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

DaleSpam

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.

stevendaryl

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?

stevendaryl

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).

PeterDonis

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).

PeterDonis

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.

PeterDonis

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.

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.

DaleSpam

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.

PeterDonis

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.

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.

PeterDonis

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.

Austin0

Quote by Austin0

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.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

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

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.

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

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???

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 [itex]d=100-vt[/itex] means that Achilles catches the tortoise at d=0 or 100-vt=0
so vt=100 and [tex]t = \frac{100}{v}[/tex]

SO clearly yiour conclusion [tex]\lim_{n\to \infty } \, t = \frac{100}{v}[/tex] 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???

DaleSpam

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.

You are making a mistake there. Pervect did not make such a description.

Oops, good catch! I definitely missed that. I need to fix that.

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.

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.

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.

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?

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.

I still disagree, the analogy is very close, but I don't understand your most recent objection.

stevendaryl

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).

--
Daryl McCullough
Ithaca, NY

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:
[tex]\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)} [/tex]
and
[tex]t=\frac{100}{v}(1-2^{-n})[/tex]

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

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

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

Austin0

Quote by Austin0

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.

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

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

Any closer???

Austin0

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???

******************************_____ So in this frame Achilles is a distance [itex]d=100-vt[/itex] behind the turtle. The definition of Zeno time, n, given is [itex]d=100/2^n[/itex].

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; [itex]d=100-vt[/itex] is not equivalent to the d' here in Zeno's frame; [itex]d'=100/2^n[/itex]. 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???
What about simultaneity???
So how can the rest of your derivation from that point be valid if this initial step is not on ??