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On the nature of the infinite fall toward the EH

by rjbeery
Tags: fall, infinite, nature
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DaleSpam
#433
Dec25-12, 02:06 PM
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Quote Quote by PeterDonis View Post
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.
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
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Dec25-12, 02:13 PM
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Quote Quote by DaleSpam View Post
since many equations of state lead to an EH it is hard to see that a patch is even possible.
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
#435
Dec25-12, 02:17 PM
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Quote Quote by PeterDonis View Post
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.
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
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Dec25-12, 04:16 PM
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Quote Quote by DaleSpam View Post
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.
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
#437
Dec27-12, 11:53 PM
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Quote by Austin0

SO it appears that your assertion that Achilles velocity is constant is based, not on calculation, but on your interpretation of the explicit statements of the classical scenario...yes???
Quote Quote by DaleSpam View Post
Yes.
Quote by Austin0
But in the classical statement it is evident that the stated constant velocity is in the frame of the ground. I.e. Zeno coordinates.
Quote Quote by DaleSpam View Post
I don't think that the "classical statement" ever explicitly introduced any coordinates. That was pervect's idea, taking the familiar statement of Zeno's paradox and using it to define a coordinate time. So I would not associate Zeno coordinates with the frame of the ground since "frame of the ground" usually indicates an inertial frame and Zeno coordinats are non-inertial.

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

Do you disagree??? What other possible frame for such a statement do you propose???
Quote Quote by DaleSpam View Post
Any inertial frame. If it is true in one inertial frame then it is true in all.
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

So when Pervect redefines Achilles velocity as non-uniform in the Zeno frame it is now ,not necessarily a logical conclusion that Achilles velocity is constant in any other frame, as no other frame was defined .
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.

Quote Quote by DaleSpam View Post
Achilles motion is inertial. That is an invariant fact which is true in all coordinate systems and does not change with pervect's introduction of Zeno coordinates. Given that his motion is inertial (frame invariant) then his velocity (frame variant) is constant in any inertial frame.
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

According to Pervect's explicit description it seems to follow that the Zeno coordinate system is not accelerating. That it would be in a state of uniform motion relative to and measured by any inertial frame. Do you disagree??
Quote Quote by DaleSpam View Post
Yes, I disagree quite strongly. The Zeno coordinate system is decidedly non-inertial. In fact, from my post 393 you can easily see that the metric in the Zeno coordinates is:

[tex]ds^2=-c^2 \left( \frac{(100-vt) ln(2)}{v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2[/tex]


This metric is clearly different from the metric in an inertial frame.
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???

Quote Quote by DaleSpam View Post
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 coordinate time wrt Achilles proper 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 coordinate time.

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 coordinate time goes to infinity Achilles proper time does not.
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
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Dec28-12, 07:41 AM
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Quote Quote by Austin0 View Post
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.
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.

Quote Quote by Austin0 View Post
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.
You are making a mistake there. Pervect did not make such a description.

Quote Quote by Austin0 View Post
If I am understanding it correctly the first term contains both Zeno coordinate time and also Achilles coordinate time yes?
Oops, good catch! I definitely missed that. I need to fix that.

Quote Quote by Austin0 View Post
3) What is your definition of inertial.
Lack of accelerometer reading? Disregarding g both Achilles and the Zeno frame are inertial by this standard.
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.

Quote Quote by Austin0 View Post
what is your basis for this strong assertion that the Zeno frame is non-inertial???
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.

Quote Quote by Austin0 View Post
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.
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.

Quote Quote by Austin0 View Post
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.
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?

Quote Quote by Austin0 View Post
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 .
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.

Quote Quote by Austin0 View Post
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???
I still disagree, the analogy is very close, but I don't understand your most recent objection.
stevendaryl
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Dec28-12, 08:06 AM
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Quote Quote by Austin0 View Post
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???
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
#440
Dec28-12, 08:07 AM
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Quote Quote by Austin0 View Post
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.
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.

Quote Quote by Austin0 View Post
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??
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
#441
Jan1-13, 10:51 PM
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Quote by Austin0

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.
Quote Quote by DaleSpam View Post
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?
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.

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 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

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 .
Quote Quote by DaleSpam View Post
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.
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

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???
Quote Quote by DaleSpam View Post
I still disagree, the analogy is very close, but I don't understand your most recent objection.
Any closer???
Austin0
#442
Jan1-13, 11:01 PM
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Quote Quote by DaleSpam View Post
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.
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???

******************************_____
Quote Quote by DaleSpam View Post
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 coordinate time wrt Achilles proper 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 coordinate time.

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 coordinate time goes to infinity Achilles proper time does not.
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 ??
stevendaryl
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Jan2-13, 07:46 AM
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Quote Quote by Austin0 View Post
Well you still have that v in the rhs of your equation. What does it represent??
It doesn't represent anything--it's just a number that is used to describe the relationship between two coordinate systems, and also happens to be the speed of Achilles in one of the coordinate systems.

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}{100-vt})[/itex]

You are asking what the physical interpretation of the noninertial coordinates are--coordinates 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.
DaleSpam
#444
Jan2-13, 07:47 AM
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Quote Quote by Austin0 View Post
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.
I assumed that the distance to the Tortoise was the spatial coordinate for Achilles, but it is true that we never actually introduced a method to assign spatial coordinates elsewhere. That would require the introduction of a simultaneity convention and a spatial metric elsewhere. It could be done, but would require some more work.

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.

Quote Quote by Austin0 View Post
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.
For coordinates non-inertial just means that the metric is not the Minkowski metric, as demonstrated. There is no requirement that a coordinate system correspond with some observer's state of motion.

Quote Quote by Austin0 View Post
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???
Constant motion relative to Achilles, yes. In other words, the coordinate acceleration of Achilles would be 0.

Quote Quote by Austin0 View Post
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???
I am not sure, but it sounds like you want the coordinate acceleration of Achilles, which is easy enough to solve. From post 393 we already found that Achilles' worldline in the Zeno coordinates is given by [itex]d = 100 \; 2^{-n}[/itex], so Achilles' coordinate acceleration is the second derivative wrt n which is [itex]a = 100 \, 2^{-n} ln(2)^2[/itex].

If this is not what you had intended, then could you be more explicit about what you want calculated?

Quote Quote by Austin0 View Post
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..)
I agree, the same thing happens in SC. The SC coordinate time is increasing at a rate which is not possible through the effects of motion for any local observer. It is only by the use of a simultaneity convention and a distant observer that SC time is related to any observer's proper time. We haven't defined either of those for Zeno coordinates, but we certainly could do so.

Quote Quote by Austin0 View Post
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???
Only the coordinate time speeds up exponentially, physical clocks do not. Similarly with a free faller passing a stream of shell observers in SC.

Quote Quote by Austin0 View Post
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?
Clocks measure proper time, not coordinate time. There is no mechanism for coordinates. Coordinates are a mathematical mapping from events in the manifold to R4. They are not physical. That is the whole point.
DaleSpam
#445
Jan2-13, 09:31 AM
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Quote Quote by Austin0 View Post
Well you still have that v in the rhs of your equation. What does it represent??
As stevendaryl mentioned, it is just a parameter for the metric. Like M in the Schwarzschild metric. In fact, this is an unintentional similarity.

Quote Quote by Austin0 View Post
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???
Yes, it is the same. Pervect only transformed the time coordinate.

Quote Quote by Austin0 View Post
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???
It isn't a distance, it is a coordinate. Coordinates and distances are not the same thing. In this case, the coordinate is numerically equal to a distance in an inertial frame, but it is still a coordinate not a distance.

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 one-to-one, but no physical restrictions. The connection to physical measurements, like clocks and rods, is done through the metric.

Quote Quote by Austin0 View Post
What about simultaneity???
You are correct, I have not defined a simultaneity convention nor any coordinates off of Achilles' worldline. However, since we are only interested in events on Achilles' worldline it is hard to see why it would matter. If you like, the easiest thing will be to take the standard simultaneity convention of Achilles' inertial frame, however that will make the analogy a bit less direct since Achilles is anlogous to a free-falling local observer and the SC simultaneity convention does not correspond to the standard simultaneity convention of a free-falling local observer.

You could make some remote non-inertial 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.
mpv_plate
#446
Jan6-13, 08:06 AM
P: 87
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?
DaleSpam
#447
Jan6-13, 08:11 AM
Mentor
P: 16,947
Quote Quote by mpv_plate View Post
I just read through this whole thread
That is impressive! It's a big thread.

Quote Quote by mpv_plate View Post
Is it that simple, or I'm missing something?
Yes, it is that simple. Thanks for the excellent summary.
harrylin
#448
Jan7-13, 07:36 AM
P: 3,181
Quote Quote by mpv_plate View Post
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?
I have yet to catch up with the last two weeks, but yes there is more - for relativity of simultaneity as in SR is quite innocent compared with "will it really happen or not". And if I now correctly understand this matter then the answer to that question (and thus also to the question of this thread) is not accessible to us. This was also somewhat discussed in http://physicsforums.com/showthread.php?t=656240.

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.
DaleSpam
#449
Jan7-13, 07:46 AM
Mentor
P: 16,947
Quote Quote by harrylin View Post
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.
I am not sure which specific topic you are refering to by "this matter", but the whole point of expressing a physical theory in terms of a mathematical framework is precisely in order to ensure that the conclusions/predictions follow logically from the premises/postulates. You just seem to have difficulty with the mathematical framework which enforces the logic. That is a natural part of learning a challenging topic, but it does not in any way indicate a deficit in the logic of the theory.
Austin0
#450
Jan7-13, 07:44 PM
P: 1,162
Quote by Austin0

Well you still have that v in the rhs of your equation. What does it represent??
Quote Quote by stevendaryl;4215692 1)
It doesn't represent anything--2) it's just a number that is used to describe3) the relationship between two coordinate systems, and also happens to be 4) the speed of Achilles in one of the coordinate systems. .
Well i asked a perfectly cogent and relevant question. 1)you deny it is a valid question . then 2) you immediately contradict yourself and present two different possible reasonable answers 3),and 4) but both your answers seem questionable.

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??

Quote Quote by stevendaryl View Post
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).
I have no trouble with the idea of an arbitrary non-linear coordinate system.
In fact, back at my second post I brought up this possibility

Quote by Austin0
Having done so it appears that it was not explicitly stated that the intervals were equivalent. And in fact they would not correspond to time on any normal clock with a constant rate.
So are you talking about an arbitrary clock that speeds up over time ??
DaleSpam denied this back then but it appears that that is exactly the case here,,,, DO you now agree??

Quote Quote by stevendaryl View Post
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 be an inertial Cartesian coordinate system for some region of spacetime.
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).
this appears to me to be a generalization of the concept of transformation between relative frames. Is this correct??
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.

Quote Quote by DaleSpam View Post
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]
Don't you agree that to assert an equivalence between coordinates or values between two frames in relative motion you need to transform the values from one frame to the other.
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=100-vt[/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]100-vt=100/2^n[/itex] over time when the systems are not only in relative motion but one of them is non-linear??

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.

Quote Quote by stevendaryl View Post
In the case DaleSpam is talking about,

[itex]X'(x,t) = x[/itex]
[itex]T'(x,t) = log_2(\dfrac{100}{100-vt})[/itex]
Quote Quote by stevendaryl View Post
You are asking what the physical interpretation of the noninertial coordinates are--coordinates 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. .
in another thread you stated that gravitational time dilation could be eliminated by a coordinate choice remember??
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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