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

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.

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'

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

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).

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 [itex]d=100-vt[/itex] behind the turtle. The definition of Zeno time, n, given is [itex]d=100/2^n[/itex]. Substituting in and simplifying we get the following transform between the inertial frame and Zeno coordinates:
[tex]n=log_2 \left( \frac{100}{100-vt} \right)[/tex]

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

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

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.

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.

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

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.

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.)

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

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

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?

Degeneracy of matter.

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.

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.

They transform one coordinate to the other chart.

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

You identify events by their coordinates.

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.

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.

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.

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.

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.

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.

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.

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

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?

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.

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.

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.