# [bi|sur|in]jective coordinates

Discussion in 'Physics & Math' started by RJBeery, Sep 29, 2010.

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1. ### RJBeeryNatural PhilosopherValued Senior Member

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4,189
The reason for my continued questioning, przyk, is that I've asked (what I feel are) some pretty basic, logical questions about black holes to many people that I would assume would be able to supply straight-forward, well established, community-accepted answers. Here's a version of my logical conundrum:
=========================================
#1 Do you agree that all frames outside of the BH calculate that no mass crosses the EH using Schwarzschild coordinates (or, more specifically, mass crosses the EH at t=infinity)?
#2 Do you agree that the EH does not expand until mass has crossed the EH (i.e. backreaction)?

If you concur with #1 and #2, then run the clock backwards in your mind and describe to me how this theoretical black hole formed in the first place...
=========================================
To this point I've received many responses including the following:

1) Backreaction occurs before EH crossing (as CptBork asked above)
3) Vaidya metric
4) Kruskal coordinates
5) The finite proper time of the in-falling body proves the BH's existence
6) "A very famous man told me so" (this is my favorite)
7) Speculative "mystery mechanisms"
8) "Here, check out my class notes" (implying, to me, that they feel the answer was explained to them in class but they didn't understand it either)

...and in each case, upon further probing, the person making the defense admits that they don't completely understand it themselves (or they quietly go away). In fact, you're the first responder that has doggedly tried to explain your defense (with patience and maturity I might add! Well done, sir).

Anyway, it appears to me that you're trying to make the case that, Schwarzschild calculations be damned, Kruskal "proves" that in-falling bodies do so in finite outside observer's time. How a coordinate change, rather than an additional physical mechanism, explains this dichotomy I still don't understand. This strikes me as a violation of something for which I have no name but I suspect is very fundamental to the mathematicians here. A coordinate change should not affect the inertial calculation of the time required for a free-falling body to cross the EH!

Your response to this is that the phrase "time required" to do something is simply a label and apparently doesn't actually measure anything..? Again, this bewilders me. Let me ask you this: in Kruskal coordinates, is V given the label of "time component" arbitrarily? Could Kruskal just as easily given U the label "time component", or was V chosen for a reason that ties V's behavior specifically and solely to the temporal dimension? I have more questions but I think I'll wait for you to respond to this.

BTW, regardless of whether I think this is a valid explanation of BHs or just unintentional obfuscation, I'm learning a lot and rather enjoy the exchange.

3. ### Guest254Valued Senior Member

Messages:
1,056
And once again, you continue to make a complete fool of yourself. And this is exactly why people think you're an idiot. This paragraph cements the fact you have absolutely no clue how general relativity works (in particular, you don't understand the notion of coordinate time). And yet you clearly believe you're probing the limits of people's understanding, rather than it be that you don't understand the theory at hand. The notion of coordinate time (i.e. the "t" in Schwarzschild coordinates) simply corresponds to a coordinate which induces a vector field $V=\partial_t$ such that $V^a V^b g_{ab} \sim \pm 1$ (depending on choice of orientation). In no way does this mean that the coordinate "t" corresponds to the the true proper time of an observer somewhere. Similarly, with Kruskal time coordinate "v". But clearly, you're unaware of these basic facts, because you've clearly spent zero time trying to learn and understand Einstein's relativity.

The whole point of Einstein's theory is that it is independent of coordinates! This is, literally, one of the first things taught to people who sit a course on GR. The fact that you are so woefully ignorant of all these basic facts of GR, whilst at the same time believe you're probing the limits of the theory, cements your utter incompetence and ignorance.

And to add fuel to the fire, you demonstrate once more how utterly ignorant you are of basic GR. Kruskal time, Schwarzschild time... Dave's time "coordinate" would simply refer to a coordinate that corresponds to a vector field $V$ such that $V^a V^b g_{ab} \sim \pm 1$. To compute the proper time experienced by an observer, you need to construct a relationship between the coordinate time and the proper time of that observer at a point. As you have been told, the proper time of a stationary observer far from the event horizon of a black hole coincides with the Schwarzschild time. But hey, you're the guy probing the limits of relativity: show this is true.

Here's a great exercise: do the standard computation in Schwarzschild coordinatres that reveals an observer outside a black hole will never see a test particle cross the event horizon. Now do precisely the same exercise using Kruskal coordinates. The first few stages of this exercise are trivial - no more than standard GR and the chain rule. If you can show us all you're willing to learn basic GR enough for you to do the easy bits, we shall all help you do the tougher bits.

I've said this before, but I didn't see it happen. Let's see the armchair physicist do some physics.

5. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,189
Guest, you have been fully assessed and found to be lacking anything of value to me. There is a reason I addressed my post specifically to Przyk. I know you want attention but I'll say it again...TAKE A HINT. I will ignore all further posts from you in this thread; continue to post and I'll officially "ignore" you.

7. ### Guest254Valued Senior Member

Messages:
1,056
One second we're brothers and you're all keen for my help, but when your brother points out you don't seem to know anything about the topic in hand, you want to run away!

How sad, though. Another request for the arm chair physicist to do some physics, and again we're all left wanting!

8. ### Green DestinyBannedBanned

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1,211
How can you say something like this, then say something completely contradictory like this:

Only if you were utterly disingenuous.

9. ### Guest254Valued Senior Member

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1,056
Could you explain the contradiction? Use small words, so you don't confuse yourself.

10. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,189
Green Destiny, don't feed the troll.

11. ### Green DestinyBannedBanned

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1,211
Your behaviour is contradictory. You couldn't be possibly treating him like a brother, if you can throw insults about him around.

Or is that how you actually do treat your brothers?

12. ### Guest254Valued Senior Member

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1,056
And still no physics from the arm chair physicist! Come on, don't be so predictable!
I hate to break this to you: RJBeery isn't my brother!

13. ### Green DestinyBannedBanned

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1,211
I know you aren't. I thought you meant it on a friendly-basis.

But then, I should have expected less from you.

14. ### Guest254Valued Senior Member

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1,056
Aww bless! You mean you can't tell when someone is treating you with contempt?!

15. ### Green DestinyBannedBanned

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1,211
No attempt, as usual, to have a constructive criticism. One without the attitude. No attempts at all.

And RJ - I don't know if you ever come on here when not logged in, but it becomes frustrating after a while having to see the ignorant posts made by Guest even when you do have the twit on ignore.

16. ### AlphaNumericFully ionizedRegistered Senior Member

Messages:
6,699
You're the guy who says he'll leave then he doesn't. You're the guy who says he 'investigates' the Dirac equation but can't do calculus and doesn't know anything about the Pauli matrices other than they are matrices. You're the guy who gets upset when people say his 'pet theory' is nonsense but then notices you mixed up scalars and vectors.

Guest, whatever his style of posting, at least clearly knows his stuff. You and RJ clearly don't and RJ is reacting here like you do in other threads, not wanting to hear from people who understand something that he clearly doesn't.

You and he are obviously viewing one another as kindred spirits, deluding yourself into thinking you're testing the limits of people's understanding and probing the edges of physics, but you aren't. Both of you are just deluding yourself and wasting your own time. You can call Guest a troll and put both of us on ignore but at the end of the day you're wasting your own time and you're learning nothing. If you both are not interested in actual understanding but just want to spout buzzwords and hope to deceive people then you're going the right way about it. If you want to actually understand something then you need to put in some effort.

Guest is clearly willing to help if RJ bothers to put in any effort. RJ clearly isn't bothered, just like you might learn something if you stopped BS'ing about your own understanding.

Then leave. You're obviously not here for honest informed discussion, the forums won't lose anything other than a buzzword spouting BS machine.

17. ### funkstarratsknufValued Senior Member

Messages:
1,390
If I may be allowed an addition: Green Destiny is the guy who'll flat out deny his obvious (and poorly done) plagiarism (where he fails to get basic set builder notation right), and then later admit that he doesn't have a clue about the notation to delimit an interval, which appears in his plagiarized post!

(Yeah, yeah, evil conspiracy etc.)

18. ### Green DestinyBannedBanned

Messages:
1,211
First I never said anything about a conspiracy. And If you want to talk about basic notation, how about quarkhead not knowing $d^4x$ notation? That about nearly, or probably just as bad. That kind of notation is taught very early on; I was seriously surprised when he asked it, because he was going about regurgitating a textbook.

Eh how about that one? Or shall we try and rake up a few posts of alphanumerics were he's got something wrong, or maybe you have a few we can talk about?

Don't give me this shit, seriously. I can do just as good.

19. ### Green DestinyBannedBanned

Messages:
1,211
Oh, here he goes, like a knight in shining armour Guest.

P-ft pathetic.

20. ### Green DestinyBannedBanned

Messages:
1,211
Oh and, just a little wake-up call. It's very difficult to plagiarize physics, because you are constantly regurtitating the same old equations. You can only plagiarize if you pawn it off as your own.

Nice try though. Nice to see some else take Guests corner.

21. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,189
Wow this thread got nicely derailed overnight...
[repost]
The reason for my continued questioning, przyk, is that I've asked (what I feel are) some pretty basic, logical questions about black holes to many people that I would assume would be able to supply straight-forward, well established, community-accepted answers. Here's a version of my logical conundrum:
=========================================
#1 Do you agree that all frames outside of the BH calculate that no mass crosses the EH using Schwarzschild coordinates (or, more specifically, mass crosses the EH at t=infinity)?
#2 Do you agree that the EH does not expand until mass has crossed the EH (i.e. backreaction)?

If you concur with #1 and #2, then run the clock backwards in your mind and describe to me how this theoretical black hole formed in the first place...
=========================================
To this point I've received many responses including the following:

1) Backreaction occurs before EH crossing (as CptBork asked above)
3) Vaidya metric
4) Kruskal coordinates
5) The finite proper time of the in-falling body proves the BH's existence
6) "A very famous man told me so" (this is my favorite)
7) Speculative "mystery mechanisms"
8) "Here, check out my class notes" (implying, to me, that they feel the answer was explained to them in class but they didn't understand it either)

...and in each case, upon further probing, the person making the defense admits that they don't completely understand it themselves (or they quietly go away). In fact, you're the first responder that has doggedly tried to explain your defense (with patience and maturity I might add! Well done, sir).

Anyway, it appears to me that you're trying to make the case that, Schwarzschild calculations be damned, Kruskal "proves" that in-falling bodies do so in finite outside observer's time. How a coordinate change, rather than an additional physical mechanism, explains this dichotomy I still don't understand. This strikes me as a violation of something for which I have no name but I suspect is very fundamental to the mathematicians here. A coordinate change should not affect the inertial calculation of the time required for a free-falling body to cross the EH!

Your response to this is that the phrase "time required" to do something is simply a label and apparently doesn't actually measure anything..? Again, this bewilders me. Let me ask you this: in Kruskal coordinates, is V given the label of "time component" arbitrarily? Could Kruskal just as easily given U the label "time component", or was V chosen for a reason that ties V's behavior specifically and solely to the temporal dimension? I have more questions but I think I'll wait for you to respond to this.

BTW, regardless of whether I think this is a valid explanation of BHs or just unintentional obfuscation, I'm learning a lot and rather enjoy the exchange.

22. ### QuarkHeadRemedial Math StudentValued Senior Member

Messages:
1,620
I apologize to one and all for perpetuating this off-topic rubbish, but I really cannot let the following pass without comment
First, unlike you, I feel no shame in admitting my ignorance. Neither do I feel any shame in admitting that I may not be the sharpest knife in the drawer, unlike you.

But, having had a little help from temur and przyk and a lot of guidance from some rather advanced math books I figured it. (remember books? These are the things that have cardboard pages, large print and lots of pictures of people called Mr, Grumpy, Mr. Pretentious etc).

So now I call you out. Explain in your own words what the notation you refer to means. Since it is so "elementary" this should present you with no difficulty. But be warned, someone here will know if you are copy/pasting!!

23. ### przyksquishyValued Senior Member

Messages:
3,182
If you're ignoring the effect that the mass itself has on the surrounding space-time geometry, then yes. If not, I don't know the result off-hand.

As I just said, I don't know the result off-hand. But I'd guess the result depends on the nature of the mass. If the mass occupies a finite volume (as the case of a collapsing star modelled as a continuous fluid), then I'd expect you simply have the mass continually falling in while the event horizon continually expands outward (as I recall, this is what happens in the case of a collapsing star, at least in a Kruskal-like chart). If the mass is a "point particle" (infinite density), then the mass itself is surrounded by its own event horizon and you're solving the problem of two merging black holes. This is something I've never personally studied, so anything I say about it would be pure guesswork.

This isn't a completely invalid response. The class notes will contain a mathematically consistent solution describing eg. a star collapsing into a black hole. That solution simply couldn't exist if the formation of a black hole were truly a logical impossibility. So it's already a given that black holes can form according to GR - all that's left is to reconcile that solution with your sense of intuition.

In general, if you're wondering how something for which a mathematical description exists can be "possible", it's your intuition that's going to have to adapt. The problem with a theory such as GR is that it doesn't conform to the set of intuitions you'll have, and there are all sorts of ways you can get your initial impressions about it wrong. If someone gives up and tells you "Go read a textbook!", it isn't an appeal to authority. The person saying it most likely knows the difference between memorisation and understanding just as well as you do. They're probably saying it because a) your intuitive understanding of the problem at hand is obviously wrong, b) they've judged that identifying and correcting all the implicit assumptions you're making would take up more of their time than they're willing to invest in you, c) you'd quickly identify them yourself if you started at the beginning and studied a mathematical treatment of the subject, and d) you'd no longer need to ask for special attention every time you thought of something apparently paradoxical in GR. You'd have all the tools you needed to resolve them yourself, or at least the basic concepts and vocabulary necessary to understand someone else's explanation with minimal risk of confusion.

I was in the same position as you a few years ago, before I studied GR. I had my own intuitive difficulty with black holes: I was wondering how a charged black hole could produce an electromagnetic field around it. Electromagnetic fields propagate at the speed of light - how can they "escape" the black hole? I even posted this once here and didn't receive any helpful reply. Instead of accusing the posters of being textbook regurgitating parrots incapable of original thought, I simply dropped the matter and decided I probably had my intuitive understanding of the problem wrong and that I'd get back to it once I'd formally studied GR. I never for a moment believed I might have found some sort of fatal flaw in general relativity. Now in retrospect I can say I had the right attitude. I've resolved the apparent paradox to my own satisfaction, and yet I doubt I could easily explain it to a four year younger version of myself - let alone my grandmother.

No, I'm saying that the question of when an in-falling body crosses the event horizon according to an outside observer isn't a meaningful one. You need to explain what convention you're using to attribute the horizon crossing event a time on the outside observer's clock. And I don't know any such convention that wouldn't be arbitrary. That's why, when you raise the question, I keep substituting it with different questions formulated in terms of concepts such as trajectories (eg. of light signals) and light cones, which are better defined (ie. independent of the choice of coordinate system).

For example, take the fact that an in-falling object eventually becomes irretreivable to an outside observer, which I illustrated on a Kruskal chart. This result isn't meaningful just because I happen to like the Kruskal chart. It's meaningful because I was careful to pick a question whose answer was independent of any particular choice of coordinates. I then used the Kruskal chart to illustrate the result only because light cones have a particularly simple appearance in that chart. Proper times and light cones are invariant concepts, so a complete calculation in some other chart is guaranteed to produce the same result.

Depends on the context. In general, it's only meaningful to consider the "clock time" of an event occurring at the same place as the clock (ie. the event has to occur on the clock's world line). Attributing a time on a clock to an event that takes place somewhere else isn't so straightforward.

No. General relativity provides a clear distinction between space-like, time-like, and light-like vectors (and therefore coordinates) at every point. It inherits these notions from SR. From the perspective of any locally inertial frame at a point, a particle following the Kruskal V coordinate (ie. U = cst) would be travelling at less than the speed of light as it passed that point. A point following the U coordinate (V = cst) would conversely be moving faster than light. Basically, if you imagine drawing the "coordinate grid", a material point could travel along the time-like coordinate lines but not the space-like ones.

Last edited: Oct 8, 2010