## On the nature of the "infinite" fall toward the EH

 Quote by DaleSpam 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?

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 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.

 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.

 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.

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 Quote by grav-universe 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 http://www.physicsforums.com/showpos...8&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).

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 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

 Quote by DaleSpam 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.

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 Quote by zonde 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.

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 Quote by PAllen 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.

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 Quote by zonde 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.

 Quote by zonde 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.

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 Quote by zonde 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.

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 Quote by zonde 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?

 Quote by 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.
 Quote by PAllen 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???

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 Quote by Austin0 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.
 Quote by Austin0 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.
 Quote by Austin0 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.
 Quote by Austin0 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.
 Quote by Austin0 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.

 Quote by stevendaryl 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?????

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 Quote by Austin0 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.

 Quote by PAllen 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???

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 Quote by Austin0 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.

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 Quote by Austin0 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.

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 .
 Quote by DaleSpam 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??
 Quote by DaleSpam 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 ??
 Quote by DaleSpam 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?
 Quote by DaleSpam 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??