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

 Quote by pervect 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 http://www.physicsforums.com/showpos...4&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.

Mentor
 Quote by Austin0 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.

Mentor
 Quote by Austin0 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.

Mentor
 Quote by Nugatory Is there any reason to take the Schwarzchild 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.

 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 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.

Recognitions:
Gold Member
 Quote by DaleSpam 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.

 Quote by DaleSpam 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?

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

Recognitions:
Gold Member
 Quote by PAllen 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".

 Quote by PAllen 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

Recognitions:
Gold Member
 Quote by pervect 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.

Mentor
 Quote by Austin0 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

Mentor
 Quote by zonde 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.

 Quote by zonde 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.

 Quote by zonde Degeneracy of matter.
And what would cause matter to become degenerate at the horizon?

Recognitions:
Gold Member
 Quote by DaleSpam Yes, but if you don't understand how it was derived then you don't understand under what conditions 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.

 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

Blog Entries: 9
Recognitions:
Gold Member
 Quote by zonde 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.

Mentor
 Quote by zonde 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.

 Quote by zonde 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.

 Quote by zonde 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.

 Quote by zonde 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?

Recognitions:
Gold Member
 Quote by DaleSpam 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?

 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

Mentor
 Quote by zonde 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.

 Quote by zonde 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.

 Quote by zonde 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.

 Quote by zonde 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).

 Quote by zonde 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.

Recognitions:
Gold Member
 Quote by DaleSpam 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 modelled as pressure and that does not seem quite right to me.
See here - Degenerate matter

 Mentor Again, how would degeneracy do anything to prevent a horizon. Degeneracy doesnt magically make any mass or energy disappear, so the curvature will remain.