What is the Relationship Between Proper Time and Coordinate Time in Black Holes?

[rjbeery]

I enjoy studying QM, but my knowledge of black holes is basically limited to the high school explanations as depicted below.

As a body approaches the event horizon the gravitational forces on that body accelerate it to a velocity approaching c. The math implies that mass crossing the event horizon is actually moving faster than the speed of light. This is expressly forbidden by special relativity, but I had always assumed that general relativity somehow provided the answer via curved space-time that allowed such speeds to be obtained "in appearance only"...or something.

After reading a couple of (non grad-level) books on GR, the problem has not been resolved for me. Sticking strictly to my SR knowledge, my interpretation of black holes has been that they never actually form; as matter begins its collapse from a neutron star, for example, that matter becomes "frozen in time" as its velocity approaches c. I'm pretty sure the SR time dilation math shows that the "outside world" clocks move to infinity during a black hole's formation, yet we seem to readily postulate that black holes currently exist...

Can someone that understands black holes please explain?

 

[malawi_glenn]

They will form, but we will never see it. When the matter has passed the horizon according to its own proper time, no light signal can reach us. You have to distinguish from a what falling observer will experience and what global fix observers will see by register light signals from that falling observer.

 

[rjbeery]

Malawi: thanks, but I'm not asking what we would "see" as observers. I'm asking "when" black holes would form as calculated from outside the black hole region itself and, unless GR provides an answer beyond my simple SR understanding, I calculate "when" to be never.

I conclude that I am naive, and I want someone to explain how we can so easily accept that black holes currently exist if their formation takes an infinite amount of time from our perspective.

 

[George Jones]

As malawi_glenn has noted the surface of a sufficiently massive collapsing star quickly passes through an event horizon. Observers who are outside the star (even far outside) can pass through the event horizon by heading towards the star. A hovering observer outside the star doesn't see the star's surface pass through the event horizon, and doesn't see plunging observer's pass through the horizon. So what?

Consider the attached spacetime diagram for special relativity. The axes are coordinate axes for a particular inertial reference frame, the black curved is the worldline of observer A who undergoes constant acceleration, the red dashed line is a lightlike asymptote for A's worldline, and the straight worldline is the worldline of observer B who moves with constant velocity with respect to the inertial reference frame.

Observer A never sees B or any other observer, cross the lightlike asymptote. Do you see why? Does mean that B never crosses the red dashed line, and that events above the asymptote don't exist, never happen?

 

[rjbeery]

George: this isn't a question about SEEING anything. I'm not claiming that an event does not happen just because it cannot be observed from some perspective. I'm asking WHEN the unobservable event of crossing the event horizon happens from the outsider's perspective (because I think the answer is that it "never" happens, and I want someone to explain why this answer is wrong). I had expected a definitive explanation to what seems to me to be an elementary question of such a foundational cosmological subject.

 

[malawi_glenn]

"As a body approaches the event horizon the gravitational forces on that body accelerate it to a velocity approaching c. The math implies that mass crossing the event horizon is actually moving faster than the speed of light."

Can you can you tell where you read this? Because it is not true. It is only being accelerated to local speed of light as measured by a global, distant, observer. The falling observes own speed, will be smaller. You have to differ between the falling observers proper time and proper motion with the global and distant observers proper time.

Now your answer is correct, and I said so in my first reply. However, you said that you did not wanted to know when we SEE the black hole form, but in the last post here you are telling us "I'm asking WHEN the unobservable event of crossing the event horizon happens from the outsider's perspective", but that is the same thing... are you aware of this?

So now let's come to your attitude, you want someone to tell you why are wrong? How do you know that you are wrong??

Secondly, we have given you a definite explanation, it is just that you don't understand that we have given you it...

 

[rjbeery]

Malawi: I understand that seeing and calculating are essentially the same thing (i.e. Relatively-speaking, observing a slowly moving clock and calculating its time dilation are equivalent), but I was making a distinction between "seeing" and "calculating" only because I anticipated that someone was going to say that the body which attempted to cross the event horizon only APPEARED to be frozen in time from out perspective but that this is not what really happens at all. What you are saying is that this "freezing in time" actually IS what occurs from the outside perspective which is my whole point. (On a side note, I find it unusual that you think I have an attitude when I'm presuming that I am wrong on something and asking others to help me )

Here is the same question in a different wording: if, from OUR frame of reference here on Earth, the time that it takes matter to cross an event horizon of any distant black hole is infinite, then how can we realistically postulate that black holes currently exist from our frame? In other words if the "creation time" of a black hole from a given frame is infinite how can they exist at all in that frame?

 

[malawi_glenn]

we will still notice their gravitational effects...

 

[rjbeery]

Yes, I do not question that. But I question whether or not an event horizon, or even a singularity, is ever created.

 

[malawi_glenn]

Yes, I do not question that. But I question whether or not an event horizon, or even a singularity, is ever created.

yes they do, but not in our reference frame, recall that space and time is relative in Einsteins theories.

 

[rjbeery]

Yes, I should've been clear on this. I question whether or not an event horizon or a singularity are ever created from our frame of reference as outside of the effective sphere of influence of the black hole. If it is true that black holes cannot be created from our frame, then how can we postulate that they currently exist from our frame?

 

[malawi_glenn]

Yes, I should've been clear on this. I question whether or not an event horizon or a singularity are ever created from our frame of reference as outside of the effective sphere of influence of the black hole. If it is true that black holes cannot be created from our frame, then how can we postulate that they currently exist from our frame?

I guess that this is more of a philosophical question, and who has made that claim anyways?

 

[diazona]

If you fell in, there's every reason to expect that you would find a singularity (apart from the obvious problem of, y'know, dying ;-). But if you stay well outside the black hole... I guess you could say an event horizon would be created after an infinite time. (Which is different from never, believe it or not.) It's kind of meaningless to ask about the existence of a singularity from outside the black hole, though, because we could never observe such a thing. The volume inside a black hole's event horizon is a "black box" from anywhere outside it.

Note that although it may take an infinite amount of time (from an external frame) for a black hole's mass to be compressed to the Schwarzschild radius (which is the radius an event horizon would have, if it exists), it takes only a short time for the mass to be compressed close to that radius, and that's really all you need. The gravitational effect at some distance from the black hole is the same whether the mass is actually inside the Schwarzschild radius or not. And since any radiation emitted from close above the Schwarzschild radius undergoes an immense gravitational redshift, the black hole becomes invisible long before it fully collapses.

 

[rjbeery]

diazona: Excellent answer, thank you so much. I understand that, "functionally", the difference between the Schwarzschild radius ever being reached or not is irrelevant to the outside observer, but it seems to me that other apparent problems are resolved if we assume that the singularity is never formed:

1) Relativity holds without any kind of space-time inversion mumbo jumbo (which is the type of response I was expecting)
2) Avoiding information loss does not require any special form of Hawking radiation
3) The world may yet be homeomorphic(?)

I'm not saying this is the way it is, and I'm not saying that you agree with me fully, I'm just appreciative of you directly answering my question and acknowledging my logic.

 

[George Jones]

George: this isn't a question about SEEING anything.

Yes, it is.

I'm not claiming that an event does not happen just because it cannot be observed from some perspective.

It seems to me that this is exactly what you're claiming.

I'm asking WHEN the unobservable event of crossing the event horizon happens from the outsider's perspective (because I think the answer is that it "never" happens, and I want someone to explain why this answer is wrong).

It's not that it never happens, it's that a certain class of observers never sees it happen. Other observers do see it happen. This is completely analogous to my special relativity example. How are the two situations different?

Malawi: I understand that seeing and calculating are essentially the same thing (i.e. Relatively-speaking, observing a slowly moving clock and calculating its time dilation are equivalent), but I was making a distinction between "seeing" and "calculating" only because I anticipated that someone was going to say that the body which attempted to cross the event horizon only APPEARED to be frozen in time from out perspective but that this is not what really happens at all. What you are saying is that this "freezing in time" actually IS what occurs from the outside perspective which is my whole point.

What is the difference between "APPEARED to be frozen" and "actually IS what occurs from the outside perspective"?

from OUR frame of reference here on Earth

What does "frame of reference" mean in general relativity?

Relativity holds without any kind of space-time inversion mumbo jumbo

Contrary to a popular myth,there is no "space-time inversion" inside a black hole.

The world may yet be homeomorphic(?)

I think you have something in mind, but I don't that "homeomorphic" is the term for it. Two topoplogical spaces are homeomorphic if there exists a continuous bijection between them whose inverse is also a continuous bijection.

See also the article "Evidence for the Black Hole Event Horizon,"

http://arxiv.org/abs/astro-ph/0310692.

 

[skeptic2]

But if you stay well outside the black hole... I guess you could say an event horizon would be created after an infinite time. (Which is different from never, believe it or not.)

Maybe not. If the black hole evaporates in finite time but the event horizon forms after an infinite time, I think you could say the event horizon never forms.

 

[rjbeery]

I guess that this is more of a philosophical question, and who has made that claim anyways?

Who has made the claim that black holes currently exist? I thought I was the bold one asking if it was possible that they do not!

George: the universe is not homeomorphic if a singularity is allowed to form. The rest of your post has an argumentative whiff to it and I don't feel like playing; besides, others in this thread have understood my question perfectly and responded accordingly.

 

[malawi_glenn]

Who has made the claim that black holes currently exist? I thought I was the bold one asking if it was possible that they do not!

George: the universe is not homeomorphic if a singularity is allowed to form. The rest of your post has an argumentative whiff to it and I don't feel like playing; besides, others in this thread have understood my question perfectly and responded accordingly.

now you are mixing such general statements with physical statements. Where is the argument that black holes have been formed in our frame of reference? There is a difference in claiming that black holes exists and that they have been formed in our frame of reference. You have to understand this.

 

[rjbeery]

OK, fair enough...so when Astrophysicists speculate about the giant black hole that exists in the center of the Milky Way they are actually referring to the black hole that does not currently exist but will form after an infinite amount of time has passed (from our perspective)? Something tells me that this is not popular sentiment.

 

[malawi_glenn]

OK, fair enough...so when Astrophysicists speculate about the giant black hole that exists in the center of the Milky Way they are actually referring to the black hole that does not currently exist but will form after an infinite amount of time has passed (from our perspective)? Something tells me that this is not popular sentiment.

welcome to the world of relativity!

 

[skeptic2]

George,

Could you explain the diagram in post #4? It seems to me if the red dashed line is the light like asymptote for A's world line, then it and A's world line should start at the same point, for instance (0,0). That way as A accelerates the slope of his world line would decrease and approach that of the red line.

"Observer A never sees B or any other observer, cross the light like asymptote. Do you see why? Does mean that B never crosses the red dashed line, and that events above the asymptote don't exist, never happen?"

Frankly no, I don't see why.

 

[malawi_glenn]

George,

Could you explain the diagram in post #4? It seems to me if the red dashed line is the light like asymptote for A's world line, then it and A's world line should start at the same point, for instance (0,0). That way as A accelerates the slope of his world line would decrease and approach that of the red line.

"Observer A never sees B or any other observer, cross the light like asymptote. Do you see why? Does mean that B never crosses the red dashed line, and that events above the asymptote don't exist, never happen?"

Frankly no, I don't see why.

one of the observers will fall outside the light cone of the other, which means that lines connecting them would represent signals which travels faster than the speed of ligth.

This is just standard SR space time diagram procedure.

 

[skeptic2]

one of the observers will fall outside the light cone of the other, which means that lines connecting them would represent signals which travels faster than the speed of ligth.

This is just standard SR space time diagram procedure.

This is just standard SR space time diagram procedure.[/QUOTE]

A standard SR space time diagram has time for its vertical axis labeled and space is by the horizontal axis. The world line of an object located at (0,0) and not moving in a spatial direction would continue vertically along the vertical axis. A beam of light originating at (0,0) would follow the red dashed line. Objects whose world line has a slope greater than that of the red line are traveling slower than light and those whose world line is closer to horizontal than the red line are traveling faster than light. Objects above the red line are inside the light cone and those below it are outside. Is this what you mean by a standard SR space time diagram?

The diagram that was posted however has the world line of A outside its own light cone. B starts out outside the light cone but then crosses into it. Whether A sees B is not dependent on A's light cone but on B's light cone.

malawi, do you have a different understanding of that diagram?

 

[rjbeery]

I agree with Skeptic2, I couldn't make heads or tails of the diagram posted but I do understand Minkowski spacetime diagrams just fine...

 

[George Jones]

A standard SR space time diagram has time for its vertical axis labeled and space is by the horizontal axis.

Yes.

The world line of an object located at (0,0) and not moving in a spatial direction would continue vertically along the vertical axis.

Yes.

A beam of light originating at (0,0) would follow the red dashed line.

Yes.

Objects whose world line has a slope greater than that of the red line are traveling slower than light and those whose world line is closer to horizontal than the red line are traveling faster than light.

Yes.

Objects above the red line are inside the light cone and those below it are outside.

Yes and no.

The red dashed line is part of a light cone for an event at (0,0), but worldlines for observers do not have to go through this event. For example, if you're here on Earth and I'm near Alpha Centauri (possibly moving with respect to you) at t=0 for you, my worldline does not go through (0,0) of your coordinates. What does my worldline look like?

The diagram that was posted however has the world line of A outside its own light cone. B starts out outside the light cone but then crosses into it.

A lightcone for an observer always starts at an event on the observer's worldline. Take any event on A's worldline, and draw a light cone starting at this event. As t increases, A's worldline never leaves this lightcone. Take any event on B's worldline, and draw a light cone starting at this event. As t increasdes, B's worldline never leaves this light cone. The dashed line isn't part of any light cone for A, but it is part of a light cone for B, the light cone that starts at the point of intersection of the dashed line and B's worldline.

More on what A sees in another post (maybe tomorrow). Keep asking questions.

 

[rjbeery]

With respect, George, your posted picture serves little purpose to this thread. I would be surprised if someone claimed that events happening outside of their light cones "do not exist", which seems to be what you were trying to refute. As an aside, I think the introduction of the red dashed asymptotic line was confusing. Was it necessary for your point?

 

[ZikZak]

With respect, George, your posted picture serves little purpose to this thread. I would be surprised if someone claimed that events happening outside of their light cones "do not exist", which seems to be what you were trying to refute. As an aside, I think the introduction of the red dashed asymptotic line was confusing. Was it necessary for your point?

What he was attempting to get you to work out for yourself was that an accelerating observer observes an event horizon form behind him. The curved line is the accelerating observer. The straight line is an inertial observer who falls through the accelerating observer's Rindler horizon (the dashed red line). At that point, the accelerating observer is no longer within the inertial observer's future light cone at all: no signal, including light, can ever reach him from there. Just as there is a region near a black hole from which signals can never reach a different class of observers.

The Rindler horizon (the dashed red line in that diagram) is a real event horizon in the reference frame of the accelerating observer, including substantially all the properties of a Schwarzschild horizon, including the fact that in this frame, time near it dilates to such a degree that nothing is ever observed to fall through it.

That being the case, your argument would then amount to questioning whether the inertial observer exists after he falls through some other observer's Rindler horizon, to which the obvious answer is "yes."

 

[rjbeery]

ZikZak: I understand now, thanks for clarifying, but doesn't this presume that the constant acceleration of A to a velocity approaching "infinity"* is possible? I am not doubting the mathematics of black holes, just the application of them in reality...

* By velocity approaching "infinity" I mean of course non-zero mass velocity approaching arbitrarily close to the speed of light.

 

[ZikZak]

ZikZak: I understand now, thanks for clarifying, but doesn't this presume that the constant acceleration of A to a velocity approaching "infinity"* is possible? I am not doubting the mathematics of black holes, just the application of them in reality...

* By velocity approaching "infinity" I mean of course non-zero mass velocity approaching arbitrarily close to the speed of light.

No, it assumes nothing but constant proper acceleration. Velocity makes no difference. There is no such thing as absolute velocity anyway.

In future, if you mean to say "speed of light," please say "speed of light" or "c," not "infinity."

 

[Dmitry67]

Yes, I do not question that. But I question whether or not an event horizon, or even a singularity, is ever created.

EVER is an observer-dependent thing, so the question is not correct.

 

[Naty1]

"But I question whether or not an event horizon, or even a singularity, is ever created."

Well if you mean " Is there incontrovertible proof that black holes and singularities inside them exist", the answer is "no"...

but most physicts believe that to be true based on theory and indirect experimental observations.

As others have noted, the existence/observation of an event horizon is relative...analogous to "who is moving, you or me?"...

Kip Thorne's BLACK HOLES AND TIME WARPS book provides detailed explanations of event horizons...a few excerpts that might help the poster:

General relativity insists that, if one falls into a black hole, one will encounter nothing at the event horizon except (continued) spacetime curvature.

And the black holes birth...as seen from outside, the implosion slowed and became frozen (in time) at the critical circumference, but as seen by someone on the stars surface (riding the implosion inward) the implosion did not freeze at all...The stars surface shrank right through the critical circumference and on inward...

..That the implosion freezes forever as measured in the statric external frame but continues rapidly on past the freezing point as measured in the frame of the stars surface was extremely hard to comprehend...

 

[skeptic2]

George,

Not being a physicist I didn't recognize your diagram. Please accept my apologies.

 

[rjbeery]

Well I didn't mean to make this a contentious thread. I'm not defending some radical idea that black holes don't exist. However, most Relativity-related responses PRESUME that the black hole already exists in order for there to be an applicable frame in which crossing the event horizon is possible.

Again, I don't disagree with the math. For the most part I think I understand GR theory as it relates to black holes, and my problem is pretty simple:

If a Physicist believes that a singularity exists somewhere out in deep space today, then when did the singularity form? This is not an attack on science or something. I am looking at the math involved, and to me it appears that in our frame of reference the singularity forms after an infinite amount of time. In other words, in my mind there is a logical disconnect between accepting the Schwarzschild metric and also claiming that singularities exist today.

I was hoping someone would help me understand why this is not the case but this has not happened yet. (And this may be my fault in not comprehending the answer so please try again!)

 

[malawi_glenn]

If a Physicist believes that a singularity exists somewhere out in deep space today, then when did the singularity form? This is not an attack on science or something. I am looking at the math involved, and to me it appears that in our frame of reference the singularity forms after an infinite amount of time. In other words, in my mind there is a logical disconnect between accepting the Schwarzschild metric and also claiming that singularities exist today.

This "when" in observer dependent..

we can measure the effect of BH's..

 

[ZikZak]

If a Physicist believes that a singularity exists somewhere out in deep space today, then when did the singularity form?

Your question is ill posed. There is no absolute "when" for anything in Relativity. Observers disagree on which set of events constitutes which moment in time. That is why it is called "spacetime."

Physicists believe that singularities exist NOT "somewhere out in deep space," but rather "somewhere out in deep spacetime." The particular locations in spacetime are inaccessible unless you are willing to pass through the event horizon.

But to answer the ill-posed question: not at any time given on your watch here on Earth. But that fact is irrelevant to its existence. The coordinate system containing your watch is separated by an enormous amount of curvature from it. You are insisting on being told the longitude of the north pole, and claiming that since we cannot provide it, it cannot exist anywhere.