On the nature of the infinite fall toward the EH

[PAllen]

To discuss scenario like this we would have to have some idea how we would model occupied and available quantum states as we add more particles to given ensemble of particles. Or what happens with occupied and available quantum states as two ensembles of degenerate matter approach each other.

Your assumptions seems to be that particles affect occupancy of quantum levels only over short distance.
I assume that occupancy of quantum level drops as inverse square law as we go further from the particle.

So we have new fundamental law of physics: the "stellar exclusion principle" that prevents gathering too many stars in the same large region??! Remember, the EH forms before there is any singularity or even any high density anywhere in the formative collapsing mass.

 

[PeterDonis]

Your assumptions seems to be that particles affect occupancy of quantum levels only over short distance.
I assume that occupancy of quantum level drops as inverse square law as we go further from the particle.

So you're saying that quantum effects play a non-negligible part in the dynamics of stars that are separated by light-years? That, for example, quantum interactions between the Sun and Alpha Centauri affect the relative motion of those two stars?

 

[zonde]

This is relevant for the formation of the singularity, not for the formation of the EH. The singularity is an infinitely dense object, but an EH can form at arbitrarily low densities. For example, see Susskind's 12th lecture on GR () from about 2:00 to about 2:03 (of course the whole series is good).

I.e. your assumption "Formation of EH relies on idea that gravitating object can get more compact" is not correct. The formation of the singularity relies on that, but not the EH. The EH can form with simply a very large amount of non-compact material and you do not need a singularity in order to obtain an EH.

So again, what would prevent the formation of the EH? Degeneracy won't do it, that would only prevent the formation of the singularity.

There are two ways how to arrive at situation where EH is supposed to form.
First, we can add more matter to given gravitating object while it's radius is not increased too much by this addition.
Second, we can make given gravitating object more compact while it's mass is not reduced too much by this compactification.

I guessed that you was talking about the second scenario. If you are considering first scenario and want arguments concerning this scenario in particular please say it so that I don't have to guess.

 

[zonde]

So we have new fundamental law of physics: the "stellar exclusion principle" that prevents gathering too many stars in the same large region??! Remember, the EH forms before there is any singularity or even any high density anywhere in the formative collapsing mass.

New? Why new? I am just extrapolating existing law.

 

[zonde]

So you're saying that quantum effects play a non-negligible part in the dynamics of stars that are separated by light-years? That, for example, quantum interactions between the Sun and Alpha Centauri affect the relative motion of those two stars?

No, I am not talking about dynamics of stars but about dynamics of particles.
So what I say is that if we have two fairly degenerate stars approaching each other then whey would melt first and after that will start to evaporate. Or alternatively will fall into pieces depending on homogeneity of star.

If particles can't remain in their quantum states they can't maintain their collective structure. Kind of obvious IMO.

 

[Dale]

I guessed that you was talking about the second scenario. If you are considering first scenario and want arguments concerning this scenario in particular please say it so that I don't have to guess.

I am considering any scenario where an EH forms. If there are multiple ways for an EH to form then a mechanism for preventing EH formation has to prevent all of them.

In general an EH forms whenever there is enough mass inside the Schwarzschild radius. That can happen at any density, so a mechanism which prevents high densities, like degeneracy, simply cannot prevent EH formation in general.

 

[stevendaryl]

I suggest you to reformulate your question. Because there is a problem with it as it is stated. As you refer to pre-existing event horizon you imply that it is formed as a result of runaway gravitational collapse i.e. you are begging the question. I already raised the issue in post #402. So DaleSpam agreed that we should talk about hypothetical formation of event horizon instead.

I'm not begging the question. I'm asking you a question. Why do you believe that degeneracy has anything to do with the formation of an event horizon? You can certainly make up your own theory, but there is nothing in General Relativity that would suggest that. If you're not talking about General Relativity, then what are you talking about?

 

[stevendaryl]

No, I am not talking about dynamics of stars but about dynamics of particles.
So what I say is that if we have two fairly degenerate stars approaching each other then whey would melt first and after that will start to evaporate. Or alternatively will fall into pieces depending on homogeneity of star.

If particles can't remain in their quantum states they can't maintain their collective structure. Kind of obvious IMO.

If you are making up your own theory of gravity, then I think this is not the appropriate place to talk about it. If you are talking about mainstream physics, then it is well understood that degeneracy prevents further collapse for any star less massive than the Chandrasekhar limit (described here: http://en.wikipedia.org/wiki/Chandrasekhar_Limit).

 

[PeterDonis]

it is well understood that degeneracy prevents further collapse for any star less massive than the Chandrasekhar limit (described here: http://en.wikipedia.org/wiki/Chandrasekhar_Limit).

Small technical point: the Chandrasekhar limit applies to white dwarfs, i.e., to objects in which electron degeneracy is the primary mechanism resisting compression. The analogous limit for neutron stars, where neutron degeneracy is the primary mechanism, is the Tolman-Oppenheimer-Volkoff limit:

http://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_limit

Conceptually, both limits work the same, but the details are different because of the different types of fermions involved (neutrons vs. electrons).

 

[PeterDonis]

So what I say is that if we have two fairly degenerate stars approaching each other then whey would melt first and after that will start to evaporate. Or alternatively will fall into pieces depending on homogeneity of star.

Do you have any actual argument for why this would happen? Why would a degenerate star suddenly start melting? If the two degenerate stars collide with each other, then I could see matter being ejected from the collision; but if the stars are just free-falling towards each other, what difference would that make to their internal structure? The quantum states inside the star don't "know" that the two stars are approaching each other, unless they actually collide.

 

[PeterDonis]

Formation of EH relies on idea that gravitating object can get more compact without any change to physical laws.

Huh? This makes no sense. The physical laws involved are the Einstein Field Equation and the equation of state for the matter. It is well known that there are a range of reasonable equations of state that allow a gravitating object to get compact enough to form an EH; there are both analytical solutions and numerical simulations that show this. The laws certainly don't need to "change" at any point during the process.

 

[PeterDonis]

As you refer to pre-existing event horizon you imply that it is formed as a result of runaway gravitational collapse i.e. you are begging the question

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.

 

[Dale]

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]

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.

 

[Dale]

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]

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]

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?

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.

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?

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.

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??

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:

$$ds^2=-c^2 \left( \frac{(100-vt) ln(2)}{v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2$$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?

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.

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 $d=100-vt$ means that Achilles catches the tortoise at d=0 or 100-vt=0
so vt=100 and $$t = \frac{100}{v}$$

SO clearly yiour conclusion $$\lim_{n\to \infty } \, t = \frac{100}{v}$$ 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?

 

[Dale]

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.

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.

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.

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.

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.

SO clearly yiour conclusion $$\lim_{n\to \infty } \, t = \frac{100}{v}$$ 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.

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?

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.

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]

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

 

[Dale]

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.

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:
$$\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)}$$
and
$$t=\frac{100}{v}(1-2^{-n})$$

Substituting the second equation in on the rhs of the first equation and simplifying we get
$$\frac{dn}{dt}=\frac{2^n v}{100 ln(2)}$$

So the metric in post 414 should be:
$$ds^2=-c^2 \left( \frac{100 ln(2)}{2^n v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2$$

Which again is clearly not the Minkowski metric of an inertial frame, thereby demonstrating that the Zeno coordinates are non-inertial.

 

[Austin0]

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.

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 .

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?

I still disagree, the analogy is very close, but I don't understand your most recent objection.

Any closer?

 

[Austin0]

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:
$$\frac{dn}{dt}=\frac{v}{(100-vt) ln(2)}$$
and
$$t=\frac{100}{v}(1-2^{-n})$$

Substituting the second equation in on the rhs of the first equation and simplifying we get
$$\frac{dn}{dt}=\frac{2^n v}{100 ln(2)}$$

So the metric in post 414 should be:
$$ds^2=-c^2 \left( \frac{100 ln(2)}{2^n v} \right)^2 dn^2 + dx^2 + dy^2 + dz^2$$

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?

******************************_____

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.

So in this frame Achilles is a distance $d=100-vt$ behind the turtle. The definition of Zeno time, n, given is $d=100/2^n$.

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; $d=100-vt$ is not equivalent to the d' here in Zeno's frame; $d'=100/2^n$. 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]

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 $(x,t)$ be an inertial Cartesian coordinate system for some region $R$ of spacetime. Let $T'(x,t), X'(x,t), X(x',t'), T(x',t')$ be any four differentiable functions such that for any pair $(x,t)$ describing a point in $R$,

$X(X'(x,t), T'(x,t)) = x$
$T(X'(x,t), T'(x,t)) = t$

Then within region $R$, you can use coordinates $x', t'$ defined by

$x' = X'(x,t)$
$t' = T'(x,t)$

As far as GR is concerned, $(x',t')$ can be used just as well as (x,t).

In the case DaleSpam is talking about,

$X'(x,t) = x$
$T'(x,t) = log_2(\dfrac{100}{100-vt})$

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.

 

[Dale]

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.

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.

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.

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 $d = 100 \; 2^{-n}$, so Achilles' coordinate acceleration is the second derivative wrt n which is $a = 100 \, 2^{-n} ln(2)^2$.

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

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.

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.

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.

 

[Dale]

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.

So in this frame Achilles is a distance $d=100-vt$ behind the turtle. The definition of Zeno time, n, given is $d=100/2^n$.

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; $d=100-vt$ is not equivalent to the d' here in Zeno's frame; $d'=100/2^n$. is it?

Yes, it is the same. Pervect only transformed the time coordinate.

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.

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]

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?

 

[Dale]

I just read through this whole thread

That is impressive! It's a big thread.

Is it that simple, or I'm missing something?

Yes, it is that simple. Thanks for the excellent summary.

 

[harrylin]

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 https://www.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.

 

[Dale]

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 referring 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]

Quote by Austin0

Well you still have that v in the rhs of your equation. What does it represent??

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??

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??

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 $(x,t)$ be an inertial Cartesian coordinate system for some region $R$ of spacetime. Let $T'(x,t), X'(x,t), X(x',t'), T(x',t')$ be any four differentiable functions such that for any pair $(x,t)$ describing a point in $R$,

$X(X'(x,t), T'(x,t)) = x$
$T(X'(x,t), T'(x,t)) = t$

Then within region $R$, you can use coordinates $x', t'$ defined by

$x' = X'(x,t)$
$t' = T'(x,t)$

As far as GR is concerned, $(x',t')$ 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.

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:
$$n=log_2 \left( \frac{100}{100-vt} \right)$$

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 : $d=100-vt$ in the A frame and $d=100/2^n$ 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 $100-vt=100/2^n$ 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.

In the case DaleSpam is talking about,

$X'(x,t) = x$
$T'(x,t) = log_2(\dfrac{100}{100-vt})$

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??

 

[Austin0]

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.

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.

It is not a matter of spatial coordinates not being assigned elsewhere, because there is no means ,with the given information, to assign coordinates to Achilles himself after the initial instant either, is there?

Quote by Austin0

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.

Please take note of what I actually said . I didn't say coordinates (Which we don't actually have) but distances between the two bodies in inertial motion.which is all that is actually given to work with.
The point was not about coordinates but about inferring a state of motion from the observations.
As far as that goes is there necessarily any rigid constraint besides the signs of the signature and the Pythagorean theorem for a valid inertial metric?
Quote by Austin0

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.

yes and if Achilles is defined as inertial and has zero coordinate acceleration in the Zeno frame then it would follow that the Zeno frame was also inertial (in uniform motion)YES?? Which is what I said.Quote by Austin0

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 , so Achilles' coordinate acceleration is the second derivative wrt n which is .

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

No I am not talking about the coordinate acceleration of Achilles in the Zeno frame which is indeterminable as far as I can see.
If you disagree please explain.
I am talking about what possible motion of the Zeno frame could make the observed relationship between Achilles and the tortoise occur.

Explicitly,, the decreasing rate of the decrease of the distance between them or comparably,, the decrease in the relative velocity between them..

Another perspective is; what possible state of motion of the Zeno frame as charted from the Achilles frame could accomplish this.


It is clear that if the Zeno frame is actually inertial in motion (constant) then an arbitrary clock rate could easily effectuate those observations. Yes?
That this is a state of motion and condition that would be consistent with the Zeno observations. Agreed?

Quote by Austin0

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 by Austin0

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.

IMO you are incorrect here. In this scenarion we are talking about a system of arbitrarily scaled clocks. Equivalent to the clocks in the GPS system which are artificially calibrated for synch purposes. A physical mechanism.
In the GPS case the artificial rate is constant. In the Zeno case the rate is increasing but the principle is the same.

In the Sc case the static clocks are natural but incrementally decreasing in rate towards the center but that isn't relevant. In the Zeno case we can assume that either all system clocks are identical and exponentially increasing in rate or that the system clocks have increasing rates along the path of Achilles but in either case they must be mechanically operating at different rates , yes?

Quote by Austin0

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.

So you think the calibrated GPS clocks are measuring proper time??

I am not following you here.
If the sole definition of time is that which clocks measure then time has no existence or meaning independent of clocks yes? Coordinates are measured and assigned by physical clocks yes?? All coordinates to events in the manifold are determined and assigned by actual clocks at the actual locations.
All calculations of coordinate times at specific locations are related to actual or hypothetical physical clocks and what they would indicate for proper time at hypothetical events at those locations, yes?
I understand the difference between proper time intervals as measured by a single clock and calculated time intervals between clocks at disparate locations but any such calculated coordinate time interval, in the end corresponds to the times read on physical clocks (even if hypothetical), agreed??

 

[Austin0]

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.

Well the M in the Sc metric represents specific things yes?? Either mass or a distance. Or is this incorrect?

so if this v represents a velocity , it is the velocity of what relative to the Zeno frame? And how did you arrive at it??

Quote by Austin0 View Post

So in this frame Achilles is a distance behind the turtle. The definition of Zeno time, n, given is .

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; is not equivalent to the d' here in Zeno's frame; . is it?

Yes, it is the same. Pervect only transformed the time coordinate.

Well I can't speak for Pervect's thought processes but what was actually defined was not a simple transformation of Achilles time.
It was a series of observations which described an obvious similarity to the Sc case.

Zeno time was not a function of Achilles time or Achilles velocity or position. It was limited to a function of the numerical value of the distance between Achilles and the tortoise ,divorced from position.

As I have been trying to make clear these conditions (observations) could be consistent with any number of possible coordinate time/ clock configurations, rates etc in the Zeno frame. Ditto Zeno simultaneity conventions. Obviously these would result in different transformations in each case , yes??
so this would seem to be a classic catch 22. There is not enough info to infer a Zeno metric, Without a defined Zeno metric you cannot derive a valid transformation . Without a valid transformation you cannot derive a Zeno metric.


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.

Well in the Achilles frame the distance 100-vt is identical to the tortoise x coordinate it's true but $d=100/2^n$ is clearly a fraction of a distance (a dx') and after the initial instant there is no basis for determining a coordinate location for either Achilles or the tortoise,,yes?SO it can't be a coordinate.
SO then you are claiming equivalence of a coordinate with an interval ,no??
How does whether it is a distance or a coordinate relate to equivalence anyway?.

If x or dx = 100 and x' or dx'= 100 does this mean x =x' or dx=dx' ?

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.

How does GR enter into this question. As far as this analogy and discussion is concerned the exercise is taking place in flat spacetime. Otherwise none of the participants are actually inertial are they?

And what does any of this have to do with whether or not $100-vt=100/2^n$ is a valid equivalence of values between two frames??

Isn't it just making an unsupported assumption of equivalence ,deriving a transformation from that which then makes it equivalent ex post facto?

There seems to be some confusion as to the meaning of an ananolgy. As i understand it , it is taking one situation ( the primary) A , which is simpler or more known and unambiguous and applying inferences or conclusions gained from it to another similar situation B.
In this case from Z to Sc. This is essentially a one way street. If you start taking inferences, principles and conclusions from B and inserting them into A you have simply negated any value as an analogy and made it a tautology.

The point of this analogy as I see it revolves around the question of the finite proper time of the Sc infaller. In the Sc context this seems to be a point of contention for a couple of reasons.

1) Due to the various effects of curved Spacetime the infaller disappears from outside observation even before reaching the EH so there is no possibility of empirical observation of the time at the EH
2) The necessity of integrating proper time and applying the theorem of limits or convergence makes this calculation somewhat less than conclusive in the minds of many.

SO the scenario was recast in the realm of ordinary observable reality where the results by implication could be definitvely determined by observation.Where Achilles would incontrovertibly reach the Tortoise horizon in finite proper time.

So now the finite Achilles proper time is a given and the infinite Zeno time has now become the questionable reality.

So the analogy has been turned upside down. Now Zeno time is analogous to the abstract infinite regression of Achilles motion in the original scenario except it is now ,an equally abstract, infinite progression. And just like Achilles DOES catch up with the tortoise,,,, the Zeno time on the Zeno clocks will in fact register some FINITE time when he does so.

So this is just another abstraction that has no correspondence to the real world. In this case the time is an arbitrary scaling which could not actually occur on real clocks, so are you suggesting that the Sc metric which is based on real natural clocks is equally divorced from reality?
That the Sc metric relating time to clocks is not in correspondence to the real world ?
To me the real point and resolution of Zeno's scenarios is that abstract mathematical and logical ideas do not necessarily work in the actual universe

It has come up on several occasions in this and related threads; the concept of mapping a finite set to an infinite one, with the idea that this was possible and taking place in the Sc question.

Well others may disagree but to me this proposition seems to be self evidently impossible in the real world. It can only occur in the abstract realm of mathematics where virtually anything is possible. But unless you remove any meaning from the word infinity I think the reasoning to accomplish this miracle must always ultimately be incorrect and simply distracting from the obvious mutual exclusivity and contradiction inherent in the premise. Just MHO

I certainly agree that spacetime is a singular continuum with a unique set of events and no coordinate substitution can alter that but I am somewhat confused when you turn around and because you don't like certain events in the Sc coordinates you then change them by switching to SK coordinates. And your proposal that because they do occur is those alternate coordinates they must happen in all coordinate systems ignores the arbitrarity of this choice. I.e . The same reasoning should apply regarding the negative event in the SC coords.
If it doesn't happen there it doesn't happen in any system. Note I am not claiming anything about the reality of events near the horizon but simply commenting on the reasoning behind taking a rigid position either way.

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.

We can assume the standard simultaneity convention in Achilles (inertial) frame no problem but for the Zeno frame there is not enough information to define one. WHich is a part of my point. In the absense of this information it is just making assumptions without basis. And to redefine the scenario with the relevant parameters explicit would be a completely different case and much of this discussion would have been unnecessary IMO

 

[stevendaryl]

Quote by Austin0

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.

I think you're just being argumentative at this point. I didn't deny that what you asked was a valid question, I answered your question as best as I could.

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,

Why do you say that? You have one set of coordinates, $(x,t)$, and in those coordinates, the path of Achilles is $x = vt$

You make a coordinate change to $(x',t')$ given by:

$x′=x$
$t' = log_2(\dfrac{100}{100 - vt}) = log(\dfrac{100}{100 - vt})/log(2)$

To compute coordinate velocity in the coordinates $(x',t')$, you use:

$\dfrac{dx'}{dt'} = \dfrac{\dfrac{\partial x'}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial x'}{\partial t}}{\dfrac{\partial t'}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial t'}{\partial t}}$For the coordinate change that we're talking about:
$\dfrac{\partial x'}{\partial x} = 1$

$\dfrac{\partial x'}{\partial t} = 0$

$\dfrac{\partial t'}{\partial x} = 0$

$\dfrac{\partial t'}{\partial t} = \dfrac{v}{log(2) (100 - vt)}$

$\dfrac{dx}{dt} = v$

So we have:

$\dfrac{dx'}{dt'} = log(2) (100 - vt)$

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?.

I have no idea what you are talking about. We know Achilles' velocity components in the $(x,t)$ coordinate system, we know how to transform to the $(x',t')$ coordinate system, so we can easily derive Achilles' velocity components in the $(x',t')$ coordinate system. Why do you think there's a problem?

I have no trouble with the idea of an arbitrary non-linear coordinate system.

Then you should understand how to transform velocity components in one coordinate system to velocity components in another.

SO in this case we are given : $d=100-vt$ in the A frame and $d=100/2^n$ in the Z frame.

The word "frame" doesn't mean anything in this case. There are two different
COORDINATE SYSTEMS in use. They aren't two different frames.

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??

No, it's not axiomatic. It's possible that in transforming between two coordinate systems. the synchronization convention changes, but not always.

It follows that the distances , the spatial intervals in the two frames cannot be congruent also Yes??

No. It depends on the transformation. For this particular transformation,
the standard of length is not changed. The standard of simultaneity is not
changed. But the standard for time interval is changed.

in another thread you stated that gravitational time dilation could be eliminated by a coordinate choice remember??

Yes, it's always possible to find a LOCAL coordinate system in which SR is approximately valid (no "gravitational time dilation").

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.

No, that's not true at all. A time coordinate is just a real number associated with points in spacetime. It's convenient if you have a clock that gives the time, but it's not necessary.

When some switches from rectangular coordinates $(x,y)$ to polar coordinates $(r,\theta)$, do you think it's necessary to perform an artificial mechanical adjustment to all your rulers, so that they can measure $r$ and $\theta$? No, of course not, because you can compute $r$ and $\theta$. You don't need a measuring rod that directly measures $r$ and $\theta$.

Changing the time coordinate is no different. You don't need a clock that computes the new time coordinate, you just need to be able to compute the time coordinate from whatever information you have from normal clocks.

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?

No, that's not correct.

Such artificial scaling is in fact used in the GPS system right?? Those clocks physically increment at a different rate yes??

I don't know for GPS whether the clocks are altered, or whether the times are manipulated afterward. It's conceptually the same, although one or the other might be more convenient.

 

[stevendaryl]

There seems to be some confusion as to the meaning of an ananolgy. As i understand it , it is taking one situation ( the primary) A , which is simpler or more known and unambiguous and applying inferences or conclusions gained from it to another similar situation B.

Yes, and it's clear that you don't understand the simpler case, either, so it was a failed tactic.

The point of the Zeno case is that we have one coordinate system, $(x,t)$ that is used to describe a region of spacetime

$0 \leq x \leq 100$
$0 \leq t \leq 100/v$

We have a second coordinate system, $(x',t')$ that describes the SAME region of spacetime as follows:

$0 \leq x' \leq 100$
$0 \leq t' < \infty$

The coordinate transformation $t' = log_2(\dfrac{100}{100 - vt})$ maps a finite interval of time coordinate $t$ to an infinite interval of time coordinate $t'$.

That's the Zeno case. It's just a change of coordinates. The fact that $t' \rightarrow \infty$ as Achilles approaches the Tortoise does NOT imply that Achilles never reaches the Tortoise, it just implies that that event is not covered by the $(x',t')$ coordinate system.

The Schwarzschild case is ALSO just a change of coordinates. It ALSO maps a finite interval of time in one coordinate system (the freefalling coordinate system) to an infinite interval of time in another coordinate system (the Schwarzschild coordinate system). The fact that Schwarzschild time goes to infinity as an observer approaches the horizon does NOT imply that the observer reaches the horizon. It just implies that that event is not covered by Schwarzschild coordinates.

The analogy is just about perfect.

 

[stevendaryl]

We can assume the standard simultaneity convention in Achilles (inertial) frame no problem but for the Zeno frame there is not enough information to define one.

What are you talking about? For one thing, there is no "Zeno frame". It's an alternative COORDINATE system. It's just like using polar coordinates instead of rectangular coordinates. It doesn't imply that anybody is in a different "frame".

The person wanting to use Zeno coordinates has the exactly the same information as the person wanting to use inertial cartesian coordinates. He can just take the inertial cartesian coordinates and perform a mathematical transformation to get the Zeno coordinates. Why do you believe that there is any problem in doing this?

WHich is a part of my point. In the absense of this information it is just making assumptions without basis.

There are no assumptions being made. They're just using a different coordinate system.

It's really just like polar coordinates. Do you need additional assumptions in order to be able to use polar coordinates? No, you can just COMPUTE $r$ and $\theta$ from $x$ and $y$, if you know the latter, using the transformations:

$r = \sqrt{x^2 + y^2}$
$\theta = arctan(y/x)$

The "Zeno time" is just a mathematical function of the time in the inertial coordinate system. No assumptions are needed to use it.