Debate: Do black holes exist?

No, Dinosaur, I knew exactly what you meant. On the surface you appeared to be telling your friend not to waste time on a fool, when you were actually just saving him from embarrassment and raising the white flag. Around the same time, one of you sent a PM to AlphaNumeric who ran through the thread, cherry-picking any phrase he could retort. Like I said, I'm flattered that you asked for AN's help, and I appreciate it, too, because AN is well-educated. If there is a problem with my BH analysis, and AN can show this to me, I will have learned something and I'll admit it.

I expect none of you to be capable of the same.

 

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No one sent me a PM. No one gives that much of a hoot about you. You aren't scaring people, you're not even offering particularly interesting discussion.

Ah, now its clear. You're just inventing an imaginary world to stroke your ego. No one considers you threatening. In fact, having read your reply to my reply I consider you woefully ignorant on relevant things and a little deluded about your ability. You used to be like this but I thought you'd improved. Didn't you learn anything from when you first joined here and got slapped around?

What analysis? You pulled one equation from somewhere, which was wrong and then you demonstrated you don't understand even the most basic concepts in vector calculus, never mind have a working grasp of GR. You obviously didn't understand the things I linked to. Are you claiming you did? Have a look on ArXiv for papers on black holes and you'll see a lot more mathematics. Your 'analysis' involved a single incorrect equation. That isn't an analysis, its a laughable attempt to come off as more knowledgeable than you are. If you think I'm wrong in my assessment of you then please increase the level of specific mathematical detail in your posts so we can have a precise and detailed discussion which isn't tainted by arm waving. If you can't present an argument which involves quantitative methods don't bother.

I'm not going to feed your delusions of understanding GR, either you demonstrate you know some GR or you get labelled (again) as yet another deluded hack.

I think my hypocrisy meter just exploded

I don't doubt that....

uiThere's your first problem. I am questioning the existence of the BH while you are PRESUMING it and forcing the math to fit. [/quote]No, the maths isn't changed to fit. There's a unique form for the metric. Birkhoff's theorem says the black hole which has no charge or spin is only parameterised by its mass. At most you have 3 parameters, mass, angular momentum and charge. There's no room to 'fiddle' things.

Where did I or anyone else 'switch metrics'? The same metric is used throughout. You seem to be under the false impression that changing coordinates is changing the metric, which is a very very basic misunderstanding. This isn't a short coming in GR understanding, its a short coming in physics 101 understanding. You accuse Ben of having trouble admitting his limits but you're doing even worse. You're now trying to act as if you have a grasp of what is going on but you don't. There's no metric changing, only rewriting the same metric in different coordinates. This is perfectly valid. It doesn't mean there's some 'sleight [sic] of hand', even flat space needs it sometimes (polar coordinates are not valid everywhere) when you are in regions where the coordinate choice is a bad one. The event horizon is not a problem mathematically, it only seems that way if all you know is the standard spherical coordinate formulation. Go into Eddington-Finklestein or Kruskal coordinates and the event horizon is not a problem. It is a coordinate singularity, not a physical one. You can see this by computing coordinate independent quantities and seeing they are never invalid on the event horizon. Well I say 'you' but obviously you can't.

So because you don't understand the role coordinate choices play in any kind of differential geometry (even things like polar coordinates in flat space or latitude/longitude on a globe have issues) you don't understand the methods used in GR and you obviously don't understand the pdfs I linked to (come on, be man enough to say it and stop being a hypocrite by saying Ben can't admit his mistakes). You clearly don't understand 'the math problem', as its clear the problem is not with the maths but with your grasp of the maths. The lecture notes I linked to aren't something I wrote, they are notes I transcribed from a lecture course I attended. The guy who did the course is an ex-PhD student of Hawking and now works along side him. If you think you have some argument against the material I've provided then you're arguing against the entire GR community and I suggest you try submitting your 'argument' to a journal if you think Ben and I are not up the the task.

 

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I am the student, please teach me. I defer to you.

Also, when I make errors please correct them so that I may learn more.

 

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Just to make it clear, you're now admitting to not understanding this stuff? Or do you think you do and you're asking me to demonstrate that I do and that you're incorrect?

Was there some reason you haven't replied to anything I actually said? I'll make it easier for you, please answer these direct questions.

Do you understand that black hole metrics are very specific, that you can't just make up any old metric for them? Do you understand that event horizons are not physical singularities? Do you understand that changing coordinates doesn't change the underlying metric, that it is nothing more than a different description of the same thing? Do you understand that spherical polar coordinates are a bad arbitrary choice of how to describe a black hole space-time? Do you understand what \(\kappa\) is?

If you don't understand or don't agree with any of these please elaborate on what you don't get or what you don't agree with and why.

As I just said and as I said in my previous post, there's a very short list of metrics which represent a black hole. In fact, in 3+1 dimensional space-time there's only a 3 parameter family. The most general metric is the Kerr-Newman metric. Set \(\alpha=Q=0\) and you get the Schwarzchild metric. If you consider only 3+1 dimensional asymptotically flat space-time then by considering that KN metric you consider every possible black hole configuration which can exist in such a space-time.

The fact you're unaware of this hardly does your claims about black holes and GR any favours, as it means you're ignorant of a very simple qualitative result which can be found easily by reading anything about black holes which isn't pop science. Compounded by the fact you don't understand the difference between changing coordinates and changing metrics I'm wondering if you did any reading before asserting your claims.

From an observer A at \(r_{1}>2M\) it will seem as if observer B at \(r_{2} \in (2M,r_{1})\) will be acting slower and always above the event horizon at 2M. However, the 'image' becomes weaker and weaker as less and less photons go from B to A and they are more and more redshifted. In the absence of quantum mechanics this could continue indefinitely. If you put in quantum mechanics then you have black hole thermodynamics, the black hole emits radiation. Any stellar sized black hole will have a very very very low temperature but as time passes the universe will cool, get colder than the black hole and then the radiation will slowly drain the black hole of mass, heating up as it does so. Eventually the combined effect of A's 'image' getting fainter and fainter and colder and colder (red shifting is cooling in regards to thermal spectra) will mean the image is lost in the background glow of the black hole, just as a candle's light would be lost in the glow of the Sun. Time progresses, the black hole heats up more and goes out with a huge flash of gamma rays. You can thus view A as either having been eaten by the black hole and then converted into thermal emissions or as never having got to the event horizon but been vaporised by the emissions. Either way the dynamics and outcomes are the same.

The point of view of a distant observer can be formulated in terms of spherical coordinates, as they are unable to view anything on or in the event horizon. All visible objects move along continuous paths in a consistent manner. From the point of view of the in falling person spherical coordinates are not a good choice for r=2M in the same way that 90 degrees latitude is not a good map coordinate. In order to have a description which allows you to describe smoothly how something goes from above the event horizon to below it you just pick a clever choice of coordinates. Such as Kruskal coordinates or Eddington-Finkelstein coordinates which are valid in the region above, on and under the event horizon. The reason they aren't used all the time is they are not very convenient if you're only considering things above the event horizon and they aren't very physically intuitive. But there's no change of metric, only a rewriting, in the same way polar coordinates and Cartesian coordinates are both ways to describe a flat plane. And even in that case r=0 is a coordinate problem as it makes \(\theta\) undefined.

As in your original post you say

According to whom does an infalling mass move at a speed approaching c? A distance observe never sees the speed approach c and the in falling person always 'sees' light over taking him. At no point does anyone measure themselves to be moving faster than light nor does anyone else see the in falling person overtake light.

This is something which you can see if you know how to do the quantitative stuff. Qualitative understanding only goes so far before you're utterly wrong and that point happens a lot lot sooner for stuff like GR or QFT than it does for high school physics. Just because physics taught in high school seems to be agreeing with intuition doesn't mean the actual physics models you're spoon fed a kiddy version of are intuitive.

 

Yes, I know nothing; if I'm ever right it's dumb luck, like The Pelican Brief. I'll say whatever it takes for you to help me understand the problem in the logic of my OP analysis, if it exists.

It's because your fucking novellas exhaust me. My average post is probably 5 lines; I'll bet yours is 40. I am making a sincere effort to address your entire post on this reply, though.

1. Yes; I erred when I used the term. I meant coordinate system and I apologize for the confusion.
2. Yes, I know what a singularity is; indeterminate/infinite values do not necessitate a singularity do they? (sincere question)
3. Yes and no. I get the concept, but are Kruskal coordinates, for example, defined at A? From what I've seen, shifting to Kruskal just moves the area of contention away from the EH to the infinite observer, not eliminating the problem. This is what I was referring to in post 59 which might be clearer now that you know I meant to type "coordinate system".
4. Yes, I get the concept but I still believe you cannot make the problem of indeterminability go away by switching between coordinate systems.

As an aside, the following is the reason I was pushing Ben's buttons:

Ben's condescending dismissal pissed me off in the Higgs thread. And didn't even have the balls to put his money where his mouth is. Furthermore, he came to this thread with the following tripe:

Fine...I showed him some calculations in post 28. Here is his response:

More condescension, even when it is clear that he's the one mistaken on this point. I asked him to give me a frame, with calculations, that would allow an observer to see A cross over the EH. His response?

So...what he demands from me (which I attempted to provide) he does not feel obligated to do in return. This is, of course, because no such frame exists, but his insistence that I am the one that "just doesn't know" what I'm talking about is maddening. I don't know everything, but here he is demanding that I'm wrong by using logic that I DO KNOW is wrong. OF COURSE I don't claim to have the answers, that's why I posted the OP in the first place! If I knew for a fact that I was right, would I more likely come here to get abused or write a paper on it?

By the way, BenTheMan, if you're reading this I really apologize for letting my maturity level fall off a bit in this thread. I made a comment about you not having common sense and it was uncalled for, which seemed to doom further civil discussion. Anyway, enough about Ben.

This is going off on a tangent, a bit, but I answered this in my statement: the mass's material can be calculated to experience a velocity...meaning the mass would experience accelerative forces approaching infinity as I outlined in post 28. You say (or someone did, maybe Ben?) that this formula doesn't work. Fine, forget it. I won't even ask what the appropriate formula is. Moot point, not needed to answer the OP. I just didn't want you claiming I didn't answer all of your questions.

I see my description as a pithy version of yours. My point is, I don't believe we are disagreeing on anything qualitatively.
Furthermore, I say:

If we both concur that no frame would see nor calculate B crossing the EH, then maybe you would agree with my logic above. It is so simple to me, if A calculates that B takes an infinite time to cross the EH, yet the BH would evaporate in finite time, then B never makes it across the EH. Using a "mathematical induction"-type reasoning, the same could be said of all material, hence the EH and therefore BH never come to fruition. This is the basis of my question...

 

The less mistakes people make the shorter my posts.

I apologise for being bothered enough to explain myself.

Yes, Kruskal coordinates don't allow for a modelling of 'spacial infinity', just like spherical coordinates don't allow for a modelling of the event horizon. This is not a sign of some problem or a slight of hand or anything, its a fundamental property of non-trivial manifolds. Some spaces (by which I mean mathematical spaces, not just space-time) do not admit a global set of coordinates due to non-triviality in their structure. A Mobius strip for instance requires at least two coordinate charts. As does a sphere. If you don't use more than one chart you get stuck dealing with coordinate singularities, where your chart fails. This isn't a problem because the very definition of a manifold means that it must allow for a consistent patching of charts, so that in areas where they overlap you can change from one to the other and then continue. Spherical coordinates are good outside the event horizon, Kruskal are good near, on and just under it and then spherical coordinates are good again. The required structure for this to be done consistently is built into differential geometry (all differential geometry, not just that used by GR) at the most basic level. Every GR book opens with sentences like "Let space-time be a manifold M which....". Once they say 'manifold' they are able to transform between coordinate charts without worry.

You obviously don't get the concept as you just contradicted the concept in the very same sentence. You've not had any first hand experience doing this sort of stuff so you don't realise just how its done. Do you think rotating coordinate axes is dubious? They are a change of chart. How about translations?

Read section 1.2 of this. I know you won't follow the specifics but maybe you'll get the concept.

At no point does the infalling object overtake light. At no point does it experience infinite force. It doesn't travel at light speed when it enters the event horizon.

Its common to think in Newtonian terms, in that if light cannot escape an event horizon then surely anything falling into it will hit light speed at it. The reason light can't escape is space-time curvature is too high, which is different from the acceleration felt. The 'surface gravity' of a Schwarzchild black hole of mass M is \(\kappa = \frac{1}{4M}\), so it gets weaker as the mass increases (due to the way the radius scales like \(R \propto M\) rather than \(R \propto \sqrt[3]{M}\) as you'd expect for an normal object of constant density. The very fact its dependent on mass at all means that the fact light can't escape has little or nothing to do with forces per-say.

I provided you with the formula. You even commented on something which made it clear you'd looked at the link.

There's more to physics than qualitative understanding.

If no event horizon forms then there wouldn't be a specific distance below which no object is observed. The fact A never sees B get to r=2M is evidence of an EH. Yes, technically the forming of the EH isn't seen directly by A because the time dilation means it takes forever before A 'sees' a fully formed black hole but A can still deduce it is there. One way would be to compute various orbital mechanics and deduce its too dense to be anything else but that assumes GR is something A trusts. Alternatively A can notice that no object ever gets closer to the centre than a specific value (which A can work out to be around r=2M, as A can work out the mass from orbital mechanics again, thus testing GR's predictions). A sees B falling in but B slows and slows such that he is never below r=2M, no matter how long A waits. That and the fact the effect blocks the light coming from stars at the other side of the sky demonstrates light can't get out either.

Yes, those aren't direct observations of a fully formed black hole but you can infer all the relevant properties and directly observe how light can't go through it. If you're next to a BH with a ship you can do things like test the Penrose mechanism and other things GR says should be going on.

 

didn't i post something like that in the religious forum?(physics/God)..lol

you guys are great..i've been reading along and i understand a few more things about black holes.

makes me understand why my 'black hole at the center of the planet' idea is only for hollywood..

on the one point;
so matter/information could still exist at a point between the event horizon and the center ?

on another;
the pics i have seen from discover magazine, showing the stars orbiting around the black hole at the center of our galaxy,they show no stationary stars.


also remember guys..
the debate is not for the debaters..

 

Au contraire, I am suggesting that it is a sign of a problem. The common response to pointing out that things are ill-defined at the EH is that, with the proper coordinate system, there is no problem. This is like claiming a bubble under Saran Wrap does not exist because we can always scoot it somewhere else. Black holes are fascinating mathematical structures, and I'm not minimizing the work of people that spend their lives studying them, I am just suggesting they are non-physical.

You don't need to tell me reasons why we think BHs exist via observation. My OP stated:

The "black hole region" would behave almost identically to what we call a black hole. It's the difference between asymptotically approaching the EH, but never reaching it, vs blasting through and causing the EH to expand.

How about this...True or false: Back-reaction only occurs as mass is crossing the EH, not before. Think about this, give me your answer, and I believe the rest will be easy.

 

The issue of not being able to globally define a set of valid coordinates has nothing to do with the physics of a black hole, it is because the black hole leads to a non-trivlal manifold structure to space-time. Precisely the same thing happens for circles and spheres of any dimension. For instance, if you use an angle to define a position on a circle then a some point you have a jump in your coordinate, like going from \(\theta = 2\pi\) to \(\theta = 0\).

Manifolds are structures which look locally like Euclidean (flat) space. Globally they can look completely different. This local property means that locally you can put a coordinate chart on it. The Earth is a sphere but if you're only considering a map of a city then you can use Cartesian coordinates, yet you can't use Cartesian coordinates for the entire Earth (you get that funny warping thing on world maps where Greenland is larger than Europe when really it isn't).

By your logic considering the Earth as a sphere is not valid because you can't give a a single set of coordinates which are valid everywhere. This issue isn't a problem, in fact its a very powerful mathematically tool because it leads to a number of important physical things like the Aharonov-Bohm effect. Topological defects play a huge role in quantum field theory and have applications in some developing technology to do with quantum computers, superconductors, superfluids and nanotechnology.

This basic mathematical property of manifolds and the fact it appears everywhere in physics should be known to you but instead of doing some reading and finding out whether or not you grasp the topic you're simply asserting things you don't know anything about.

Back reactions happen everywhere. A back reaction is when the presence of an object in a background gravitational field causes the gravitational field to change. Generally these effects are so small they can be ignored, as a satellite's mass isn't going to alter the gravitational field of the Sun enough to be worth worrying about. Hence why gravity books talk about 'test masses', where you simply work out geodesics of the original space-time and have the object move along them. If the object is large enough then it'll warp the space-time in a significant way and you get effects you can't ignore.

Two black holes spirally into one another is the ultimate example, as they are significant masses and alter the gravitational field in a way you just can't ignore. It was only in the last few years that numerical methods were developed to do this complicated GR problem, as you couldn't use normal methods. The back reaction happens all the time and once the black holes get close enough its extremely difficult to model even with supercomputers.

The wording of your question implies you didn't realise this. I suggest you do some GR before pretending you know all about it. You called Ben a few names for supposedly sticking his neck out and you're doing exactly the same. I hate to break it to you but you simply lack the knowledge and experience to pass any comment on these sorts of things. I know you don't want to accept that but tough.

 

Chapter 5 eh? Yeah I apologized for saying Ben had no common sense but "he started it!"

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True or false: back-reaction, as is applies to expanding the event horizon of a black hole, does not occur before mass crosses the existing event horizon.

 

Depending on the size of the object the event horizon will be slightly modified before the object reaches it, with larger objects making a larger modification from further away. Quite what it looks like I don't know because its an extremely difficult problem. Two black holes, which have spherical event horizons on their own, will not be perfect spheres until they touch. That would be what "No back reaction till it reaches the event horizon" implies. Instead they'd merge into one another and quickly settle down into a new spherical event horizon whose area satisfies \(A \geq A_{1}+A_{2}\), as per the 2nd law of black hole mechanics.

Do you understand what I said about coordinates? Do you understand that a lack of a global set of coordinates is not to do with a problem with the mathematics, else you have to deny the notion of a sphere! Do you have any first hand experience with this stuff, on which you're basing your claims? Or are you just reading wordy explanations on Wikipedia and then going with your inexperienced gut felling about things?

 

Bludgeoning him with your intellect will teach him nothing. There is more to being a teacher than facts and details otherwise you're just showing off. Explain it to him in a way that he can understand. Otherwise you're talking for ego's sake or worse to actually degenerating him for having ideas.

 

Aren't you the guy who claims to have his own theory of physics?

Did you miss the earlier part of the thread where I have tried to explain things to him? RJ is talking about things which are typically not taught until the 3rd or 4th year of a degree (if at all). The notion of global vs local properties of a space and coordinate transformations in regions where charts 'patch' over one another are not trivial things. You complain I'm making it too high level but its catch 22. RJ (and many people who like to talk about quantum or relativity stuff) wants to talk about advanced stuff but doesn't want to put in the effort to actually understand it. Any real discussion should be done on a high level, with lots of detail but RJ can't engage in that because he hasn't done the ground work. If I provide too much detail he or you complain. If I provide no detail he'll ignore what I say as it seems I don't justify myself.

He posted an equation about acceleration. He then demanded people provide an alternative expression if he's wrong. I provided it, along with a link to considerable amounts of lecture notes. Lecture notes from a course which covered the things he's complaining about. I doubt he understood them but that isn't my fault.

If someone wants to find out what advanced physics does then fine. But don't think you can say "This is done wrong" if you haven't put in the time to understand the details, just because you've heard something you don't like or don't understand.

If RJ has put in the effort to learn some of the details he's hiding it well. If he hasn't put in the effort then its not my fault he wants to talk about details which are, quite frankly, beyond him.

I'm willing to help people who help themselves. I'm not going to spoon feed people every single thing, they have to do the majority of the effort themselves. The lecture notes I linked to are 4th year stuff, it took me 3.5 years working full time on maths to get to doing them. Even if I were willing to spoon feed people I'm not going to be able to get them up to speed to do that material. If someone isn't willing to put in the effort for themselves why should I be willing to do it for them?

 

Indeed I have my own ideas. But I consider all points of view equally with no malice. Can you say the same?

Are you telling me when sublty fails it justifies the use of blunt trauma?

But you can't just jump to that assumption no more than you can throw all the math and concepts at someone and expect them to understand. That's the use of Meta-messages and they aren't used to be understood.

Then isn't that the clue that this discussion shouldn't be taking place?

Hey, you're right but you have to understand as an intellectual how this looks. This is almost like arguing with an infant. It just comes off as mean spirited.

Let me ask you this.
Do you think that the knowledge that he has is useless? If not then you do have a common frame of reference and so obviously he has SOME working knowledge. He's well above me that's for sure. Give him credit for that because he's at least gone that far...But you don't inspire the next generation whether a teacher or not like this. Why discourage him is all I'm saying. Just...think about it.

P.S I'm just glad he has an interest...

 

Saquist: Thanks for your support; you really don't need to defend me though, I'm a big boy. I also believe that AlphaNumeric gives me more credit than he will admit. We have an understanding - he acts pompous while I feign humility.:cheers:

Again, you are presuming the black holes exist in the first place, which I question. Regardless, I think you are mistaken that the mathematical structures of the EHs would not remain spherical until they touch. Maybe we need to define "touch"? Maybe we could reduce the complexities here and study the creation of a BH using 2 simple hydrogen atoms?

"When" would they merge? The same goes for defining "quickly". I believe you are imagining this happen in your mind but forgetting "when" the mathematical model says these things would take place (from an outsider's perspective).

Yes, I read what you wrote about coordinates. I can use a Polar/Spherical coordinate system for the Earth to eliminate all indeterminate regions. I do not believe there is such a substitute for studying BH's.

 

I often find myself biting my tongue when I read threads like this. Often, it's more cringe-worthy than anything.

This indicates you have absolutely no idea what you're on about. Most notably, it indicates you completely misunderstand the most basic concepts in differential geometry. And since differential geometry is absolutely central to Einstein's general relativity, it doesn't put you on a solid footing.

Take the sphere S^2. You have your spherical coordinates \((\theta,\phi)\). Please give me the coordinates of the north pole.

 

oh...good...it's Guest...AlphaNumeric's "mini-me".

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Are you saying that because theta can take any value when Phi=90 that the north pole is indeterminate? Ah, I see the confusion

My spherical coordinates are valid everywhere. There is no area of indeterminability on the sphere.

Now please go back to lurking...I frankly don't like talking to you and any future posts from you will be ignored.

 

Not so sure I much understanding the original post, but I have heard that black holes probably aren't really singularity points, but rather a spinning disk. Provided the gravity field is so great that light can't escape, the black hole is still defined to actually exist, even though not really a singularity.

Anybody have any thoughts as to as the way our universe curves in upon itself, that the universe could be a black hole? As that probably doesn't work, how far then can the metaphor hold or be useful for? And by what term or name is this concept called? Or is it just some silly sci-fi idea?

 

Shocking! You have the issue handed to you on a plate, and yet you still fail to understand it. You are woefully out of your depth.

 

Most of the time. I've yet to see an idea put forth on a forum which was worth writing up and publishing though.

When the message just isn't getting through it would seem that being pleasant fails.

Its not like RJ and I have only just crossed paths. We've both been here long enough for people to get a feel for whether each of us is willing and/or able to put in the effort. My view is that RJ isn't, he's neither willing nor able. If he wasn't trying to talk about such things as globally definable coordinates on non-trivial manifolds then it'd be fine but he is making claims about topics he doesn't have understanding of.

Why are you saying this to me when I'm not the instigator?

When the infant spends all his time jumping up and down in the corner yelling and won't sit down when told then its time to be a little firmer.

I've yet to see him display any relevant knowledge which can't be got from Wiki in 30 minutes of reading.

I didn't say they'd be spherical precisely until they touch. I said they'd affect one another in a very complicated way. Very far from one another they have a pretty much spherical shape but close together the very complicated rules governing their behaviour is not easily described. And my replies about black holes are from the point of view of what GR says about them. There's no point in saying "If we ignore any model of gravity and what they say then what do black holes behave like?" because you end up with random speculation.

The highly non-linear dynamics means that its not something you can give an analytic expression for. It'll depend on their initial masses, charges, angular momentum, position and velocities.

If you think 'quickly' isn't precisely defined enough then might I suggest you get yourself a book on general relativity and work through it so that you understand the underlying principles and can work out the results yourself.

If you, Saquist or anyone else find replies by me or anyone else to be lacking in detail then get off your backside and find out the information for yourself. That's what I do. This week I've been trying to get my head around a particular result and I didn't initially understand the textbook example, so I spent 2 days and a lot of bits of paper working through it (and getting it wrong many times) till I understood it. Effort, it works wonders sometimes.

You read but you didn't understand. Spherical coordinates are not globally valid and Guest tried to get you to think but you failed. If latitude is measured by \(\theta\) and longitude by \(\phi\) such that the equator is \(\theta = \frac{\pi}{2}\) and Greenwich is at \(\phi = 0\) what value of \(\phi\) is the North pole? If the coordinates are valid then each point on the sphere is represented by one and only one pairing \((\theta,\phi)\).

Coordinates must be related by bijective maps, where you can map a single point in one coordinates to another single point in another. When certain projection methods are used to make global maps they map the North and South poles to lines. Thus one of the coordinate systems is not globally valid. You're wrong and you're now making excuses. You complain something something up with black holes but you excuse your own failure to provide globally valid coordinates on a sphere. Hell, you don't even know what the requirements for valid coordinates are!

Do you think I and other mathematicians/physicists are unaware of spherical coordinates? I specifically said a sphere doesn't have globally valid coordinates because its a simple example. It has a non-trivial topology, which is an enormous area of mathematics which examines these things. Topological invariants like homotopy groups take up the entire careers of people and not only are you unaware of such things but you are daft enough to think you are in a position to make vast claims about things you haven't studied and which you are incapable of understanding the details as you lack any experience and any knowledge of the required things. You can't even do Euclidean space vector calculus, never mind differential geometry on non-trivial manifolds. No doubt Saquist is thinking "He's using big words again" but I'm not, its just you don't spend each and everyday doing this stuff, unlike mathematicians.

If you think I'm wrong about the spherical coordinates feel free to write to a mathematics journal and point out that a sphere is trivial topology and actually spherical coordinates, which are covered in every basic course on geometry, are in fact globally valid. Let me know when you get published....