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

by rjbeery
Tags: fall, infinite, nature
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 Quote by harrylin Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics. My earlier comment on his views hasn't changed:
What is patchwork?

Btw, you are in dangerous territory. If patchwork is something mentioned in those lecture notes then it is part of mainstream physics and your claim that it isn't good physics would therefore be quite speculative.

Furthermore, all of the comments on manifolds and coordinates in that section apply in simple spacetimes too. Like flat or constant curvature.
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 Quote by DaleSpam What is patchwork?
In my response to this a few posts ago, I assumed he meant using more than one coordinate patch to describe a spacetime.

 Quote by DaleSpam If patchwork is something mentioned in those lecture notes then it is part of mainstream physics
Which it is if my assumption above is correct.
P: 3,179
 Quote by PeterDonis We've already stipulated that he can, because he can detect tidal gravity (as can Eve'). But given that, why would he ever assume he was moving in a straight line in the first place? Maybe I should expound a bit more on what I'm looking for here. The standard view of this scenario is that the two cases are exactly parallel: in both cases, the accelerated observer (Eve, Eve'), because of her proper acceleration, is unable to observe or explore a region of spacetime that the free-falling observer (Adam, Adam') can. The physical criterion that distinguishes them is clear, and is the same in both cases (zero vs. nonzero proper acceleration). You are claiming that, contrary to the above, the cases are different: Adam is "privileged" in the first case, but Eve' is in the second. So I'm looking for some criterion that picks out Adam in the first case, but picks out Eve' in the second; in other words, something that applies to Adam but not Eve, and applies to Eve' but not Adam'. The only criterion I have so far is "moves in a straight line according to my chosen coordinates", but that only pushes the problem back a step: what is it that applies to the coordinates of Adam but not Eve, *and* to those of Eve' but not Adam'? I haven't seen an answer yet.
You made that view sufficiently clear; and I thought that I made the opposing view also sufficiently clear - but apparently not. I'll try to explain once more, but will subsequently let it rest - in case you forgot, my intended role here was just that of a curious but critical reporter, but suddenly people start to argue with the reporter and asking him questions.

The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect). I noted that in case that one or both are unable to do so (for example Adam only has a simple accelerometer and no windows), that could make them like bees that fly against a window. Surely you'll agree that nature can't care less if they did not predict the window, and the window is not "unphysical" if the bee didn't notice it before hitting it.

The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it. This weaker version of Einstein's equivalence principle remains, in the form cited in the first post of http://www.physicsforums.com/showthread.php?t=656240. Also, special Relativity is the theory of flat spacetime, without equivalence principle. That enables the use of universal ("global") descriptions such as Minkowski space-time for negligible effects of gravitation on "clocks and "rulers" (that's extremely handy for solving Langevin's original "twin" example!) and similarly universal descriptions such as Schwarzschild space-time for negligible effects of velocity; the two systems can be combined to globally account for both effects. That is de facto how the ECI frame is constructed.

From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno. Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).

That's enough philosophy. It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view".
P: 329
 Quote by harrylin Patchwork is in my eyes not good physics.
Here you are either very confused or deliberately using the word "patchwork" out of the context of manifold charts.
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 Quote by harrylin the physical criterion that experimentally distinguishes the two cases is clear (tidal effect).
Note that tidal effects are arbitrarily small at the EH if the mass of the BH is sufficiently large. So the physical criterion can be made arbitrarily small.
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 Quote by harrylin my intended role here was just that of a curious but critical reporter, but suddenly people start to argue with the reporter and asking him questions.
Because if the reporter is going to report about a theory, we want to make sure he reports accurately. He can add an editorial about how he doesn't really think certain aspects of the theory are "good physics", but that goes on the editorial page, not the news page. Nobody said the reporter's job was easy.

 Quote by harrylin The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect).
That distinguishes the cases, but it doesn't help in deciding which observer should be privileged in each case. Eve and Adam both measure zero tidal gravity; Eve' and Adam' both measure nonzero tidal gravity. Nothing in that helps to pick out Adam vs. Eve, or Eve' vs. Adam.

 Quote by harrylin The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it.
I agree that the view you state here is mainstream, but it's not the criterion we need, because it only helps to distinguish the two spacetimes (flat vs. curved); it doesn't help in picking out which observer is privileged in each (Adam vs. Eve, Eve' vs. Adam'). See above.

 Quote by harrylin Schwarzschild space-time for negligible effects of velocity
I don't understand this; Schwarzschild spacetime can handle any velocity. Unless you mean that the central gravitating body is at rest?

 Quote by harrylin the two systems can be combined to globally account for both effects.
No "combination" is necessary; Schwarzschild spacetime by itself can handle the regions with negligible gravity, since the metric coefficients go to the Minkowski values as r -> infinity.

 Quote by harrylin That is de facto how the ECI frame is constructed.
See my note above about "combination"; the ECI frame doesn't have to combine a Minkowski spacetime and a Schwarzschild spacetime. It's just a Schwarzschild-type chart centered on the Earth whose time coordinate is rescaled to the rate of proper time on the geoid.

With the word "combination" you may be thinking of the fact that the ECI is also a sort of "local inertial frame" for the Earth in its orbit about the Sun. This is true (with some technicalities), but note the word "local"; it is certainly not any kind of "combination" of a global Minkowski frame with a global Schwarzschild frame. If we look at the Solar System as a whole, the global frame is a Schwarzschild frame centered on the Sun.

 Quote by harrylin From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno.
But in so far as these coordinates are "pseudo", it's not because there is a "pseudo gravitational field" in them. It's because they go to infinity at the Rindler horizon, yet the spacetime itself is finite there. Exactly the same criticism applies to Schwarzschild coordinates for a black hole.

 Quote by harrylin Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).
Sure, but that doesn't help with the issues you're having with the SC chart and black hole horizons, because it only differentiates between spacetimes, not between observers. See above.
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 Quote by harrylin It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view".
Have you read my post #123 yet?
P: 1,412
 Quote by harrylin The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it. This weaker version of Einstein's equivalence principle remains, in the form cited in the first post of http://www.physicsforums.com/showthread.php?t=656240. Also, special Relativity is the theory of flat spacetime, without equivalence principle. That enables the use of universal ("global") descriptions such as Minkowski space-time for negligible effects of gravitation on "clocks and "rulers" (that's extremely handy for solving Langevin's original "twin" example!) and similarly universal descriptions such as Schwarzschild space-time for negligible effects of velocity; the two systems can be combined to globally account for both effects. That is de facto how the ECI frame is constructed. From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno. Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).
As I said earlier, "fictitious gravitational fields" have nothing to do with this topic, at all. It's simply a matter of coordinate systems, and whether or not they cover the entire manifold.
P: 1,412
 Quote by harrylin Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics.
What do you mean by that? On the contrary, all physics involves splitting up the world into pieces that can be analyzed in (approximate) isolation.
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P: 7,207
 Quote by harrylin If you mean the SR twin paradox: once more, that is very different as the (t, t') sets are finite and agree with each other. It's different however with Einstein's GR twin paradox which is much more interesting and relevant as background for this topic. It would distract too much from this topic to discuss it here, but I encourage you to study it. I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.
The funny thing is, you're basically using the same reasoning that Zeno did - without apparently even realizing this fact, not even when it's pointed out!

Let's compare what happens issue by issue.

In Zeno's paradox, we have two times. Let's let the "real" time be represented by tau, and zeno time by Z.

In the infalling black hole case, we have proper time tau, and Schwarzschild time t

The mapping from t to tau that we worked out previously for the Schwarzschild case in great detail is:

$$t = \tau-4\,\left(-3 \, \tau \right)^{\frac{1}{3}}+4\,\ln \left[ \left(-3 \, \tau \right)^{\frac{1}{3}}+2 \right] -4\,\ln \left[\left(-3 \, \tau \right)^{\frac{1}{3}}-2 \right]$$

(You can probably find this in a textbook if you want to check my math).

The characteristic features of this mapping is that t increases monotonically with tau, and that infinite range of tau only covers a finite range of t.

This is due specifically to the term

$$-4 \ln \left[\left(-3 \, \tau \right)^{\frac{1}{3}}-2 \right]$$

This is rather complicated, the argument will be clearest if we assume this term, which is the one that approaches infinity, is the dominant term near the event horizon, in which case we can solve for $\tau$ assming that this is the only term that matters.

$$\tau \approx \frac{1}{3} \, \left[\exp^{-t/4} - 2\right]^3$$

and we see that $\tau$ approaches -8/3 as t-> infinity (which is when the horizon is reached).

In the Zeno paradox, the mapping is something like

$$\tau = a(1-\exp^{z/b})$$
where a and b are some constants

where $\tau$ approaches some constant a as Z approaches infinity

And we see the issue -in both cases, even though t (in one case), Z (in the other case) cover infnite ranges, $\tau$ does not.

So, essentially Zeno never assigns a label, Z to some events. And he concludes from this that these events don't exist.

And you assume the same thing - because you never assign a coordinate "t" to some events, you assume they don't happen.

And this conclusion is just as unjustified when you do it, as when Zeno does it.

So the not-very-complicated moral of the story is that because you can choose ANY coordinates you want, you need to be careful in your interpretation of the results. Specifically, it's possible to choose coordinates like Zeno did, that exclude important regions from analysis, because the coordinates don't label physically significant events. However, not giving something a label doesn't make it not exist, any more than closing your eyes does. At least not for most defitnitions of "existence".
 PF Patron P: 3,931 Pervect, that's a showstopping reply ! Harrylin, pay heed.
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 Quote by harrylin The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect).
The issue is not whether you can distinguish flat spacetime from curved spactime; of course, you can, by looking at the curvature tensor. The issue is that you seem to think that because an event is not covered by the Schwarzschild coordinate system, then that event never happens. What reason is there for believing that, as opposed to realizing that the Schwarzschild coordinates don't cover the entire manifold?

It is easy to demonstrate that it is possible to choose coordinates that leave out part of the manifold. What reason do you have for thinking that's NOT the case with Schwarzschild coordinates? (It PROVABLY is the case, so what I'm really asking you is why you seem to believe something that is provably false.)

 I noted that in case that one or both are unable to do so (for example Adam only has a simple accelerometer and no windows), that could make them like bees that fly against a window. Surely you'll agree that nature can't care less if they did not predict the window, and the window is not "unphysical" if the bee didn't notice it before hitting it.
I don't understand the point of your story. What you seem to be thinking is that one of the observers is "correct" and the other is "wrong", and you have to look to clues such as accelerometers to figure out which is which. That is a completely wrong way to think about it. ANY coordinate system can be used to describe events within a chart. There is no "correct" coordinate system or "incorrect" coordinate system. But a coordinate system only works within a chart. It can't possibly describe events that are NOT in its chart.

So in Rindler coordinates, someone sees a dropped object asymptotically approach the location X=0 as time T → ∞. The correct interpretation of this situation isn't: "Rindler coordinates are wrong. Cartesian coordinates are right." The correct interpretation is "The event of the object crossing the 'event horizon' at X=0 is not an event covered by the Rindler coordinates". Rindler coordinates are perfectly fine for describing any events taking place within its chart, but it can't possibly describe events outside that chart.

The same thing is true of an object crossing the event horizon in Schwarzschild coordinates. That event is not covered by Schwarzschild coordinates. Schwarzschild coordinates are perfectly good for describing events within its chart, but can't be used to describe events outside its chart. It's not a question of whether the "hovering observer" is correct and the "infalling observer" is wrong, or vice-verse. The only issue is whether the event of crossing the horizon is in fact covered by this coordinate system or that coordinate system.

 The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it.
This doesn't have anything to do with "real" versus "pseudo" gravitation! It has to do with whether a coordinate system covers the entire manifold, or not. An easy way to prove that it does not is to show that there is a second coordinate system that has an overlapping chart with the first coordinate system, yet includes points that are not covered by the first. That's been done, with Schwarzschild coordinates.

 This weaker version of Einstein's equivalence principle remains...
This doesn't really have anything to do with the equivalence principle.

 From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno.
There is no such thing as "pseudo coordinates". The only issue about coordinates is what region of spacetime do they cover, and are there regions that are not covered by them.

 Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).
It doesn't have anything to do with gravitational fields, pseudo or otherwise.

 That's enough philosophy. It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view".
If you look at KS coordinates, the metric looks like this:
$\dfrac{4 R_s^3}{r} e^{-r/R_s}(dV^2 - dU^2)$
where $R_s$ is the Schwarzschild radius, and $r$ is the Schwarzschild radial coordinate. This metric is defined everywhere, except at the singularity $r=0$. The "time" coordinate is $V$. The event horizon in these coordinates consists of all points with $V^2 = U^2$. So an object can certainly cross the event horizon at a finite value for the time coordinate $V$.

Now, to see that this is describing the SAME situation as the Schwarzschild black hole, you note that the "patch" with $U > 0$ and $-U < V < +U$ describes exactly the same region of spacetime as the Schwarzschild patch $r > R_s$ and $-\infty < t < +\infty$, and the "patch" with $1 > V > 0$ and $-V < U < +V$ describes exactly the Schwarzschild patch with $0 < r < R_s$ and $-\infty < t < \infty$. But the KS coordinates also covers the boundary between these two regions, the event horizon.
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 Quote by stevendaryl If you look at KS coordinates, the metric looks like this: $\dfrac{32 R_s^3}{r} e^{-r/R_s}(dV^2 - dU^2)$ where $R_s$ is the Schwarzschild radius
Quick pedantic note: if you write the K-S line element this way, in terms of $R_s$, then the coefficient in front is $4 R_s^3 / r$. The 32 is there if you write it in terms of M:

$$\dfrac{32 M^3}{r} e^{-r/2M}(dV^2 - dU^2)$$
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 Quote by pervect The funny thing is, you're basically using the same reasoning that Zeno did - without apparently even realizing this fact, not even when it's pointed out! Let's compare what happens issue by issue. In Zeno's paradox, we have two times. Let's let the "real" time be represented by tau, and zeno time by Z.
That is an interesting comparison. One difference seems to be that "Zeno time" is the result of an arbitrary mathematical mapping, that doesn't have any physical significance. The coordinate time in Schwarzshild coordinates on the other hand, can be interpreted as the proper time of a clock at infinity, which is a observable physical quantity.
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 Quote by A.T. That is an interesting comparison. One difference seems to be that "Zeno time" is the result of an arbitrary mathematical mapping, that doesn't have any physical significance. The coordinate time in Schwarzshild coordinates on the other hand, can be interpreted as the proper time of a clock at infinity, which is a observable physical quantity.
It's not really the proper time of a clock at infinity - it's still a coordinate time. I'd describe it as the coordinate time of a static observer, with the coordinate clocks normalized to run at the same rate as proper clocks at infinity.

It seems rather strange to me to ignore the readings of actual, physical clocks (proper time) in favor of some abstract coordinate time, but it seems all-too-common. My speculation is that this is based on a desire for the "absolute time" of Newtonian physics.

Static observers do have _some_ physical significance where they exist , which is outside the event horizon. This significance is derived mostly form the Killing vector field of their timelike worldlines. The Killing vector still exists at the event horizon, but it's null, so it doesn't represent any sort of "observer".

The coordinate system of static observers, where they exist, has about the same relevance to an infalling observer as the coordinate system of some "stationary" frame to somoene rapidly moving. Which in my opinion is "not very much". But I suppose opinions could vary on this point, it's not terribly critical.

The biggest difference here, and another significant underlying issue, is that static observers cease to exist at the event horizon. This makes their coordinates there problematic, as you're trying to defie a coordinate system for an observer that doesn't exist anymore. This isn't any sort of breakdown in physics - it's a breakdown of the concept of static observers.

For any actual physical observer, the horizon will always be approaching them at "c" - because any physical observer will have a timelike worldline, and the horizon is a null surface. This isn't really very compatabile with the event horizon as a "place". This is why space-time diagrams that represent the event horizon as a null surface (such as the Kruskal or penrose diagram) are a good aid to understanding the physics there, and why Schwarzschild coordinates are not.

Another sub-issue (of many) is the absolute refusal of certain posters to even consider any other coordinate systems other than Schwarzschild as having any relevance to the physics. Which gives rise to severe problems, as Schwarzschild coordinates are ill-behaved at the event horizon, for the reasons I've previously aluded to (the non-existence of static observers upon which the coordinate system is based).

This ill behavior is hardly any secret - pretty much ANY textbook is going to tell you the same thing.
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 Quote by A.T. That is an interesting comparison. One difference seems to be that "Zeno time" is the result of an arbitrary mathematical mapping, that doesn't have any physical significance. The coordinate time in Schwarzshild coordinates on the other hand, can be interpreted as the proper time of a clock at infinity, which is a observable physical quantity.
Coordinate time always represents a simultaneity convention, which is arbitrary by definition. I.e. The way that readings on different clocks are compared is arbitrary. In the case of SC the simultaneity convention is additionally labeled to correspond with the rate of a distant clock. So the coordinate time in SC is not just proper time of that distant clock, it also necessarily involves the arbitrary simultaneity convention.

We can always do the same thing with Zeno time by judicious choice of our reference clock and our simultaneity convention. For instance, we can use a Rindler-like simultaneity convention. As long as our reference clock asymptotically approaches the worldline of the light pulse from the arrow reaching the target then that event will be at infinite coordinate time. By varying the acceleration of the reference clock we can adjust the spacing of the time coordinate between the other points on the arrows path.
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 Quote by PeterDonis Quick pedantic note: if you write the K-S line element this way, in terms of $R_s$, then the coefficient in front is $4 R_s^3 / r$. The 32 is there if you write it in terms of M: $$\dfrac{32 M^3}{r} e^{-r/2M}(dV^2 - dU^2)$$
Thanks, I changed it.
P: 1,412
 Quote by A.T. That is an interesting comparison. One difference seems to be that "Zeno time" is the result of an arbitrary mathematical mapping, that doesn't have any physical significance. The coordinate time in Schwarzshild coordinates on the other hand, can be interpreted as the proper time of a clock at infinity, which is a observable physical quantity.
The relationship between proper time $\tau$ and Schwarzschild coordinate $t$ for a clock at rest in the Schwarzschild coordinates is:

$dt = d\tau/\sqrt{1-R_s/r}$

I don't immediately see any simple physical interpretation for $dt$ at finite values of $r$.

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