, if not outright error.

 

[tensordyne]

I can not help but agree in its entirety with the post above.

 

[Ziggurat]

Maybe, but neither of you have given a sufficiently precise definition of what you mean by boundary mappings.

It's quite possible I haven't.

Consider the cylindrical manifold I described above, in which charts f and g both use -5 < y < +5. Now reverse g's y coordinates from the mapping I described, without changing the metric form. (If I'm not mistaken, that gives you an infinitely wide Möbius strip instead of a cylinder.) Do you want that to count as a change of boundary mappings, even though (by the definition of chart) neither chart includes a boundary?

Yes, I would count that as different.

Note also that even if you want to stitch the two sides together without flipping it over (so you form a tube), where you stitch them still makes a difference. That is, if y=5, x=0 touches y=-5,x=0, you will get a different tube than if you touch y=5, x=0 to y=-5, x=1. In the former case, the seam will lie along the length of the tube, in the latter case the seam will spiral around the tube, and the tube will have a fatter diameter.

The ranges of the coordinates are arbitrary

Only to a degree. I gave a simple example: identical metrics with different ranges of coordinates, one producing a plane and the other producing a cone. Some changes can be accommodated with the same metric without changing the geometry, but many cannot, and in all cases that range still carries important information.

Both you and sol invictus appear to be assuming that all charts use the same coordinate system and metric form.

No, I'm not making that assumption. But there are some things I haven't made explicit: in particular, I would include within the whole "boundary conditions" category whether or not the metric covers the entire space.

When the metric form is stated in coordinate form, that metric form is chart-dependent, which is exactly the same thing as saying it's coordinate-dependent.

Of course. But I don't think that's the point of contention here. Perhaps I'm misreading what tensordyne is trying to say, but it seems like he's taking a very extrinsic geometry viewpoint, whereas sol and I are saying that an intrinsic geometry viewpoint is actually sufficient (even if an extrinsic viewpoint is useful).

 

[tensordyne]

OK, here is what I am saying, coordinate variables are not just variables. Each coordinate variable, r, t, what have you, also comes with a method of determining it (even if it is a lookup function of points versus coordinate variable values). If I give the form of the metric in terms of coordinate variables (let's only consider one patch, seems to me most people debating on this post are aware of the relevant topological issues associated with charts, atlases, manifolds and that kind of thing, so let's just get over that now) over an open subset of a manifold, then we can talk about how the coordinate system in that patch of space works.

It is true, there is no intrinsic geometric meaning to using one coordinate system over any other in a given region of space (chart), or even one set of charts over any other system of charts. However one decides to partition up a space, one should keep in mind though that the coordinate variables are not just variables acting in a certain range.

Associated with each coordinate variable is a method of determining it, and hence of determining where a point is relative to each coordinate in a coordinate system. If I give the form of a metric in a given region of space in terms of coordinate variables and do not also specify how those coordinate variables are ultimately to be determined from the points in the space, I have not really fully specified the geometry.

The above has been my only and main point for the last couple of posts. This is something I think that sometimes is forgotten. I could say x and y are coordinate variables for a metric over the (x,y)-values such that

F(x,y) < 0.

Now the form of the metric over that region defined by the inequality above is

ds² = dx² + x² dy².

If I have properly interpreted some posts in the past, what I have given should be enough to tell us everything about the geometry (in the chart).

The only problem is, I did not say how x and y are determined from the points on the manifold yet. x and y could be determined as per Cartesian Coordinates, which would be one geometry for that form of the metric, or x and y could be determined like how Polar Coordinates are determined, which would give another geometry, or etc...

I hope this makes the point; which is really not that altogether of a complicated point. I mean seriously, if one does not specify how coordinate variables are determined from points on the manifold, then you can not even say where on a manifold some point should be given a coordinate, so how are you supposed to know what the geometry is really like?

Additionally, I should have given the form of F if I really wanted to be specific.

I agreed with W.D.Clinger's post so readily because the link it had in it was to the way that coordinate systems and manifolds interact. Good consistent use of Differential Geometry is not a bad thing either.

 

[W.D.Clinger]

Consider the cylindrical manifold I described above, in which charts f and g both use -5 < y < +5. Now reverse g's y coordinates from the mapping I described, without changing the metric form. (If I'm not mistaken, that gives you an infinitely wide Möbius strip instead of a cylinder.) Do you want that to count as a change of boundary mappings, even though (by the definition of chart) neither chart includes a boundary?

Yes, I would count that as different.

Note also that even if you want to stitch the two sides together without flipping it over (so you form a tube), where you stitch them still makes a difference. That is, if y=5, x=0 touches y=-5,x=0, you will get a different tube than if you touch y=5, x=0 to y=-5, x=1. In the former case, the seam will lie along the length of the tube, in the latter case the seam will spiral around the tube, and the tube will have a fatter diameter.

I appreciate your explanation.

Let me use this to illustrate something that's clear to you and to sol invictus, but might not be so clear to, uhm, Farsight.

Suppose the y-range of g were 230 < y < 240 instead of -5 < y < 5. Then we'd be talking about whether y=5, x=0 touches y=230, x=0. Formally, we're talking about whether g(f ^-1<0,5>) = <0, 230>.

Both you and sol invictus appear to be assuming that all charts use the same coordinate system and metric form.

No, I'm not making that assumption. But there are some things I haven't made explicit: in particular, I would include within the whole "boundary conditions" category whether or not the metric covers the entire space.

I think the word "boundary" is a wee bit confusing here because the overlap maps are defined on open sets, not boundaries.

Perhaps I'm misreading what tensordyne is trying to say, but it seems like he's taking a very extrinsic geometry viewpoint, whereas sol and I are saying that an intrinsic geometry viewpoint is actually sufficient (even if an extrinsic viewpoint is useful).

It isn't entirely clear to me whether tensordyne knows the relevant theorem, so I'll state it here. With the basic definitions I've been using,

Theorem. The full atlas of a differentiable manifold completely determines its topology.​

I thought tensordyne's point was that metric forms, by themselves, do not determine the full atlas.

That hardly matters for the Schwarzschild metric, because a single chart covers the entire manifold outside the event horizon except for a set of measure zero, and spherical symmetry takes care of that set of measure zero. I think that's where the "boundary" language comes from.

When the metric is extended down the central singularity, as with Lemaître or Painlevé-Gullstrand coordinates, people like, uhm, Farsight may get the impression that the original form of the Schwarzschild metric (to which they may have become emotionally attached) has been abandoned. Maybe it would help to reassure them that a "coordinate transformation" is just an overlap map, defined by composing charts as in the example above.

On second thought, I don't think that would help.

Seriously, these issues become pretty confusing when you're trying to relate Schwarzschild coordinates to Kruskal-Szekeres coordinates, and the most seriously confused people (such as, uhm, Stephen J Crothers), do indeed talk as though Schwarzschild and Kruskal-Szekeres coordinates are the only ones they've seen.

 

[sol invictus]

Now it seems the metric is gaining requirements for greater specificity that it did not originally have.

Nope. I specified that from the beginning.

It is flat Euclidean eh?

Yes.

Locally about some (u,v) specified point I could make the geometry pseudo-euclidean, highly curved Euclidean, flat Euclidean, etc. etc. Hell, u and v could be complex valued (octonians anyone?), as I did not specify that either.

I hope this makes my point.

It doesn't, because it's flat-out wrong. You cannot make it curved, it's flat. You cannot make it pseudo-Euclidean, it's Euclidean. If u and v are either complex or octonians, that's not a real metric and hence nonsense.

 

[sol invictus]

Both you and sol invictus appear to be assuming that all charts use the same coordinate system and metric form. That assumption is often seen in the research literature on relativity, but its ubiquity doesn't mean it's the right way to think about manifolds. When the metric form is stated in coordinate form, that metric form is chart-dependent, which is exactly the same thing as saying it's coordinate-dependent. Physicists tend to assume there's always just one set of coordinates that covers the entire manifold, but that's just as wrong as assuming the entire manifold is covered by a single chart.

Physicists never assume the coordinates cover the entire manifold - there are plenty of examples we're very familiar with where that doesn't happen. What we do tend to assume is that the metric is analytic, at least away from singularities. That allows one to extend just about any metric "maximally", thereby defining the entire manifold (that's how the Kruskal extension of the Schwarzschild coordinates works, for instance).

But none of that is necessary in tensordyne's example. In failing to specify boundary conditions/identifications, s/he hasn't fully specified the metric. If we assume u and v run from -infinity to infinity, it's fine - but we have to assume something like that to compute the geodesic distance between two points. Regardless of that, in all cases that metric is locally flat Euclidean.

tensordyne has drawn our attention to the importance of implicit assumptions and notational conventions. The moral of the story, as I see it, is that the basic definitions are important, and the informality with which these matters are often discussed can lead to ambiguity and misunderstanding, if not outright error.

Well, that part is certainly true. But it goes without saying. For instance, perhaps tensordyne speaks an alternate form of English where every word means its opposite. Assuming that isn't the case is on the same footing as assuming his use of mathematical symbols is in accord with convention.

 

[sol invictus]

Now the form of the metric over that region defined by the inequality above is

ds² = dx² + x² dy².

If I have properly interpreted some posts in the past, what I have given should be enough to tell us everything about the geometry (in the chart).

Yep - again with the caveat that you need to specify any finite ranges, identifications, or boundary conditions you're imposing.

The only problem is, I did not say how x and y are determined from the points on the manifold yet. x and y could be determined as per Cartesian Coordinates, which would be one geometry for that form of the metric, or x and y could be determined like how Polar Coordinates are determined, which would give another geometry, or etc...

Nonsense. That metric is again flat Euclidean locally, regardless of "how" the coordinates were determined.

I hope this makes the point; which is really not that altogether of a complicated point. I mean seriously, if one does not specify how coordinate variables are determined from points on the manifold, then you can not even say where on a manifold some point should be given a coordinate, so how are you supposed to know what the geometry is really like?

If the manifold comes with some special marked points - points that are identified by some non-geometric characteristic or label - then yes, you would need to specify that. But if the manifold is defined purely by its geometry, the metric contains all that information.

For instance, in the metic above, if y goes from -infinity to infinity and x goes from 0 to infinity, the space is a locally flat Euclidean "cone" with a singularity at x=0. There's only one special point, and it's the point x=0. You can determine that immediately from the metric.