On the nature of the "infinite" fall toward the EH

PeterDonis

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

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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

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 .


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 any one 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???

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

"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

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.

DaleSpam

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.

DaleSpam

Yes.

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.

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

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.

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:
[tex]ds^2=-c^2 \left( \frac{(100-vt) ln(2)}{v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2[/tex]

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

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.

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

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

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

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

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.

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 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.

DaleSpam

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 (http://www.youtube.com/watch?v=fVqYlSNqSQk) 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.

PAllen

So we have new fundamental law of physics: the "stellar exclusion principle" that prevents gathering too many stars in the same large region??!! Remember, the EH forms before there is any singularity or even any high density anywhere in the formative collapsing mass.

PeterDonis

So you're saying that quantum effects play a non-negligible part in the dynamics of stars that are separated by light-years? That, for example, quantum interactions between the Sun and Alpha Centauri affect the relative motion of those two stars?

zonde

There are two ways how to arrive at situation where EH is supposed to form.
First, we can add more matter to given gravitating object while it's radius is not increased too much by this addition.
Second, we can make given gravitating object more compact while it's mass is not reduced too much by this compactification.

I guessed that you was talking about the second scenario. If you are considering first scenario and want arguments concerning this scenario in particular please say it so that I don't have to guess.

zonde

New? Why new? I am just extrapolating existing law.

zonde

No, I am not talking about dynamics of stars but about dynamics of particles.
So what I say is that if we have two fairly degenerate stars approaching each other then whey would melt first and after that will start to evaporate. Or alternatively will fall into pieces depending on homogeneity of star.

If particles can't remain in their quantum states they can't maintain their collective structure. Kind of obvious IMO.