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On the nature of the infinite fall toward the EH 
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#433
Dec2512, 02:06 PM

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#434
Dec2512, 02:13 PM

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PF Gold
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#435
Dec2512, 02:17 PM

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#436
Dec2512, 04:16 PM

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#437
Dec2712, 11:53 PM

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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 noninertial.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 noninertial. Quote by Austin0 WHich is why I said Quote by Austin0 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. Unsupported assertion that my interpretation is wrong and yours is fact. 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. 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 noninertial??? your initial premise here [itex]d=100vt[/itex] means that Achilles catches the tortoise at d=0 or 100vt=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 nonuniformity is all temporal. DO you disagree ? If so what possible motion?? In this case then, the temporal nonuniformity 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 nonapplicable. Or do you still disagree??? 


#438
Dec2812, 07:41 AM

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So, yes, it is an assumption that Achilles' motion is an inertial, that assumption is part of the original wellknown 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. 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 noninertial under the second definition. I hadn't originally noticed that you were mixing a worldline and a coordinate system in your question above. 


#439
Dec2812, 08:06 AM

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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 


#440
Dec2812, 08:07 AM

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[tex]\frac{dn}{dt}=\frac{v}{(100vt) ln(2)} [/tex] and [tex]t=\frac{100}{v}(12^{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 noninertial. 


#441
Jan113, 10:51 PM

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Quote by Austin0
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. You have asserted that the Zeno frame is noninertial 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 nonlinearly. 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 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 


#442
Jan113, 11:01 PM

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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??? ******************************_____ 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=100vt[/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 nonuniform motion??? What about simultaneity??? So how can the rest of your derivation from that point be valid if this initial step is not on ?? 


#443
Jan213, 07:46 AM

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I think that you are having trouble grasping the idea of an arbitrary, noninertial, curvilinear coordinate system (as opposed to an inertial, Cartesian coordinate system). An inertial Cartesian coordinate system is set up in some standard way (for example, using light signals to measure distances and using a standard clock to measure time, and using the Einstein synchronization convention for synchronizing distant clocks). But you can use any convention you like to set up a coordinate system. Let [itex](x,t)[/itex] be an inertial Cartesian coordinate system for some region [itex]R[/itex] of spacetime. Let [itex]T'(x,t), X'(x,t), X(x',t'), T(x',t')[/itex] be any four differentiable functions such that for any pair [itex](x,t)[/itex] describing a point in [itex]R[/itex], [itex]X(X'(x,t), T'(x,t)) = x[/itex] [itex]T(X'(x,t), T'(x,t)) = t[/itex] Then within region [itex]R[/itex], you can use coordinates [itex]x', t'[/itex] defined by [itex]x' = X'(x,t)[/itex] [itex]t' = T'(x,t)[/itex] As far as GR is concerned, [itex](x',t')[/itex] can be used just as well as (x,t). In the case DaleSpam is talking about, [itex]X'(x,t) = x[/itex] [itex]T'(x,t) = log_2(\dfrac{100}{100vt})[/itex] You are asking what the physical interpretation of the noninertial coordinates arecoordinates don't HAVE a physical interpretation, or they don't need to, anyway. They're just a way of identifying points in spacetime. They're just names, but names chosen in a "smooth" way, so that you know that nearby points will have names that are close together as numbers. 


#444
Jan213, 07:47 AM

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However, since the only thing of interest in the scenario is Achilles I don't see the need. If you want to do more complicated scenarios which are still analogous to the SC horizon then I would recommend going to Rindler coordinates. There the analogy is even closer. If this is not what you had intended, then could you be more explicit about what you want calculated? 


#445
Jan213, 09:31 AM

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You made similar comments about time and clocks in your previous post. Perhaps this is the root of your problem. In GR time coordinates are not readings on some clock and spatial coordinates are not measurements on some rod. They are mathematical functions which map open subsets of events in spacetime to open subsets of points in R4. They have some mathematical restrictions like being smooth, continuous, and onetoone, but no physical restrictions. The connection to physical measurements, like clocks and rods, is done through the metric. You could make some remote noninertial observer and give a simultaneity convention that maps his coordinates to Zeno time. This would make the analogy better, but it seems like a lot of effort. 


#446
Jan613, 08:06 AM

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I just read through this whole thread and it seems that it's all about relativity of simultaneity.
In the infaller's reference frame is his passage through the horizon simultaneous with some finite well defined event at the distant observer's ship (like when the distant observer is muttering to himself: "Well, it's been 2 hours since his jump, let's go home"). While the distant observer, if using Schwarzschild coordinates, does not connect these 2 events as simultaneous. In SC the infaller's passege through EH is in infinite future for the distant observer, but this distant observer can use different coordinates where the infaller's passage through EH lies in finite future. He has the choice of different coordinates because in curved spacetime the simultaneity convention is not given unambiguously. Is it that simple, or I'm missing something? 


#447
Jan613, 08:11 AM

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#448
Jan713, 07:36 AM

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It appears that some people (e.g Austin and Dalespam) are still trying to argue about this matter in this thread; I wish them good luck as to me there doesn't seem to be a possible way of deciding who is right based on logic. 


#449
Jan713, 07:46 AM

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#450
Jan713, 07:44 PM

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Quote by Austin0
Working from the information defined by pervect it is not possible to derive a velocity for the Achilles frame in the Zeno frame as far as i can see, Likewise it is not possible to define a velocity for Achilles himself in the Zeno frame. So again I ask what is the velocity referring to that could be a valid part of the Zeno metric???. And how do you arrive at it?? In fact, back at my second post I brought up this possibility Quote by Austin0 if this is so i don't see the relevance. This particular case is not about setting up a system from the ground but working within the constraints of defined relationships and partial definitions without a completely defined system for Zeno .We can assume a standard inertial system for Achilles but we have only some data from observations in Zeno frame to go by. If in fact you do not already have the correct transform functions, the T,X,T' and X' in your generalization you cannot simply assume the equivalence between some values in both frames and derive a valid transform from that . There has to be some relevant basis for the equivalence from first principles to justify such an identity and substitution. Wouldn't you agree?? SO in this case we are given : [itex]d=100vt[/itex] in the A frame and [itex]d=100/2^n[/itex] in the Z frame. Is the 100 in the A frame equivalent to the 100 in the Z frame??? Assuming that at A time t =0 Achilles is at x=0 and the tortoise is at x=100 and at Z time n= 0 Achilles is at x'=0 and the tortoise is at x'=100. isn't it axiomatic that if these events are simultaneous in the A frame that they cannot be simultaneous in the Z frame?? It follows that the distances , the spatial intervals in the two frames cannot be congruent also Yes?? So if the intervals dx,t=0 and dx', t'=0 are not equivalent, even initially when you can assign coordinates to the positions in the Z frame, how do you justify the equivalence [itex]100vt=100/2^n[/itex] over time when the systems are not only in relative motion but one of them is nonlinear?? Where you do not have a basis to even determine coordinate positions in the Z frame for A and the tortoise or relate times in that frame to the A frame?? It appears to me that to make this assumption of equivalence is unfounded and circular. I.e.,,to determine if these are equivalent requires a valid transformation so to use them to derive a transformation then makes them equivalent circularly. I asked you if you were talking about an arbitrary scaling of clock periodicity and you agreed, correct? So then we are talking about a physical interpretation of clock rates. AN artificial mechanical adjustment to the workings of the mechanism. What could be clearer than that?? In this case this means a mechanistic device that exponentially increases the rate at which the hands spin or the LED increments or whatever means that is used to actually indicate the measure of time,,,, CORRECT???? Such artificial scaling is in fact used in the GPS system right?? Those clocks physically increment at a different rate yes?? 


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