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

 Quote by zonde 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.
I am considering any scenario where an EH forms. If there are multiple ways for an EH to form then a mechanism for preventing EH formation has to prevent all of them.

In general an EH forms whenever there is enough mass inside the Schwarzschild radius. That can happen at any density, so a mechanism which prevents high densities, like degeneracy, simply cannot prevent EH formation in general.

 Quote by 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.
I'm not begging the question. I'm asking you a question. Why do you believe that degeneracy has anything to do with the formation of an event horizon? You can certainly make up your own theory, but there is nothing in General Relativity that would suggest that. If you're not talking about General Relativity, then what are you talking about?

 Quote by 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.
If you are making up your own theory of gravity, then I think this is not the appropriate place to talk about it. If you are talking about mainstream physics, then it is well understood that degeneracy prevents further collapse for any star less massive than the Chandrasekhar limit (described here: http://en.wikipedia.org/wiki/Chandrasekhar_Limit).

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 Quote by stevendaryl it is well understood that degeneracy prevents further collapse for any star less massive than the Chandrasekhar limit (described here: http://en.wikipedia.org/wiki/Chandrasekhar_Limit).
Small technical point: the Chandrasekhar limit applies to white dwarfs, i.e., to objects in which electron degeneracy is the primary mechanism resisting compression. The analogous limit for neutron stars, where neutron degeneracy is the primary mechanism, is the Tolman-Oppenheimer-Volkoff limit:

http://en.wikipedia.org/wiki/Tolman%...3Volkoff_limit

Conceptually, both limits work the same, but the details are different because of the different types of fermions involved (neutrons vs. electrons).

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 Quote by zonde 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.
Do you have any actual argument for why this would happen? Why would a degenerate star suddenly start melting? If the two degenerate stars collide with each other, then I could see matter being ejected from the collision; but if the stars are just free-falling towards each other, what difference would that make to their internal structure? The quantum states inside the star don't "know" that the two stars are approaching each other, unless they actually collide.

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 Quote by zonde Formation of EH relies on idea that gravitating object can get more compact without any change to physical laws.
Huh? This makes no sense. The physical laws involved are the Einstein Field Equation and the equation of state for the matter. It is well known that there are a range of reasonable equations of state that allow a gravitating object to get compact enough to form an EH; there are both analytical solutions and numerical simulations that show this. The laws certainly don't need to "change" at any point during the process.

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 Quote by zonde 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
Since there are already known solutions of the EFE with various equations of state that show runaway gravitational collapse, assuming it is possible is not begging the question.

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 Quote by PeterDonis Since there are already known solutions of the EFE with various equations of state that show runaway gravitational collapse, assuming it is possible is not begging the question.
I think that he is objecting to the equations of state, in which case it is begging the question. However, I think it is clear that his proposed patch to the equations of state does not accomplish his goal, and since many equations of state lead to an EH it is hard to see that a patch is even possible.

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 Quote by DaleSpam since many equations of state lead to an EH it is hard to see that a patch is even possible.
Exactly. We don't know enough about the strong nuclear force and QCD to be able to derive the exact equation of state for neutron star matter from first principles, so any equation of state we use is an assumption. We can only debate about which equations of state are "reasonable"; but since as you say, many equations of state lead to an EH forming, it would take a very impressive argument to show that *all* of them are too "unreasonable". I certainly don't see any such argument being made in this thread.

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 Quote by PeterDonis since as you say, many equations of state lead to an EH forming, it would take a very impressive argument to show that *all* of them are too "unreasonable". I certainly don't see any such argument being made in this thread.
Agreed, particularly for supermassive black holes where the densities required are well within the "ordinary" range in which we have lots of data and experience and very well-validated equations of state.

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 Quote by DaleSpam Agreed, particularly for supermassive black holes where the densities required are well within the "ordinary" range in which we have lots of data and experience and very well-validated equations of state.
Yes, good point; the neutron star case, where we don't have very good knowledge of the actual equation of state, is only one of many possibilities.

Quote by Austin0

 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???
 Quote by DaleSpam Yes.
Quote by Austin0
 But in the classical statement it is evident that the stated constant velocity is in the frame of the ground. I.e. Zeno coordinates.
 Quote by DaleSpam 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.

Well I agree that Zeno did not explicitly define a coordinate frame ibut he did implicitly define Achilles motion in the terms of the ground.I.e. Achilles successively caught up with a previous position of the tortoise which would naturally be a spatial point on the ground.
So in this context the ground would be an inertial frame. And Pervects statements could validly be interpreted in this context. In which case it would be Achilles motion which was non-inertial.Such an interpretation would be perfectly consistent with Pervects statements right up to total zeno time being infinite. Yes???.
so you are circularly inserting an assumption that Zeno coordinates are non-inertial.

Quote by Austin0

 Do you disagree??? What other possible frame for such a statement do you propose???
 Quote by DaleSpam Any inertial frame. If it is true in one inertial frame then it is true in all.
Yes it is possible to assume an interpretation of an abstract unspecified inertial frame however unlikely that was what was assumed by Zeno .
WHich is why I said

Quote by Austin0

 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 .
this.

Clearly I did not suggest that my interpretation was the only possible one but only pointed out that it was also not precluded and other interpretations were not exclusive or preferred.

 Quote by DaleSpam 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.
As opposed to your unequivocal statement of "invariant fact" which is actually not the result of inevitable logic but in the end really no more than edict.
Unsupported assertion that my interpretation is wrong and yours is fact.

Quote by Austin0

 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??
 Quote by DaleSpam 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.
1) this metric is based on your a priori definition of Achilles motion as inertial and Zeno coordinates as non-inertial so is disregarding Pervects description of Achilles non-uniform motion in an inertial system.

2) Could you explain this metric? It is true it does not look like an inertial metric but it also does not resemble the Sc metric either.

If I am understanding it correctly the first term contains both Zeno coordinate time and also Achilles coordinate time yes? How does that work ? it appears a bit circular no??

It also appears that it is based on a constant velocity term in the Zeno frame , how is this possible???

3) What is your definition of inertial.
Lack of accelerometer reading? Disregarding g both Achilles and the Zeno frame are inertial by this standard.

Constant motion. As observed from all inertial frames both Achilles and Zeno frames are in uniform coordinate motion yes? So are equivalent.

As far as I know inertial frames are simply defined by uniform rectilinear motion without explicit reference to time flow so what is your basis for this strong assertion that the Zeno frame is non-inertial???

 Quote by DaleSpam 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.
SO as you have declared Achilles motion inertial then it follows that his velocity is constant and time rate uniform so:
your initial premise here $d=100-vt$ means that Achilles catches the tortoise at d=0 or 100-vt=0
so vt=100 and $$t = \frac{100}{v}$$

SO clearly yiour conclusion $$\lim_{n\to \infty } \, t = \frac{100}{v}$$ is directly equivalent to your initial premise [tex t = \frac{100}{v}[/tex] without any of your intermediate steps and is classically circular reasoning. A tautology if you like.

Also: Given your declaration of Achilles inertial motion, as far as I can see there is no possible state of accelerated motion of the Zeno frame that could effectuate the observations of Achilles motion as defined by Pervect.

SO unless you can come up with such a description I propose that Zeno motion is also inertial i.e. constant and the non-uniformity is all temporal. DO you disagree ? If so what possible motion??

In this case then, the temporal non-uniformity could not be actual dilation , meaning change of physical processes etc. as there is no known physics to explain this kind of exponential increase of time rate concurrent with the decrease in coordinate velocity of the inertial Achilles .

SO this leaves arbitrary mechanical clock rate as the only possible scenario consistent with your own conditions and assumptions.

Just as I suggested early on and you rejected with your tautological definition.

Or do you disagree and have an alternative explanation???

so the Zeno clocks speed up exponentially but Zeno observers do not ..

But this seems to me to mean that finite proper time on Achilles clock could not possibly mean infinite time on a mechanically calibrated actual physical clock. SO the analogy is completely non-applicable.

Or do you still disagree???

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 Quote by Austin0 Well I agree that Zeno did not explicitly define a coordinate frame ibut he did implicitly define Achilles motion in the terms of the ground.I.e. Achilles successively caught up with a previous position of the tortoise which would naturally be a spatial point on the ground. So in this context the ground would be an inertial frame. And Pervects statements could validly be interpreted in this context. In which case it would be Achilles motion which was non-inertial.
Again, defining new coordinates does not change any invariants. The fact that Achilles' motion is inertial is an invariant, therefore it cannot change by the introduction of new coordinates. You cannot change the invariants without changing the physics, the scenario.

So, yes, it is an assumption that Achilles' motion is an inertial, that assumption is part of the original well-known scenario. Pervect's definition of a coordinate system does not change that assumption since it is invariant, and an "interpretation" of pervect's comments which assumes that specifying coordinates also changes invriants is simply a mistake.

 Quote by Austin0 1) this metric is based on your a priori definition of Achilles motion as inertial and Zeno coordinates as non-inertial so is disregarding Pervects description of Achilles non-uniform motion in an inertial system.
You are making a mistake there. Pervect did not make such a description.

 Quote by Austin0 If I am understanding it correctly the first term contains both Zeno coordinate time and also Achilles coordinate time yes?
Oops, good catch! I definitely missed that. I need to fix that.

 Quote by Austin0 3) What is your definition of inertial. Lack of accelerometer reading? Disregarding g both Achilles and the Zeno frame are inertial by this standard.
Yes. That is the standard definition in GR.

EDIT: I later realized that there may be some lingering confusion about the meaning of inertial. When we are talking about a worldline then inertial does mean zero proper acceleration (zero accelerometer reading). When we are talking about a coordinate system then inertial means that the metric is the Minkowski metric in those coordinates. These are both the standard definitions in GR. So Achilles' worldline is inertial under the first definition, and the Zeno coordinates are non-inertial under the second definition. I hadn't originally noticed that you were mixing a worldline and a coordinate system in your question above.

 Quote by Austin0 what is your basis for this strong assertion that the Zeno frame is non-inertial???
The metric in any inertial frame is the standard Minkowski metric. Of course, I need to fix the metric above in order to show that the time term doesn't simplify to -1.

 Quote by Austin0 SO clearly yiour conclusion $$\lim_{n\to \infty } \, t = \frac{100}{v}$$ is directly equivalent to your initial premise [tex t = \frac{100}{v}[/tex] without any of your intermediate steps and is classically circular reasoning. A tautology if you like.
Yes. Which is why pervect and I thought that the analogy was obvious. The coordinate system was explicitly, deliberately, and purposely designed so that that limit would go to infinity as Achillies reached the Tortoise.

 Quote by Austin0 Also: Given your declaration of Achilles inertial motion, as far as I can see there is no possible state of accelerated motion of the Zeno frame that could effectuate the observations of Achilles motion as defined by Pervect.
I don't know what you mean here. What does "effectuate the observations" mean? Achilles' motion and the Tortoise's motion are inertial, so what accelerated motion are you talking about?

 Quote by Austin0 In this case then, the temporal non-uniformity could not be actual dilation , meaning change of physical processes etc. as there is no known physics to explain this kind of exponential increase of time rate concurrent with the decrease in coordinate velocity of the inertial Achilles .
What are you talking about here? This is a coordinate system, it is just mathematical labeling, not any physical process nor any physical explanation. That is the point. I don't understand what you mean by "actual dilation" and "change of physical processes"? It seems contrary to the principle of relativity.

 Quote by Austin0 But this seems to me to mean that finite proper time on Achilles clock could not possibly mean infinite time on a mechanically calibrated actual physical clock. SO the analogy is completely non-applicable. Or do you still disagree???
I still disagree, the analogy is very close, but I don't understand your most recent objection.

 Quote by Austin0 But this seems to me to mean that finite proper time on Achilles clock could not possibly mean infinite time on a mechanically calibrated actual physical clock. SO the analogy is completely non-applicable. Or do you still disagree???
I disagree. The analogy with Schwarzschild coordinates is almost exact. In both cases, you have a local inertial coordinate system, according to which it takes a finite amount of time for the traveler to move from point A to point B, and there is a second coordinate system, with a nonlinear relationship to the first, according to which it takes an infinite amount of time for the traveler to move from point A to point B. What are you saying is the difference?

Actually, there is a difference having to do with causality, but it doesn't come into play in anything you've said so far: For the Schwarzschild case, events after the traveler crosses the event horizon are inaccessible to the distant observer, while in the Zeno cases, there are events after Achilles crosses the finish line that are accessible to the distant observer (although they can't be given a time coordinate in the coordinate system of the distant observer).

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 Quote by Austin0 2) Could you explain this metric? It is true it does not look like an inertial metric but it also does not resemble the Sc metric either.
You are correct, it is not the same as the SC metric. The Zeno coordinates are defined on a flat spacetime, so there will always be some difference there. It is an analogy, not a derivation.

 Quote by Austin0 If I am understanding it correctly the first term contains both Zeno coordinate time and also Achilles coordinate time yes? How does that work ? it appears a bit circular no??
OK, thanks for pointing out my mistake. Unfortunately, it is too late to go and edit the post, so I hope anyone who refers to it in the future notices this update. Anyway, from post 393 we have:
$$\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)}$$
and
$$t=\frac{100}{v}(1-2^{-n})$$

Substituting the second equation in on the rhs of the first equation and simplifying we get
$$\frac{dn}{dt}=\frac{2^n v}{100 ln(2)}$$

So the metric in post 414 should be:
$$ds^2=-c^2 \left( \frac{100 ln(2)}{2^n v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2$$

Which again is clearly not the Minkowski metric of an inertial frame, thereby demonstrating that the Zeno coordinates are non-inertial.

Quote by Austin0

 Also: Given your declaration of Achilles inertial motion, as far as I can see there is no possible state of accelerated motion of the Zeno frame that could effectuate the observations of Achilles motion as defined by Pervect.
 Quote by DaleSpam I don't know what you mean here. What does "effectuate the observations" mean? Achilles' motion and the Tortoise's motion are inertial, so what accelerated motion are you talking about?
Yes I am referring to the Zeno frame which you have declared is non-inertial (I.e. accelerated).

Pervect has here given a series of events. Or at least relationships as there seems to be no determinable velocities or explicit spatial coordinates to be derived from this information.

 Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise. At a zeno time of 1, Achilles is 50 meters behind the tortise. At a zeno time of 2, Achillies is 25 meters behind the tortise At a zeno time of n, Achillies is 100/(2^n) meters behind the tortise. Then, as n goes to infinity, Achillies is always behind the tortise. So, in "zeno time", Achilles never does catch up with the tortise, even as "zeno time" appoaches infinity.
SO we have these times and relative distances and the premise that both Achilles and the tortoise are inertial with which to synthesize a coordinate system and metric.

You have asserted that the Zeno frame is non-inertial so the question is what possible state of motion of that frame could make possible those observed distances between two bodies in uniform motion.

Maybe an example would help you visualize:
If the observations in the Zeno frame supported a picture of linear decrease in distance between Achilles and the tortoise this would indicate a constant motion of the Zeno frame also , agreed???

If the observed decrease in distance, itself increased in rate , this would support a conclusion of positive parallel acceleration of the Zeno frame.I.e. Zeno frame increasing it's velocity relative to A and the tortoise.

But according to Pervect the decrease in relative distance between Achilles and the tortoise is decreasing over time non-linearly.
SO what possible motion (acceleration) of the Zeno frame could make this possible????

My conclusion is that there is no possible acceleration that could do this alone and therefore the observations in the Zeno frame could only be possible if the Zeno time rate was increasing at a rate not possible through the effects of motion ( Lorentz effects..)

Quote by Austin0

 In this case then, the temporal non-uniformity could not be actual dilation , meaning change of physical processes etc. as there is no known physics to explain this kind of exponential increase of time rate concurrent with the decrease in coordinate velocity of the inertial Achilles .
 Quote by DaleSpam What are you talking about here? This is a coordinate system, it is just mathematical labeling, not any physical process nor any physical explanation. That is the point. I don't understand what you mean by "actual dilation" and "change of physical processes"? It seems contrary to the principle of relativity.
Put simply:
Achilles is passing a stream of Zeno clocks and observers. Do you think Achilles sees everything in the Zeno frame speed up exponentially or only the clocks???

If you think everything speeds up (actual dilation) then what is your explanation of the physics behind this???
This would be to a certain extent possible if Achilles and the tortoise were racing at relativistic speeds in a circle in a stationary Zeno frame but I doubt the exponential increase would be possible even with accelerating racers.

If you think it is only the clocks, an arbitrary coordinate choice, then you are talking about a mechanism to accomplish this radical increase in rate in actual physical clocks correct?

Quote by Austin0

 But this seems to me to mean that finite proper time on Achilles clock could not possibly mean infinite time on a mechanically calibrated actual physical clock. SO the analogy is completely non-applicable. Or do you still disagree???
 Quote by DaleSpam I still disagree, the analogy is very close, but I don't understand your most recent objection.
Any closer???

 Quote by DaleSpam You are correct, it is not the same as the SC metric. The Zeno coordinates are defined on a flat spacetime, so there will always be some difference there. It is an analogy, not a derivation. OK, thanks for pointing out my mistake. Unfortunately, it is too late to go and edit the post, so I hope anyone who refers to it in the future notices this update. Anyway, from post 393 we have: $$\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)}$$ and $$t=\frac{100}{v}(1-2^{-n})$$ Substituting the second equation in on the rhs of the first equation and simplifying we get $$\frac{dn}{dt}=\frac{2^n v}{100 ln(2)}$$ So the metric in post 414 should be: $$ds^2=-c^2 \left( \frac{100 ln(2)}{2^n v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2$$ Which again is clearly not the Minkowski metric of an inertial frame, thereby demonstrating that the Zeno coordinates are non-inertial.
Well you still have that v in the rhs of your equation. What does it represent??
The only definition of v actually expressed is in the Achilles frame so that does not seem like it could be that ,right?
So how do you define v in the Zeno frame and what does it apply too???

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 Quote by DaleSpam 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.
So in this frame Achilles is a distance $d=100-vt$ behind the turtle. The definition of Zeno time, n, given is $d=100/2^n$.

You have stated that although Achilles and the tortoise are inertial, the Zeno frame is not, so how do you arrive at your identity here to justify your substitution and simplification. The d here in Achilles frame; $d=100-vt$ is not equivalent to the d' here in Zeno's frame; $d'=100/2^n$. is it???
Having invoked relativistic principles in this classic scenario how can you now directly equate a distance in one frame with that in another which is not only moving at a relative velocity but which is in non-uniform motion???