I am the first in the world

[Farsight]

Sorry, I missed a few posts:

I seem to remember having this conversation once before, but I could be thinking of someone else. I think I remember you suggesting some time back that gravity could be described in terms of a sort of "refractive index" of space.

It probably was me, referring to Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime

I hope one of the professionals here will correct me if I'm wrong, but if I understand this correctly your hypothesis implies that gravity could be described completely by a single scalar field. If so then your wrong to say "it's that simple", since AFAIK there are no successful scalar theories of gravity.

It doesn't follow. Check out Einstein's 1920 Leyden Address where he says things like this:

"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. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..."

For example, Nordström (who knew what he was doing) tried to formulate such a theory back in 1913. It suffers with a number of problems:

• It predicts that light is not bent by gravitational fields (in contradiction to observation).
• It predicts that the perihelion of Mercury's orbit will lag by 7 arcseconds per year, instead of advancing by 43 (contradicting observation).
• It gets the Shapiro time delay prediction wrong (contradicting observation once more).

Bit of a straw man then, isn't it?

Since this matter is anything but "simple", perhaps you could actually show that your theory fares better than Nordström's when applied to these classical tests?

No. It isn't my theory, it's Einstein's. I'm just reminding a few people what he actually said, and putting some evidence in front of their nose. And it is simple. The speed if light isn't constant, and light shone straight up from the event horizon doesn't curve, and it doesn't get out.

Aw, I'm going to bed. Come on guys, raise your game. Somebody please give me an intelligent sincere discussion here instead of all this... squealing.

 

[ben m]

Oh slings and arrows. I don't need a coordinate system to see two optical clocks losing synchronisation, I just look at the numbers.

Numbers are coordinates. When you say "the light never hits the mirror", you mean that you tried to label the spacetime coordinates of the light-hitting-the-mirror, using a particular clock, and you found that it happens at t2=infinity.

(ETA: Or, rather, that's what Einstein told you to do. Did you do something else? Then you did it wrong.)

 

[W.D.Clinger]

Farsight's like the guy who's got his atlas open to a chart that shows only the 48 contiguous states, and is using that chart to deny the existence of Hawaii and Alaska. When we tell him the next page contains a chart of all 50 states plus Mexico and Canada, he refuses to look.

I'm not, you are.

No. You must be confused.

I'm willing to use Schwarzschild charts, Lemaître charts, Eddington-Finkelstein charts, Kruskal-Szekeres charts, and any other charts that will help me to understand the spacetime manifold around a black hole.

You, however, have refused to look at any charts that don't use Schwarzschild coordinates.

I'm showing you the variable speed of light,

No. You must be confused.

It was I who showed you, in exercises 22 and 23, that the coordinate speed of light depends on the chart you're using. Had you done your homework, you'd know the speed of light near the event horizon is different when observed by a observer at infinity (Schwarzschild coordinates) than by an observer who's free-falling into the black hole (Lemaître coordinates).

The Schwarzschild coordinates and Lemaître coordinates both yield correct descriptions of the submanifolds on which they are defined, and Lemaître coordinates are defined everywhere the Schwarzschild coordinates are defined, so the variable speed of light is a coordinate-dependent effect.

You're the one who's been denying that coordinate-dependence.

sol invictus explained the coordinate-dependent variable speed of light almost two years ago, in the "How gravity works" thread:

Now, as everyone knows, one of the laws of physics according to special relativity is that the speed (and velocity) of light in vacuum is constant. Therefore, according to general relativity, the speed of light measured in any sufficiently small vacuum region is constant.

However: we're not always restricted to small laboratories. We can measure the speed of light over long distances, over distances where the gravity field varies a lot. What then?

The answer is a little bit subtle, because "speed" is no longer uniquely defined (hence the quotes). To measure speed you need to measure a distance and a time, then take the ratio. But it turns out that neither distance nor time is uniquely defined over such long distances, and depending on your definition you could find either that the speed of light is constant or that it isn't.

In particular, the answer depends upon the chart you choose to use. For a faraway observer who wants to regard spacetime as static outside the event horizon of a black hole, Schwarzschild charts are natural. Schwarzschild charts say the speed of light converges to zero as the event horizon is approached (exercise 23). For observers who are free-falling into the black hole, Lemaître charts are more natural, and they say the speed of light remains locally constant through the event horizon and all the way down to the central singularity (that's exercise 22).

you're refusing to look, preferring instead to wallow in mathematical abstraction and erudition

No. You must be confused.

You're the one who refuses to look at Lemaître charts.

Apart from exercise 8, which you don't need to prove, the 23 exercises I gave you involve no math beyond high school algebra and differential calculus.

If those prerequisites have prevented you from doing your homework, I'm sorry. General relativity does rely on some fairly advanced mathematics, so you can't hope to understand much of anything about general relativity without making an effort to learn some math.

and to believe that for a stopped observer a stopped clock carries on ticking.

No. You must be confused.

To a faraway observer, a clock that's falling into a black hole appears to run slower as it approaches the event horizon. That's because the faraway observer observes its ticking only by means of light (or other communications) emitted from the clock. To an observer moving with the clock, the clock continues to tick at its usual pace. If it's a good clock, so the time between ticks appears constant to the co-moving observer, only a finite number of ticks will occur before the clock passes through the event horizon (exercises 20, 21, 22, and 23). Because of the time dilation observed by an observer at infinity (or sufficiently far away for Schwarzschild charts to give a good approximation to what he observes), that finite number of ticks gets smeared out over the rest of eternity. That's why the faraway observer might deduce that time stops at the event horizon.

To the co-moving observer, time doesn't even slow down at the event horizon (exercise 22). The clock continues to tick normally even when the clock's inside the event horizon, but the photons it emits cannot escape past the event horizon to inform a faraway observer of the clock's ticking. Those photons will soon follow the clock and its unfortunate co-moving observer as they are pulled down into the singularity at the center of the black hole.

And please, spare me your not-qualified to judge pomposity.

I went back and read the Relativity+ thread. I took notes.

I saw a lot of stuff that resembled your post of a few minutes ago:

Aw, I'm going to bed. Come on guys, raise your game. Somebody please give me an intelligent sincere discussion here instead of all this... squealing.

Are you sure you want to discuss the pomposity of telling people they aren't qualified to judge?

You have no counter-argument, and it shows.

No. You must be confused.

My counter-argument shows up here:

• Starting with the most basic definitions, I explained that a chart is not the manifold.
• I applied those definitions to the familiar example of a 2-sphere (which approximates the surface of the earth).
• I gave examples of what happens as coordinates approach a coordinate singularity. Those examples show why we should suspect Schwarzschild's r=2M singularity of being a mere coordinate singularity.
• I used Lemaître coordinates to prove that Schwarzschild's r=2M singularity is a mere coordinate singularity.
• I pointed out that your argument rests on two related errors:
  • You assume Schwarzschild charts cover the entire spacetime manifold.
  • You assume there is some global, observer-independent notion of time that could justify your use of the word "yet".
  (By the way, you repeated your "yet" error at the end of the first paragraph of your post #166.)
• I identified five recent mistakes that you made because you haven't done your homework.
• I also pointed out that anyone who's proficient in high school algebra and differential calculus can work this out for themselves well enough to identify the major mistakes you've been making.

General relativity is a difficult subject. Once upon a time, people like Lemaître published research papers about the very matters we're discussing.

It's okay to make beginner's mistakes when you're first learning a difficult subject. No one expects an amateur or student to get everything right.

It's also okay to give up on a difficult subject when you discover you just don't have the mathematical and/or scientific background required to understand it. Life is short. You can't learn everything.

It's not okay to pretend to understand a difficult subject while making beginner's mistakes and ignoring expert correction. That pretense and willful ignorance is what distinguishes cranks and crackpots from students and amateurs.

 

[ctamblyn]

Um, I did say The author of this paper says "matter can indeed fall across the event horizon within a finite time" but disagreeing with that really isn't some wild hypothesis..

The author disagrees with you, and concludes that frozen stars cannot exist in our universe. Do you understand the argument made in that paper?

 

[ctamblyn]

Sorry, I missed a few posts:

It probably was me, referring to Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime

No, that isn't quite what you were saying. You were suggesting that GR could be understood in terms of variation in the scalar parameter Z₀ ("it's as simple as that", you said). You cannot reproduce GR that way - it is mathematically and logically impossible.

Bit of a straw man then, isn't it?

No, as it turns out. You have proposed a scalar theory of gravitation which is therefore inequivalent to GR.

No. It isn't my theory, it's Einstein's.

See above. Einstein flirted briefly with a scalar theory of gravitation, but abandoned it in 1914 for various reasons. Just as well, because it failed to match experimental data.

Interestingly, one of the main reasons he didn't like the scalar theory was that it didn't meet his "general covariance" requirements. It singled out a privileged class of coordinate systems, which Einstein was not keen on at all:
http://en.wikipedia.org/wiki/Scalar_theories_of_gravitation#Einstein.27s_scalar_theory

I'm just reminding a few people what he actually said, and putting some evidence in front of their nose. And it is simple. The speed if light isn't constant, and light shone straight up from the event horizon doesn't curve, and it doesn't get out.

See above. Your idea of gravity being as "simple" as a variation in Z₀ demonstrably contradicts GR, and your attachment to a particular coordinate systems (in the special, idealised case of the Schwarzschild metric) is leading you into conceptual errors.

 

[sol invictus]

What part of I'm not choosing any coordinate system did you fail to understand?

That part that isn't true, namely "I'm not choosing any coordinate system".

What's real is light moving through space.

Right. Like the light our intrepid observer (remember her?) watches bouncing back and forth at exactly the same rate it always does, as she crosses the horizon.

 

[ben m]

You have no counter-argument, and it shows.

Look in the mirror, ben. You have no counterargument, remember?

It only makes it even more patently obvious that you've got no counter-argument whatsoever. LOL.

You keep using that word. I do not think it means what you think it means.

 

[Roboramma]

Would you like to reiterate his pretty clear counter-argument? Didn't think so.

The the "stopping of light" is a relic of a particular coordinate system, and that this "stopping of light" at the event horizon doesn't happen in the coordinate system in which the infalling observer is taken to be at rest.

Given this, and the fact that GR is coordinate invariant, the idea that the light "stops" is rather meaningless.

Furthermore ben m shows that the mathematics of GR can show you exactly how much proper time is experienced by that infalling observer as he crosses the event horizon, and that value is finite.

While I'm sure I've missed some nuance here, this seems like a pretty clear counter-argument to me. You may not think it's valid (though I can't find any flaws myself), but to deny that it exists is ridiculous.

 

[Farsight]

All: there's a few posts I haven't replied to, but they're maybe old news now. Let me know if there's any you particularly want me to get back to you on.

Yes, that's what we've all been saying. In the coordinate system of an external observer, objects arrive at the event horizon only when t=infinity. That's the easily-computed, well-known GR prediction for this quantity in this coordinate system. The statement that "stuff never falls in" is a statement about the experience of coordinate system O, not an invariant statement about spacetime. And the paper explicitly points out a coordinate system, different than O, that sees stuff fall in. You have an astounding ability to ignore that.

I really am not ignoring anything. I'm the guy who isn't. Don't forget that a coordinate system doesn't experience things, it's people who do that. Like sol said, coordinate systems are human conventions, they have no bearing on reality. It's similar for spacetime. It isn't something that's actually out there, it's a mathematical space. What's actually out there is space and motion through it. That's why Einstein gave the equations of motion. Sure, you can conceive a coordinate system where the infalling observer crosses the event horizon in "finite proper time", but it always comes back to a stopped observer and a stopped clock. That t=infinity isn't something you should ignore.

 

[Farsight]

Numbers are coordinates. When you say "the light never hits the mirror", you mean that you tried to label the spacetime coordinates of the light-hitting-the-mirror, using a particular clock, and you found that it happens at t2=infinity.

(ETA: Or, rather, that's what Einstein told you to do. Did you do something else? Then you did it wrong.)

What Einstein told me is that the speed of light varies:

1911: If we call the velocity of light at the origin of co-ordinates c_o, then the velocity of light c at a place with the gravitation potential Φ will be given by the relation c = c_oo(1 + Φ/c²).
1912: On the other hand I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of constant gravitational potential.
1913: I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis.
1915: the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned.
1916: In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when die Ausbreitungs-geschwindigkeit des Lichtes mit dem Orte variiert.

...and that time is an emergent property of motion. In turn that tells me that the t=infinity is really a c=0. I really don't think it's me who has done something wrong here. Because all the transformations in the world can't make a stopped clock tick.

 

[Reality Check]

Why are you ignoring the coordinate systems where there is no event horizon effects

Like sol said, coordinate systems are human conventions, they have no bearing on reality. .

Like sol actually said:

That's exactly the point. Coordinate systems are human conventions, they have no bearing on reality. You can choose any coordinates you like - and it's only in the coordinate system you insist on always using that light slows down or time stops at the horizon. In an infinite number of other coordinate systems, all equally as valid as yours, nothing unusual happens at the horizon.

(emphasis added)

The reality that coordinate systems have no effect on is the reality that is described by GR (which is expressed in coordinate-free mathematics).

There are an infinite number of coordinate systems where an observer will measure that light slows down as you approach an event horizon and that nothing can get to the event horizon (light never stops).

There are an infinite number of coordinate systems where an observer will measure that nothing happens as you approach, cross and go past the event horizon.

The questions are

1. Why are you obsessing with one of the infinite number of coordinate systems where clocks slow as they approach the event horizon?
2. Why are you ignoring (or are just ignorant of) the infinite number of coordinate systems where clocks just tick along as they pass through the event horizon?

 

[Reality Check]

...snipped the restatement of the obvious...
In turn that tells me that the t=infinity is really a c=0.

In turn that tells us that you are ignorantly treating infinity as a number.

"t=infinity" can never happen. This is because we can add any number to your "t=infinity" and get another t that is also "t=infinity".

Infinity is the concept of increasing without limit.

The proper way to state this is that t gets bigger and bigger and c gets closer and closer (but never equals) 0.

 

[Farsight]

No. You must be confused. I'm willing to use Schwarzschild charts, Lemaître charts, Eddington-Finkelstein charts, Kruskal-Szekeres charts, and any other charts that will help me to understand the spacetime manifold around a black hole.

You, however, have refused to look at any charts that don't use Schwarzschild coordinates.

Not so. There is no confusion on my part. I'm the one who's given the page from MTW where the Kruskal-Szekeres coordinates are is depicted, see post #135.

It was I who showed you, in exercises 22 and 23, that the coordinate speed of light depends on the chart you're using. Had you done your homework, you'd know the speed of light near the event horizon is different when observed by a observer at infinity (Schwarzschild coordinates) than by an observer who's free-falling into the black hole (Lemaître coordinates).

We all know that different observers measure different speeds, but we all know that it is nevertheless the one same thing being measured, and thus the different measurements result from those observers being subject to different conditions. We also know that even a single observer can see that light really does move at different speeds, as per my post
#134. Since we also all know that vertical light emitted at the event horizon does not slow down and does not curve and does not escape a black hole, we can all work out that you haven't done your homework. And we can all see that you have evaded the issue and posed "exercises" instead.

The Schwarzschild coordinates and Lemaître coordinates both yield correct descriptions of the submanifolds on which they are defined, and Lemaître coordinates are defined everywhere the Schwarzschild coordinates are defined, so the variable speed of light is a coordinate-dependent effect.

No, it is real. It's no effect. You can sit there watching two optical clocks at different elevations losing synchronisation, and you know full well that parallel-mirror light clocks will also lose synchronisation:

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

You're the one who's been denying that coordinate-dependence.

I'm the one reminding you that coordinate systems are human conventions, that light moves, and that you can see that it moves at different speeds at different elevations.

sol invictus explained the coordinate-dependent variable speed of light almost two years ago, in the "How gravity works" thread:

Now, as everyone knows, one of the laws of physics according to special relativity is that the speed (and velocity) of light in vacuum is constant. Therefore, according to general relativity, the speed of light measured in any sufficiently small vacuum region is constant.

However: we're not always restricted to small laboratories. We can measure the speed of light over long distances, over distances where the gravity field varies a lot. What then?

The answer is a little bit subtle, because "speed" is no longer uniquely defined (hence the quotes). To measure speed you need to measure a distance and a time, then take the ratio. But it turns out that neither distance nor time is uniquely defined over such long distances, and depending on your definition you could find either that the speed of light is constant or that it isn't.

See above. Sol will now agree that it isn't constant, because he knows that he can see that it isn't constant. And if he begs to differ, ask him to imagine that these two light beams are racehorses. Then ask him if they're moving at the same speed:

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

In particular, the answer depends upon the chart you choose to use. For a faraway observer who wants to regard spacetime as static outside the event horizon of a black hole, Schwarzschild charts are natural. Schwarzschild charts say the speed of light converges to zero as the event horizon is approached

I'm glad you agree. Please note that spacetime is static. There is no motion through spacetime, because it's an all-times "block universe" mathematical space. We draw worldlines in it, but objects do not move through it.

For observers who are free-falling into the black hole, Lemaître charts are more natural, and they say the speed of light remains locally constant through the event horizon and all the way down to the central singularity (that's exercise 22).

I don't care what you think is "natural". What matters here is that those observers are subject to gravitational time dilation that goes to infinity at the event horizon, and that infinity cannot be wished away by putting yourself in their shoes because underlying that infinity is a c=0. There are no infinities in nature, Clinger. Every time you meet one, narrow your eyes, be suspicious, be sceptical.

No. You must be confused.

No I'm not confused. I am crystal clear about this.

You're the one who refuses to look at Lemaître charts. Apart from exercise 8, which you don't need to prove, the 23 exercises I gave you involve no math beyond high school algebra and differential calculus. If those prerequisites have prevented you from doing your homework, I'm sorry. General relativity does rely on some fairly advanced mathematics, so you can't hope to understand much of anything about general relativity without making an effort to learn some math.

Yawn. Spare me your intellectual arrogance and your you cannot hope to understand it erudition. Because I do.

No. You must be confused.

Nope.

To a faraway observer, a clock that's falling into a black hole appears to run slower as it approaches the event horizon. That's because the faraway observer observes its ticking only by means of light (or other communications) emitted from the clock. To an observer moving with the clock, the clock continues to tick at its usual pace. If it's a good clock, so the time between ticks appears constant to the co-moving observer, only a finite number of ticks will occur before the clock passes through the event horizon (exercises 20, 21, 22, and 23). Because of the time dilation observed by an observer at infinity (or sufficiently far away for Schwarzschild charts to give a good approximation to what he observes), that finite number of ticks gets smeared out over the rest of eternity. That's why the faraway observer might deduce that time stops at the event horizon.

This is... banal. It's kid's stuff, Clinger. Time doesn't stop because time isn't moving in the first place. It's light moving. That's what a light clock clocks up. Not the flow of time. So when the clock has stopped light has stopped, so your co-moving observer can't even see. What, you think a stopped observer and a stopped clock cancel each other out? And the clock carries on ticking? Somebody please spare me from this cargo-cult pseudoscience that traduces relativity.

To the co-moving observer, time doesn't even slow down at the event horizon (exercise 22). The clock continues to tick normally even when the clock's inside the event horizon, but the photons it emits cannot escape past the event horizon to inform a faraway observer of the clock's ticking. Those photons will soon follow the clock and its unfortunate co-moving observer as they are pulled down into the singularity at the center of the black hole.

Clinger, you're parroting. Shut up, think it through. Then get back to me.

I went back and read the Relativity+ thread. I took notes.

I saw a lot of stuff that resembled your post of a few minutes ago:

​

Are you sure you want to discuss the pomposity of telling people they aren't qualified to judge?

Pompous is the word, Clinger.

No. You must be confused.

Change the record.

My counter-argument shows up here:

• Starting with the most basic definitions, I explained that a chart is not the manifold.
• I applied those definitions to the familiar example of a 2-sphere (which approximates the surface of the earth).
• I gave examples of what happens as coordinates approach a coordinate singularity. Those examples show why we should suspect Schwarzschild's r=2M singularity of being a mere coordinate singularity.
• I used Lemaître coordinates to prove that Schwarzschild's r=2M singularity is a mere coordinate singularity.
• I pointed out that your argument rests on two related errors:
  • You assume Schwarzschild charts cover the entire spacetime manifold.
  • You assume there is some global, observer-independent notion of time that could justify your use of the word "yet".
  (By the way, you repeated your "yet" error at the end of the first paragraph of your post #166.)
• I identified five recent mistakes that you made because you haven't done your homework.
• I also pointed out that anyone who's proficient in high school algebra and differential calculus can work this out for themselves well enough to identify the major mistakes you've been making.

General relativity is a difficult subject. Once upon a time, people like Lemaître published research papers about the very matters we're discussing.

It's okay to make beginner's mistakes when you're first learning a difficult subject. No one expects an amateur or student to get everything right.

It's also okay to give up on a difficult subject when you discover you just don't have the mathematical and/or scientific background required to understand it. Life is short. You can't learn everything.

It's not okay to pretend to understand a difficult subject while making beginner's mistakes and ignoring expert correction. That pretense and willful ignorance is what distinguishes cranks and crackpots from students and amateurs.

Yawn. You're boring us to death. You haven't given a counter-argument, just exercises. They fool nobody. And pretence is what distinguishes quacks from sincere contributors.

 

[Farsight]

No, that isn't quite what you were saying. You were suggesting that GR could be understood in terms of variation in the scalar parameter Z₀ ("it's as simple as that", you said). You cannot reproduce GR that way - it is mathematically and logically impossible...

No, as it turns out. You have proposed a scalar theory of gravitation which is therefore inequivalent to GR...

See above. Einstein flirted briefly with a scalar theory of gravitation, but abandoned it in 1914 for various reasons. Just as well, because it failed to match experimental data...

Interestingly, one of the main reasons he didn't like the scalar theory was that it didn't meet his "general covariance" requirements. It singled out a privileged class of coordinate systems, which Einstein was not keen on at all:
http://en.wikipedia.org/wiki/Scalar_theories_of_gravitation#Einstein.27s_scalar_theory...

See above. Your idea of gravity being as "simple" as a variation in Z₀ demonstrably contradicts GR, and your attachment to a particular coordinate systems (in the special, idealised case of the Schwarzschild metric) is leading you into conceptual errors...

Specious straw-man garbage. You skip from a non-constant vacuum impedance to a scalar theory of gravity, and then declare it to be impossible. This is distasteful, CT. Let me lay it on the line for you. It concerns the thing you didn't respond to. Only this time I will bold the important bit:

Check out Einstein's 1920 Leyden Address where he says things like this:

"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. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..."

Inhomogeneous space, CT. Not curved spacetime. Light curves when the space it moves through is not homogeneous. It veers like a car veers when it encounters gravel at the side of the road.

 

[Farsight]

That part that isn't true, namely "I'm not choosing any coordinate system".

I'm not. FFS sol, how many times have I got to say this? You don't need a coordinate system to see that a clock clocks up the motion of light, and that two clocks at different elevations lose synchronisation because the speed of of light varies with gravitational potential because the properties of space vary:

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

Right. Like the light our intrepid observer (remember her?) watches bouncing back and forth at exactly the same rate it always does, as she crosses the horizon.

Where gravitational time dilation goes infinite but time doesn't flow so it's the speed of light going to zero, so she sees zip because light hasn't got to her eyes yet. Come on sol. Of all the guys here I know you've got a brain. Use it. Forget what you've been taught at Sunday School, forget what MTW has been telling people for decades, think for yourself.

 

[Reality Check]

Not so. There is no confusion on my part. I'm the one who's given the page from MTW where the Kruskal-Szekeres coordinates are is depicted, see post #135.

That just illustrates your inability to understand a point we have been making. There is no event horizon in Kruskal-Szekeres coordinates

In general relativity Kruskal–Szekeres coordinates, named for Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity

You know about Kruskal-Szekeres coordinates but are still ignoring that there is no event horizon in them .

 

[Farsight]

The the "stopping of light" is a relic of a particular coordinate system, and that this "stopping of light" at the event horizon doesn't happen in the coordinate system in which the infalling observer is taken to be at rest.

Get with the program, Robo. Sol, tell him about Coordinate systems are human conventions, they have no bearing on reality.

Given this, and the fact that GR is coordinate invariant, the idea that the light "stops" is rather meaningless.

It isn't meaningless at all, it's what this discussion is all about. The speed of light varies, Einstein said it, and you can see it. The Shapiro delay isn't called a delay for nothing. But you can't see time moving, and from that you can work out that when gravitational time dilation goes infinite, light stops, for real. Then there is no coordinate system. If you beg to differ, try measuring distance and time when light isn't moving. You can't, that's it. Switching to cloud-cuckoo coodinates isn't going to make it start moving, not in a zillion years. Not ever.

Furthermore ben m shows that the mathematics of GR can show you exactly how much proper time is experienced by that infalling observer as he crosses the event horizon, and that value is finite.

Phooey. I said to ben your simple proof was wrong. When 2M/r = 1 at r = 2M you've got a division by zero in [latex]\[ \frac{dt}{d\tau} = \frac{e}{1-2M/r} \][/latex]. That's an undefined result, an infinity. But you just sail on past it. The finite proper time demands an infinite coordinate time, so it hasn't happened yet, and it never ever will. And even a blind man can see the problem with it. Note that ben has not replied, and has evaded my challenge to repeat another expression that I said I'd knock down like the rest of 'em. Instead he became abusive, calling me names. He's trying to blind you with mathematical smoke and mirrors. Don't be fooled by it.

While I'm sure I've missed some nuance here, this seems like a pretty clear counter-argument to me. You may not think it's valid (though I can't find any flaws myself), but to deny that it exists is ridiculous.

It's no counter-argument at all. It's sophistry. If he had a counter-argument he'd be giving it in clear robust fashion that everybody could understand. Ain't gonna happen.

 

[Farsight]

Like sol actually said:

(emphasis added)...

The reality that coordinate systems have no effect on is the reality that is described by GR (which is expressed in coordinate-free mathematics).

There are an infinite number of coordinate systems where an observer will measure that light slows down as you approach an event horizon and that nothing can get to the event horizon (light never stops).

There are an infinite number of coordinate systems where an observer will measure that nothing happens as you approach, cross and go past the event horizon.

The questions are

1. Why are you obsessing with one of the infinite number of coordinate systems where clocks slow as they approach the event horizon?
2. Why are you ignoring (or are just ignorant of) the infinite number of coordinate systems where clocks just tick along as they pass through the event horizon?

Because like sol said, coordinate systems are human conventions, they have no bearing on reality. I don't "obsess" with one of the infinite number of coordinate systems. I focus on what's there. Light moving. Or not. Come on, RC, you're nearly there with this. Look, imagine that you and I are watching a movie. Then for some reason, the movie goes slo-mo. Hey look we say, the movie's going slower. But then ben and clinger, who are in the movie, swear blind that it isn't going slower at all. They hold up clocks and demonstrate that their proper time is unaffected. Then the movie stops. If they could talk, they would swear blind that the movie's going at the same rate as before and that their proper time to the event horizon is finite. Only they can't. Because they're stopped too.

It's really simple when you think about it. Think about light rising vertically from the event horizon too. Light isn't slowed down by gravity. And if it's vertical it doesn't curve. So why doesn't it get out?

 

[Reality Check]

Phooey. I said to ben your simple proof was wrong. When 2M/r = 1 at r = 2M you've got a division by zero in

. That's an undefined result, an infinity. But you just sail on past it.

Phooey. You said that to Vorpal.
You are still wrong because Vorpal wrote

Plugging them into the geodesic equation gives

Which can be directly integrated (with t' = dt/dτ as variable) to give

where e is some constant of integration. Plug that into g_μνu^μu^ν = -1 for the Schwarzschild metric, and voila:

In other words, radial freefall is exactly Newtonian in Schwarzschild r and proper time τ. And so the proper time to reach the horizon (or the r = 0) is finite.

Thus a free-falling observer sees no event horizon.

Another bit of bad math: You state that an undefined result is infinity. That is wrong. An undefined result is an undefined result. Infinity has a definition.

N.B. r is not the radial distance. It is the Schwarzschild r (circumference of a circle centered on the star divided by 2π). It is zero at the center (the singularity).

 

[Farsight]

...In turn that tells us that you are ignorantly treating infinity as a number...

I'm not. I'm telling you that it isn't an infinity, it's a zero. A c=0. That's what it tends to. Like I said a page or two back, it's like an inverse version of Zeno's arrow, where it takes the arrow a second to cover half the distance to the target. It never pierces the target. Think it through.

Guys: I smell desperation here, and sense capitulation. It's signalled by the evasion and abuse. I think one of you will crack and say I think Farsight's got a point soon. Then the floodgates will open, and then will come catharsis. You will have your OhMyGawd moment. It'll feel good, like the way you feel after a tear-jerker movie. And then I can rest easy knowing that you've got something out of this conversation. I know I have. Aw, it's my guilty pleasure. Yeah., I know I'm like a cage-fighter working out toddlers, but it's kinda fun all the same. LOL. Ciao for now!