# On the nature of the infinite fall toward the EH

by rjbeery
Tags: fall, infinite, nature
PF Gold
P: 5,027
 Quote by Mike Holland Very nice in theory, but it doesn't work in practice if you believe that physics is universal. How is that free-fall observer, using Rain coordinates, going to explain that contracting mass that is accelerating towards him while he is motionless? Where are the rockets that are making it accelerate? Gravity and acceleration may give the same answers, but if there is a heavy mass present, then gravity wins over acceleration.as an explanation - or at least as part of the explanation where both are involved.
The principal of equivalence is local. It will show, for example, that within a moderately large capsule in free fall, physics is the same as in 'empty space'. Or that from top to bottom of building on a planetary surface, you have the same behavior as an accelerating rocket. If you go global, it doesn't apply in GR because there are neither extended uniform gravitational fields, nor global inertial frames.

However, the coordinate dependence of gravitational time dilation is shown by something like Lemaitre coordinates, which reproduce all measurements of any other coordinates, but display no gravitational time dilation (in the sense of dilation as a function of position). The key physical point is that measured time dilation is always done between two clocks; or between an emitter and receiver. It is thus a feature of two world lines and signals between them. The measurements are coordinate independent, and gravitational redshift or time dilation are not necessary to compute any such measurement.

Note also that the general way to define clock and redshift comparison between two world lines applies perfectly well to the vicinity of co-orbiting neutron stars. Meanwhile, there is no way to even define gravitational time dilation for such an inherently non-static field.

Gravitational time dilation is a useful concept for static spacetime - but it is not a general feature of GR, and it is never necessary to use. J. L. Synge, in his classic book on general relativity, argued against using it at all - because one universal method may be used in all cases (kinematic, cosmologic, and in strong, non-static geometry) instead.
PF Gold
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 Quote by DaleSpam That much is true. I think that the bulk of the argument stems from a misunderstanding or mistrust of the basic mathematical framework of GR.
I was talking about scientific method, not about math.

 Quote by DaleSpam I think you have this backwards. The predictions are all invariants, so all coordinate systems agree. We pick a coordinate system so that the calculation of those invariants is easy.
You see the problem is that our observations are not expressed as invariants but as coordinate dependant physical quantities instead. So if you want to compare predictions with observations you would have to convert your invariants into coordinate dependant physical quantities.

 Quote by DaleSpam What would stop it? I mean, not the singularity, but the horizon.
As far as my understanding of GR goes this is out of scope of GR.
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 Quote by zonde I was talking about scientific method, not about math.
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.

 Quote by zonde You see the problem is that our observations are not expressed as invariants but as coordinate dependant physical quantities instead. So if you want to compare predictions with observations you would have to convert your invariants into coordinate dependant physical quantities.
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.

 Quote by zonde As far as my understanding of GR goes this is out of scope of GR.
OK, so considering all other mainstream physics theories as well. What would prevent the formation of a horizon?
PF Gold
P: 1,376
 Quote by PAllen And what stops it for a supermassive BH, where densities are quite low at SC radius? It's clear what you will see from afar (the cluster of stars slowing, effectively freezing, and forming a black object at essentially SC radius).
Stellar blackholes are formed in violent explosions - this is quite close to direct observation.
What you are telling is (mainstream) speculation. The closest thing to something like that as I can imagine is galactic collisions.

 Quote by PAllen But for someone orbiting one of the stars in the interior, what do you think is experiences? Are we (from afar) not allowed to ask that just because we can't see it?
You can ask of course. But it does not mean you can get testable answer.
I for example can ask what people experience when they die. So what?

As I see speculations about BHs relies on one important thing that GR takes as a postulate. That is that laws of physics are independent of (Newtonian) gravitational potential. If we assume that this assumption holds without bonds then we have no reason to assume that anything will happen with a clock falling into the hypothetical BH.
But I don't buy the idea about assumptions holding without bonds. And that takes it out of domain of GR.
PF Gold
P: 5,027
 Quote by zonde Stellar blackholes are formed in violent explosions - this is quite close to direct observation. What you are telling is (mainstream) speculation. The closest thing to something like that as I can imagine is galactic collisions.
No, galaxies are believed to contain supermassive central black holes, 10 billion or more sun's worth in some cases.
 Quote by zonde You can ask of course. But it does not mean you can get testable answer. I for example can ask what people experience when they die. So what?
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.
 Quote by zonde As I see speculations about BHs relies on one important thing that GR takes as a postulate. That is that laws of physics are independent of (Newtonian) gravitational potential. If we assume that this assumption holds without bonds then we have no reason to assume that anything will happen with a clock falling into the hypothetical BH. But I don't buy the idea about assumptions holding without bonds. And that takes it out of domain of GR.
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.)
 Emeritus Sci Advisor P: 7,599 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.
P: 1,162
Quote by pervect

 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 in my opinion, the confusion arises by taking "zeno time", which is analogous to the Schwarzschild coordinate time "t", too seriously. While it is correct to say that as t-> infinity Alice never reaches the event horizon, just as Achilles never reaches the tortise in zeno time, it still happens. It's just that that event hasn't been assigned a coordinate label.

Quote by Austin0 View Post
 On the other hand, 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.
 Quote by DaleSpam I can calculate it explicitly if you like, but it is exceedingly well-founded.
Well this whole post of yours is nothing more than a repetitive bald assertion that you are right and I am wrong without content or justification so yes some hint as to the math you are referring to would be appropriate.

Where in the stated parameters :

 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.
is the mathematical basis for the derivation of time dilation . I.e. justification of its insertion into a classical scenario???

Quote by Austin0 View Post
 So your statements ---"Neither correspond to the proper time on the falling/Achilles' clock" and " For Zeno time, the proper time on Achilles' clock between Zeno t=1 and t=2 is indeed smaller than between Zeno t=0 and t=1" are both simply unwarranted assertions without validity. Simply entering the desired conclusion as an assumption
 Quote by DaleSpam See above. It isn't an assumption. It falls out of the math quite naturally.

Quote by Austin0 View Post
 Explicitly as Zeno time goes to infinity so does Achilles'
 Quote by DaleSpam No, Achilles' proper time is finite as Zeno coordinate time goes to infinity. I thought that it would be obvious, but apparently it isn't.

Quote by Austin0 View Post
 Yes their coordinate velocity is reducing but in the Zeno system a la Pervect there is no reason that Achilles proper velocity would not also decrease.
 Quote by DaleSpam Achilles' proper velocity is clearly constant.

Quote by Austin0 View Post
 SO I will again state my opinion that the analogy doesn't really apply. Zeno time does not demonstrate a small finite time on Achilles clock. Do you still disagree??
 Quote by DaleSpam Yes, I disagree. I think that the math is so unfamiliar to you that you have a whole bunch of mistaken beliefs about how this works out. To me it is pretty obvious that none of the claims you made in your previous post are correct.
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 Quote by Austin0 Well this whole post of yours is nothing more than a repetitive bald assertion that you are right and I am wrong without content or justification so yes some hint as to the math you are referring to would be appropriate.
I will work it out in full and post it either later tonight or early tomorrow. I am sorry that it isn't obvious to you from pervect's description, but I think when you are unfamiliar with the math that you would be better served to simply ask for a detailed derivation instead of asserting that well qualified individuals like pervect are wrong or implying that they are acting deceptively.
Thanks
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 Quote by Austin0 Where in the stated parameters is the mathematical basis for the derivation of time dilation. I.e. justification of its insertion into a classical scenario???

It sounds as if we still haven't clearly explained the Zeno time analogy... Let's try a different tack...

When we say that there is time dilation between two observers, what are we saying in coordinate-independent terms? We are saying that:
1) There are two points on A's worldline; call them A1 and A2. Call the proper time between them ΔA.
2) There are two points on B's worldline; call them B1 and B2. call the proper time between them ΔB.
3) A claims, using some more or less reasonable definition of "simultaneous", that A1 and B1 are simultaneous and that A2 and B2 are simultaneous.
4) Now A calculates the ratio ΔA/ΔB. If that ratio comes out to be greater than unity, then A says that B's clock is running slow because of time dilation. Obviously this result depends on the simultaneity convention used to choose the endpoints B1 and B2 as well as the metric distance between them on B's worldline.

The standard SR definition of time dilation is contained in this more general definition; you just use the obvious and only sensible simultaneity convention, namely that all events on a line of constant t in a given frame are simultaneous in that frame, and you'll get the SR time dilation formula.

This is the only definition of time dilation that can be made to work in GR, although this is somewhat obscured by the need to do the calculations in SOME coordinate. Note that in GR the choice of simultaneity convention is arbitrary, and that if you cannot draw null geodesics from B1 to A1 and from B2 to A2 there's no reason to even prefer one convention over another.

The same method even applies in a classical scenario (although it is trivial and uninteresting). There's only one possible simultaneity convention, that defined by the Newtonian absolute time, and the ratio ΔA/ΔB always comes out to one, so there's no reason to mess with any of this coordinate-independent description.

But that is the point of the Zeno time analogy. We pick a deliberately absurd time coordinate instead of the obvious Newtonian one; it's so absurd that we cannot assign any time coordinate to event B2 ("the arrow hits the wall"), and then we calculate in this coordinate system that the arrow cannot hit the wall. Of course we know that the arrow does in fact hit the wall, so we know that something is wrong with the coordinate system and that the ratio of zeno time to arrow time is not telling us anything.

And it's the same way with the Schwarzchild time coodinate. The ratio of A's Schwarzchild time coordinate to proper time on B's worldline serves only to mislead. The interesting quantity is the ratio of proper time between any two points on B's world line and any two points on A's worldline.
P: 1,162
 Quote by Nugatory But that is the point of the Zeno time analogy. We pick a deliberately absurd time coordinate instead of the obvious Newtonian one; it's so absurd that we cannot assign any time coordinate to event B2 ("the arrow hits the wall"), and then we calculate in this coordinate system that the arrow cannot hit the wall. Of course we know that the arrow does in fact hit the wall, so we know that something is wrong with the coordinate system and that the ratio of zeno time to arrow time is not telling us anything. And it's the same way with the Schwarzchild time coodinate. The ratio of A's Schwarzchild time coordinate to proper time on B's worldline serves only to mislead. The interesting quantity is the ratio of proper time between any two points on B's world line and any two points on A's worldline.
So according to what you are saying here , Pervects stated conditions are to be taken as outside of Newtonian uniform time
 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.
so then have an implicit assumption of time dilation. That Achilles clock is running at a different rate and his velocity is constant.
Well of course given these conditions everything else is obvious. But then you have simply rewritten Zeno's paradox completely. Simply stuck Zeno's and Achilles names on the conditions of free fall in Sc coordinates.

And those conditions are not derivable from the stated Zeno time as above ,alone.

Since everyone basically agrees there is no merit in the logic in the classic Zeno argument, then by association and implication anyone considering the possible validity of the Sc case is obviously silly, right???
What other point was there as you simply made the scenarios identical (I.e. completely different from the classic argument).???

If those assumptions had been explicitly stated by Pervect then it would have been quite obvious that Zeno time was explicitly dilated and outside any classical context .
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P: 3,467
 Quote by Austin0 If those assumptions had been explicitly stated by Pervect then it would have been quite obvious that Zeno time was explicitly dilated and outside any classical context.
Aaargh... I'm still not being clear enough.... Zeno time is not "explicitly dilated" because it's a coordinate time so doesn't "dilate" - dilation is a statement about the ratio between two amounts of proper time, not coordinate time.

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).
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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:

 As we will see, this is an illusion, and the light ray (or a massive particle) actually has no trouble reaching r = 2GM. But an observer far away would never be able to tell. If we stayed outside while an intrepid observational general relativist dove into the black hole, sending back signals all the time, we would simply see the signals reach us more and more slowly. This should be clear from the pictures, and is confirmed by our computation of &)1/&)2 when we discussed the gravitational redshift (7.61). As infalling astronauts approach r = 2GM, any fixed interval &)1 of their proper time corresponds to a longer and longer interval &)2 from our point of view. This continues forever; we would never see the astronaut cross r = 2GM, we would just see them move more and more slowly (and become redder and redder, almostas if they were embarrassed to have done something as stupid as diving into a black hole). The fact that we never see the infalling astronauts reach r = 2GM is a meaningful statement, but the fact that their trajectory in the t-r plane never reaches there is not. It is highly dependent on our coordinate system, and we would like to ask a more coordinate independent question (such as, do the astronauts reach this radius in a finite amount of their proper time?). The best way to do this is to change coordinates to a system which is better behaved at r = 2GM. There does exist a set of such coordinates, which we now set out to find. There is no way to “derive” a coordinate transformation, of course, we just say what the new coordinates are and plug in the formulas. But we will develop these coordinates in several steps, in hopes of making the choices seem somewhat motivated
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)

 [for a m=2 black hole, with a horizon at r=2m = 4] $$r = {3}^{2/3} \left( -\tau \right) ^{2/3}$$ $$t = \tau-4\,\sqrt [3]{3}\sqrt [3]{-\tau}+4\,\ln \left( \sqrt [3]{3}\sqrt [3]{-\tau}+2 \right) -4\,\ln \left( \sqrt [3]{3}\sqrt [3]{-\tau}-2 \right)$$ presents the trajectory t(tau) and r(tau) for the case of a black hole where m=2.
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.
P: 1,162
 Quote by Nugatory Aaargh... I'm still not being clear enough.... Zeno time is not "explicitly dilated" because it's a coordinate time so doesn't "dilate" - dilation is a statement about the ratio between two amounts of proper time, not coordinate time. 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).
Actually I misspoke. It is Achilles' time which is dilated within the context of Pervects conditions if we add the condition that Achilles' velocity is constant.
I really just meant that time dilation was in effect within the stated conditions and coordinates

And yes I am quite aware of the meaning of dilation and it is the ratio of rates or intervals of two different clocks. In a recent post I made the simple statement that time dilation was inherently relative. Self evidently true for exactly this reason. It is meaningless applied to a single clock. Like the term length contraction or the word faster. It intrinsically requires and implies a comparison.
But somehow I got a bunch of flack from several people telling me I was wrong.
???????????

I am not convinced that Sc coordinates are necessarily preferred or correct. I am still just learning their subtleties and details and trying to synthesize a logically coherent structure up to the horizon. My exception to this analogy was purely logical. You all may be ultimately right about Sc coords and the horizon but this use of Zeno added nothing of logical probative value to the debate and was actually misleading in it's subtle reframing of Zeno.
P: 1,162
 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.
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 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.
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P: 16,985
 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.
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 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.
P: 1,162
 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.

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