Notions of simultaneity in strongly curved spacetime

[PeterDonis]

But in GR transformations between coordinates don't have to preserve metric intact. That's how Lorentz interval is left the same, right?

No. GR coordinate transformations leave the metric intact; at least, they do in the normal meaning of that term, that geometric invariants are preserved. The metric may *look* different, as a formula, in a different coordinate chart; for example, the metric of the exterior vacuum region of Schwarzschild spacetime looks different in Painleve coordinates than it does in Schwarzschild coordinates. But if you calculate any geometric invariant, such as the length of a curve, in different coordinate charts, you will get the same answer.

 

[pervect]

I would like get this point better and as I understand you are confident about your understanding of that point.

Transformations between inertial SR coordinates preserve Lorentz interval. And they don't change metric just as well.
But in GR transformations between coordinates don't have to preserve metric intact. That's how Lorentz interval is left the same, right?

If you have two events in space-time, everyone who can see both events agrees on the Lorentz interval between them. So you don't really need to focus overmuch on the coordinates, the Lorentz interval doesn't depend on your coordinate choices.

In Newtonian physics you used to be able to say that about distance. For instance, if you were doing plain plane geometry, you might not use coordinates at all, but Euclid's axioms - though you could use analytic geometry as a fill-in.

In relativistic physics distance is no longer an invariant, but the Lorentz interval is.

Distance is a geometric invariant of Newtonian physics the Lorentz interval is a geometric invariant of special and general relativity.

As far as the metric tensor goes, in some abstract sense it's always the same geometrical object, but the components in any given coordinate system do change as you change the coordinates.

 

[harrylin]

I honestly haven't seen this in what I've seen of the reputable physics community (which is not a lot as I am not an academic). There is certainly a lot of disagreement, but it doesn't look to me like disagreement about which coordinate systems are valid. It looks to me like disagreement on which physical principles should be retained, and which discarded, when they conflict. [...]

By mere chance the first paper on this topic that I read (which was very recently), was Vachaspati's paper in Physical Review D. He first summarizes the standard solution based on Schwartzschild's "map". Just as Oppenheimer who found that "it is impossible for a singularity to develop in a finite time", also Vachapati concludes that "it takes an infinite time for objects to fall into a pre-existing black hole as viewed by an asymptotic observer".
This looks so easy to verify that I also did this today on a piece of paper with a pocket calculator.

However, Hamilton claims - based on other maps - that it is possible for a singularity to develop. Maybe he believes in multiple universes? The contradiction is shouting at me, but I have only read a few papers and seen a few web sites, so I don't know how other people interpret this - except for Einstein, who regarded it as a problem that cannot occur, and Vachaspati who argues the same on other grounds.

If I understand it correctly and calculated it correctly, then this is like my Earth map illustration, which I also imagined for this purpose. I will thus elaborate on that, because I continue to see an unsolved issue repeated again and again and I want to get to the bottom of it.

The first Earth cartographer - let's call him Schwarzy - makes a map according to which nobody can get North of the North pole. Schwarzy would agree that you cannot literally see someone walk North of the North pole; however that completely misses the point. According to Schwarzy's map it can never happen.

Then comes along a second cartographer - let's call him CalCross - who makes a map based on Mercator that eliminates that impossibility, so that on his map that "unphysical" limit is removed and people can walk on beyond the North Pole. According to his map the events that cannot happen following Schwarzy, happen smoothly, without any problem.

In my world (but perhaps not yours), those maps contradict each other. I have no doubt that Schwarzy's map is perfectly conform Flatland's Earth Science, and Mercator's map is just a conformal copy of the same; but I wonder about CalCross's map, which is some kind of an extension of the last. And that brings me to the logical request:

I thought it would be useful to do something Peter Donis suggested in another thread. That is, to look classically at the complete geometry of a collapsing shell of matter, which is like the Krauss case. I found a couple of references, and have done some order of magnitude calculations based on the arxiv paper (adjusted for a matter shell rather than a null dust shell).

The follwoing gives a Kruskal diagram (and other coordinates) for a collapsing shell. [..] [/url]

I think that there is no doubt that Schwarzschild etc (incl. Einstein, O-S and Vachaspati) used a valid GR model. However I don't know at all the physics behind a Kruskal diagram. And these models look to me just as contradictory - and roughly in the same way - as the Earth maps in my example.
Please clarify what reference system the Kruskal diagram portrays. Also, I wonder if it is a valid reference system according to Einstein's GR, or only according to the mathematical equations that are used in Einstein's GR; or if it is in fact no reference system at all, but more a kind of transformation map. In the last case, the same questions would need to be asked for every point on that map.

 

[PAllen]

By mere chance the first paper on this topic that I read (which was very recently), was Vachaspati's paper in Physical Review D. He first summarizes the standard solution based on Schwartzschild's "map". Just as Oppenheimer who found that "it is impossible for a singularity to develop in a finite time", also Vachapati concludes that "it takes an infinite time for objects to fall into a pre-existing black hole as viewed by an asymptotic observer".
This looks so easy to verify that I also did this today on a piece of paper with a pocket calculator.

Look carefully at his statement: "as viewed by and asymptotic observer". This wording is not accidental. An asymptotic observer is one who is 'infinitely far' from the BH. SC type time coordinate represents proper time only for this observer (that is, times they assign to events along a world line at asymptotic infinity). For any other observer, to get clock time, you integrate along some trajectory. For free faller, using these coordinate or any other, you would find finite clock time to reach the event horizon. If you fill in the space time hole in these coordinates (e.g. using SC interior type coordinates) you can continue the infall world line to the singularity in finite additional time as well.

However, Hamilton claims - based on other maps - that it is possible for a singularity to develop. Maybe he believes in multiple universes? The contradiction is shouting at me, but I have only read a few papers and seen a few web sites, so I don't know how other people interpret this - except for Einstein, who regarded it as a problem that cannot occur, and Vachaspati who argues the same on other grounds.

Classically, there is no contradiction. What you have is the analog of the situation I described of removing a disk from the pole of a 2-sphere. You have a coordinate chart that only covers the 2-sphere minus the disk, versus other charts the cover the whole sphere. They agree on the parts they both cover. The incomplete chart simply cannot make predictions about the region it doesn't cover.

Krauss, et. al. then provide a reason to consider the missing region of spacetime irrelevant - that quantum mechanics says bodies on the incomplete world lines actually end by evaporation before reaching the event where the incomplete chart chops them.

Einstein's argument (from a valid calculation) is considered invalid. No one, on any side BH related debates uses it any more. The calculation showed matter particles would have to go the speed of light before reaching SC radius to maintain stability. The correct conclusion is that then matter can't be stable inside a critical radius; if the particles cannot exceed local c, they must proceed with collapse. Einstein argued that 'something' must stop this state from occurring. He provided no basis for this something. You may say he simply believed something must stop this from happening. Krauss et.al. effectively provide a basis for this.

But there are no contridictions between maps. Classically, you just have different coverage by different maps.

If I understand it correctly and calculated it correctly, then this is like my Earth map illustration, which I also imagined for this purpose. I will thus elaborate on that, because I continue to see an unsolved issue repeated again and again and I want to get to the bottom of it.

No, it is not like your case. It is like a map that covers the whole sphere versus a map that is missing a disk.

The first Earth cartographer - let's call him Schwarzy - makes a map according to which nobody can get North of the North pole. Schwarzy would agree that you cannot literally see someone walk North of the North pole; however that completely misses the point. According to Schwarzy's map it can never happen.

This is where you are misunderstanding things. The correct analogy here is that this map does not include a little disk around the north pole. It agrees with complete maps on the distance to this disk boundary (finite proper time for infallers, computed same for SC coordinates as all others). However, the disk is simply not covered by this map. This map assigns an infinite value of some coordinate to lines approaching the disk; however, computing distance along these lines (proper time for infallers), it agrees with any other map that the distance to the disk boundary is finite.

Then comes along a second cartographer - let's call him CalCross - who makes a map based on Mercator that eliminates that impossibility, so that on his map that "unphysical" limit is removed and people can walk on beyond the North Pole. According to his map the events that cannot happen following Schwarzy, happen smoothly, without any problem.

This is because one map has a hole that the other one fills.

In my world (but perhaps not yours), those maps contradict each other. I have no doubt that Schwarzy's map is perfectly conform Flatland's Earth Science, and Mercator's map is just a conformal copy of the same; but I wonder about CalCross's map, which is some kind of an extension of the last. And that brings me to the logical request:

I think that there is no doubt that Schwarzschild etc (incl. Einstein, O-S and Vachaspati) used a valid GR model. However I don't know at all the physics behind a Kruskal diagram. And these models look to me just as contradictory - and roughly in the same way - as the Earth maps in my example.

All the maps agrees on every computation of an observable, for the events they have in common. One map is incomplete. Others are complete (include more of space time - world lines don't end for no reason, on a topological hole).

The unique contribution of Krauss et.al. is to provide a proposed physical reason to prefer the incomplete map: that the incomplete map already 'covers' too much. The real world, with quantum effects, diverges from classical near the edges of the incomplete map, so that even the very edge of the incomplete map becomes irrelevant. (Of course this is conditional on their debated quantum analysis).

Please clarify what reference system the Kruskal diagram portrays. Also, I wonder if it is a valid reference system according to Einstein's GR, or only according to the mathematical equations that are used in Einstein's GR; or if it is in fact no reference system at all, but more a kind of transformation map. In the last case, the same questions would need to be asked for every point on that map.

I believe I have covered this above. Kruskal, GP, Lemaitre, etc. are simply maps that cover more events. Every computed measurement in them agrees with SC for the events included in both. SC assigns infinite coordinate values at a boundary of its coverage, the others do not, but all measurements right up to this edge agree in all coordinates (that infaller's clocks pass finite time reaching the edge; that distant observers never see/detect anything reaching the edge = EH).

I cannot respond to what you call Einstein's GR versus other GR. Only you know what you mean by this. There is one GR. Over the course of his life, Einstein changed his mind several times over which predictions of it are physically plausible, but this isn't different theories but beliefs about applicability of predictions to the real world. For example, the theory has a cosmological constant that may or may not be zero. First Einstein thought a value of zero was implausible; then he decided it was physically preferred; now it appears small positive value is most plausible. Einstein first accepted, then rejected, then accepted the prediction of gravitational waves by GR. Einstein's position on black holes amounted to the belief that they weren't physically plausible. However, classically, there is no way to remove them as predictions without something as artificial as: events not seen by a chosen class of observer do not exist.

Almost nobody believes the classical description of BH appllies to our universe. There is much disagreement about what occurs instead.

There are a wide range of GR predictions that people differ on the likelihood of their corresponding to our universe: white holes, closed time like curves, naked singularities, alcubierre drive, etc. There are varying strong reasons for doubting them.

 

[rjbeery]

When you have extended an infalling worldline all the way to t = infinity by your clock, that worldline still only has a *finite* length. "Length" for worldlines means proper time, and the proper time for an object to fall to the horizon is finite.
...
This also shows why the region of spacetime covered by the SC time slicing can't be the entire spacetime: what happens to the worldlines once they reach the horizon? They have only covered a finite length, and spacetime is perfectly smooth and well-behaved at the horizon: the curvature is finite, there is nothing there that would stop the objects from going further. The only physically reasonable conclusion is that they *do* go further; that is, that there is a region of spacetime on the other side of the horizon, where the infalling objects go, but which can't be covered by the SC time slicing.

I'm not sure this is the only physically reasonable conclusion. The BH in question would have to exist for an eternity from the distant observer's perspective before the objects would cross the EH, yet current theory suggests that BHs would evaporate in finite time from that same distant observer's perspective. (Without examining how that EH came to exist in the first place!) this would suggest to me that infalling objects are destroyed and emitted as Hawking Radiation before they cross the EH.

 

[PeterDonis]

By mere chance the first paper on this topic that I read (which was very recently), was Vachaspati's paper in Physical Review D. He first summarizes the standard solution based on Schwartzschild's "map". Just as Oppenheimer who found that "it is impossible for a singularity to develop in a finite time", also Vachapati concludes that "it takes an infinite time for objects to fall into a pre-existing black hole as viewed by an asymptotic observer".
This looks so easy to verify that I also did this today on a piece of paper with a pocket calculator.

The "time" you refer to is time according to a distant observer, i.e., an observer who is spatially separated from the infalling matter. Did you also calculate the proper time experienced by an observer who is *not* spatially separated from the infalling matter? That is, an observer who is falling in along with it? O-S show that that proper time is finite. Have you really stopped to think about what that means?

However, Hamilton claims - based on other maps - that it is possible for a singularity to develop. Maybe he believes in multiple universes?

No, he is just recognizing that the proper time for an infalling observer to reach the horizon is finite, and realizing what that means. All the proofs you refer to, which say that "it is impossible for a singularity to form in a finite time", are proofs that a singularity cannot form *in the region of spacetime where the distant observer's time coordinate is finite*. They are *not* proofs that that region of spacetime is the only region of spacetime that exists. In fact, it is easy to show that there *must* be another region of spacetime, below the horizon, that is not covered by the distant observer's time coordinate. That must be true *because* the proper time for an infalling observer to reach the horizon is finite. Have you considered this at all?

The contradiction is shouting at me, but I have only read a few papers and seen a few web sites, so I don't know how other people interpret this

Have you not been reading all the posts I and others have made explaining "how other people interpret this"? Please, before you keep bringing this up, take some time to seriously consider what I said above, and what I'm going to say below.

The first Earth cartographer - let's call him Schwarzy - makes a map according to which nobody can get North of the North pole. Schwarzy would agree that you cannot literally see someone walk North of the North pole; however that completely misses the point. According to Schwarzy's map it can never happen.

Then comes along a second cartographer - let's call him CalCross - who makes a map based on Mercator that eliminates that impossibility, so that on his map that "unphysical" limit is removed and people can walk on beyond the North Pole. According to his map the events that cannot happen following Schwarzy, happen smoothly, without any problem.

In my world (but perhaps not yours), those maps contradict each other.

This is actually a good analogy, but not for quite the reason you think. (I'll give my version of the analogy below.) A Mercator projection doesn't actually include the North Pole; it maps the finite distance from the equator to the North Pole, on the actual globe, to an *infinite* vertical distance on the flat map. Actual maps using the Mercator projection, on finite-sized sheets of paper, don't reach all the way to the North Pole; they are cut off at some latitude short of 90 degrees North. So in your analogy, CalCross's map does *not* show that you can walk North of the North Pole; instead, it shows (or appears to show) that it would take an infinite time to reach the North Pole, because the distance to it looks infinite.

So let's try a different version of the analogy. Schwartz and CalCross both live on the equator right where it crosses the prime meridian. CalCross makes a map, using the Mercator projection, and claims, based on that map, that the distance to the North Pole is infinite, so nobody can ever reach the North Pole; it would take an infinite amount of time. Therefore, CalCross claims, there is nothing beyond the North Pole, since any such place would have to be "further away than infinity".

Schwartz, however, has a mathematical model based on the Earth being a sphere (he can't draw his model undistorted on a flat map, but he can work with it mathematically), which says that the distance to the North Pole is finite, and that if you walk there and then continue walking, the Earth's surface continues on just fine. Explorers are sent north along the prime meridian; which of the two (CalCross and Schwartz) will be proved right, and which will be proved wrong?

Obviously this case is not exactly like the case of Schwarzschild spacetime, because the North Pole is not a "horizon"; the explorers can turn around and come back, bringing their data with them. But CalCross's coordinates, in which the distance to the North Pole looks infinite, even though it really isn't, are very much like Schwarzschild coordinates, in which the "distance" (which in this case is time, since we are looking in a timelike direction) to the horizon looks infinite, even though it really isn't.

I think that there is no doubt that Schwarzschild etc (incl. Einstein, O-S and Vachaspati) used a valid GR model.

Yes, they did. Their model is valid in the same way that CalCross's map of the Earth is valid; you can use CalCross's map to calculate the length of any curve on the Earth's surface you like, as long as the curve doesn't include one of the poles. Similarly, you can use the standard SC exterior coordinates to calculate the length (proper time) of any worldline in Schwarzschild spacetime you like, as long as the worldline doesn't cross the horizon. Both maps are correct within their limited scope, but they are limited in scope.

However I don't know at all the physics behind a Kruskal diagram.

The Kruskal diagram is probably not the best place to begin if you are trying to understand how GR models a black hole spacetime. I would start with either ingoing Painleve coordinates or ingoing Eddington-Finkelstein coordinates instead. That said, I'll make some comments about the Kruskal diagram below.

Please clarify what reference system the Kruskal diagram portrays.

What do you mean by "reference system"? It is true that there is no observer whose worldline is the "time" axis (i.e., vertical axis) of the Kruskal diagram; but there's no requirement in GR that that be true for a valid coordinate chart. (Strictly speaking, it's not a requirement even in SR; you can describe flat spacetime in some wacky coordinate chart whose "time axis" isn't the worldline of any observer.) The Kruskal chart is a coordinate chart; it's a mapping of points (events) in spacetime to 4-tuples of real numbers $( V, U, \theta, \phi )$, such that the metric on the spacetime can be written in this form:

$$ds^2 = \frac{32 M^2}{r} e^{-r / 2M} \left( - dV^2 + dU^2 \right) + r^2 \left( d \theta^2 + sin^2 \theta d \phi^2 \right)$$

Here V is the "time" coordinate (vertical axis) and U is the "radial" coordinate (horizontal axis) in the Kruskal diagram. (Note that I've used units in which G = c = 1.) The "r" that appears in this line element is not a separate coordinate in this chart; it is a function of U and V, which is used for convenience to make the line element look simpler and to make clear the correspondence with the Schwarzschild chart. An example of the diagram is here:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

Note that this diagram is for the "maximally extended" Schwarzschild spacetime, which is not physically realistic. If we drew a similar diagram of the spacetime of the O-S model (the modern version which completes the O-S analysis by carrying it beyond the point where the horizon forms), it would include a portion of regions I and II in the diagram on the Wikipedia page, plus a non-vacuum region containing the collapsing matter. DrGreg posted such a diagram in the thread on the O-S model here:

https://www.physicsforums.com/showpost.php?p=4164435&postcount=64

(I know you've already seen this, but I want to be clear about exactly which diagrams I'm referring to.)

A key fact about the Kruskal diagram that makes it so useful is that the worldlines of radial light rays are 45 degree lines, just as they are in a standard Minkowski diagram in flat spacetime. (You should be able to see this from looking at the line element above; if you can't, please ask. Being able to "read off" such things from a line element is a very useful skill.) That makes it easy to look at the Kruskal diagram and see the causal structure of the spacetime--which events can send light signals to which other events.

The other useful thing about the Kruskal diagram is that it let's you see how standard SC coordinates are distorted. Look at the dotted lines through the origin of the diagram, fanning out into region I; these are lines of constant Schwarzschild time t. See how they all intersect at the origin? That's why SC coordinates become singular at the horizon, which on this diagram is represented by the 45 degree line U = V (i.e., the one going up and to the right), and which therefore includes the origin. What look to the distant observer like "parallel" lines of constant time are actually *converging* lines. And what looks to the distant observer like an infinite "length" (i.e., time) to the horizon is actually a finite length (this can be easily calculated in the Kruskal chart, just take any timelike curve that intersects the horizon and integrate the above line element--the easiest curve is one with U = constant, so the only nonzero differential is dV).

As far as whether the Kruskal chart is "valid", of course it is. You can find a correspondence between it and the SC chart (or any other chart) in the same way you can find a correspondence between the standard latitude/longitude "chart" on the Earth's surface and a Mercator chart. But if one chart only represents a portion of the spacetime (as the SC exterior chart does), then there will only be a correspondence with other charts on that portion of the spacetime.

But how do we know that the other portions of spacetime shown on the Kruskal chart "really exist"? Because the Einstein Field Equation says so. When you solve the EFE for the case of a spherically symmetric vacuum, and make sure your solution is complete, what you get is the spacetime shown in the Kruskal chart. When you solve the EFE for the case of a spherically symmetric vacuum surrounding collapsing matter, what you get is a portion of regions I and II of the Kruskal chart, as shown in DrGreg's diagram. There is no way to solve the EFE and only get region I; such a solution is incomplete, just as the original O-S solution was incomplete.

Also, I wonder if it is a valid reference system according to Einstein's GR, or only according to the mathematical equations that are used in Einstein's GR; or if it is in fact no reference system at all, but more a kind of transformation map.

I'm not sure what you think the difference is between all these things. See my comments above; perhaps they will help to either clear up your confusion or at least clarify your questions.

 

[PAllen]

I'm not sure this is the only physically reasonable conclusion. The BH in question would have to exist for an eternity from the distant observer's perspective before the objects would cross the EH, yet current theory suggests that BHs would evaporate in finite time from that same distant observer's perspective. (Without examining how that EH came to exist in the first place!) this would suggest to me that infalling objects are destroyed and emitted as Hawking Radiation before they cross the EH.

This is a quantum argument, not a classical one. It is basically the same one Krauss et. al. make. A problem is that there is no consensus on this. Both before, and after in specific answer to it, other researchers find that quantum corrections and evaporation to not prevent the event horizon from forming in finite time for observers falling with the collapse. The two school's of thought, then, differ on how quantum mechanics solve the 'information problem' for black holes:

1) There is no problem. Evaporation saves the day in time. You don't even need to worry about a quantum treatment of a horizon that doesn't exist.

2) Evaporation doesn't save the day. There is problem. You do need to worry about a quantum treatment of a horizon. The solution is some type (many proposals) of a quantum black hole analog - that shares many predictions to a classical BH, but differs in various details, and has no singularity. This object also eventually evaporates.

 

[rjbeery]

This is a quantum argument, not a classical one. It is basically the same one Krauss et. al. make. A problem is that there is no consensus on this. Both before, and after in specific answer to it, other researchers find that quantum corrections and evaporation to not prevent the event horizon from forming in finite time for observers falling with the collapse. The two school's of thought, then, differ on how quantum mechanics solve the 'information problem' for black holes:

1) There is no problem. Evaporation saves the day in time. You don't even need to worry about a quantum treatment of a horizon that doesn't exist.

2) Evaporation doesn't save the day. There is problem. You do need to worry about a quantum treatment of a horizon. The solution is some type (many proposals) of a quantum black hole analog - that shares many predictions to a classical BH, but differs in various details, and has no singularity. This object also eventually evaporates.

That's interesting, thank-you. In my limited experience, discussions on Black Holes seem to presume their existence before examining their properties. Let's not discuss if and when objects can cross the EH for a moment; rather, let's discuss "when" an existing BH formed in the past from the distant observer's perspective. Classically, it's eternity! Infinite time in the future to grow and infinite time in the past to be created. And what to do with the singularity?

From a layman's perspective it seems that BH's introduce more problems than they solve, particularly when we have a theory for resolving the issue (i.e. Hawking Radiation)

 

[PAllen]

That's interesting, thank-you. In my limited experience, discussions on Black Holes seem to presume their existence before examining their properties. Let's not discuss if and when objects can cross the EH for a moment; rather, let's discuss "when" an existing BH formed in the past from the distant observer's perspective. Classically, it's eternity! Infinite time in the future to grow and infinite time in the past to be created. And what to do with the singularity?

The only objective statement, classically, that can be made about the distant observer is that they never see a BH finish forming (for a collapse, they see a dark ball just bigger than where EH would be calculated to be; the matter inside the collapsing body has apparently vanished).

When you try to go from here to 'when' a BH formed, (classically or otherwise) you have a problem. This gets right back to special relativity as Pervect has reminded several times. There is no objective meaning to now at a distance. Depending on what simultaneity convention you use, you can say, for a distant observer, the BH never forms; or that it formed 3PM yesterday. Neither statement has any physical content. So, "a BH never forms for distant observer" is a statement with no meaning in classical GR. The very similar statement "a BH is never seen to finish forming by distant observer" is a physical and indisputable statement.

From a layman's perspective it seems that BH's introduce more problems than they solve, particularly when we have a theory for resolving the issue (i.e. Hawking Radiation)

The issue is that a successful theory predicts they readily form (in sense above) from reasonable initial conditions. Observationally, the evidence piles up that things exist which have all the properties of GR black holes that can be verified from a distance. So the problem must be dealt with. What, exactly, is really there remains in dispute and will for some time (more observational evidence is coming all the time; quantum gravity theory will eventually progress).

 

[PeterDonis]

In my limited experience, discussions on Black Holes seem to presume their existence before examining their properties.

Their existence is not "presumed"; it is shown by solving the Einstein Field Equation for a spherically symmetric vacuum spacetime. The solution makes it clear that there *is* and event horizon and a black hole region inside it, and that objects *can* cross the EH in a finite proper time (i.e., a finite time according to a clock that is falling in with the object). Of course this is a classical solution and doesn't take quantum effects into account; we can't fully take quantum effects into account because we don't have a theory of quantum gravity yet. See my comments at the end of this post.

Infinite time in the future to grow

Infinite *coordinate* time according to Schwarzschild coordinates. But, as I've explained in previous posts in this thread, Schwarzschild coordinates become "infinitely distorted" at the horizon; they make finite lengths, like the finite length of an infalling worldline that crosses the horizon, look like infinite lengths.

and infinite time in the past to be created.

AFAIK nobody claims that the full, maximally extended solution, which includes a white hole that is "infinitely far in the past" according to Schwarzschild coordinate time (which has the same limitations here as it does in the future direction, see above), is physically reasonable. The physically reasonable solution includes a collapsing object (such as a star) in the past, not a white hole. That object collapsed at a finite time in the past, even according to Schwarzschild coordinate time.

And what to do with the singularity?

Do you mean the actual, physical singularity at r = 0? Or do you mean the coordinate "singularity" at the horizon"? The latter is not a "real" singularity; it's an artifact of the infinite distortion of Schwarzschild coordinates at the horizon. The former *is* a real singularity, and does show a limitation of classical GR. See below.

we have a theory for resolving the issue (i.e. Hawking Radiation)

Actually, we don't have a full theory that resolves the issue. "The issue" is really three issues; following on from what PAllen said, they are:

(1) When we take quantum effects into account, do they prevent a horizon from forming at all? In other words, does some quantum process cause any collapsing object that is predicted by classical GR to form a horizon and a black hole, such as a sufficiently massive star, to instead get turned completely into outgoing radiation *before* the horizon forms?

(2) If the answer to #1 is "no", do quantum effects at least prevent a singularity of infinite spacetime curvature from forming at r = 0 when the outer surface of the collapsing object reaches that point?

(3) If the answer to #1 is "no", regardless of what the answer to #2 is, can we at least be sure that quantum effects, such as Hawking radiation, prevent any information from being lost behind the horizon? In other words, even if objects do fall into the black hole and get destroyed in the singularity at r = 0, is their information still converted into Hawking radiation so it gets preserved?

We don't know the correct answer to any of these questions at this point. My understanding of our current "best guess" is that the answer to #1 is "no", and the answers to #2 and #3 are "yes". But we don't know for sure.

 

[rjbeery]

This also shows why the region of spacetime covered by the SC time slicing can't be the entire spacetime: what happens to the worldlines once they reach the horizon? They have only covered a finite length, and spacetime is perfectly smooth and well-behaved at the horizon: the curvature is finite, there is nothing there that would stop the objects from going further. The only physically reasonable conclusion is that they *do* go further; that is, that there is a region of spacetime on the other side of the horizon, where the infalling objects go, but which can't be covered by the SC time slicing.

Their existence is not "presumed"; it is shown by solving the Einstein Field Equation for a spherically symmetric vacuum spacetime. The solution makes it clear that there *is* and event horizon and a black hole region inside it, and that objects *can* cross the EH in a finite proper time (i.e., a finite time according to a clock that is falling in with the object). Of course this is a classical solution

With respect, when I'm philosophically discussing the existence of black holes I'm speaking about the realm of reality, not mathematical models. You said that the "only physically reasonable conclusion" was that they existed, while PAllen pointed out that the true answer is ambiguous at best. My personal opinion is that they do not exist in reality and current theory (as I understand it) cannot objectively conclude otherwise.

 

[rjbeery]

When you try to go from here to 'when' a BH formed, (classically or otherwise) you have a problem. This gets right back to special relativity as Pervect has reminded several times. There is no objective meaning to now at a distance. Depending on what simultaneity convention you use, you can say, for a distant observer, the BH never forms; or that it formed 3PM yesterday. Neither statement has any physical content. So, "a BH never forms for distant observer" is a statement with no meaning in classical GR. The very similar statement "a BH is never seen to finish forming by distant observer" is a physical and indisputable statement.

In reality we are not allowed the luxury of an infinite past. If a calculation shows that the EH must have formed prior to the Big Bang, I find this problematic.

 

[PeterDonis]

With respect, when I'm philosophically discussing the existence of black holes I'm speaking about the realm of reality, not mathematical models. You said that the "only physically reasonable conclusion" was that they existed

Just to clarify: I said that's true according to classical theory. But we know classical theory has limitations.

while PAllen pointed out that the true answer is ambiguous at best.

Because we don't know what the effect of quantum corrections to the classical theory is. They may prevent the horizon from forming, or they may not, as I said.

My personal opinion is that they do not exist in reality and current theory (as I understand it) cannot objectively conclude otherwise.

Yes, there are plenty of people who have that opinion. My personal opinion is basically the same as what I said the current "best guess" is: horizons do form, but they eventually evaporate away. But nobody knows for sure.

 

[PeterDonis]

In reality we are not allowed the luxury of an infinite past. If a calculation shows that the EH must have formed prior to the Big Bang, I find this problematic.

No calculation shows that; even the classical calculations, that show an EH forming, don't show it forming prior to the Big Bang.

 

[rjbeery]

B) Consider an observer that is distant and hovering into eternal past, but at some moment free falls into the BH (late enough so they hit the singularity). For this observer, both past-only and future-only conventions include both interior and exterior events. However, past only covers only a portion of spacetime - ending with the past of the termination of free fall world line on the singularity. A future only simultaneity covers all of space time, and is thus also a causal inclusive simultaneity.

In my opinion, it seems clearly desirable to favor causal inclusive simultaneity; and thus it is unfortunate that so much attention is paid to SC time slice simultaneity, which is exclusively a past-only simultaneity.

Going back to your OP, wouldn't the analysis of a white hole lead to the opposite conclusion?

 

[PAllen]

Going back to your OP, wouldn't the analysis of a white hole lead to the opposite conclusion?

Did you read a few sentences earlier:

"Now consider these for the Oppenheimer-Snyder spacetime (asymptotically flat; collapsing space time region; interior and exterior SC regions eventually). I choose this for qualitative plausibility and to avoid the white hole region (the notions certainly apply to full SC geometry)."

There is no white hole in this scenario. As noted, I could apply the definitions to a WH case, but then the results would be different. I wasn't interested in doing so, because I don't consider a WH plausible. GR itself requires white holes originate in the past without cause; while black holes are predicted (classically) to form from plausible starting conditions.

In case you were asking me to apply the concepts to the white hole case, I am not interested. There is more than enough confusion about collapse to BH; I don't want distraction from the white hole case.

 

[rjbeery]

No calculation shows that; even the classical calculations, that show an EH forming, don't show it forming prior to the Big Bang.

If an EH is shown to form at all it would be shown to occur after the BB by definition. I'm talking about starting with a black hole of mass M+A, where A is the mass of an object which has fallen past the EH, and calculating "when" from the distant observer's perspective that object crossed the EH. If the object takes a local eternity to cross the EH falling in then it takes a local eternity to cross the EH coming out.

 

[PAllen]

If an EH is shown to form at all it would be shown to occur after the BB by definition. I'm talking about starting with a black hole of mass M+A, where A is the mass of an object which has fallen past the EH, and calculating "when" from the distant observer's perspective that object crossed the EH. If the object takes a local eternity to cross the EH falling in then it takes a local eternity to cross the EH coming out.

I understand what you are asking except the part about coming out. Nothing comes out unless you are talking about quantum evaporation.

As for the rest:

- Classically, an infalling body merges with the pre-existing BH and expands its actual event horizon in finite (short) time locally for the infalling body; and reaches the singularity of the pre-existing BH in finite local time. The infaller does have an objective basis to correlate local and distant events, because they can keep receiving signals from outside until the moment they reach the singularity. They can see a specific distant clock time (in theory) as of the moment they reach the singularity.

- From a distant observers point of view, I keep repeating the question cannot be answered as worded; even similar questions in SR cannot be answered. You can answer when will a distant observe see the above happening? Then there is an answer: never. Because this physical answer is never, it follows that there is no objective answer to when A crossed the horizon for the distant observer. They can make an infinite number equally defensible answers, one of which is never.

 

[rjbeery]

GR itself requires white holes originate in the past without cause; while black holes are predicted (classically) to form from plausible starting conditions.

From a distant observers point of view, I keep repeating the question cannot be answered as worded; even similar questions in SR cannot be answered. You can answer when will a distant observe see the above happening? Then there is an answer: never. Because this physical answer is never, it follows that there is no objective answer to when A crossed the horizon for the distant observer. They can make an infinite number equally defensible answers, one of which is never.

Logic shows this is a contradiction. Take the BH mentioned above of mass M+A, where A is the mass of an object *having already passed* the EH from a distant observer's perspective. Note the time = $$T_0$$Now turn the clock back $$T_{-1}, T_{-2}, T_{-3}$$...until the object of mass A is no longer beyond the EH at $$T_{-x}$$ (and I don't care if we're using Schwarzschild metric for the observer's calculations, for example, or we simply move backwards in time until he *sees* the object, as you said)

What are we left with? At $$T_{-x}$$ we have an object outside of the BH, and at $$T_0$$ that object has crossed over the EH in finite time according to the distant observer. The conclusion is that observing the object crossing back out of the BH as we turn the clock backwards will never happen from the distant observer's perspective, certainly not within the finite age of the Universe.

 

[PAllen]

Logic shows this is a contradiction. Take the BH mentioned above of mass M+A, where A is the mass of an object *having already passed* the EH from a distant observer's perspective.

This is already a statement whose meaning is rejected by relativity. All else is irrelevant. Because the distant observer can never see it happening, they can never say the know it happened. Maybe it did, maybe it didn't.

Maybe I misunderstand your intent. It is absolutely possible for a distant observer to assign remote times in a consistent way such that they consider the object to have crossed the horizon in finite time. They can also consistently assign remote times so that never happens. It will never be possible to verify one assignment over another precisely because event horizon crossing will never be seen.

 

[rjbeery]

This is already a statement whose meaning is rejected by relativity. All else is irrelevant. Because the distant observer can never see it happening, they can never say the know it happened. Maybe it did, maybe it didn't.

Exactly. This is equivalent to saying "maybe the black hole exists, maybe it does not." Point being black holes do not even necessarily exist in Relativity. I'd be curious to see a time-reversed Kruskal analysis of objects crossing (back out of) the EH.

 

[PAllen]

Exactly. This is equivalent to saying "maybe the black hole exists, maybe it does not." Point being black holes do not even necessarily exist in Relativity. I'd be curious to see a time-reversed Kruskal analysis of objects crossing (back out of) the EH.

Yes, you can say a distant observer can never know an actual BH (rather than an 'almost BH') has formed. They can only compute that if GR is true, then for the matter at the 'almost BH' they do see, there is no alternative to a BH forming, in finite time for that matter. If GR is false in the near BH domain (as is likely to some degree), then there are alternatives.

However, because GR is unambiguous that a BH forms in finite time for the infalling matter, and new matter falls in in finite time for the infalling matter, it must be said that GR predicts black holes. That this prediction is not verifiable from the outside, doesn't make it not a prediction of GR. For example, QCD says quarks will never be detected in isolation. They are still a prediction of QCD.

The real 'way out' is that quantum gravity changes the classical GR predictions.

 

[PeterDonis]

I'd be curious to see a time-reversed Kruskal analysis of objects crossing (back out of) the EH.

If you are referring to a white hole, it's already present in the Kruskal diagram. The white hole is region IV on the Kruskal chart, as shown for example on the Wikipedia page:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

If you consider a timelike free-fall trajectory that starts at the past singularity (the hyperbola at the bottom of region IV), emerges from the white hole (i.e., crosses from region IV into region I), rises to some finite radius r at Kruskal time V = 0, then falls back into the black hole (crosses from region I into region II), and finally ends up at the future singularity (the hyperbola at the top of region II): such an object's trajectory is time-symmetric; the part before V = 0 is the exact time reverse of the part after V = 0.

If, however, you are referring to a spacetime where a BH forms from the collapse of a massive object, then evaporates away, I haven't seen a Kruskal-type diagram of that case, but I have seen Penrose diagrams of the most obvious way to model it (which not everyone agrees is the correct model, but it's a good starting point for discussion). See, for example, the diagram here:

http://en.wikipedia.org/wiki/Black_hole_information_paradox

Compare with the Penrose diagrams on this page:

http://www.pitt.edu/~jdnorton/teaching/HPS_041/chapters/black_holes_picture/index.html

The Penrose diagram corresponding to the Kruskal diagram I linked to above is in the section "Conformal Diagram of a Fully Extended, Schwarzschild Black Hole". The Penrose diagram corresponding to the classical GR model of a collapsing massive object (like a star) is in the section "A Conformal Diagram of a Black Hole formed from Collapsing Matter".

Note that in *none* of the diagrams, other than the Kruskal diagram and the Penrose diagram corresponding to it, does the white hole appear. In the evaporation diagram, Hawking radiation escapes as the hole evaporates, but there is still a black hole interior region and a singularity, and anything that gets inside the horizon is still doomed to be destroyed in the singularity, according to this model. The big open question is, if this model is *not* correct (which most physicists in the field now seem to think it is not, since it leads to the loss of quantum information), what replaces it? There are a lot of suggestions, but no good answer yet.

 

[rjbeery]

Yes, you can say a distant observer can never know an actual BH (rather than an 'almost BH') has formed. They can only compute that if GR is true, then for the matter at the 'almost BH' they do see, there is no alternative to a BH forming, in finite time for that matter. If GR is false in the near BH domain (as is likely to some degree), then there are alternatives.

However, because GR is unambiguous that a BH forms in finite time for the infalling matter, and new matter falls in in finite time for the infalling matter, it must be said that GR predicts black holes. That this prediction is not verifiable from the outside, doesn't make it not a prediction of GR. For example, QCD says quarks will never be detected in isolation. They are still a prediction of QCD.

The real 'way out' is that quantum gravity changes the classical GR predictions.

PAllen, I appreciate your maturity in acknowledging other (albeit subjective) points of view. The usual response is an emotional defense of BHs as a matter of fact...

 

[harrylin]

This discussion is growing a bit over my head, especially concerning time (my time, not Schwartzschild t, although it's almost the same ); I intended to quickly move on from a simple illustration to show that there is an issue, to a concrete physics discussion involving clocks and light rays. However it is interesting for me and perhaps also for invisible onlookers. I'll try to group things piece-wise and only discuss the essentials.

[..] there are no contridictions between maps. Classically, you just have different coverage by different maps.

Different coverage means to me in the context of relativity, from the same reference system - from the same "perspective". However, what I was referring to was the mapping between different reference systems, of a time τ to a time t>∞, as follows:

[..] [O-S] talk about t>∞. That doesn't make sense to me, which is what I had in mind with my remark that it looks like they didn't fully think it through. And that's not so strange, as their results were new.

[..] I think [O-S] didn't fully explore the question of what the region of spacetime with "t > infinity" would look like. But just contemplating the existence of such a region is not a contradiction. Check my latest post in the simultaneity thread. [..]

As I clarified earlier, a "region of spacetime" is for me merely a mathematical tool for calculations of, as Einstein put it, "clocks and rods". t>∞ has as physical meaning a possible clock that indicates t>∞. That makes as little sense to me as v>∞.
On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics. Thus you would need to make a strong case with the following if your intention is to convince me (but I hope that that is not what you are trying to do):

The "time" you refer to is time according to a distant observer, i.e., an observer who is spatially separated from the infalling matter. Did you also calculate the proper time experienced by an observer who is *not* spatially separated from the infalling matter? That is, an observer who is falling in along with it? O-S show that that proper time is finite. Have you really stopped to think about what that means?
[..]
"it is impossible for a singularity to form in a finite time", are proofs that a singularity cannot form *in the region of spacetime where the distant observer's time coordinate is finite*. [..] In fact, it is easy to show that there *must* be another region of spacetime, below the horizon, that is not covered by the distant observer's time coordinate. That must be true *because* the proper time for an infalling observer to reach the horizon is finite. Have you considered this at all?

First of all, I can't find anything that explains how you map a time τ to a time t>∞, as O-S suggest, and make physical sense of it. You must have considered this, and you suggested that you did, but I do not see that you clarified that essential point. It is a simultaneity that looks completely impossible to me.

Secondly, of course I considered the fact that O-S map t->∞ to τ->a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees. I did not make a plot of it, but I did not see an issue with that. In the O-S model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!), an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ₀. As I picture it (for I have not seen a description of it), for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. In fact, it was a discussion based on a blog including that aspect with more than 100 posts that was the first thing that I read about this topic (http://blogs.discovermagazine.com/badastronomy/2007/06/19/news-do-black-holes-really-exist/)

Have you not been reading all the posts I and others have made explaining "how other people interpret this"? [..]

Sorry: I did not see any explanation for the inside region that made any sense to me, or that explains to me how it cannot contradict Schwarzschild's model. Perhaps some others who asked similar questions were convinced, but I did not see that happen (and of course, there is no use to try to convince anyone about which model is "right"; this is just a discussion of models). Perhaps there is another post that I overlooked?

 

[PAllen]

Different coverage means to me in the context of relativity, from the same reference system - from the same "perspective". However, what I was referring to was the mapping between different reference systems, of a time τ to a time t>∞, as follows:

I'm not quite sure what you mean by reference system. In GR there is no such thing a global frame of reference - there are only local frames of reference. As a result, you cannot discuss global issues in frames of reference in GR. Instead, for global issues you either use coordinate systems or coordinate free geometric methods (e.g. Plane geometry without coordinates).

Two coordinate systems are just two different sets of labels attached to an overall space time. It can happen that they don't cover all the same region of spacetime. However, they are just relabelings of the same geometry for coverage in common. You obviously can't use a particular coordinate system for a part of the geometry it doesn't cover.


As for coodinate infinities, let me try an example. Start with a flat plane with Euclidean metric (distance given by ds^2= dx^2 + dy^2). Now define coordinates u and v as:

u=1/x , v = 1/y ; the metric (distance formula) expressed in these will be different, such that all lengths, angles and areas computed in cartesian coordinates are the same with computed with u and v - using the transformed metric.

Note that u and v become infinite as you approach the x or y axis. However, no computation or measurement is different from cartesian coordinates (when you use the transformed metric). But you can't directly do a computation involving any point on or line crossing the x or y-axis in these coordinates. You can compute the length of a line approaching the x-axis and get a finite value limit value; you can continue it on the other side and get a finite value for its length, limiting from the other side.

The ininite value of u and v has no geometric meaning, because coordinates are interpreted through the metric.

The behavior of the t coordinate in SC coordinates is just like this. It has meaning only through the metric for computation of 'proper time' which is what a clock measures. If you compute proper time for an infalling clock, you get a finite value for it to reach the EH. If you continue it over the EH using, e.g. interior SC coordinates, you get an additional finite proper time from the EH to the singularity.

As I clarified earlier, a "region of spacetime" is for me merely a mathematical tool for calculations of, as Einstein put it, "clocks and rods". t>∞ has as physical meaning a possible clock that indicates t>∞. That makes as little sense to me as v>∞. On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics. Thus you would need to make a strong case with the following if your intention is to convince me:

No, t means nothing. It is not a reading on any clock. To get a reading on a clock, you have to specify the clock (world line) and compute proper time (clock time) along it.

You will find, that for a static clock (stationary with respect to the spherical symmetry), very far from the center, SC coordinate time matches clock time for that clock. It doesn't match clock time for other clocks. The closer you get the the EH, the less this t coordinate has anything to do with what clocks measure. Just like with my u coordinate above, u becoming infinite says nothing about what a ruler will measure.

First of all, I can't find anything that explains how you map a time τ to a time t>∞, as O-S suggest, and make physical sense of it. You must have considered this, and you suggested that you did, but I do not see that you clarified that essential point. It is a simultaneity that looks completely impossible to me.

Hopefully, my explanations above have helped a little. As for simultaneity, let's see if I can exploit my u,v example more. In a plane, I can propose, as an analog of simultaneity: both on a line parallel to the cartesian x axis. Then the points (x,y)=(-1,1) and (x,y)=(1,1) are 'simultaneous'. However, in u,v coordinates, the horizontal line connecting them goes through v=-∞ and v=∞. But I should still be able to call them simultaneous.

Secondly, of course I considered the fact that O-S map t->∞ to τ->a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees. I did not make a plot of it, but I did not see an issue with that. In the O-S model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!), an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ₀. As I picture it (for I have not seen a description of it), for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. As a matter of fact, it was a similar discussion on the other blog that was the first thing that I read about this.

This is not what SC or O-S geometry predicts. They predict that an infaller will see the external universe going at a relatively normal rate, with no extreme red or blueshift. There will be optical distortions, analogous to Einstein rings. The infaller sees perfectly SR physics locally, until they hit the singularity. If you declare their world line to end at some arbitrary point, (e.g. the EH), there is no possible local physics explanation for it.

Sorry: I did not see any explanation for the inside region that made any sense to me, or that explains to me how it cannot contradict Schwarzschild's model. Perhaps some others who asked similar questions were convinced, but I did not see that happen (and of course, nobody needs to convince anyone; this is just a discussion of models). Perhaps there is another post that I overlooked?

Well, we have tried and tried.

 

[PeterDonis]

Different coverage means to me in the context of relativity, from the same reference system - from the same "perspective".

But some perspectives may simply not be able to cover all of spacetime; they may be limited in scope. Do you admit this possibility?

As I clarified earlier, a "region of spacetime" is for me merely a mathematical tool for calculations of, as Einstein put it, "clocks and rods".

That's not quite how I'm using the term. A "spacetime" is a geometric object, like the surface of the Earth. A "region of spacetime" is a portion of that geometric object, like the western hemisphere on the Earth. It's not a "mathematical tool"; it's a part of a mathematical model, true, but I'm trying to convey the fact that the mathematical model is of something "real" and physical.

t>∞ has as physical meaning a possible clock that indicates t>∞.

No, it does *not*. Either you haven't been reading carefully or I (and PAllen) haven't been explicit enough. We are *not* saying that you can assign a "t" coordinate greater than infinity to events behind the horizon. We are saying you can't assign a "t" coordinate *at all* to events behind the horizon. (Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer.) It's like saying you can't assign a real square root to a negative number; it simply can't be done.

This goes back to what I said above; you appear to be assuming that it must somehow be possible to assign a well-defined "t" value to every single event, everywhere in spacetime. You can't. That's just the way it is. If you want to describe events at or inside the horizon, you simply can't use the "t" that the distant observer uses. It just can't be done. If you can't admit or can't grok this possibility, then probably further discussion is useless unless/until you can. It's not easy, I agree; it took me quite some time to wrap my mind around it. But it's critical to understanding the standard classical GR model of black holes.

On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics.

That wasn't my intent, and I don't think it was the intent of PAllen. We are not trying to make philosophical points; we are trying to help you see the possibility of a kind of mathematical model that you hadn't seen before, and therefore of a kind of physical spacetime that you hadn't considered before. That model may or may not represent the actual spacetime of a black hole, because of the quantum issues that have been brought up many times in this and other threads. But it quite certainly does represent a *consistent* classical model of a black hole. That's what we're trying to help you see: that the model is consistent and represents something physically possible within the limits of classical theory.

First of all, I can't find anything that explains how you map a time τ to a time t>∞, as O-S suggest

You don't. See above. What you do is recognize that at the instant when an infalling observer crosses the horizon, his $\tau$ is *finite*, not infinite; therefore we can construct a *different* coordinate chart that maps *finite* values of some "time" coordinate T to the finite values of his $\tau$ that occur on his worldline after he has crossed the horizon, i.e., after the value $\tau_0$ that his clock reads at the instant he reaches the horizon. The simplest such chart is the Painleve chart, where the coordinate time T is simply equal to $\tau$. But there are others.

Those events inside the horizon, the ones with $\tau > \tau_0$, do *not* have well-defined "t" values at all, if "t" is the time coordinate of a distant observer. They simply can't be mapped in the distant observer's chart.

You must have considered this, and you suggested that you did, but I do not see that you clarified that essential point.

I've tried to clarify it more above; but I see from your next comment that one more thing needs to be clarified:

It is a simultaneity that looks completely impossible to me.

That's because it is. There is *no* simultaneity that both (1) assigns "t" coordinates to events outside the horizon in such a way that t goes to infinity as the horizon is approached, *and* (2) assigns well-defined "t" coordinates from the same set of surfaces of simultaneity to events inside the horizon. If you are willing to take another look at the Kruskal chart, I can try to explain why (though I think I already tried to in a previous post in this thread or one of the others that's running). But first I need to know if you can grok the possibility of such a thing at all; that seems to me to be a major stumbling block at this point.

Secondly, of course I considered the fact that O-S map t->∞ to τ->a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees.

If you agree with this, that's great. I wasn't sure, because if you realize this, it seems to me like a simple step to the reasoning I gave above (what you call a here, I called $\tau_0$ there). But of course that's just the way it seems to me; obviously it doesn't seem that way to you. But I think this is where attention needs to be focused.

In the O-S model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!),

Schwarzschild may indeed have thought that. He was using still another coordinate chart, one in which his radial coordinate "R" went to *zero* at the horizon. But that would take us way too far afield.

an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ₀.

That's not really what O-S said. A finite value of $\tau$ means a finite amount of time elapsed on the infalling observer's clock; there's no room there for his clock time to "slowly never reach a certain value". To the observer, if the infall time is 1 day (which was the order of magnitude of the value O-S calculated for the collapse of a sun-like star), he will experience 1 day, just like you will experience 1 day between now and this time tomorrow, and to him there will be nothing abnormal happening.

Sorry: I did not see any explanation for the inside region that made any sense to me, or that explains to me how it cannot contradict Schwarzschild's model. Perhaps some others who asked similar questions were convinced, but I did not see that happen (and of course, there is no use to try to convince anyone about which model is "right"; this is just a discussion of models). Perhaps there is another post that I overlooked?

I'm pretty sure you have read all the relevant posts; evidently they didn't make things click for you. I've given it another try above.

 

[PAllen]

No, it does *not*. Either you haven't been reading carefully or I (and PAllen) haven't been explicit enough. We are *not* saying that you can assign a "t" coordinate greater than infinity to events behind the horizon. We are saying you can't assign a "t" coordinate *at all* to events behind the horizon. (Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer.) It's like saying you can't assign a real square root to a negative number; it simply can't be done.

This goes back to what I said above; you appear to be assuming that it must somehow be possible to assign a well-defined "t" value to every single event, everywhere in spacetime. You can't. That's just the way it is. If you want to describe events at or inside the horizon, you simply can't use the "t" that the distant observer uses. It just can't be done. If you can't admit or can't grok this possibility, then probably further discussion is useless unless/until you can. It's not easy, I agree; it took me quite some time to wrap my mind around it. But it's critical to understanding the standard classical GR model of black holes.

Here I would like to express a slightly different interpretation. Given a specific rule for relating t for one observer to other events, you may not be able to assign a t coordinate at all to all events. Specifically, I don't particularly like this statement: "Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer." I don't agree this statement has well defined meaning. "Correspond" is just another word for simultaneity convention. If you insist simultaneity requires two way communication, this is true. However, I have proposed several simultaneity rules based on the one way causal connection from exterior to interior events, that, IMO assign a time to interior events corresponding to time for the distant observer. In effect, they simply delegate the correspondence between distant and interior events to the interior observer, who 'sees' the causal relation. This gets to the thrust of this thread as I conceived it:

If my wife gives birth to Judy and Jill, and Jill stays nearby and Judy goes to Africa, and I never hear from Judy again (unless I think Judy died), I have the expectation that there is simultaneity between events for Judy and for Jill. Their mutual causal connection to me gives me this expectation. Even more so if I believe Judy is getting my birthday cards (damn that she doesn't respond).

This concept can be formalized using the one of the procedures I outlined to say: I consider (though I can't verify it) that the singularity of that collapse formed at 3 pm today for me.

It almost seems you are saying there is a physically preferred chart for the distant observer. I don't accept this. I only accept that locally, there clear preference for Fermi-Normal coordinates; but globally? None. And the specific simultaneity I proposed a few times corresponds exactly, locally, to Fermi normal coordinates for an asymptotic observer - it diverges from this further away.

 

[PeterDonis]

Here I would like to express a slightly different interpretation. Given a specific rule for relating t for one observer to other events, you may not be able to assign a t coordinate at all to all events. Specifically, I don't particularly like this statement: "Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer." I don't agree this statement has well defined meaning.

You're correct, I should have specified that by "time for the distant observer" I meant the "natural" time coordinate he would choose, i.e., Schwarzschild coordinate time. I meant that time coordinate specifically because that's the one that seems to be causing all the trouble. I fully agree that other choices of time coordinate are possible that match the distant observer's proper time (at least to a good enough approximation) and also assign finite time values to events on and inside the horizon. Painleve time itself is one example; as r goes to infinity, Painleve time and Schwarzschild coordinate time get closer and closer to each other.

It almost seems you are saying there is a physically preferred chart for the distant observer.

There is in a weak sense: Schwarzschild coordinate time is the only time coordinate in the exterior region with both of the following properties:

(1) The integral curves of the time coordinate are also integral curves of the timelike Killing vector field;

(2) The surfaces of constant time are orthogonal to these integral curves.

Painleve time has property #1, but not #2. Kruskal "time" has neither.

I agree this is a weak sense of "preferred", but it's important to note that it is these two properties together that make Schwarzschild coordinate time seem so "natural"; it *seems* like the "natural" extension of Fermi Normal coordinates along the distant observer's worldline, because it *seems* like properties #1 and #2 are the "right" ones for a "natural" coordinate chart to have. Only when you start looking close to the horizon do you start running into problems with this "natural" extension.

And the specific simultaneity I proposed a few times corresponds exactly, locally, to Fermi normal coordinates for an asymptotic observer

I'm not sure I agree with "exactly" here; I think the only global time coordinate that can correspond "exactly" to Fermi normal coordinates (by which I mean surfaces of constant global time are also *exactly* the same as surfaces of constant local Fermi normal coordinate time) is Schwarzschild coordinate time. This is because of property #2 above, which must be satisfied by any global time coordinate whose simultaneity corresponds exactly, locally, to the simultaneity of Fermi normal coordinates on the distant observer's worldline. I don't think the simultaneity you proposed satisfies that property, for the same reasons that Painleve coordinate time doesn't: any surface of simultaneity that crosses the horizon can't be orthogonal to integral curves of the timelike Killing vector field. The simultaneity you proposed is *approximately* the same far away from the hole, but the correspondence is not exact for any finite value of r.

 

[PAllen]

I agree this is a weak sense of "preferred", but it's important to note that it is these two properties together that make Schwarzschild coordinate time seem so "natural"; it *seems* like the "natural" extension of Fermi Normal coordinates along the distant observer's worldline, because it *seems* like properties #1 and #2 are the "right" ones for a "natural" coordinate chart to have. Only when you start looking close to the horizon do you start running into problems with this "natural" extension.

But I only see Fermi-Normal as natural locally in GR. Even in SR, I see it as natural globally only for inertial observers (where it becomes Minkowski coordinates). For a non-inertial observer in SR, at a distance, I see many other simultaneity conventions that are physically based that avoid numerous seeming absurdities of Fermi-Normal carried too far. For example, Radar simultaneity has the feature of approaching Fermi-Normal locally, but has far more natural global properties for a highly non-inertial observer. While it is not universal, the idea that Fermi-Normal coordinates are only natural locally is a common one in GR.

I'm not sure I agree with "exactly" here; I think the only global time coordinate that can correspond "exactly" to Fermi normal coordinates (by which I mean surfaces of constant global time are also *exactly* the same as surfaces of constant local Fermi normal coordinate time) is Schwarzschild coordinate time. This is because of property #2 above, which must be satisfied by any global time coordinate whose simultaneity corresponds exactly, locally, to the simultaneity of Fermi normal coordinates on the distant observer's worldline. I don't think the simultaneity you proposed satisfies that property, for the same reasons that Painleve coordinate time doesn't: any surface of simultaneity that crosses the horizon can't be orthogonal to integral curves of the timelike Killing vector field. The simultaneity you proposed is *approximately* the same far away from the hole, but the correspondence is not exact for any finite value of r.

A more accurate description, I agree, would be exactly matches in the limit for an asymptotic observer. Concretely, there exists a sufficiently distant observer where my proposed simultaneity matches Fermi-Normal to one part in 10^50 for one light year (for example). Formally, the relation is more like Radar locally converging to Fermi-Normal for arbitrary non-inertial observers in SR.

 

[zonde]

This is false. You detect readily in SC coorinates that there is a hole in space time. You integrate proper time along an infall trajectory and find that proper time stops at a finite value (unlike for various other world lines). You ask, what stops the clock?

No, I ask what SC coordinates (+ SC type interior coordinates) this event has got. And it doesn't got any because infinity is not a coordinate. And event without coordinates does not exist (if coordinate system covers the whole space-time).

There is no local physics to stop the clock - tidal gravity may be very small; curvature tensor components are finite. The infinite coordinate time is not a physical quantity in GR. Einstein spoke of rulers and clocks, as Harrylin likes to point out. This clock stops for no conceivable local reason. If you add SC interior coordinates, and use limiting calculations, you smoothly extend this world line to the real singularity (with infinite curvature). All of this is exactly as if you chopped a disk around the pole from a sphere - you would find geodesics ending for no reason.

You mean that SC coordinates has hole because there is no interior coordinates? Well, we add SC type interior coordinates (with simultaneity defined using round-trip of signal at light speed), but this worldline has nothing much to do with these coordinates if it already extends toward infinite future in SC exterior coordinates.

There is no physical observable, anywhere in SR or GR, that depends on simultaneity convention at all. This is part of what Pervect was saying above. Belief that simultaneity convention has physical consequence reflects complete, total, misunderstanding of SR and GR.

Don't know what to make about "physical observables" but surely there are physical quantities that depend on simultaneity convention.
Not sure about GR but I am certain about my understanding of SR.

As for flat spacetime, the Rindler example Dr. Greg has posted beautiful pictures of, is relevant. The belief that there is no hole in SC exterior coordinates is 100% equivalent to the belief that most of the universe doesn't exist because a uniformly accelerating rocket can't see it.

Hmm, I believe Rindler coordinates do not extend to infinity in every direction.

And isn't Rindler coordinates (and horizon) more an analogue of eternal BH rather than collapsing mass (forming BH)?

And besides you have to take into account that rocket can't remain in state of uniform acceleration for indefinite time. And observer on that rocket would not observe rather static picture of other matter in the same state of uniform acceleration.
But as far as we know we can remain in a state of gravitational acceleration for indefinite time and we observe a lot of matter in the same state.

 

[PeterDonis]

But I only see Fermi-Normal as natural locally in GR. Even in SR, I see it as natural globally only for inertial observers (where it becomes Minkowski coordinates). For a non-inertial observer in SR, at a distance, I see many other simultaneity conventions that are physically based that avoid numerous seeming absurdities of Fermi-Normal carried too far. For example, Radar simultaneity has the feature of approaching Fermi-Normal locally, but has far more natural global properties for a highly non-inertial observer. While it is not universal, the idea that Fermi-Normal coordinates are only natural locally is a common one in GR.

I understand all this, and I agree with it. I'm talking about what I think is a typical thought process among people who come to post on PF asking questions about black holes; it seems to me that to these people, SC coordinates are "privileged", and I'm speculating as to the reasons why.

A more accurate description, I agree, would be exactly matches in the limit for an asymptotic observer.

Ok, good, we're in agreement.

 

[PAllen]

No, I ask what SC coordinates (+ SC type interior coordinates) this event has got. And it doesn't got any because infinity is not a coordinate. And event without coordinates does not exist (if coordinate system covers the whole space-time).

The path of an outer edge infall particle has finite proper time integrated to the SC radius. If you declare it stops there, you have a hole in spacetime. You have a geodesic ending with finite 'interval', where curvature is finite.

If you imagine the surface of such particles infalling, and you don't allow it to proceed to Sc radius, you have, geometrically, a hole: proper time on this surface is finite, area is finite, but if you stop it from continuing, it is geometrically a hole. Geometry is defined by invariants, not coordinate quantities. Consider the example I gave to harrylin several posts back of a horizontal geodesic in the plane in (u,v) coordinates. u coordinate goes to infinity on both sides of coordinate singularity, but it is still nothing but a geodesic in the flat plane.

Don't know what to make about "physical observables" but surely there are physical quantities that depend on simultaneity convention.
Not sure about GR but I am certain about my understanding of SR.

Nope. Einstein was very clear that simultaneity is purely a convention, not an observable. There is no observation or measurement in SR that changes if you use a different one than the standard one (but you have to change the metric as well; it is no longer eg. diag(+1,-1,-1,-1) if you use a funky convention.

Hmm, I believe Rindler coordinates do not extend to infinity in every direction.

so what? The point is that the trajectory of an object dropped from the rocket has coordinate time approaching infinity as it approaches, say, x=0. Proper time is finite. If you take these as the 'natural' coordinates for a rocket, what do you make of this? If you use two way signals for simultaneity, the event of the dropped object reaching x=0 never becomes simultaneous to an event for the rocket. So, should the rocket conclude the universe ends, or consider using a different simultaneity convention to look at the further history of the dropped object? This is analagous to the choice of using different simultaneity that allows analysis of events smoothly over a horizon.

And isn't Rindler coordinates (and horizon) more an analogue of eternal BH rather than collapsing mass (forming BH)?

Again, so what? You asked for flat space analog of issues under discussion: coordinate infinities and simultaneity conventions.

And besides you have to take into account that rocket can't remain in state of uniform acceleration for indefinite time. And observer on that rocket would not observe rather static picture of other matter in the same state of uniform acceleration.
But as far as we know we can remain in a state of gravitational acceleration for indefinite time and we observe a lot of matter in the same state.

I don't see that this is relevant.

 

[harrylin]

[..] I'm talking about what I think is a typical thought process among people who come to post on PF asking questions about black holes; it seems to me that to these people, SC coordinates are "privileged", and I'm speculating as to the reasons why. [..]

I did not yet see what you speculated, and it will be most useful to tell you my thinking about this without knowing what you thought about the thinking of me and others. Then we can compare it to your speculation.

So here's my thought process: Schwartzschild's solution is the one that I heard about in the literature, and it happens to be the one that I happened to stumble on in the first papers that I read about this topic, dating from 2007 and 1939. It is obviously a valid reference system according to GR, and it is "privileged" in the same sense as inertial frames and centre of mass systems are "privileged": it allows for the most simple mathematics, so simple that one doesn't need to be an expert to understand it. Thus it is a natural choice in a public discussion about predictions of GR. And the way I understand the first paper that I read about this, it's probably all I will ever need to understand this topic.

 

[harrylin]

I may not be able to fully catch up with this thread - I'm reading this at work while I should be doing something very different ... But here's a quick unrelated point:

[..]Schwarzschild may indeed have thought that. He was using still another coordinate chart, one in which his radial coordinate "R" went to *zero* at the horizon. But that would take us way too far afield.

Thanks again - I quickly went through his 1916 papers and got puzzled by them, just on that issue!