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

Pervect, that's a showstopping reply !

Harrylin, pay heed.

 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:

$$\dfrac{32 M^3}{r} e^{-r/2M}(dV^2 - dU^2)$$

 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.

Mentor
 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.

 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.

 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 $r$.

 Quote by stevendaryl But it has the same "punch line" as the paradox of the infalling observer. Using one time coordinate, the runner starts running at time t=0 and crosses the finish line at time t=1 (say). But you can set up a different time coordinate, t', with the mappings: t=0 → t'=0 t=1/2 → t'=1 t=3/4 → t'=2 etc. (in general, t' = log2(1/(1-t))) Clearly, as t' runs from 0 to ∞, the runner never reaches the finish line. That's simply an artifact of the choice of coordinates.
It sounds as if you want to hear my opinion about how convincing that illustration may be for your arguments about the nature of Schwarzschild's physics. I won't let myself be pulled again in arguments, but will give minimal advice. t coordinates represent of course clocks (literal or virtual) and together with space coordinates they allow to calculate for example the speed of a runner or of light between different points. So, if in Zeno's story there is something to support the assumption of an effect on runner speed (similar to Schwarzschild's effect on the speed of light due to gravitation from matter), then that illustration may be helpful to explain your view.

PS. I see that A.T. gave a similar clarification:
 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. [..]
which was however obscured by what he said next (and that was probably sufficiently clarified by others).

Mentor
 Quote by harrylin t coordinates represent of course clocks (literal or virtual)
No, proper time represents clocks. Coordinate time represents a simultaneity convention.

 Quote by harrylin It sounds as if you want to hear my opinion about how convincing that illustration may be for your arguments about the nature of Schwarzschild's physics. I won't let myself be pulled again in arguments, but will give minimal advice. t coordinates represent of course clocks (literal or virtual)
No, it really doesn't. The time coordinate $t$ is related to the time $\tau$ shown on a standard clock at a constant value for $r$ by:
$t = \tau/\sqrt{1-r/R_s}$

The factor of $\sqrt{1-r/R_s}$ has no direct physical significance. $\tau$ is directly measurable. $t$ has no physical significance; it's just chosen to make the metric expression look as simple as possible.

 ...and together with space coordinates they allow to calculate for example the speed of a runner or of light between different points.
That's true of any coordinate system. You seem to think that there is something special about Schwarzschild coordinates, that they reflect reality in a way that other coordinates don't, but I can't get any kind of idea why you think that. Any coordinate system, as I have said, can be used equally well to describe physics within a patch. No coordinate system says anything at all about the physics outside of that patch.

 So, if in Zeno's story there is something to support the assumption of an effect on runner speed (similar to Schwarzschild's effect on the speed of light due to gravitation from matter), then that illustration may be helpful to explain your view.
The Schwarzschild coordinates are not derived by computing the effect of gravity on light speed!!! It is derived by looking for a vacuum solution of Einstein's equations that is spherically symmetric. You are making up a physical meaning to Schwarzschild coordinates that isn't there.

Radially moving light has a certain "coordinate speed" in Schwarzschild coordinates:
$v = 1-2GM/(c^2 r)$

It has a different "coordinate speed" in Kruskal-S-whatever coordinates:
$v = 1$

You seem to think that there is a deep physical significance to the first speed, but not to the second. But you're just making things up. You're not getting that from GR. GR does not give any significance to one coordinate system over another. If you want to make up your own theory, go ahead, but from the context of GR, what you're saying makes no sense.

 Quote by harrylin I won't let myself be pulled again in arguments...
In other words, you have no interest in actually defending the statements you've made? Why post anything, if you don't want people to respond to your statements?

What you're posting seems to be nonsensical. You seem to be giving a physical significance to a completely arbitrary choice. Schwarzschild coordinates are chosen for CONVENIENCE. With that choice, the metric looks the simplest. For you to go from that to the conclusion that Schwarzschild coordinates reflect reality in a way that other coordinates do not is just making stuff up. It's not part of GR. In creating GR, Einstein explicitly REJECTED the idea that some coordinates reflect reality more than other coordinates. So you're not talking about GR, you're talking about your own theory, which has an unspecified relationship with GR.

Mentor
 Quote by stevendaryl For you to go from that to the conclusion that Schwarzschild coordinates reflect reality in a way that other coordinates do not is just making stuff up. It's not part of GR. In creating GR, Einstein explicitly REJECTED the idea that some coordinates reflect reality more than other coordinates.
That, I think, is the key point of the whole thread and all of its predecessors.

 Mentor I would like to expand further on the idea of coordinate time vs proper time. I have stated above that coordinate time represents a simultaneity convention. If you set coordinate time to some fixed value then you get a continuous and smooth set of events which forms some 3D hypersurface. In order to qualify as a time coordinate, this hypersurface must be spacelike everywhere, but otherwise there is no restriction to the shape of the hypersurface. This surface represents a set of all events that happened at the same time, which is, by definition, a simultaneity convention. In contrast, proper time is only defined along a timelike worldline. If you set proper time to some fixed value, instead of getting a set of events, you get a single event. Geometrically, a fixed proper time is a point in the manifold whereas a fixed coordinate time is a hypersurface in the manifold. Now, assuming that we have a valid time coordinate and assuming that the coordinate system is well defined along some timelike worldline, then it is always possible to transform to a closely related coordinate system where the coordinate time matches the proper time along that worldline, but the hypersurfaces of simultaneity are unchanged. So, here you can say that SC represent the time of a distant clock using the Schwarzschild simultaneity convention, but you can easily make KS-like coordinates that also represent the time of the same clock using the KS simultaneity convention. So that is not a distinguishing feature of SC, i.e. it doesn't make SC uniquely represent the viewpoint of a distant observer. This implies that whether or not an object falls across the EH according to a distant observer is simply a matter of convention.

 Quote by stevendaryl In other words, you have no interest in actually defending the statements you've made? Why post anything, if you don't want people to respond to your statements? [..]
"Making statements" is not the question; explanations can be helpful, but it's weird to have to explain things to those who are supposed to provide the answers - I still have questions similar to the OP here concerning black holes. To my surprise, when I first asked about black holes I found myself drawn into philosophical discussions. And I noticed that people started debating their philosophical views. That is a waste of time for me; I don't want to waste time but to increase knowledge. Similarly, the philosophy forum has now been closed because it consumed too much time of the mentors.
 [..] Schwarzschild coordinates are chosen for CONVENIENCE. [...]
Yes of course - I made a similar remark in an earlier thread (probably it was in the "simultaneity" thread).

 Quote by harrylin "Making statements" is not the question; explanations can be helpful, but it's weird to have to explain things to those who are supposed to provide the answers
I don't think you've asked any very specific questions. Maybe I missed them. It seemed to me that you were making the claim that an infalling observer never reaches the event horizon. You were making another claim that there was a contradiction between the description of the situation as described by the coordinate system of the distant observer and the coordinate system of the infalling observer. Those seemed to be claims, not questions.

 Quote by harrylin I still have questions similar to the OP here concerning black holes.
I looked back through your posts, and I didn't see a single question. So what are your questions about black holes?