On the nature of the infinite fall toward the EH

[Austin0]

Aaargh... I'm still not being clear enough... Zeno time is not "explicitly dilated" because it's a coordinate time so doesn't "dilate" - dilation is a statement about the ratio between two amounts of proper time, not coordinate time.

Is there any reason to take the Schwarzschild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space?
(IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason - there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates).

Actually I misspoke. It is Achilles' time which is dilated within the context of Pervects conditions if we add the condition that Achilles' velocity is constant.
I really just meant that time dilation was in effect within the stated conditions and coordinates

And yes I am quite aware of the meaning of dilation and it is the ratio of rates or intervals of two different clocks. In a recent post I made the simple statement that time dilation was inherently relative. Self evidently true for exactly this reason. It is meaningless applied to a single clock. Like the term length contraction or the word faster. It intrinsically requires and implies a comparison.
But somehow I got a bunch of flack from several people telling me I was wrong.
??

I am not convinced that Sc coordinates are necessarily preferred or correct. I am still just learning their subtleties and details and trying to synthesize a logically coherent structure up to the horizon. My exception to this analogy was purely logical. You all may be ultimately right about Sc coords and the horizon but this use of Zeno added nothing of logical probative value to the debate and was actually misleading in it's subtle reframing of Zeno.

 

[Austin0]

Austin0 was asking for some more detailed math. I'd suggest looking at Caroll's GR lecture notes online.

I'll also add that while Caroll's online notes are perfectly fine (one can't trust everything online, Caroll's online notes are drafts of a book by a physics profesor that was later published. From my POV the main advantage of them is that they're free).

I could also quote similar statements from some of my other GR textbooks, (i.e. Caroll is not an isolated occurrence among textbooks). However, I think it would be better from a pedagogical point of view if interested people went out and found their own textbooks if they don't like Caroll (though I can't think of any valid reason for not liking Caroll).


But - onto Caroll:



These just the words - the actual calculation consists of solving for the trajectory of the worldline. If one does this in the usual format, one doesn't even have to integrate the length of the worldline to get proper time, instead one solves the geodesic equations to find t(tau) and r(tau).

One can then observe directly that the event horizion is reached at a finite tau, even though t(tau) is infinite.

If one is willing to just take the limit as r approaches the event horizon one can do this all in Schwarzschild coordinates. This is even observable. One can say that's it's possible to observe the limiting sequence of proper time as one approaches the event horizon from outside, and observe that the limit is finite.

To go futher and carry the trajectory smoothly through the event horizon, one needs coordinates that are better behaved, which is what Caroll does next.

The point of the Zeno analogy is to demonstrate a simple example of how a coordinate time can be infinite while the time actually measured on a clock is finite.

Specifically, zeno time is infinite, while as far as Achilles is concrned, there's a finite time at which he passes the tortise.

I'm afraid I don't understand the difficulties people are having in understanding the analogy. It could be my fault, sometimes I "leap ahead' too far when I write.

The way you demonstrate that the proper time on an infalling clock is actually finite rigorously is that you calculate it.

Post #12 in this thread
https://www.physicsforums.com/showpost.php?p=4185014&postcount=12

(and a later post after it, #13)



One can see that at tau = -8/3 , which is finite, r=4 so one is at the event horizon. Furthermore, t(tau) is infinite because of one of the ln(...) terms.

To verify this is a solution one needs to demonstrate that said trajectory satisfies the geodesic equations. You'll find them in my post #12, Caroll's GR lecture notes, for starters.

The idea behind the Zeno analogy isn't to "prove" anything - that's what textbooks are for. The idea behind the Zeno analogy is to illustrate how t can be infinite and tau can be finite in a simple, easy-to-understand example.

WEll, the Zeno analogy does prove one thing. It demonstrates that just because you have a time coordinate t going to infinity doesn't prove that something doesn't happen. It's an example of how t going to infinity can be the result of a poor choice of coordinates. It's a counterexample to the argument "t goes to infinity, therefore it can't happen".

Historically, I do believe that the "tortise coordinate" was named after the tortise in Zeno's paradox, but I haven't seen anything really detailed on this in textbooks. There was something in Scientific American about it a long time ago as well, I think.

You are here demonstrating the validity of the Schwarzschild conclusion.

I do understand the math processes and reasoning behind this. Integrating proper time is not difficult to grasp , certainly not after SR
Now that I understand that your statement of Zeno time was with the expectation that it was assumed Achilles' proper velocity was constant even though it decreased in Zeno's frame then of course the situations are effectively identical.
Of course this means that this adaptation is no clearer or more persuasive than the original Sc scenario.
I have never said that the infaller doesn't reach the horizon in some relatively short proper time on its clock.I have questioned the assertion that this does not transform to
some tremendously distant future time in the frame of the distant observer.
This seems to call into question the Sc coordinates not only in the immediate vicinity of the horizon but effectively throughout the system. How or why a system which is empirically verified within a certain range of the domain would become totally unreliable (pathological ;-) ) in another part.

 

[Dale]

there is , in Pervect's stated conditions, absolutely no foundation or justification for an inference or assertion that Achilles' clock does not run at the same rate as Zeno's.
...
Explicitly as Zeno time goes to infinity so does Achilles'

Consider the inertial frame where Achilles is at rest. In this frame the turtle's worldline is given by (t,100-vt) where v is the relative velocity between Achilles and the turtle. So in this frame Achilles is a distance d=100-vt behind the turtle. The definition of Zeno time, n, given is d=100/2^n. Substituting in and simplifying we get the following transform between the inertial frame and Zeno coordinates:
n=log_2 \left( \frac{100}{100-vt} \right)

Taking the derivative of Zeno coordinate time wrt Achilles proper time we get
\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)} \neq 1
So Achilles' clock does not run at the same rate as Zeno coordinate time.

Taking the inverse transform we get
t=\frac{100}{v}(1-2^{-n})
so
\lim_{n\to \infty } \, t = \frac{100}{v}
So as Zeno coordinate time goes to infinity Achilles proper time does not.

 

[Dale]

this use of Zeno added nothing of logical probative value to the debate and was actually misleading in it's subtle reframing of Zeno.

Saying that it added nothing is one thing, but saying it is misleading is accusatory and untrue. It is, as I think is now established, a valid and close analogy in many respects. The fact that the parallels escaped you at first doesn't make it misleading or deceptive in any way.

 

[Dale]

Is there any reason to take the Schwarzschild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space?
(IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason - there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates).

Excellent point. It highlights the real reason for picking any coordinate system: ease of computation. That is true in all branches of physics, not just GR.

 

[Austin0]

Consider the inertial frame where Achilles is at rest. In this frame the turtle's worldline is given by (t,100-vt) where v is the relative velocity between Achilles and the turtle. So in this frame Achilles is a distance d=100-vt behind the turtle. The definition of Zeno time, n, given is d=100/2^n. Substituting in and simplifying we get the following transform between the inertial frame and Zeno coordinates:
n=log_2 \left( \frac{100}{100-vt} \right)

Taking the derivative of Zeno time wrt Achilles time we get
\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)} \neq 1
So Achilles' clock does not run at the same rate as Zeno's.

Taking the inverse transform we get
t=\frac{100}{v}(1-2^{-n})
so
\lim_{n\to \infty } \, t = \frac{100}{v}
So as Zeno time goes to infinity Achilles time does not.

Yes this is fine . But it is based on an assumption of a constant v in Achilles' frame ,,,,yes? You are not deriving either the time dilation or the constant v from the stated Zeno time parameters alone.
so according to Nugatory I get that it was supposed to be understood implicitly that that was a given but everything i said was clearly within the context of what Pervect actually outlined.

 

[zonde]

The math is what the theory uses to make testable predictions for the scientific method. If you do not understand the math then you do not understand the theory well enough to address it with the scientific method. Hence the disagreements.

I can evaluate if prediction is scientifically testable even without knowing how it was derived.

This is simply false. All experimental measurements are invariants. If they were not invariant then you could always construct a paradox of the form "Dr. Evil builds a bomb which is detonated iff device X measures Y, device X measures Y under coordinate system A, but Z under coordinate system B. Therefore the bomb explodes in one coordinate system but not in the other."

Two different coordinate systems may disagree on the meaning of the measurement, e.g. they may disagree whether or not the rod is accurately measuring length, but they must agree on what value is measured.

There is observer A who is using coordinate system K and there is observer B who is using coordinate system K'. Now observer A observes event X but observer B observes event X'. How do they find out if event X and event X' is the same event?

OK, so considering all other mainstream physics theories as well. What would prevent the formation of a horizon?

Degeneracy of matter.

 

[zonde]

True, but this is not the the only case of physical theories including untestable predictions. To better understand a theory (and its limits), it is useful to understand what a theory predicts for such things. GR + known theories of matter (classically) predict continued collapse. GR must be modified in some way to avoid this.

I believe we can make untestable extrapolations of the theory for educational purposes - to make the explanations more colorful. But then confirmation of the theory is still based on testable things. And if we have any doubt about the theory then it needs to address only the things within limits of testability.

Say we address hypothesis of runaway collapse only to the limits of "frozen star".

Fine - you agree that GR must be modified to get the result you want. What you call laws being affected by something like Newtonian potential is a fundamental violation of the principle of equivalence, which is built in (as a local feature) to the math and conceptual foundations of GR. Note, for gravity to be locally equivalent to acceleration, a direct consequence is that free fall must have locally the same physics everywhere. (Otherwise, observing what happens inside a (small) free falling system would locally distinguish gravity from corresponding acceleration.)

Yes

 

[zonde]

There's growing experimental evidence for the existence of event horizons. Basically, black hole candidates are very black, and don't appear to surface features.

WHen matter falls onto a neutron star, the surface heats up and re-radiates. The spectra signature is rather distinctive, also there are "type 1 x ray bursts".

Black hole candidates do not appear to have any such "surface" features, and it's already very difficult to explain by any means other than an event horizon how they can suck in matter without , apparently re-radiating anything detectable.

For the details, see

See for instance http://arxiv.org/pdf/0903.1105v1.pdf

and check for other papers by Naryan in particular.

Yes, this is a good argument. Thanks for the paper. I will read it.

Minor point. This is not experimental evidence. This is observational evidence. We have no control over conditions.

 

[Dale]

Yes this is fine . But it is based on an assumption of a constant v in Achilles' frame ,,,,yes?

Yes, that is a standard part of Zeno's paradox. See the second sentence of the description here:

http://en.wikipedia.org/wiki/Zeno's_paradoxes#Achilles_and_the_tortoise

 

[Dale]

I can evaluate if prediction is scientifically testable even without knowing how it was derived.

Yes, but if you don't understand how it was derived then you don't understand under what conditions it is logically implied by the things that have been tested.

Furthermore, that objection doesn't apply to event horizons. The predictions about what happens at the horizon can be tested. Signals from the test cannot reach us here since we are outside its future light cone, but we are also outside the future light cone of many other experiments of things that I am sure you would agree are testable.

There is observer A who is using coordinate system K and there is observer B who is using coordinate system K'. Now observer A observes event X but observer B observes event X'. How do they find out if event X and event X' is the same event?

They transform one coordinate to the other chart.

Degeneracy of matter.

And what would cause matter to become degenerate at the horizon?

 

[zonde]

Yes, but if you don't understand how it was derived then you don't understand under what [STRIKE]conditions[/STRIKE] assumptions it is logically implied by the things that have been tested.

My replacement.

Now the way I wrote it, if we can't test prediction we can't find out if assumptions hold. But if we can't find out that then the derivation is not very interesting.

Furthermore, that objection doesn't apply to event horizons. The predictions about what happens at the horizon can be tested. Signals from the test cannot reach us here since we are outside its future light cone, but we are also outside the future light cone of many other experiments of things that I am sure you would agree are testable.

Test is when we do something and then we learn something about the thing we did.
It's action and feedback. If you leave out feedback (or learning) part it's not a test.

They transform one coordinate to the other chart.

And then you compare coordinates of two events, right? You identify events by their coordinates. So you can't get away just by using invariants.

And what would cause matter to become degenerate at the horizon?

You are begging the question. If we talk about event horizon then we imply that BH can form as a result of runaway gravitational collapse. So there is no point asking what will prevent runaway gravitational collapse.

 

[PeterDonis]

You identify events by their coordinates.

No, you identify events by what happens at them, and what happens at them is expressed in terms of invariants. You can express those invariants without even choosing a coordinate chart; coordinate charts are a convenience, not a necessity.

 

[Dale]

My replacement.

Now the way I wrote it, if we can't test prediction we can't find out if assumptions hold. But if we can't find out that then the derivation is not very interesting.

I am fine with that replacement. It doesn't change my point any.

For example, I can have direct evidence of the value of the fine structure constant from my lab today, and I can have direct evidence of the value of the fine structure constant from your lab yesterday, since signals from both experiments can reach me here and now. I cannot have any direct evidence of the value of the fine structure constant in my lab tomorrow because a signal from such an experiment cannot possibly reach me here and now.

However, if I assume that the laws of physics are homogenous then the value of the fine structure constant in my lab tomorrow is logically implied by that assumption and the experimental evidence of its value here today and there yesterday. Furthermore, while we cannot gather any direct evidence of its value here tomorrow we can design experiments that would be sensitive to violations in our assumption of homogeneity. Taken together those can give us strong empirical evidence of something for which we cannot gather data.

Similarly for the event horizon. In this case the assumption is the Einstein equivalence principle. That and all the rest of the laws of physics as we know them imply that events at and beyond the horizon do exist. The evidence that we have supporting GR and the standard model as well as the evidence we have supporting the Einstein equivalence principle, taken together, are good evidence for the existence of the interior of the EH.

Test is when we do something and then we learn something about the thing we did.
It's action and feedback. If you leave out feedback (or learning) part it's not a test.

Yes, I understand that, and was assuming that. Even with that restriction predictions about what happens at the horizon can be tested. You can learn about the tests at and beyond the horizon as long as you are at or beyond the horizon yourself.

And then you compare coordinates of two events, right? You identify events by their coordinates. So you can't get away just by using invariants.

Sure you can. Coordinates are not the only way to identify events. Events are more primitive than coordinates, they are points in the manifold, i.e. geometric objects independent of coordinates.

You are begging the question. If we talk about event horizon then we imply that BH can form as a result of runaway gravitational collapse. So there is no point asking what will prevent runaway gravitational collapse.

You are correct, I was begging the question of the existence of the horizon. However, I was not trying to ask about the horizon but about your claim regarding degeneracy, so let me rephrase:

And what would cause matter to become degenerate during gravitational collapse and prevent a horizon from forming?

 

[zonde]

I am fine with that replacement. It doesn't change my point any.

For example, I can have direct evidence of the value of the fine structure constant from my lab today, and I can have direct evidence of the value of the fine structure constant from your lab yesterday, since signals from both experiments can reach me here and now. I cannot have any direct evidence of the value of the fine structure constant in my lab tomorrow because a signal from such an experiment cannot possibly reach me here and now.

However, if I assume that the laws of physics are homogenous then the value of the fine structure constant in my lab tomorrow is logically implied by that assumption and the experimental evidence of its value here today and there yesterday. Furthermore, while we cannot gather any direct evidence of its value here tomorrow we can design experiments that would be sensitive to violations in our assumption of homogeneity. Taken together those can give us strong empirical evidence of something for which we cannot gather data.

Similarly for the event horizon. In this case the assumption is the Einstein equivalence principle. That and all the rest of the laws of physics as we know them imply that events at and beyond the horizon do exist. The evidence that we have supporting GR and the standard model as well as the evidence we have supporting the Einstein equivalence principle, taken together, are good evidence for the existence of the interior of the EH.

So basically your argument is that it is not reasonable to expect sudden breakdown of equivalence principle. So if we test equivalence principle to further and further limits and it holds just as well then our confidence grows that it won't break at even further limits, right?

Yes, I understand that, and was assuming that. Even with that restriction predictions about what happens at the horizon can be tested. You can learn about the tests at and beyond the horizon as long as you are at or beyond the horizon yourself.

Hmm, let me rephrase my statement. We can't falsify prediction of event horizon. If prediction about event horizon is false then we of course can't appear at event horizon.

And more down to Earth objection to that. I am not sure it is a valid test when an experimenter should become part of the experimental setup. Say we can reason that it is possible to test if there is life after death - just kill yourself and you will find out.

Sure you can. Coordinates are not the only way to identify events. Events are more primitive than coordinates, they are points in the manifold, i.e. geometric objects independent of coordinates.

Yes, events are more primitive than coordinates. But how does this make a point about invariants identifying events?

And I want to add that while we might try to identify events by other means than coordinates we can uniquely identify events only by coordinates.

For example, when you write a paper you put at the end references. And references are expressed as when and where the paper was published. Even title is optional. Well we have one invariant - name of the author. But it would be possible to find the paper even without the author.

And what would cause matter to become degenerate during gravitational collapse and prevent a horizon from forming?

It's just an observation that there is such a thing. Well I have some speculations about the cause but I am not sure you want to know them as I suppose you want arguments not explanations. And in that case it goes as far as observations.

 

[Dale]

So basically your argument is that it is not reasonable to expect sudden breakdown of equivalence principle. So if we test equivalence principle to further and further limits and it holds just as well then our confidence grows that it won't break at even further limits, right?

Yes. We have physical laws that have been tested to reasonable levels of accuracy (GR and SM) and we have an assumption that has also been tested to reasonable levels of accuracy (EEP). Together they imply the existence of events on the horizon and inside. It certainly is possible that further testing will falsify one or more of those, but until such tests are available, the position with the best empirical support is the standard one.

In order to believe otherwise you must reject an assumption or a law for which we currently have empirical support and insert an alternative law or assumption for which we do not have any specific empirical support.

Hmm, let me rephrase my statement. We can't falsify prediction of event horizon. If prediction about event horizon is false then we of course can't appear at event horizon.

True, but we could falsify GR's prediction of a horizon. If the horizon doesn't behave exactly how GR says it does then GR's prediction is falsified. It is true that we could always make a different theory with horizons elsewhere, but it wouldn't be GR as we know it.

And more down to Earth objection to that. I am not sure it is a valid test when an experimenter should become part of the experimental setup. Say we can reason that it is possible to test if there is life after death - just kill yourself and you will find out.

I think that is a valid test for life after death. But, since I will eventually have that test forced upon me, I personally am not inclined to pursue it further at this time

However, I don't think that tests of the EH fall into that same category. I.e. I would assume that the experimental test for the EH would involve some clocks and some signal receivers and emitters and perhaps some devices to measure tidal gravity. The experimenter wouldn't be any part of that. But, as with all experiments, in order to learn about the outcome the experimenter must be in the future light cone of the experiment. That requires crossing the EH also.

And I want to add that while we might try to identify events by other means than coordinates we can uniquely identify events only by coordinates.

There is only one event on the worldline of the center of my watch where its proper time reads 12:48 pm Dec. 22, 2012. That event is uniquely identified by the invariant description just given (specified worldline and specified proper time).

It's just an observation that there is such a thing. Well I have some speculations about the cause but I am not sure you want to know them as I suppose you want arguments not explanations. And in that case it goes as far as observations.

So there is no empirical support for your position. You just have an aesthetic aversion to the idea of an EH and so, since it doesn't sit well with you, you are just making stuff up.

Btw, matter degeneracy won't stop the horizon from forming. It may be degenerate, but as long as it has mass it will curve spacetime.

 

[zonde]

So there is no empirical support for your position. You just have an aesthetic aversion to the idea of an EH and so, since it doesn't sit well with you, you are just making stuff up.

Well as I know at least electrons in metals are degenerate.
From wikipedia article about Fermi-Dirac statistics:
"Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. For example, the electronic heat capacity of a metal at room temperature seemed to come from 100 times fewer electrons than were in the electric current.[3] It was also difficult to understand why the emission currents, generated by applying high electric fields to metals at room temperature, were almost independent of temperature."

But usually degeneracy of matter is modeled as pressure and that does not seem quite right to me.
See here - Degenerate matter

 

[Dale]

Again, how would degeneracy do anything to prevent a horizon. Degeneracy doesn't magically make any mass or energy disappear, so the curvature will remain.

 

[PeterDonis]

But usually degeneracy of matter is modeled as pressure and that does not seem quite right to me.

What else would you model it as? It's true that degeneracy pressure doesn't arise kinetically (i.e., it's independent of temperature), but so what? The effect at the classical level is the same: the material resists being compressed. That's what "pressure" is, from the standpoint of the stress-energy tensor: resistance to compression.

 

[Austin0]

Quote by Austin0 View Post

Yes their coordinate velocity is reducing but in the Zeno system a la Pervect there is no reason that Achilles proper velocity would not also decrease.

Achilles' proper velocity is clearly constant.
.

I can calculate it explicitly if you like, but it is exceedingly well-founded.
Achilles' proper velocity is clearly constant.
.

Quote by Austin0 View Post

Yes this is fine . But it is based on an assumption of a constant v in Achilles' frame ,,,,yes?

Yes, that is a standard part of Zeno's paradox. See the second sentence of the description here:

http://en.wikipedia.org/wiki/Zeno's_...d_the_tortoise .

SO it appears that your assertion that Achilles velocity is constant is based, not on calculation, but on your interpretation of the explicit statements of the classical scenario...yes? Yes I am aware it is a part of the classical paradox as I mentioned in my initial post

In the first case (Zeno) as the distance incrementally reduces, the velocity of Achilles remains constant. So for each reduction in distance, the time for the next reduction in distance becomes shorter.
.

But in the classical statement it is evident that the stated constant velocity is in the
frame of the ground. I.e. Zeno coordinates.
Do you disagree? What other possible frame for such a statement do you propose?

So when Pervect redefines Achilles velocity as non-uniform in the Zeno frame it is now ,not necessarily a logical conclusion that Achilles velocity is constant in any other frame, as no other frame was defined .

In Zeno coordinate time the time for the next reduction is constant, by definition. So the Zeno coordinate velocity in fact reduces.

It is the proper time which reduces. And the velocity in some unspecified inertial coordinate system which remains constant..

According to Pervect's explicit description it seems to follow that the Zeno coordinate system is not accelerating. That it would be in a state of uniform motion relative to and measured by any inertial frame. Do you disagree??

So if Achilles is in non- uniform motion (accelerating) as measured in the Zeno frame how do you propose that it is measured as uniform (inertial) in anyone of those other inertial frames?

So what is the basis ,in the classical description, for your assumption of constant velocity for Achilles ?

What unspecified inertial frame ?

Without a valid basis for an assumption of constant velocity there is no basis for calculating a different time rate for Achilles either, is there?

Saying that it added nothing is one thing, but saying it is misleading is accusatory and untrue. It is, as I think is now established, a valid and close analogy in many respects. The fact that the parallels escaped you at first doesn't make it misleading or deceptive in any way.

It was not that the parallels escaped me or the math was too complex it was purely a question of logic and applicability.
I certainly never thought for a moment there was deception on Pervects part.

OTOH wouldn't you agree that the original is easily and unambiguously falsified by empirical demonstration? As simple as getting up and catching up to a friend.

Wouldn't you also agree that creating an association between the two cases seems to imply that the Sc case is equally unambiguously false??

But isn't the amended Zeno case now as unfalsifiable in the real world as the Sc scenario?
As ambiguous??
Do you think that if Achilles started out in Zeno's time with Pervect's conditions he would have caught the tortoise by now in our frame (Zeno coordinates)??

 

[zonde]

Again, how would degeneracy do anything to prevent a horizon. Degeneracy doesn't magically make any mass or energy disappear, so the curvature will remain.

"Degeneracy of matter" does not tell us why it happens. It just tells that it happens.

Look, Pauli exclusion principle says that no two identical fermions can occupy the same quantum state. It does not tell us what would happen if two identical fermions would try to occupy the same quantum state. Currently we have no idea why the nature behaves that way.

And there is still some room for interpretation. QM gives quite abstract definition for "quantum state". From wikipedia article about quantum state:
"In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the state vector."

Well, we consider particles to be physical entities but quantum state is defined as mathematical entity. So it seems that Pauli exclusion principle is not very rigorous. This leaves (at least for me) the question open how we should model quantum state in real space (space-time).

 

[zonde]

What else would you model it as? It's true that degeneracy pressure doesn't arise kinetically (i.e., it's independent of temperature), but so what? The effect at the classical level is the same: the material resists being compressed. That's what "pressure" is, from the standpoint of the stress-energy tensor: resistance to compression.

I would model it as a slipping away from the trap and not as a resistance to the trap. Let's say it this way - degenerate matter can not be contained.

 

[Dale]

"Degeneracy of matter" does not tell us why it happens. It just tells that it happens.

Look, Pauli exclusion principle says that no two identical fermions can occupy the same quantum state. It does not tell us what would happen if two identical fermions would try to occupy the same quantum state. Currently we have no idea why the nature behaves that way.

And there is still some room for interpretation. QM gives quite abstract definition for "quantum state". From wikipedia article about quantum state:
"In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the state vector."

Well, we consider particles to be physical entities but quantum state is defined as mathematical entity. So it seems that Pauli exclusion principle is not very rigorous. This leaves (at least for me) the question open how we should model quantum state in real space (space-time).

OK. I am fine with all of this, but I am still missing the connection with how any of this prevents the formation of the EH. I could see it preventing the formation of the singularity, but not the horizon.

 

[Dale]

SO it appears that your assertion that Achilles velocity is constant is based, not on calculation, but on your interpretation of the explicit statements of the classical scenario...yes?

Yes.

But in the classical statement it is evident that the stated constant velocity is in the frame of the ground. I.e. Zeno coordinates.

I don't think that the "classical statement" ever explicitly introduced any coordinates. That was pervect's idea, taking the familiar statement of Zeno's paradox and using it to define a coordinate time. So I would not associate Zeno coordinates with the frame of the ground since "frame of the ground" usually indicates an inertial frame and Zeno coordinats are non-inertial.

Do you disagree? What other possible frame for such a statement do you propose?

Any inertial frame. If it is true in one inertial frame then it is true in all.

So when Pervect redefines Achilles velocity as non-uniform in the Zeno frame it is now ,not necessarily a logical conclusion that Achilles velocity is constant in any other frame, as no other frame was defined .

Achilles motion is inertial. That is an invariant fact which is true in all coordinate systems and does not change with pervect's introduction of Zeno coordinates. Given that his motion is inertial (frame invariant) then his velocity (frame variant) is constant in any inertial frame.

According to Pervect's explicit description it seems to follow that the Zeno coordinate system is not accelerating. That it would be in a state of uniform motion relative to and measured by any inertial frame. Do you disagree??

Yes, I disagree quite strongly. The Zeno coordinate system is decidedly non-inertial. In fact, from my post 393 you can easily see that the metric in the Zeno coordinates is:
ds^2=-c^2 \left( \frac{(100-vt) ln(2)}{v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2

This metric is clearly different from the metric in an inertial frame.

So if Achilles is in non- uniform motion (accelerating) as measured in the Zeno frame how do you propose that it is measured as uniform (inertial) in anyone of those other inertial frames?

Again, his motion is inertial in all frames, that is an invariant which follows directly from the original description and is not changed by the introduction of any coordinate system. The Zeno coordinates are non-inertial and therefore it is no surprise that he is accelerating in the Zeno frame and not accelerating in any inertial frame.

OTOH wouldn't you agree that the original is easily and unambiguously falsified by empirical demonstration? As simple as getting up and catching up to a friend.

Wouldn't you also agree that creating an association between the two cases seems to imply that the Sc case is equally unambiguously false??

Yes, that is the whole point of the analogy.

However, let's be careful about exactly the way in which the original is false. The original is correct in its description of all events up to (but not including) the event where Achilles catches up with the turtle. Where it fails is if it asserts anything about events at or beyond that point. Similarly with SC, SC is correct in its description of all events up to (but not including) the EH. Where it fails is if it asserts anything about events at or beyond the EH.

 

[stevendaryl]

"Degeneracy of matter" does not tell us why it happens. It just tells that it happens.

Look, Pauli exclusion principle says that no two identical fermions can occupy the same quantum state. It does not tell us what would happen if two identical fermions would try to occupy the same quantum state. Currently we have no idea why the nature behaves that way.

I don't understand why degeneracy would have any relevance to black hole event horizons. Are you thinking that matter falling toward the event horizon would run out of states, and so the Pauli exclusion principle would prevent a collection of Fermions from falling further? If that's what you're thinking, then that's not correct. Nothing special happens at the event horizon that would force matter to become degenerate.

 

[PAllen]

"Degeneracy of matter" does not tell us why it happens. It just tells that it happens.

Look, Pauli exclusion principle says that no two identical fermions can occupy the same quantum state. It does not tell us what would happen if two identical fermions would try to occupy the same quantum state. Currently we have no idea why the nature behaves that way.

And there is still some room for interpretation. QM gives quite abstract definition for "quantum state". From wikipedia article about quantum state:
"In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the state vector."

Well, we consider particles to be physical entities but quantum state is defined as mathematical entity. So it seems that Pauli exclusion principle is not very rigorous. This leaves (at least for me) the question open how we should model quantum state in real space (space-time).

What does any of this have to do with horizon formation for millions of galactic center BH, each with mass of millions to billions of suns. The issue here is that matter density for the aggregate at SC radius is much less than stellar atmosphere density, let alone stellar centers or neutron stars. How does degeneracy even become relevant?

 

[PeterDonis]

Well, we consider particles to be physical entities but quantum state is defined as mathematical entity. So it seems that Pauli exclusion principle is not very rigorous. This leaves (at least for me) the question open how we should model quantum state in real space (space-time).

Quantum states *are* modeled using real spacetime; spacetime position is part of the description of a quantum state. The Pauli exclusion principle does not prevent two fermions of the same particle type from being in the same spin state at two different spacetime positions; it only prevents two fermions of the same particle type from being in the same spin state at the *same* spacetime position.

Actually, even that is not really the right way to say it. The Pauli exclusion principle as we have stated it is not a fundamental law; the fundamental law is that fermion wave functions are antisymmetric under particle exchange, whereas boson wave functions are symmetric. If I have a boson, say a spin-0 particle, at spacetime position x, and another spin-0 particle of the same particle type at spacetime position y, the wave function is symmetric under exchange of those two particles. But if I have a fermion in a definite spin state, say a spin-up electron, at spacetime position x, and another spin-up electron at spacetime position y, the wave function is antisymmetric (i.e., it changes sign) under exchange of those two particles.

The Pauli exclusion principle, which says that the wave function is identically zero if x = y, is an obvious consequence of the antisymmetry. However, it's not the only consequence; another consequence is that as x and y get closer together, the amplitude of the wave function decreases. That's what causes degeneracy pressure.

But all of that is below the level that GR models anyway. GR doesn't care about the microscopic details of matter; all it cares about is the stress-energy tensor. Degeneracy pressure, from the standpoint of the stress-energy tensor, works just like any other kind of pressure. The only real difference is the equation of state, i.e., the relationship between pressure and energy density.

 

[zonde]

OK. I am fine with all of this, but I am still missing the connection with how any of this prevents the formation of the EH. I could see it preventing the formation of the singularity, but not the horizon.

Formation of EH relies on idea that gravitating object can get more compact without any change to physical laws. But degeneracy of matter becomes more important at more compact configurations of matter.

What does any of this have to do with horizon formation for millions of galactic center BH, each with mass of millions to billions of suns. The issue here is that matter density for the aggregate at SC radius is much less than stellar atmosphere density, let alone stellar centers or neutron stars. How does degeneracy even become relevant?

To discuss scenario like this we would have to have some idea how we would model occupied and available quantum states as we add more particles to given ensemble of particles. Or what happens with occupied and available quantum states as two ensembles of degenerate matter approach each other.

Your assumptions seems to be that particles affect occupancy of quantum levels only over short distance.
I assume that occupancy of quantum level drops as inverse square law as we go further from the particle.

 

[zonde]

I don't understand why degeneracy would have any relevance to black hole event horizons. Are you thinking that matter falling toward the event horizon would run out of states, and so the Pauli exclusion principle would prevent a collection of Fermions from falling further? If that's what you're thinking, then that's not correct. Nothing special happens at the event horizon that would force matter to become degenerate.

I suggest you to reformulate your question. Because there is a problem with it as it is stated. As you refer to pre-existing event horizon you imply that it is formed as a result of runaway gravitational collapse i.e. you are begging the question. I already raised the issue in post #402. So DaleSpam agreed that we should talk about hypothetical formation of event horizon instead.

 

[Dale]

Formation of EH relies on idea that gravitating object can get more compact without any change to physical laws. But degeneracy of matter becomes more important at more compact configurations of matter.

This is relevant for the formation of the singularity, not for the formation of the EH. The singularity is an infinitely dense object, but an EH can form at arbitrarily low densities. For example, see Susskind's 12th lecture on GR () from about 2:00 to about 2:03 (of course the whole series is good).

I.e. your assumption "Formation of EH relies on idea that gravitating object can get more compact" is not correct. The formation of the singularity relies on that, but not the EH. The EH can form with simply a very large amount of non-compact material and you do not need a singularity in order to obtain an EH.

So again, what would prevent the formation of the EH? Degeneracy won't do it, that would only prevent the formation of the singularity.