On the nature of the infinite fall toward the EH

[A.T.]

An interesting intuitive approach for this problem is presented by Epstein in his book "Relativity Visalized". He uses space-propertime embeddings of the Schwarzshild-Metric like this one:

http://www.adamtoons.de/physics/gravitation.swf

"space" : radial Schwarzshild coordinate r
"proper time" : the proper time elapsed along the cyan world line
length of the world line : Schwarzshild coordinate time t
inflation of the "pipe" at certain r : time dialtion of a static clock at that r relative to t

For a BH the funnel would inflate infinitely into a plateau at the event horizon:

The cyan world line here can have an infinite length (coordinate time t) but will still cover a finite angular displacement around the pipe (proper time)

 

[Austin0]

Just a quick mention that in Schwarzschild coordinates, sqrt(1 - 2 m / r) = K sqrt(1 - (v'/c)^2), where v' is the speed that is locally measured by a static observer at r and K is a constant of motion, with K = 1 for a freefall from rest at infinity, so those two statements would be equivalent in terms of dt and dτ in that case.

Thank you.
As free fall from rest at infinity was in fact the context of the thread I was correct in my understanding then , right??
Am I correct in assuming that K=1 in this case is related to free fall velocity at a particular r being equivalent to the escape velocity at that location, which is the case when starting from v=0 at r=∞?

 

[grav-universe]

Thank you.
As free fall from rest at infinity was in fact the context of the thread I was correct in my understanding then , right??
Am I correct in assuming that K=1 in this case is related to free fall velocity at a particular r being equivalent to the escape velocity at that location, which is the case when starting from v=0 at r=∞?

Actually, you were correct regardless of where the object falls from or the value of K. K is the constant of motion for freefall here, so with the initial conditions v' = 0 at r = ∞, we get

K = sqrt(1 - 2 m / ∞) / sqrt(1 - (0/c)^2) = 1

and K will then remain constant for any r during freefall, even non-radially, so one can find the speed locally measured by a static observer at some other r with

v' = c sqrt[1 - (1 - 2 m / r) / K^2]

Since SR is valid locally, we have

dτ = dt' sqrt(1 - (v'/c)^2)

= dt sqrt(1 - 2 m / r) sqrt(1 - (v'/c)^2)

= dt (1 - 2 m / r) / K

Only the last statement depends upon the value of K. Vice versely if an object were thrown upward from r with the corresponding speed v' where K = 1, that would be its escape velocity also, right.

 

[Austin0]

The underlying thought process here is that there is some physically meaningful way to define a "local rate of time". Relativity doesn't necessarily say this. (I think one can make even stronger claims, but it'd start to detract from my point, so I'll refrain from now).

One can certainly say that Alice appears to freeze according to the coordinate time "t". But is this physically significant?

It might be instructive to consider Zeno's paradox. I'll use the wiki definition of the paradox.
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.

Are we therefore justified in claiming that Zeno was right, and that Achilles never catches the tortise? I don't think so, and I'd be more than a bit surprised if anyone really believed it. (I could imagine someone who likes to debate claiming they believed it as a debating tactic, I suppose - and to my view this would be a good time to stop debating and do something constructive).So in my opinion, the confusion arises by taking "zeno time", which is analogous to the Schwarzschild coordinate time "t", too seriously. While it is correct to say that as t-> infinity Alice never reaches the event horizon, just as Achilles never reaches the tortise in zeno time, it still happens. It's just that that event hasn't been assigned a coordinate label.

The analogy is actually quite apt regarding light chasing an accelerating system with only the slight modification that the tortoise has some finite constant acceleration. In which case at and beyond some magnitude of head start Achilles (light) can never catch up.

But wrt approaching an EH I think both the paradox and the related theorem that resolves the same kind of unbounded series to a finite value
1/2 + 1/4 + 1/8 + ... = 1

are not validly applicable.

In the first case (Zeno) as the distance incrementally reduces, the velocity of Achilles remains constant. So for each reduction in distance, the time for the next reduction in distance becomes shorter.

So it is obvious, even without a formal mathematical proof, that the difference between an evaluation of some large but finite number of iterations and the self evident ultimate value after an infinite number of iterations effectively disappears.

In approaching the horizon this is not true. Each reduction in distance results in a reduction in speed so increases the time interval for the next distance. Etc etc
As the speed approaches zero nearing the horizon the time approaches infinite which is clearly a whole other ball game. Or at least seems so to me.
Neither the theorem nor the paradox apply.

 

[PeterDonis]

Each reduction in distance results in a reduction in speed

The "speed" that is reduced is just a coordinate "speed". It doesn't have any physical meaning. For example, there is no observer who observes the infalling object moving at this "speed".

 

[Dale]

In the first case (Zeno) as the distance incrementally reduces, the velocity of Achilles remains constant. So for each reduction in distance, the time for the next reduction in distance becomes shorter.

In Zeno coordinate time the time for the next reduction is constant, by definition. So the Zeno coordinate velocity in fact reduces.

It is the proper time which reduces. And the velocity in some unspecified inertial coordinate system which remains constant.

In approaching the horizon this is not true. Each reduction in distance results in a reduction in speed so increases the time interval for the next distance. Etc etc
As the speed approaches zero nearing the horizon the time approaches infinite which is clearly a whole other ball game. Or at least seems so to me.
Neither the theorem nor the paradox apply.

No, the two scenarios are very closely analogous on this point. Again in SC coordinate time the time for the next reduction is constant, by definition. So as you mention the SC coordinate velocity reduces.

Similarly, the proper time reduces in the SC case, and the velocity in a local inertial frame remains constant. Exactly analogously to Zeno.

 

[zonde]

It could happen. Note, though, that here at PF, we have an educational evnvironment, not a research one. Our goal is not to advance the state of science. While this is of course an important task, it's not our goal.

And ... ? I know that. Why do you think you have to remind me that?

 

[Austin0]

In Zeno coordinate time the time for the next reduction is constant, by definition. So the Zeno coordinate velocity in fact reduces.

It is the proper time which reduces. And the velocity in some unspecified inertial coordinate system which remains constant.

No, the two scenarios are very closely analogous on this point. Again in SC coordinate time the time for the next reduction is constant, by definition. So as you mention the SC coordinate velocity reduces.

Similarly, the proper time reduces in the SC case, and the velocity in a local inertial frame remains constant. Exactly analogously to Zeno.

Actually I was talking within the context of the original statement of the paradox where the distance was the basis parameter so I didn't look at the specifics of Pervect's Zeno time .

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.

According to such a constant clock the interval between events Zeno t=1 and t=2 would be smaller than between events Zeno t=0 and t=1
Or do you disagree??

So are you talking about an arbitrary clock that speeds up over time ??

Could you explain where you get this "Again in SC coordinate time the time for the next reduction is constant, by definition." In the original, the next reduction was reducing the remaining distance to the horizon by half so how do you get a constant time interval for each of these increments?

 

[Dale]

Actually I was talking within the context of the original statement of the paradox where the distance was the basis parameter so I didn't look at the specifics of Pervect's Zeno time .

Yes, he adapted the original paradox deliberately in order to make the analogy with SC time more exact.

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.

According to such a constant clock the interval between events Zeno t=1 and t=2 would be smaller than between events Zeno t=0 and t=1
Or do you disagree??

I agree completely. Again, the whole point of the analogy is that the exact same thing happens with SC time. For Zeno time, the proper time on Achilles' clock between Zeno t=1 and t=2 is indeed smaller than between Zeno t=0 and t=1. For SC time, the proper time on the free-falling clock between SC t=1 and t=2 is also smaller than between SC t=0 and t=1.

So are you talking about an arbitrary clock that speeds up over time ??

No, I just mean that coordinate time proceeds at a rate of one coordinate second per coordinate second, by definition. It is a simple tautology. If you are using coordinate time as your standard (as the SC proponents want to do) then coordinate time is uniform, by definition, i.e. tautologically.

In SC coordinate time each successive SC coordinate time interval for the free-faller is tautologically constant. In Zeno coordinate time each successive Zeno coordinate time interval for Achilles is tautologically constant. Neither correspond to the proper time on the falling/Achilles' clock.

Could you explain where you get this "Again in SC coordinate time the time for the next reduction is constant, by definition." In the original, the next reduction was reducing the remaining distance to the horizon by half so how do you get a constant time interval for each of these increments?

The analogies diverge quantitatively, but not qualitatively. In SC coordinates each successive SC coordinate time interval does not correspond to half the distance to the horizon, that is a feature of the construction of Zeno coordinates. But in both SC and Zeno coordinates the coordinate distance traveled by the free-faller/Achilles decreases for each successive coordinate time interval. This obviously means that their coordinate velocity is reducing, which you already recognized and pointed out above.

 

[pervect]

Just a quick mention that in Schwarzschild coordinates, sqrt(1 - 2 m / r) = K sqrt(1 - (v'/c)^2), where v' is the speed that is locally measured by a static observer at r and K is a constant of motion, with K = 1 for a freefall from rest at infinity, so those two statements would be equivalent in terms of dt and dτ in that case.

As long as v actually is the speed that's locally measured by a static observer, I believe that's correct. I usually use E for K, many sources use ~E.

It wasn't clear to me how v was being defined - I should have asked. I should still ask, because it's still not clear to me how the OP is defining v, and it's very common not to use the correct formula or defintion of v.

The relation between v and the derivatives of the coordinates is moderately messy, but in https://www.physicsforums.com/showpost.php?p=602558&postcount=29

I get (and another poster also gets) in geometric units.

v= \frac{\sqrt{E^2 - (1 - \frac{2M}{r})}}{E}(The natural way to do this is via frame fields, but I choose to introduce locally Lorentz coordinates instead. THe intent was to make it easier to follow, I'm not sure how successful it was. But the intent is to use local coordinates rr and tt that agree with the local clocks and rulers.)

This expression for v is also what I get when I solve your equation for v/c (which is just v in geometric units as c is assumed to be 1).

 

[zonde]

Definitely. Changing scientists minds is the whole point of doing physics, both theoretical and experimental. Does that surprise you in any way?

It should be experimental results that can change viewpoint of overwhelming majority of the physicists not theoretical arguments. If there are different theoretical arguments they should get balanced support until it is decided from experiments or observations.

Take for example Higgs boson.

Okay there is another possibility when different viewpoints can't be supported at the same time. We can take one viewpoint as a working hypothesis and go with it for some time.

The point is that the opposition to the existence of the interior of a BH is not based on a sound understanding of the theory.

The point is that discussion between position and opposition to the existence of the interior of a BH lacks scientific basis.

It is based on an unsound elevation of a particular coordinate chart to some priveliged status.

We use some coordinate system to order our observations. In order to compare theoretical predictions with our observations we have to express theoretical predictions in a form that is convenient for that coordinate system.

 

[PAllen]

Okay there is another possibility when different viewpoints can't be supported at the same time. We can take one viewpoint as a working hypothesis and go with it for some time.

Is your view really that the interior doesn't exist or that collapse freezes? In the case of a collapsing mass, there is an interior at all times.

 

[zonde]

Is your view really that the interior doesn't exist or that collapse freezes? In the case of a collapsing mass, there is an interior at all times.

My view is is that there is no such thing as runaway gravitational collapse.

 

[Dale]

The point is that discussion between position and opposition to the existence of the interior of a BH lacks scientific basis.

That much is true. I think that the bulk of the argument stems from a misunderstanding or mistrust of the basic mathematical framework of GR.

We use some coordinate system to order our observations. In order to compare theoretical predictions with our observations we have to express theoretical predictions in a form that is convenient for that coordinate system.

I think you have this backwards. The predictions are all invariants, so all coordinate systems agree. We pick a coordinate system so that the calculation of those invariants is easy.

 

[Dale]

My view is is that there is no such thing as runaway gravitational collapse.

What would stop it? I mean, not the singularity, but the horizon.

 

[PAllen]

My view is is that there is no such thing as runaway gravitational collapse.

And what stops it for a supermassive BH, where densities are quite low at SC radius? It's clear what you will see from afar (the cluster of stars slowing, effectively freezing, and forming a black object at essentially SC radius). But for someone orbiting one of the stars in the interior, what do you think is experiences? Are we (from afar) not allowed to ask that just because we can't see it?

 

[Austin0]

I.e. " Both observers will agree on these relative elapsed times and both observers will agree that the faller has not reached the horizon."

SO in principle there is a finite point, short of the horizon, where both observers will agree that the distant clock reads 10¹²years and the inertial clock reads some relatively short time (in related threads approx. 1 day has been mentioned for freefall proper time to EH) correct?
This is a rational application of the metric as it pertains to and in both frames, agreed?


How do you manage to turn this into an idea that the free faller reaches the horizon in some relatively short time in the real world. I.e. the majority of the universe which is outside the EH and relatively static.

I'm not sure the context, but a free fall observer will never see something like 10^12 years on distant clock. As I've explained, if they start free fall from relatively far away, they will see the distant clock fall behind theirs (but not by a lot).

Taking it as given that we are not talking about visually seeing but rather calculating through the metric, how do you calculate that the distant static clock falls behind the inertial clock approaching the horizon?

Could you explain your statement above regarding time on the static clock at infinity??

DO you think that the geometry that the falling clock is passing through has no effect on the periodicity of this clock??
That it would not be red shifted relative to the distant clo9ck equivalent to a proximate static clock?

That the integrated proper times of the relative clocks would not be related by the metric?

That dt=d\tau/(1-2M/r)^1/2(1-v²/c² would not apply?

I was under the impression that it was an implicit assumption of valid coordinate systems that relative velocity was symmetric and reciprocal.
That the velocity of the faller relative to the distant observer is the same as the velocity of the distant observer relative to the faller.
Does this not hold in Sc coordinates?

 

[PAllen]

Taking it as given that we are not talking about visually seeing but rather calculating through the metric, how do you calculate that the distant static clock falls behind the inertial clock approaching the horizon?

Within limits, we are talking about seeing. In the case of supermassive BH, conditions on event horizon crossing are not extreme in any way.

Could you explain your statement above regarding time on the static clock at infinity??

DO you think that the geometry that the falling clock is passing through has no effect on the periodicity of this clock??

For a supermassive BH, there is minimal curvature at the horizon.

That it would not be red shifted relative to the distant clo9ck equivalent to a proximate static clock?

That the integrated proper times of the relative clocks would not be related by the metric?

That dt=d\tau/(1-2M/r)^1/2(1-v²/c² would not apply?

In the above, you have two limits competing. Remember, v is relative to an adjacent static observer. For any infaller, v->c as horizon is approached. The limit of the product is always finite, and for free fall from infinity represents a redshift at the horizon. Inside the horizon, this formula loses all validity because there are no static observers. However, there is a uniform approach to redshift and clock comparison that I have explained several times on this thread. Using the general method (aside: it is never necessay to use gravitational redshift - that is computational convenience for the very special case of static spacetime - which doesn't exist inside the horizon; it also doesn't exist for to co-orbiting neutron stars), redhshift perceived by an inside horizon observer remains finite, and (for free fall from far away from BH) reshifted up to singularity.

I was under the impression that it was an implicit assumption of valid coordinate systems that relative velocity was symmetric and reciprocal.

Relative velocity at a distance is undefined in GR. Only relative velocity for nearby observers is defined. Coordinate velocity is not relative velocity. It is a purely arbitrary convention.

That the velocity of the faller relative to the distant observer is the same as the velocity of the distant observer relative to the faller.
Does this not hold in Sc coordinates?

There is no such thing as relative velocity for distant observers in GR, at all. The basic issue is that if you bring one 4-velocity over a distance to another, you get a different result depending on what path you choose. That is at the core of the definition of curvature. There is no physical basis to choose one path over another. Thus curvature precludes giving meaning to relative velocity at a distance.

 

[Austin0]

Sure.

Near the surface of the Earth, the metric can be described approximately using the line element ds^2 = (1+gX)^2 dT^2 - dX^2 where X is the height above the surface, and g is the acceleration due to gravity.

In these coordinates, we can compute the "rate" \dfrac{d \tau}{dT} for a clock at rest at height X:

\dfrac{d \tau}{dT} = (1+gX)

So higher clocks (larger X) have a higher rate. In particular, if an observer at sea level sends a signal once per millisecond (according to his clock) toward an observer on top of a mountain, the arrival times for the signals will be slower than one per second, according to the clock at the top of the mountain.

Now, transform coordinates to free-fall coordinates x,t defined by:

x = (1/g + X) cosh(gT) - 1/g
t = (1/g + X) sinh(gT)

In terms of these coordinates, the metric looks like:

ds^2 = dt^2 - dx^2

This is the metric of Special Relativity. In these coordinates, there is no "gravitational time dilation". The locations of clocks have no effect on their rates. In particular, a clock at sea level will have the same rate as a clock on top of a mountain. Initially.

So, how, in terms of these coordinates, does one explain the fact that signals sent once per millisecond from an observer at sea level arrive on top of a mountain at a rate lower than that? Well, in the free-falling coordinate system, the two observers are accelerating upward. Each signal sent by the observer at sea level must travel farther than the last to reach the observer on the mountain. So the free-falling coordinate system attributes the difference in send rates and receive rates purely to Doppler shift, not to time dilation. (At least initially.)

Interesting. Wouldn't you agree that free falling (inertial ) systems of more than very limited radial extent are highly problematic for various reasons.

EG. The Born rigidity question rears its head. Differential acceleration and velocities at separated locations etc.

But ignoring these considerations for a moment:In principle measurements of static clocks at two heights could be accomplished by falling observers without necessity of signal exchange between the static clocks. Comparing elapsed times on two separated clocks for extended intervals which is what is required to measure rate which is not instantaneously determinable.

So it would seem that to the extent that observations from an infalling frame aren't too ambiguous to be meaningful they support the validity of gravitational dilation as an independent local effect of mass.

Regarding the EP ,,,I certainly consider it one of the most brilliant and productive abstract bootstraps in scientific history. And the result, the relativity of time flow due to gravity is beyond question at this point. That being said I think that it is somewhat abused in certain cases and that there are limits to its validity as an analogy .

SO the difference in local rates can be empirically demonstrated simply by relocation without need of a coordinate system beyond identical uniform rate parameterization. This is a physical fact or as close to a fact as any of our physics gets so what does it really mean to say that it is "as if" the clocks were actually moving radially upward under impulse and so the dilation isn't really due to gravity but is from relative motion as you are suggesting here?

 

[PAllen]

Interesting. Wouldn't you agree that free falling (inertial ) systems of more than very limited radial extent are highly problematic for various reasons.

EG. The Born rigidity question rears its head. Differential acceleration and velocities at separated locations etc.

For supermassive BH, these issues are non-existent at horizon. There is no more tidal forces than at the Earth's surface.

 

[Austin0]

For supermassive BH, these issues are non-existent at horizon. There is no more tidal forces than at the Earth's surface.

That is fine but is not relevant to his post which was regarding time dilation of clocks at differing altitudes . I.e not both near the surface.
I trust you are not suggesting that surrounding such a BH that an extended bar would not be subjected to stresses from the difference in g at the top and bottom?

 

[PAllen]

That is fine but is not relevant to his post which was regarding time dilation of clocks at differing altitudes . I.e not both near the surface.
I trust you are not suggesting that surrounding such a BH that an extended bar would not be subjected to stresses from the difference in g at the top and bottom?

For a supermassive black hole, something as big as the Empire State building would have no more stresses at the horizon than it does on the Earth's surface (could be made arbitrarily small, actually). The horizon is not intrinsically related to any particular amount of local curvature, stresses, etc. Only the singularity is. For stellar black holes, extreme stresses and tidal forces occur long before the horizon - e.g. approaching a neutron star.

 

[PAllen]

Regarding the EP ,,,I certainly consider it one of the most brilliant and productive abstract bootstraps in scientific history. And the result, the relativity of time flow due to gravity is beyond question at this point. That being said I think that it is somewhat abused in certain cases and that there are limits to its validity as an analogy .

But the e.p. suggest gravitational time dilation can equally be considered the same as acceleration in SR. And acceleration of two rigidly connected clocks in SR, observed in an inertial frame, differ in clock rate purely due to speed difference between the front and the back. Thus the EP says gravitational time dilation is equally subject alternate, coordinate dependent interpretations.

 

[Austin0]

Quote by Austin0

Actually I was talking within the context of the original statement of the paradox where the distance was the basis parameter so I didn't look at the specifics of Pervect's Zeno time .

Yes, he adapted the original paradox deliberately in order to make the analogy with SC time more exact.

I would say in an attempt to make it appear to apply ;-)

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.

According to such a constant clock the interval between events Zeno t=1 and t=2 would be smaller than between events Zeno t=0 and t=1
Or do you disagree??

I agree completely. Again, the whole point of the analogy is that the exact same thing happens with SC time. For Zeno time, the proper time on Achilles' clock between Zeno t=1 and t=2 is indeed smaller than between Zeno t=0 and t=1. For SC time, the proper time on the free-falling clock between SC t=1 and t=2 is also smaller than between SC t=0 and t=1.

Quote by Austin0

So are you talking about an arbitrary clock that speeds up over time ??

No, I just mean that coordinate time proceeds at a rate of one coordinate second per coordinate second, by definition. It is a simple tautology. If you are using coordinate time as your standard (as the SC proponents want to do) then coordinate time is uniform, by definition, i.e. tautologically.

In SC coordinate time each successive SC coordinate time interval for the free-faller is tautologically constant. In Zeno coordinate time each successive Zeno coordinate time interval for Achilles is tautologically constant. Neither correspond to the proper time on the falling/Achilles' clock.

While I tend to think that the term "by definition" means literally by explicit prior statement I certainly agree that your "coordinate time proceeds at a rate of one coordinate second per coordinate second," is a tautology. So essentially applies to all times. Time is uniform unless stated differently.

On the other hand, there is , in Pervect's stated conditions, absolutely no foundation or justification for an inference or assertion that Achilles' clock does not run at the same rate as Zeno's.
So your statements ---"Neither correspond to the proper time on the falling/Achilles' clock" and " For Zeno time, the proper time on Achilles' clock between Zeno t=1 and t=2 is indeed smaller than between Zeno t=0 and t=1" are both simply unwarranted assertions without validity. Simply entering the desired conclusion as an assumption

Explicitly as Zeno time goes to infinity so does Achilles'

Quote by Austin0

Could you explain where you get this "Again in SC coordinate time the time for the next reduction is constant, by definition." In the original, the next reduction was reducing the remaining distance to the horizon by half so how do you get a constant time interval for each of these increments?

The analogies diverge quantitatively, but not qualitatively. In SC coordinates each successive SC coordinate time interval does not correspond to half the distance to the horizon, that is a feature of the construction of Zeno coordinates. But in both SC and Zeno coordinates the coordinate distance traveled by the free-faller/Achilles decreases for each successive coordinate time interval. This obviously means that their coordinate velocity is reducing, which you already recognized and pointed out above.

Yes their coordinate velocity is reducing but in the Zeno system a la Pervect there is no reason that Achilles proper velocity would not also decrease.

SO I will again state my opinion that the analogy doesn't really apply. Zeno time does not demonstrate a small finite time on Achilles clock. Do you still disagree??

 

[Austin0]

For a supermassive black hole, something as big as the Empire State building would have no more stresses at the horizon than it does on the Earth's surface (could be made arbitrarily small, actually). The horizon is not intrinsically related to any particular amount of local curvature, stresses, etc. Only the singularity is. For stellar black holes, extreme stresses and tidal forces occur long before the horizon - e.g. approaching a neutron star.

I was not suggesting any special significance to BH's or the vicinity of the horizon. I understood Mike_Holland's statements regarding g dilation as being general so took Steves post in the same context..
Are you suggesting that with a system accelerating under thrust we just disregard Born rigidity and acceleration if the system is smaller than the Empire State building?
Joke. ;-)

 

[PAllen]

I was not suggesting any special significance to BH's or the vicinity of the horizon. I understood Mike_Holland's statements regarding g dilation as being general so took Steves post in the same context..
Are you suggesting that with a system accelerating under thrust we just disregard Born rigidity and acceleration if the system is smaller than the Empire State building?
Joke. ;-)

I'm just suggesting that to the extent we assume rigidity in practice for relativity large objects is achievable for modest tidal forces and accelerations, then rigidity to a similar extent may be assumed for an appropriately chosen event horizon (for free fall). [Edit: for a static observer near a horizon, proper acceleration approaches infinity, so rigidity is impossible for a static object. But for free fall, it is easy. Remember, it is the static observer that is analogous to the accelerating observer in SR.]

 

[Dale]

On the other hand, there is , in Pervect's stated conditions, absolutely no foundation or justification for an inference or assertion that Achilles' clock does not run at the same rate as Zeno's.

I can calculate it explicitly if you like, but it is exceedingly well-founded.

So your statements ---"Neither correspond to the proper time on the falling/Achilles' clock" and " For Zeno time, the proper time on Achilles' clock between Zeno t=1 and t=2 is indeed smaller than between Zeno t=0 and t=1" are both simply unwarranted assertions without validity. Simply entering the desired conclusion as an assumption

See above. It isn't an assumption. It falls out of the math quite naturally.

Explicitly as Zeno time goes to infinity so does Achilles'

No, Achilles' proper time is finite as Zeno coordinate time goes to infinity. I thought that it would be obvious, but apparently it isn't.

Yes their coordinate velocity is reducing but in the Zeno system a la Pervect there is no reason that Achilles proper velocity would not also decrease.

Achilles' proper velocity is clearly constant.

SO I will again state my opinion that the analogy doesn't really apply. Zeno time does not demonstrate a small finite time on Achilles clock. Do you still disagree??

Yes, I disagree. I think that the math is so unfamiliar to you that you have a whole bunch of mistaken beliefs about how this works out. To me it is pretty obvious that none of the claims you made in your previous post are correct.

 

[Mike Holland]

Thus the EP says gravitational time dilation is equally subject alternate, coordinate dependent interpretations.

Very nice in theory, but it doesn't work in practice if you believe that physics is universal. How is that free-fall observer, using Rain coordinates, going to explain that contracting mass that is accelerating towards him while he is motionless? Where are the rockets that are making it accelerate?

Gravity and acceleration may give the same answers, but if there is a heavy mass present, then gravity wins over acceleration.as an explanation - or at least as part of the explanation where both are involved.

 

[PAllen]

Very nice in theory, but it doesn't work in practice if you believe that physics is universal. How is that free-fall observer, using Rain coordinates, going to explain that contracting mass that is accelerating towards him while he is motionless? Where are the rockets that are making it accelerate?

Gravity and acceleration may give the same answers, but if there is a heavy mass present, then gravity wins over acceleration.as an explanation - or at least as part of the explanation where both are involved.

The principal of equivalence is local. It will show, for example, that within a moderately large capsule in free fall, physics is the same as in 'empty space'. Or that from top to bottom of building on a planetary surface, you have the same behavior as an accelerating rocket. If you go global, it doesn't apply in GR because there are neither extended uniform gravitational fields, nor global inertial frames.

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

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

Gravitational time dilation is a useful concept for static spacetime - but it is not a general feature of GR, and it is never necessary to use. J. L. Synge, in his classic book on general relativity, argued against using it at all - because one universal method may be used in all cases (kinematic, cosmologic, and in strong, non-static geometry) instead.

 

[zonde]

That much is true. I think that the bulk of the argument stems from a misunderstanding or mistrust of the basic mathematical framework of GR.

I was talking about scientific method, not about math.

I think you have this backwards. The predictions are all invariants, so all coordinate systems agree. We pick a coordinate system so that the calculation of those invariants is easy.

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

What would stop it? I mean, not the singularity, but the horizon.

As far as my understanding of GR goes this is out of scope of GR.

 

[Dale]

I was talking about scientific method, not about math.

The math is what the theory uses to make testable predictions for the scientific method. If you do not understand the math then you do not understand the theory well enough to address it with the scientific method. Hence the disagreements.

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

This is simply false. All experimental measurements are invariants. If they were not invariant then you could always construct a paradox of the form "Dr. Evil builds a bomb which is detonated iff device X measures Y, device X measures Y under coordinate system A, but Z under coordinate system B. Therefore the bomb explodes in one coordinate system but not in the other."

Two different coordinate systems may disagree on the meaning of the measurement, e.g. they may disagree whether or not the rod is accurately measuring length, but they must agree on what value is measured.

As far as my understanding of GR goes this is out of scope of GR.

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

 

[zonde]

And what stops it for a supermassive BH, where densities are quite low at SC radius? It's clear what you will see from afar (the cluster of stars slowing, effectively freezing, and forming a black object at essentially SC radius).

Stellar black holes are formed in violent explosions - this is quite close to direct observation.
What you are telling is (mainstream) speculation. The closest thing to something like that as I can imagine is galactic collisions.

But for someone orbiting one of the stars in the interior, what do you think is experiences? Are we (from afar) not allowed to ask that just because we can't see it?

You can ask of course. But it does not mean you can get testable answer.
I for example can ask what people experience when they die. So what?


As I see speculations about BHs relies on one important thing that GR takes as a postulate. That is that laws of physics are independent of (Newtonian) gravitational potential. If we assume that this assumption holds without bonds then we have no reason to assume that anything will happen with a clock falling into the hypothetical BH.
But I don't buy the idea about assumptions holding without bonds. And that takes it out of domain of GR.

 

[PAllen]

Stellar black holes are formed in violent explosions - this is quite close to direct observation.
What you are telling is (mainstream) speculation. The closest thing to something like that as I can imagine is galactic collisions.

No, galaxies are believed to contain supermassive central black holes, 10 billion or more sun's worth in some cases.

You can ask of course. But it does not mean you can get testable answer.
I for example can ask what people experience when they die. So what?

True, but this is not the the only case of physical theories including untestable predictions. To better understand a theory (and its limits), it is useful to understand what a theory predicts for such things. GR + known theories of matter (classically) predict continued collapse. GR must be modified in some way to avoid this.

As I see speculations about BHs relies on one important thing that GR takes as a postulate. That is that laws of physics are independent of (Newtonian) gravitational potential. If we assume that this assumption holds without bonds then we have no reason to assume that anything will happen with a clock falling into the hypothetical BH.
But I don't buy the idea about assumptions holding without bonds. And that takes it out of domain of GR.

Fine - you agree that GR must be modified to get the result you want. What you call laws being affected by something like Newtonian potential is a fundamental violation of the principle of equivalence, which is built in (as a local feature) to the math and conceptual foundations of GR. Note, for gravity to be locally equivalent to acceleration, a direct consequence is that free fall must have locally the same physics everywhere. (Otherwise, observing what happens inside a (small) free falling system would locally distinguish gravity from corresponding acceleration.)

 

[pervect]

There's growing experimental evidence for the existence of event horizons. Basically, black hole candidates are very black, and don't appear to surface features.

WHen matter falls onto a neutron star, the surface heats up and re-radiates. The spectra signature is rather distinctive, also there are "type 1 x ray bursts".

Black hole candidates do not appear to have any such "surface" features, and it's already very difficult to explain by any means other than an event horizon how they can suck in matter without , apparently re-radiating anything detectable.

For the details, see

See for instance http://arxiv.org/pdf/0903.1105v1.pdf

and check for other papers by Naryan in particular.

 

[Austin0]

Quote by pervect

Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise.

At a zeno time of 1, Achilles is 50 meters behind the tortise.

At a zeno time of 2, Achillies is 25 meters behind the tortise

At a zeno time of n, Achillies is 100/(2^n) meters behind the tortise.

Then, as n goes to infinity, Achillies is always behind the tortise.

So, in "zeno time", Achilles never does catch up with the tortise, even as "zeno time" appoaches infinity.



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

Quote by Austin0 View Post

On the other hand, there is , in Pervect's stated conditions, absolutely no foundation or justification for an inference or assertion that Achilles' clock does not run at the same rate as Zeno's.

I can calculate it explicitly if you like, but it is exceedingly well-founded.

Well this whole post of yours is nothing more than a repetitive bald assertion that you are right and I am wrong without content or justification so yes some hint as to the math you are referring to would be appropriate.

Where in the stated parameters :

Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise.

At a zeno time of 1, Achilles is 50 meters behind the tortise.

At a zeno time of 2, Achillies is 25 meters behind the tortise

At a zeno time of n, Achillies is 100/(2^n) meters behind the tortise.

is the mathematical basis for the derivation of time dilation . I.e. justification of its insertion into a classical scenario?
Quote by Austin0 View Post

So your statements ---"Neither correspond to the proper time on the falling/Achilles' clock" and " For Zeno time, the proper time on Achilles' clock between Zeno t=1 and t=2 is indeed smaller than between Zeno t=0 and t=1" are both simply unwarranted assertions without validity. Simply entering the desired conclusion as an assumption

See above. It isn't an assumption. It falls out of the math quite naturally.

Quote by Austin0 View Post

Explicitly as Zeno time goes to infinity so does Achilles'

No, Achilles' proper time is finite as Zeno coordinate time goes to infinity. I thought that it would be obvious, but apparently it isn't.

Quote by Austin0 View Post

Yes their coordinate velocity is reducing but in the Zeno system a la Pervect there is no reason that Achilles proper velocity would not also decrease.

Achilles' proper velocity is clearly constant.

Quote by Austin0 View Post

SO I will again state my opinion that the analogy doesn't really apply. Zeno time does not demonstrate a small finite time on Achilles clock. Do you still disagree??

Yes, I disagree. I think that the math is so unfamiliar to you that you have a whole bunch of mistaken beliefs about how this works out. To me it is pretty obvious that none of the claims you made in your previous post are correct.

 

[Dale]

Well this whole post of yours is nothing more than a repetitive bald assertion that you are right and I am wrong without content or justification so yes some hint as to the math you are referring to would be appropriate.

I will work it out in full and post it either later tonight or early tomorrow. I am sorry that it isn't obvious to you from pervect's description, but I think when you are unfamiliar with the math that you would be better served to simply ask for a detailed derivation instead of asserting that well qualified individuals like pervect are wrong or implying that they are acting deceptively.

 

[Nugatory]

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

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

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

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

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

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

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

And it's the same way with the Schwarzschild time coodinate. The ratio of A's Schwarzschild time coordinate to proper time on B's worldline serves only to mislead. The interesting quantity is the ratio of proper time between any two points on B's world line and any two points on A's worldline.

 

[Austin0]

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

And it's the same way with the Schwarzschild time coodinate. The ratio of A's Schwarzschild time coordinate to proper time on B's worldline serves only to mislead. The interesting quantity is the ratio of proper time between any two points on B's world line and any two points on A's worldline.

So according to what you are saying here , Pervects stated conditions are to be taken as outside of Newtonian uniform time

Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise.

At a zeno time of 1, Achilles is 50 meters behind the tortise.

At a zeno time of 2, Achillies is 25 meters behind the tortise

At a zeno time of n, Achillies is 100/(2^n) meters behind the tortise.

so then have an implicit assumption of time dilation. That Achilles clock is running at a different rate and his velocity is constant.
Well of course given these conditions everything else is obvious. But then you have simply rewritten Zeno's paradox completely. Simply stuck Zeno's and Achilles names on the conditions of free fall in Sc coordinates.

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

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

If those assumptions had been explicitly stated by Pervect then it would have been quite obvious that Zeno time was explicitly dilated and outside any classical context .

 

[Nugatory]

If those assumptions had been explicitly stated by Pervect then it would have been quite obvious that Zeno time was explicitly dilated and outside any classical context.

Aaargh... I'm still not being clear enough... Zeno time is not "explicitly dilated" because it's a coordinate time so doesn't "dilate" - dilation is a statement about the ratio between two amounts of proper time, not coordinate time.

Is there any reason to take the Schwarzschild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space?
(IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason - there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates).

 

[pervect]

Austin0 was asking for some more detailed math. I'd suggest looking at Caroll's GR lecture notes online.

I'll also add that while Caroll's online notes are perfectly fine (one can't trust everything online, Caroll's online notes are drafts of a book by a physics profesor that was later published. From my POV the main advantage of them is that they're free).

I could also quote similar statements from some of my other GR textbooks, (i.e. Caroll is not an isolated occurrence among textbooks). However, I think it would be better from a pedagogical point of view if interested people went out and found their own textbooks if they don't like Caroll (though I can't think of any valid reason for not liking Caroll).But - onto Caroll:

As we will see, this is an illusion, and the light ray (or a massive particle) actually has no trouble reaching r = 2GM. But an observer far away would never be able to tell. If we stayed outside while an intrepid observational general relativist dove into the black hole, sending back signals all the time, we would simply see the signals reach us more and more slowly. This should be clear from the pictures, and is confirmed by our computation of &)1/&)2 when we discussed the gravitational redshift (7.61). As infalling astronauts approach r = 2GM, any fixed interval &)1 of their proper time corresponds to a longer and longer interval &)2 from
our point of view. This continues forever; we would never see the astronaut cross r = 2GM, we would just see them move more and more slowly (and become redder and redder, almostas if they were embarrassed to have done something as stupid as diving into a black hole).

The fact that we never see the infalling astronauts reach r = 2GM is a meaningful statement, but the fact that their trajectory in the t-r plane never reaches there is not. It is highly dependent on our coordinate system, and we would like to ask a more coordinate independent
question (such as, do the astronauts reach this radius in a finite amount of their proper time?). The best way to do this is to change coordinates to a system which is better behaved at r = 2GM. There does exist a set of such coordinates, which we now set out to find. There is no way to “derive” a coordinate transformation, of course, we just say what
the new coordinates are and plug in the formulas. But we will develop these coordinates in several steps, in hopes of making the choices seem somewhat motivated

These just the words - the actual calculation consists of solving for the trajectory of the worldline. If one does this in the usual format, one doesn't even have to integrate the length of the worldline to get proper time, instead one solves the geodesic equations to find t(tau) and r(tau).

One can then observe directly that the event horizion is reached at a finite tau, even though t(tau) is infinite.

If one is willing to just take the limit as r approaches the event horizon one can do this all in Schwarzschild coordinates. This is even observable. One can say that's it's possible to observe the limiting sequence of proper time as one approaches the event horizon from outside, and observe that the limit is finite.

To go futher and carry the trajectory smoothly through the event horizon, one needs coordinates that are better behaved, which is what Caroll does next.

The point of the Zeno analogy is to demonstrate a simple example of how a coordinate time can be infinite while the time actually measured on a clock is finite.

Specifically, zeno time is infinite, while as far as Achilles is concrned, there's a finite time at which he passes the tortise.

I'm afraid I don't understand the difficulties people are having in understanding the analogy. It could be my fault, sometimes I "leap ahead' too far when I write.

The way you demonstrate that the proper time on an infalling clock is actually finite rigorously is that you calculate it.

Post #12 in this thread
https://www.physicsforums.com/showpost.php?p=4185014&postcount=12

(and a later post after it, #13)

[for a m=2 black hole, with a horizon at r=2m = 4]

r = {3}^{2/3} \left( -\tau \right) ^{2/3}
t = \tau-4\,\sqrt [3]{3}\sqrt [3]{-\tau}+4\,\ln \left( \sqrt [3]{3}\sqrt <br /> [3]{-\tau}+2 \right) -4\,\ln \left( \sqrt [3]{3}\sqrt [3]{-\tau}-2 <br /> \right)

presents the trajectory t(tau) and r(tau) for the case of a black hole where m=2.

One can see that at tau = -8/3 , which is finite, r=4 so one is at the event horizon. Furthermore, t(tau) is infinite because of one of the ln(...) terms.

To verify this is a solution one needs to demonstrate that said trajectory satisfies the geodesic equations. You'll find them in my post #12, Caroll's GR lecture notes, for starters.

The idea behind the Zeno analogy isn't to "prove" anything - that's what textbooks are for. The idea behind the Zeno analogy is to illustrate how t can be infinite and tau can be finite in a simple, easy-to-understand example.

WEll, the Zeno analogy does prove one thing. It demonstrates that just because you have a time coordinate t going to infinity doesn't prove that something doesn't happen. It's an example of how t going to infinity can be the result of a poor choice of coordinates. It's a counterexample to the argument "t goes to infinity, therefore it can't happen".

Historically, I do believe that the "tortise coordinate" was named after the tortise in Zeno's paradox, but I haven't seen anything really detailed on this in textbooks. There was something in Scientific American about it a long time ago as well, I think.

 

[Austin0]

Aaargh... I'm still not being clear enough... Zeno time is not "explicitly dilated" because it's a coordinate time so doesn't "dilate" - dilation is a statement about the ratio between two amounts of proper time, not coordinate time.

Is there any reason to take the Schwarzschild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space?
(IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason - there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates).

Actually I misspoke. It is Achilles' time which is dilated within the context of Pervects conditions if we add the condition that Achilles' velocity is constant.
I really just meant that time dilation was in effect within the stated conditions and coordinates

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

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

 

[Austin0]

Austin0 was asking for some more detailed math. I'd suggest looking at Caroll's GR lecture notes online.

I'll also add that while Caroll's online notes are perfectly fine (one can't trust everything online, Caroll's online notes are drafts of a book by a physics profesor that was later published. From my POV the main advantage of them is that they're free).

I could also quote similar statements from some of my other GR textbooks, (i.e. Caroll is not an isolated occurrence among textbooks). However, I think it would be better from a pedagogical point of view if interested people went out and found their own textbooks if they don't like Caroll (though I can't think of any valid reason for not liking Caroll).


But - onto Caroll:



These just the words - the actual calculation consists of solving for the trajectory of the worldline. If one does this in the usual format, one doesn't even have to integrate the length of the worldline to get proper time, instead one solves the geodesic equations to find t(tau) and r(tau).

One can then observe directly that the event horizion is reached at a finite tau, even though t(tau) is infinite.

If one is willing to just take the limit as r approaches the event horizon one can do this all in Schwarzschild coordinates. This is even observable. One can say that's it's possible to observe the limiting sequence of proper time as one approaches the event horizon from outside, and observe that the limit is finite.

To go futher and carry the trajectory smoothly through the event horizon, one needs coordinates that are better behaved, which is what Caroll does next.

The point of the Zeno analogy is to demonstrate a simple example of how a coordinate time can be infinite while the time actually measured on a clock is finite.

Specifically, zeno time is infinite, while as far as Achilles is concrned, there's a finite time at which he passes the tortise.

I'm afraid I don't understand the difficulties people are having in understanding the analogy. It could be my fault, sometimes I "leap ahead' too far when I write.

The way you demonstrate that the proper time on an infalling clock is actually finite rigorously is that you calculate it.

Post #12 in this thread
https://www.physicsforums.com/showpost.php?p=4185014&postcount=12

(and a later post after it, #13)



One can see that at tau = -8/3 , which is finite, r=4 so one is at the event horizon. Furthermore, t(tau) is infinite because of one of the ln(...) terms.

To verify this is a solution one needs to demonstrate that said trajectory satisfies the geodesic equations. You'll find them in my post #12, Caroll's GR lecture notes, for starters.

The idea behind the Zeno analogy isn't to "prove" anything - that's what textbooks are for. The idea behind the Zeno analogy is to illustrate how t can be infinite and tau can be finite in a simple, easy-to-understand example.

WEll, the Zeno analogy does prove one thing. It demonstrates that just because you have a time coordinate t going to infinity doesn't prove that something doesn't happen. It's an example of how t going to infinity can be the result of a poor choice of coordinates. It's a counterexample to the argument "t goes to infinity, therefore it can't happen".

Historically, I do believe that the "tortise coordinate" was named after the tortise in Zeno's paradox, but I haven't seen anything really detailed on this in textbooks. There was something in Scientific American about it a long time ago as well, I think.

You are here demonstrating the validity of the Schwarzschild conclusion.

I do understand the math processes and reasoning behind this. Integrating proper time is not difficult to grasp , certainly not after SR
Now that I understand that your statement of Zeno time was with the expectation that it was assumed Achilles' proper velocity was constant even though it decreased in Zeno's frame then of course the situations are effectively identical.
Of course this means that this adaptation is no clearer or more persuasive than the original Sc scenario.
I have never said that the infaller doesn't reach the horizon in some relatively short proper time on its clock.I have questioned the assertion that this does not transform to
some tremendously distant future time in the frame of the distant observer.
This seems to call into question the Sc coordinates not only in the immediate vicinity of the horizon but effectively throughout the system. How or why a system which is empirically verified within a certain range of the domain would become totally unreliable (pathological ;-) ) in another part.

 

[Dale]

there is , in Pervect's stated conditions, absolutely no foundation or justification for an inference or assertion that Achilles' clock does not run at the same rate as Zeno's.
...
Explicitly as Zeno time goes to infinity so does Achilles'

Consider the inertial frame where Achilles is at rest. In this frame the turtle's worldline is given by (t,100-vt) where v is the relative velocity between Achilles and the turtle. So in this frame Achilles is a distance d=100-vt behind the turtle. The definition of Zeno time, n, given is d=100/2^n. Substituting in and simplifying we get the following transform between the inertial frame and Zeno coordinates:
n=log_2 \left( \frac{100}{100-vt} \right)

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

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

 

[Dale]

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

Saying that it added nothing is one thing, but saying it is misleading is accusatory and untrue. It is, as I think is now established, a valid and close analogy in many respects. The fact that the parallels escaped you at first doesn't make it misleading or deceptive in any way.

 

[Dale]

Is there any reason to take the Schwarzschild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space?
(IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason - there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates).

Excellent point. It highlights the real reason for picking any coordinate system: ease of computation. That is true in all branches of physics, not just GR.

 

[Austin0]

Consider the inertial frame where Achilles is at rest. In this frame the turtle's worldline is given by (t,100-vt) where v is the relative velocity between Achilles and the turtle. So in this frame Achilles is a distance d=100-vt behind the turtle. The definition of Zeno time, n, given is d=100/2^n. Substituting in and simplifying we get the following transform between the inertial frame and Zeno coordinates:
n=log_2 \left( \frac{100}{100-vt} \right)

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

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

Yes this is fine . But it is based on an assumption of a constant v in Achilles' frame ,,,,yes? You are not deriving either the time dilation or the constant v from the stated Zeno time parameters alone.
so according to Nugatory I get that it was supposed to be understood implicitly that that was a given but everything i said was clearly within the context of what Pervect actually outlined.

 

[zonde]

The math is what the theory uses to make testable predictions for the scientific method. If you do not understand the math then you do not understand the theory well enough to address it with the scientific method. Hence the disagreements.

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

This is simply false. All experimental measurements are invariants. If they were not invariant then you could always construct a paradox of the form "Dr. Evil builds a bomb which is detonated iff device X measures Y, device X measures Y under coordinate system A, but Z under coordinate system B. Therefore the bomb explodes in one coordinate system but not in the other."

Two different coordinate systems may disagree on the meaning of the measurement, e.g. they may disagree whether or not the rod is accurately measuring length, but they must agree on what value is measured.

There is observer A who is using coordinate system K and there is observer B who is using coordinate system K'. Now observer A observes event X but observer B observes event X'. How do they find out if event X and event X' is the same event?

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

Degeneracy of matter.

 

[zonde]

True, but this is not the the only case of physical theories including untestable predictions. To better understand a theory (and its limits), it is useful to understand what a theory predicts for such things. GR + known theories of matter (classically) predict continued collapse. GR must be modified in some way to avoid this.

I believe we can make untestable extrapolations of the theory for educational purposes - to make the explanations more colorful. But then confirmation of the theory is still based on testable things. And if we have any doubt about the theory then it needs to address only the things within limits of testability.

Say we address hypothesis of runaway collapse only to the limits of "frozen star".

Fine - you agree that GR must be modified to get the result you want. What you call laws being affected by something like Newtonian potential is a fundamental violation of the principle of equivalence, which is built in (as a local feature) to the math and conceptual foundations of GR. Note, for gravity to be locally equivalent to acceleration, a direct consequence is that free fall must have locally the same physics everywhere. (Otherwise, observing what happens inside a (small) free falling system would locally distinguish gravity from corresponding acceleration.)

Yes

 

[zonde]

There's growing experimental evidence for the existence of event horizons. Basically, black hole candidates are very black, and don't appear to surface features.

WHen matter falls onto a neutron star, the surface heats up and re-radiates. The spectra signature is rather distinctive, also there are "type 1 x ray bursts".

Black hole candidates do not appear to have any such "surface" features, and it's already very difficult to explain by any means other than an event horizon how they can suck in matter without , apparently re-radiating anything detectable.

For the details, see

See for instance http://arxiv.org/pdf/0903.1105v1.pdf

and check for other papers by Naryan in particular.

Yes, this is a good argument. Thanks for the paper. I will read it.

Minor point. This is not experimental evidence. This is observational evidence. We have no control over conditions.

 

[Dale]

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

Yes, that is a standard part of Zeno's paradox. See the second sentence of the description here:

http://en.wikipedia.org/wiki/Zeno's_paradoxes#Achilles_and_the_tortoise