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

 Quote by stevendaryl What do you mean by "he doesn't use Schwarzschild here"? Do you mean he doesn't use Schwarzschild coordinates? That was certainly how I interpreted his statement: He doesn't say Schwarzschild coordinates, but what else would he mean by "coordinate time for a distant observer"? It seemed to me that his whole paper was from the point of view of Schwarzschild coordinates. The issue is this: You have a black hole of mass M, and a thin spherical shell of dust, also of mass M, falling inward. The question put by the author is how the position of the shell changes as a function of Schwarzschild time coordinate t. As t → ∞, the location of the shell asymptotically approaches the Schwarzschild radius, but which Schwarzschild radius? That of the original mass M, or the eventual mass, 2M?
Actually, this question has a very easy answer, from the point of view of the distant observer. Outside of the infalling shell, the effective mass is 2M, and so the usual Schwarzschild coordinates can be used with that mass. Those coordinates say that the outer surface of the shell must approach radius r = 4GM/c2 asymptotically as t → ∞. So there is never a finite coordinate value for t at which the shell is inside its own event horizon.

So I don't know what the author meant when he said that "matter from the outside must reach the interior, and it must do so in a way that is perceptible in finite coordinate time for a distant observer".

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 Quote by Mike Holland Just a little quibble. Why don't the clocks all run at the same rate? I thought that is was the time dilation that caused this situation.
In an invariant sense clocks do all run at the same rate. They all run at a rate of 1 second/light-second, in an invariant sense.

In order to make a statement that they run at different rates you already have to introduce a coordinate system with a simultaneity convention. Only then can you get clocks running at different rates (1/γ) proper-second/coordinate-second.

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 Quote by stevendaryl Just a little comment about that. I would say that proper time and Schwarzschild coordinate time are both physically meaningful, but for different reasons. Proper time is always physically meaningful, for any geometry. Schwarzschild coordinate time is physically meaningful in the context of black holes because it is a Killing vector field. Of all possible time-like coordinates in the exterior of a black hole, only the Schwarzschild time allows a time-independent metric. That's physically meaningful.
But even in this case it is the invariants*, not the coordinates which are important. The Killing vector field exists in all coordinate charts and is the same geometric field in each expressing the same symmetry in each. It is only easier to calculate in the Schwarzschild coordinates.

I think what I said in 176 still holds:
 Quote by DaleSpam That is easy. There is NO physical interpretation of ANY coordinate system (incl. SC and all of the other coordinate systems that we have discussed); what has physical interpretations are the invariants. The purpose of any coordinate system is simply to make calculations possible or even easy. In some coordinate systems the calculation of specific invariants becomes particularly easy, but even then it is the physical invariants which are easily calculated from the coordinates which have a physical interpretation, not the coordinates themselves.
*maybe I should say "covariants" instead of "invariants", but that sounds weird

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 Quote by stevendaryl What do you mean by "he doesn't use Schwarzschild here"? Do you mean he doesn't use Schwarzschild coordinates? That was certainly how I interpreted his statement: He doesn't say Schwarzschild coordinates, but what else would he mean by "coordinate time for a distant observer"?
Kevin Brown has a writing style where he often poses statements shows are wrong (partly or completely) in an extended dialog. Further along on the page he not only arrives at the conclusion of infinite SC exterior time but has some nice pictures of how it looks in a representation of complete spactime. You see that the event of the horizon passing each particle is on 'sheet' of infinite SC coordinate time.

 Quote by DaleSpam But even in this case it is the invariants*, not the coordinates which are important. The Killing vector field exists in all coordinate charts and is the same geometric field in each expressing the same symmetry in each. It is only easier to calculate in the Schwarzschild coordinates.
The Schwarzschild time coordinate is the integral of the Killing vector field. So whether you're using Schwarzschild coordinates or not, the Schwarzschild time t is physically meaningful as the integral of the Killing vector field.

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 Quote by stevendaryl The Schwarzschild time coordinate is the integral of the Killing vector field. So whether you're using Schwarzschild coordinates or not, the Schwarzschild time t is physically meaningful as the integral of the Killing vector field.
The SC t coordinate additionally introduces a simultaneity convention between the different integral curves of the Killing field that is not present in the Killing field itself and which is an arbitrary convention. I stand by my previous statements: the Killing field is physically meaningful, the coordinates are not.
 Blog Entries: 1 Recognitions: Gold Member Science Advisor Independent of the 'extra' killing vector field in the SC geometry (timelike -> static exterior; spacelike -> not static interior; 'extra' meaning in addition to the kvfs of spherical symmetry), there is a physical statement that can be made about spacetimes with horizons that is much more general than for just SC geometry (e.g. allows evolving and merging horizons, thus no timelike kvfs at all): The union of past light cones along all timelike world lines that always include future null infinity in their future light cones, fails to cover all of spacetime. [Open universe required for this statement to be have meaning] This can be physically interpreted as saying 'outside observers' never see or are influenced by any physical event on or inside a horizon. This observation also has a coordinate consequence: if your conventions for building coordinates requires an outside observer to receive a signal from an event in order to label it, any horizon and interior cannot be labeled at all in such coordinates (irrespective of where you assign infinite coordinate values). Exterior SC coordinates and generalizations of them for non-static exteriors happen to be of this class - they simply cannot assign coordinates to certain parts of spacetime. If you allow building coordinates in such a way as to label events outside observers can either receive signals from or send signals to, then you can label horizons and interiors, as well as exterior, in a single coherent coordinate system [edit: there may be issues of global topology of spacetime preventing covering all spacetime, but horizons and interiors will be accessible to such coordinate conventions.]

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 Quote by stevendaryl Schwarzschild coordinate time is physically meaningful in the context of black holes because it is a Killing vector field. Of all possible time-like coordinates in the exterior of a black hole, only the Schwarzschild time allows a time-independent metric.
This is not correct. As DaleSpam pointed out, Schwarzschild coordinate time uses the KVF plus a particular simultaneity convention. Other charts, such as Painleve and Eddington-Finkelstein, use the same KVF to define their time coordinates, so that the line element in all of them is independent of the time coordinate, but with different simultaneity conventions.

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 Quote by stevendaryl The Schwarzschild time coordinate is the integral of the Killing vector field.
I assume you mean that integral curves of the KVF are also integral curves of the Schwarzschild time coordinate. That's true, but the Schwarzschild time coordinate imposes a particular parameterization of those integral curves which is only one of many possible ones. The Painleve and Eddington-Finkelstein charts have the same integral curves for the time coordinate, but with different parameterizations. (However, there *is* something about the Schwarzschild coordinate time parameterization which is special; see the response I'm about to post to DaleSpam.)

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 Quote by DaleSpam The SC t coordinate additionally introduces a simultaneity convention between the different integral curves of the Killing field that is not present in the Killing field itself and which is an arbitrary convention.
This is true, but there is something about the SC t coordinate simultaneity convention which is special: it is the only one whose surfaces of constant time are orthogonal to the integral curves of the KVF. (Thus, the SC chart is the only chart with integral curves of the KVF as integral curves of its time coordinate, in which the line element is diagonal.) That is an invariant way of characterizing the simultaneity convention of the SC chart.

Of course, this doesn't fix any of the problems with the SC chart, such as the fact that it is singular at the horizon. It just points out that, in a curved spacetime, you probably won't be able to find a single chart that has all the properties you would like a chart to have, the way you can in flat spacetime.

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 Quote by pervect The reading on a clock is a physical measurement. It's something you can observe directly. It's about as simple as you get. It's a good thing to take as a primitive axiomatic element, one that you can't make simpler.
Certainly

 Quote by pervect An "observer" is a much more complicated mental construct. You not only have one physical clock (which you still assume keeps perfect time, or at least good enough time, as above), but you start imagining a whole network of virtual clocks. These clocks don't actually exist, but you imagine them as if they do. It's much more demanding assumption than assuming basically that "clocks exist, and you can use them to measure time".
There is certain philosophical problem with your line of reasoning. If observer doesn't exist we don't care about clocks. If clocks don't exist we still care that observer exists.

Besides you jumped from single physical measurement of single clock to statement about many clocks and "measurement of time".

 Quote by pervect So, I'll repeat it with emphasis, in the hope it get's through. (Though if the problem is linguistic, rather than one of attention in such a huge, meandering thread) the emphasis might not help. There are no static observers at the event horizon of a black hole
pervect, have you ever heard about Begging the question fallacy?
 Recognitions: Gold Member There is certain problem with the statement that falling clock will cross the event horizon in finite time. While proper time of the clock is invariant the concept of "event horizon" and therefore event of "crossing the event horizon" might turn out to be not so clearly defined and slightly more coordinate and assumptions dependant than proper time of the clock.

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 Quote by zonde Besides you jumped from single physical measurement of single clock to statement about many clocks and "measurement of time".
No, pervect didn't do that. He said that people who think SC coordinates are privileged, do that.

 Quote by zonde pervect, have you ever heard about Begging the question fallacy?
He wasn't stating an assumption, he was stating a physical prediction of GR. That prediction doesn't involve any assumptions about whether, or where, static observers exist; you find that out by solving the EFE with the appropriate constraints. Again, it's the people who think SC coordinates are privileged who are begging the question, by assuming there have to be static observers everywhere instead of actually looking at the solution of the EFE to find out.

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 Quote by zonde While proper time of the clock is invariant the concept of "event horizon" and therefore event of "crossing the event horizon" might turn out to be not so clearly defined and slightly more coordinate and assumptions dependant than proper time of the clock.
Whether they "might" or not, they aren't; the event horizon is an invariant, global feature of the spacetime, and so are any events where particular worldlines cross the horizon. So this "problem" is not a problem.

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 Quote by zonde Certainly There is certain philosophical problem with your line of reasoning. If observer doesn't exist we don't care about clocks. If clocks don't exist we still care that observer exists.
Actually, I think banishing the observer is a good idea. So I'd have to disagree with that "if the observer doesn't exist, we don't care about clocks". At least not in the sense that I'm talking about "an observer". "Observer" can have several meanings, the one you see to be suggesting is not at all the one I meant. I think the meaning I meant is made as clear as I can make it in the text. I''ll try to clarify - some.

In trying to make the exposition simple, entertaining, and easy to follow, I've probably sacrificed a lot of rigor. Quite possibly, even too much rigor. On the other hand, I've seen more rigorous explanations presented, which seem to just sail over everyone's head, or get ignored totally. (For instance when I mention Caroll's lecture notes. Or when I documented the historical shift in views on the topic in http://link.springer.com/article/10....A1022919909683

 , the noun immediately recalls to the mind this puzzling circumstance: during more than four decades since the discovery of the “Schwarzschild solution,” the overwhelming majority of the relativists harbored the conviction that the region within the “Schwarzschild radius” was physically meaningless, and strove to show that it could not be accessed from the outer space. During the subsequent four decades, after a seminal and nearly forgotten paper [1] that Synge wrote in 1950, an equally overwhelming majority of them came to the conviction that the same region was physically meaningful and accessible “without a bump” along geodesics
If this doesn't convince people that the practicing view that the event horizon is "inaccessible" is outdated, I don't know what will. This quote does take the approach of "appealing to authority", though.

So - I thought I'd try something else....to see if I could explain, not just quote the literature, but to explain the logic. Furthermore, to explain in a way that didn't require math. (If people did follow the math, in my opinion we wouldn't be having this argument. It's the math, IMO, that convinced all those physicists to change their position - not the words.)

Apparently, however, the result from my experiment was not very successful - at least to date.

I will give an example in the literature about the merits of "banishing the observer" - demonstrating that the idea is possible, that it exists in the literature, and providing the rigor and dryness that I did not provide.

http://arxiv.org/abs/gr-qc/9508043 "Precis of General Relativity"

 Quote by Misner A method for making sure that the relativity effects are specified correctly (according to Einstein’s General Relativity) can be described rather briefly. It agrees with Ashby’s approach but omits all discussion of how, historically or logically, this viewpoint was developed. It also omits all the detailed calculations. It is merely a statement of principles. One first banishes the idea of an “observer”. This idea aided Einstein in building special relativity but it is confusing and ambiguous in general relativity. Instead one divides the theoretical landscape into two categories. One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments and the data tables they provide.

I would note that the author doesn't claim that the method presented is "the one true and exclusive way" to understand relativity. Their claim is more along the line of it's a way that works, and gets you to the right answers.

The second point: Misner (and I) put coordinates in the first category, the category of the mathematical model of what is going on. This is the "map" not the "territory". We put proper time in the second category, the category of measuring instruments and what they measure.

 Besides you jumped from single physical measurement of single clock to statement about many clocks and "measurement of time".
It _was_ a big jump.

However, the whole notion of the "clocks and rods" thing was intended to be a quick and non-rigorous summary of the traditional classical notions of the observer and his coordinate system, drawn from memory. (I suspect one can find some discussion along similar lines by Einstein, certainaly one can in MTW).

I intended it to be familiar, not something new. Since this particular observer - and - coordinate based approach doesn't actually work in this case, I didn't and don't really want to put in a lot of effort in justifying it. I'm trying to say"I think this approach is basically what you are doing, and while the idea has a lot of classical history to it, it will always fail to explain black holes, because the fundamental approach contains some false assumptions.

 pervect, have you ever heard about Begging the question fallacy?
[/quote]

I just reviewed that, and I don't think I'm doing that.

Something else I should probably explain in greater detail, which is why there isn't any such thing as a stationary obserer at the event horizon. The reason is simple. The event horizion is a trapped, lightlike surface. So you can't have an "observer" there any more than you can have an "observer" sitting on a light beam.

THere's a PF Faq on why you can't have an observer ride along on a light beam. I hope this much is accepted by all, the only other thing you need to know then is that you can mark the event horizoin with a beam of light that sits there.

FYI, concerning my post http://physicsforums.com/showpost.ph...&postcount=259 , PAllen insisted:
 Quote by PAllen And again: I claim, along with others here, that there is no classical claim in the 2007 paper inconsistent with mathpages. This is based on understanding the math and background.[..]
I wrote to prof. Vachaspati to clarify if the classical findings in his paper are consistent with mathpages as PAllen thinks, while it is for me an obvious disagreement. His reply may be useful for some. I cited mathpages to him as follows:

"unavoidably [..] matter from the outside must reach the interior" because "an empty region around which matter "bunches up" outside an event horizon isn't viable", and "we arrive at a contradiction unless the value of m inside the horizon increases [..] in finite coordinate time." - http://www.mathpages.com/rr/s7-02/7-02.htm

Prof. Vachaspati comments (cited here with his permission):

Thanks for the interest. The issues you are discussing do seem to be all classical. Then, as you say, it is quite simple -- if you solve Einstein equations for the collapsing shell, it gives R=R_S only at infinite t.

It is true that t is a coordinate time but it is also the natural time coordinate for the asymptotic observer. In particular, the human life span is say ~100 years as measured in t. More to the point, however, is that the total energy of the collapsing body is emitted in some finite t, while the gravitational collapse takes infinite t.
Best,
Tanmay

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 Quote by harrylin FYI, concerning my post http://physicsforums.com/showpost.ph...&postcount=259 , PAllen insisted: I wrote to prof. Vachaspati to clarify if the classical findings in his paper are consistent with mathpages as PAllen thinks, while it is for me an obvious disagreement. His reply may be useful for some. I cited mathpages to him as follows: "unavoidably [..] matter from the outside must reach the interior" because "an empty region around which matter "bunches up" outside an event horizon isn't viable", and "we arrive at a contradiction unless the value of m inside the horizon increases [..] in finite coordinate time." - http://www.mathpages.com/rr/s7-02/7-02.htm Prof. Vachaspati comments (cited here with his permission): Thanks for the interest. The issues you are discussing do seem to be all classical. Then, as you say, it is quite simple -- if you solve Einstein equations for the collapsing shell, it gives R=R_S only at infinite t. I also asked him about his interpretation of t, and he answered: It is true that t is a coordinate time but it is also the natural time coordinate for the asymptotic observer. In particular, the human life span is say ~100 years as measured in t. More to the point, however, is that the total energy of the collapsing body is emitted in some finite t, while the gravitational collapse takes infinite t. Best, Tanmay
Interesting, but it still leaves many question muddy.

Nothing he says about the classical solution is new or unusual, per se. Even, for example: "It is true that t is a coordinate time but it is also the natural time coordinate for the asymptotic observer" is also similar to statements in mathpages (see below), for example. I see no claim that the classical part is new, in result or interpretation, by itself. Then, the key point he makes to attach more fundamental meaning to the coordinate time result is: " More to the point, however, is that the total energy of the collapsing body is emitted in some finite t, while the gravitational collapse takes infinite t." . This is strictly a quantum claim - classically there is no emitted energy. This is precisely the statement that Padmnabhan disputes in the 2009 paper.

As for mathpages, I have addressed what are superficial readings of Keven Brown's sometimes complicated presentations style. For example, in addition to statements like the following (but note the point "paradox to be resolved"):

"Nevertheless, if mass accumulates near the exterior of a black hole's event horizon the gravitational radius of the combined system must eventually increase far enough to encompass the accumulated mass, leading unavoidably to the conclusion that matter from the outside must reach the interior, and it must do so in a way that is perceptible in finite coordinate time for a distant observer, which seems to directly conflict with Item 2 (and certainly seems inconsistent with the "frozen star" interpretation). To resolve this apparent paradox requires a careful examination of the definition of a black hole, and of the behavior of the Schwarzschild time coordinate near an event horizon."

You have statements like:

"We saw that the radial position of a test particle starting at radius r = 10m and t = 0 (for example) as a function of the particle’s proper time is a simple cycloid right down to r = 0, whereas if the same trajectory is described in terms of Schwarzschild coordinate time, the infalling object traverses through infinite coordinate time in order to reach the event horizon"

and: "The event horizon is in the future of every locus of constant Schwarzschild coordinate time, all the way to future infinity. In fact, the event horizon is part of future null infinity"

"Also, the Schwarzschild time coordinate is physically significant in the sense that it is the unique time coordinate in terms of which the spherically symmetrical solution is static, i.e., the metric coefficients are independent of time. In other words, the time coordinate is a Killing vector field. The existence of a singularity in a Killing vector has global significance, being a one-way causal boundary."

There are a number of specific statements in the mathpages description that I might take exception to as poorly worded, stretching a point, etc. But, I still see nothing either in mathpages or Vachaspati's strictly classical claims inconsistent with how I summarize the mainstream (which is also similar to how textbooks and Padmanabhan summarize it):

"Everyone agrees on infinite Schwarzschild coordinate time for black hole formation. Brown, and mainstream GR since 1960 supplements this statement with the understanding that this coordinate time has a limited meaning, and that if you ask what is predicted for the infalling matter you must conclude BH formation in finite clock time of the infalling clocks. And that there are many way besides SC coordinate time by which these events can be correlated with external events."

[Edit: consistent with the above, is that other researchers interpret the only significant content of the 2007 paper is the quantum claim that "evaporation completes before collapse". Either this is true, or there is nothing to the 2007 paper.]