Notions of simultaneity in strongly curved spacetime

[PAllen]

While simultaneity conventions for inertial frames in flat spacetime (SR) are non-controversial, numerous questions, discussions, and debates in this forum indicate how confusing and controversial notions of simultaneity can be for more general cases. A couple of formal and true answers are generally unsatisfying to many:

- Simultaneity is undefinable, in any preferred way, in general. It is never observable or measurable anyway.

- You can pick any any event not in your past or future light cone to be a simultaneous event to your now. Except locally, there is no preference. (Sufficiently locally, one can argue for a preference for the Fermi-Normal simultaneity).

I thought of a possibly useful way to classify simultaneity notions for fairly general spacetimes and observers (I assume an orientable spacetime). Of critical importance is that any sensible implementation of these notions for inertial observers in flat spacetime produce the same result. However, they may differ wildly in curved spacetimes and/or for non-inertial observers. I assume, in what follows, that any observer can be considered past/future eternal unless their world line encounters a singularity.

1) It is reasonable to expect that any event in your causal past (on or inside your past light cone) is simultaneous to some event in your past.

2) It is reasonable to expect that any event in your causal future (on or inside your future light cone) is simultaneous to some event in your future. Executing (1) in various ways produces a foliation (family of simultaneity surfaces) that at least covers the union of all of your past light cones (all events that are ever in your causal past). I propose to call such foliations past inclusive if they at least cover your total causal past, but may cover more; and past only if they cover nothing except your total causal past.

Similarly, notion (2) leads to future inclusive and future only simultaneity conventions.

Finally, one may require that simultaneity be designed to cover any event in your total causal past or future. Call these causal inclusive and causal only

As mentioned, for inertial observers in flat spacetime, these are all the same, and the obvious implementation is Minkowski frames.

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).

A) Consider a distant, hovering, eternal, observer. Exterior SC type time slices represent an implementation of past-only simultaneity. No events on or inside the EH are covered. On the other hand, any future-only simultaneity implementation covers the interior, and indeed, is also a causal inclusive simultaneity. There are infinite such choices which can agree with local Fermi-Normal simultaneity.

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.

 

[arindamsinha]

Without claiming to have understood all of the above, especially the different theories/interpretations mentioned, I would like to add the question on how far we can analyze such things as 'simultaneity' based on existing GR theory when considering 'strongly curved spacetime'.

The reason for this thought stems from some statements of Einstein in his book 'The Meaning of Relativity' (6th Ed, 1955):
"In this connexion the following should be noted: The present theory of relativity is based on a division of physical reality into a metric field (gravitation) on the one hand, and into an electromagnetic field and matter on the other hand. In reality space will probably be of a uniform character and the present theory be valid only as a limiting case. For large densities of field and of matter, the field equations and even the field variables which enter into them will have no real significance. One may not therefore assume the validity of the equations for very high density of field and of matter, and one may not conclude that the ‘beginning of the expansion’ must mean a singularity in the mathematical sense. All we have to realize is that the equations may not be continued over such regions."

 

[PeterDonis]

For large densities of field and of matter, the field equations and even the field variables which enter into them will have no real significance. One may not therefore assume the validity of the equations for very high density of field and of matter

In general this is more or less the present understanding of GR; it is an "effective field theory" that is a low-energy approximation to some more fundamental theory.

However, none of that affects the predictions of GR about event horizons and black holes, at least not for BHs of sufficiently large mass (certainly any BH of stellar mass or more), because at the horizon of any such BH, and even far into its interior, there are no "large densities of field and of matter"; spacetime curvature for a hole of that size does not become large until you get close to the singularity at r = 0.

 

[arindamsinha]

In general this is more or less the present understanding of GR; it is an "effective field theory" that is a low-energy approximation to some more fundamental theory.

However, none of that affects the predictions of GR about event horizons and black holes, at least not for BHs of sufficiently large mass (certainly any BH of stellar mass or more), because at the horizon of any such BH, and even far into its interior, there are no "large densities of field and of matter"; spacetime curvature for a hole of that size does not become large until you get close to the singularity at r = 0.

Actually, this is a question which bugs me all the time. What do we consider as 'large' in terms of matter and field density? What is the cut-off point?

I realize there is no hard answer to this question, and we have to go with certain heuristics. My understanding has been that matter and field density near or within a stellar mass black hole can be considered large enough.

This may not apply to supermassive black holes, as the average matter density does tend to get closer to that of ordinary matter, but any black holes even a few order of magnitude smaller should qualify as having large matter and field densities?

 

[PAllen]

Actually, this is a question which bugs me all the time. What do we consider as 'large' in terms of matter and field density? What is the cut-off point?

I realize there is no hard answer to this question, and we have to go with certain heuristics. My understanding has been that matter and field density near or within a stellar mass black hole can be considered large enough.

This may not apply to supermassive black holes, as the average matter density does tend to get closer to that of ordinary matter, but any black holes even a few order of magnitude smaller should qualify as having large matter and field densities?

If you want to consider quantum theories or 'what really happens in our universe', those are very different questions from what classical GR predicts.

Because of the strength of evidence for supermassive collapsed object, any quantum + gravity theory must address the facts:

- during collapse, average matter density is not large at time of crossing EH
- curvature = tidal gravity is mild.

Be that is it may, in the context of beyond classical GR, the question is wide open. There are, for example, several approaches where a true horizon never forms, even for a super massive BH (fuzz balls approach from string theory is just one of half dozen such approaches). Any approach that preserves unitarity would seem (IMO) to require some relaxation of true horizon behavior.

 

[PeterDonis]

Actually, this is a question which bugs me all the time. What do we consider as 'large' in terms of matter and field density? What is the cut-off point?

As you say, there is no "hard" answer to the question, because we don't know for sure what more fundamental theory classical GR is the low energy limit of. However, the best current belief, AFAIK, is that "large" means "approaching a value of 1 in Planck units", since Planck units are the natural units of quantum gravity. In other words, curvature becomes "large" when the radius of curvature becomes small enough to be of the same order as the Planck length. This was the criterion I was using when I said that the curvature at the horizon, and even deep into the interior, of a BH of stellar mass or larger is not "large"--the radius of curvature is many, *many* orders of magnitude larger than the Planck length.

My understanding has been that matter and field density near or within a stellar mass black hole can be considered large enough.

"Field density" means radius of curvature; see above for why it's not "large" near or within a stellar mass BH. For matter density, the corresponding criterion would be the Planck density (one Planck mass per Planck length cubed). The density of collapsing matter in an idealized spherically symmetric collapse is far smaller than the Planck density until the matter has collapsed almost to r = 0 (i.e., it is not "large" at the horizon and well inside it).

This may not apply to supermassive black holes, as the average matter density does tend to get closer to that of ordinary matter, but any black holes even a few order of magnitude smaller should qualify as having large matter and field densities?

Not by the Planck criterion. By that criterion what we consider "ordinary matter" has a density of something like 10^-93. Even neutron star matter has a density of something like 10^-80 in Planck units. It takes a *lot* more than a few orders of magnitude to get from "ordinary" densities, or even neutron star densities, to "large" densities in Planck units.

 

[arindamsinha]

...the best current belief, AFAIK, is that "large" means "approaching a value of 1 in Planck units"...

OK, I wasn't aware of this. If that is the case, then I suppose nowhere in the Universe is matter and field density large, except very close to singularities within black holes.

I was going by references I have come across stating 'curvature of space is large near a big star' or a neutron star. In fact, thinking about it, that may not be quite the same thing as matter/field density (rather its rate of variation perhaps), or may be those are also relative statements.

 

[PeterDonis]

OK, I wasn't aware of this. If that is the case, then I suppose nowhere in the Universe is matter and field density large, except very close to singularities within black holes.

And very close to the Big Bang.

I was going by references I have come across stating 'curvature of space is large near a big star' or a neutron star. In fact, thinking about it, that may not be quite the same thing as matter/field density (rather its rate of variation perhaps), or may be those are also relative statements.

Those statements are using a different criterion for "large", basically comparing the matter/field density to that of "ordinary matter". Which criterion you use depends on what you want to use it for. If you want to determine at what point classical GR, as a low-energy effective field theory, starts breaking down (i.e., stops being a good approximation), the Planck unit criterion is the right one to use (at least, according to our best current understanding).

 

[arindamsinha]

And very close to the Big Bang.

Yes, there is that. Perhaps another case would be at velocities close to c? Not at all sure that is correct, just a random thought...

Those statements are using a different criterion for "large", basically comparing the matter/field density to that of "ordinary matter". Which criterion you use depends on what you want to use it for. If you want to determine at what point classical GR, as a low-energy effective field theory, starts breaking down (i.e., stops being a good approximation), the Planck unit criterion is the right one to use (at least, according to our best current understanding).

I can accept that as a good answer, to the best of our current knowledge as you mentioned.

 

[PeterDonis]

Perhaps another case would be at velocities close to c?

No. The criterion can't be frame-dependent, and "velocity close to c" is frame-dependent.

 

[arindamsinha]

No. The criterion can't be frame-dependent, and "velocity close to c" is frame-dependent.

OK. I thought it might be a wrong idea even when I posted it. The clarification helps.

 

[arindamsinha]

...Be that is it may, in the context of beyond classical GR, the question is wide open. There are, for example, several approaches where a true horizon never forms, even for a super massive BH (fuzz balls approach from string theory is just one of half dozen such approaches). Any approach that preserves unitarity would seem (IMO) to require some relaxation of true horizon behavior.

I somehow missed this post earlier. Yes, I can see that logic.

 

[harrylin]

[..]
I assume, in what follows, that any observer can be considered past/future eternal unless their world line encounters a singularity.

1) It is reasonable to expect that any event in your causal past (on or inside your past light cone) is simultaneous to some event in your past.

2) It is reasonable to expect that any event in your causal future (on or inside your future light cone) is simultaneous to some event in your future.Executing (1) in various ways produces a foliation (family of simultaneity surfaces) that at least covers the union of all of your past light cones (all events that are ever in your causal past). I propose to call such foliations past inclusive if they at least cover your total causal past, but may cover more; and past only if they cover nothing except your total causal past.

Similarly, notion (2) leads to future inclusive and future only simultaneity conventions.

Finally, one may require that simultaneity be designed to cover any event in your total causal past or future. Call these causal inclusive and causal only
[..]
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).

A) Consider a distant, hovering, eternal, observer. Exterior SC type time slices represent an implementation of past-only simultaneity. No events on or inside the EH are covered. On the other hand, any future-only simultaneity implementation covers the interior, and indeed, is also a causal inclusive simultaneity. There are infinite such choices which can agree with local Fermi-Normal simultaneity.

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.

That looks very interesting. Can you translate the above into normal English, with which I mean the kind of physicists English that Einstein and Feynman used? Then likely more people will understand what you mean and participate.

 

[PAllen]

That looks very interesting. Can you translate the above into normal English, with which I mean the kind of physicists English that Einstein and Feynman used? Then likely more people will understand what you mean and participate.

If you ask a specific question, maybe I can help. I put a lot of time into writing that up, and it is as clear and simple as I know how to make it without writing a 'book'. IMO, Einstein and Feynman would understand it perfectly and be able to discuss it.

Are you familiar with backward and forward going light cones, and their use to define causal structure of spacetime?

 

[harrylin]

If you ask a specific question, maybe I can help. I put a lot of time into writing that up, and it is as clear and simple as I know how to make it without writing a 'book'. IMO, Einstein and Feynman would understand it perfectly and be able to discuss it.

?? I suppose that they would be able to understand it and translate your mathematical English into plain English. Einstein would perhaps talk of rods and clocks, and Feynman would give colourful examples.

Are you familiar with backward and forward going light cones, and their use to define causal structure of spacetime?

Light cones, yes; their use to define "causal structure of spacetime", no. And I don't believe in "structure of spacetime" as a physical entity. Of course, I do believe in space-time events as physical occurrences.

 

[PAllen]

?? I suppose that they would be able to understand it and translate your mathematical English into plain English. Einstein would perhaps talk of rods and clocks, and Feynman would give colourful examples.

Light cones, yes; their use to define "causal structure of spacetime", no. And I don't believe in "structure of spacetime" as a physical entity. Of course, I do believe in space-time events as physical occurrences.

Though we have had little luck understanding each other, I will try one tack (it would be really helpful if you asked a specific question).

Do you think it is plausible to expect that if I compute that a physical detector somewhere in the history of the universe receives a signal from me, that I would want to assign a time coordinate to this predicted physical event?

Background: I can compute, purely using SC coordinates (exterior + interior, with limiting process over SC radius), that a physical detector with its own clock falling with (but above, in vacuum) a collapsing body will receive a signal from me at a specific finite reading on its own clock, when it has fallen through an event horizon to near the singularity. Is there some 'hand of god' that prohibits me from assigning a time coordinate to this predicted physical event, because I will never detect this event?

 

[zonde]

- Simultaneity is undefinable, in any preferred way, in general. It is never observable or measurable anyway.

- You can pick any any event not in your past or future light cone to be a simultaneous event to your now. Except locally, there is no preference. (Sufficiently locally, one can argue for a preference for the Fermi-Normal simultaneity).

So basically you are saying that simultaneity is a convention, right?
But it does not mean that you can use different conventions at the same time.

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).

A) Consider a distant, hovering, eternal, observer. Exterior SC type time slices represent an implementation of past-only simultaneity. No events on or inside the EH are covered. On the other hand, any future-only simultaneity implementation covers the interior, and indeed, is also a causal inclusive simultaneity. There are infinite such choices which can agree with local Fermi-Normal simultaneity.

With SC type time slicing there is no EH and no interior region for collapsing mass. All you can get is "frozen star". EH appears at infinite future i.e. never.

In order to have EH and interior region with SC type time slicing you have to have eternal BH.

 

[harrylin]

[..](it would be really helpful if you asked a specific question).

Regretfully it would require specific questions about nearly ALL of the cited text - and I really think that this is why there was little feedback on your first post. However, it seems that we won't need it, see next:

Do you think it is plausible to expect that if I compute that a physical detector somewhere in the history of the universe receives a signal from me, that I would want to assign a time coordinate to this predicted physical event?

In fact you are continuing the discussion that I started earlier about the theoretical possibility of assigning distant time, even putting physical clocks at distant places, thus making the discussion very concrete and physical. Evidently this is what we agree on.

However, that brings us immediately to the real sticking point that has all the time been lurking over the discussions of the last weeks:

Background: I can compute, purely using SC coordinates (exterior + interior, with limiting process over SC radius), that a physical detector with its own clock falling with (but above, in vacuum) a collapsing body will receive a signal from me at a specific finite reading on its own clock, when it has fallen through an event horizon to near the singularity. Is there some 'hand of god' that prohibits me from assigning a time coordinate to this predicted physical event, because I will never detect this event?

If you use a valid coordinate system, then there is nothing against it. The issue is about what kind of coordinate systems are valid in GR, and if perhaps contradictory mapping models can be made that match the mathematics of GR (but perhaps not all equally well matching the foundations), thus resulting in contradictory predictions.

We know that this can happen with Earth maps; however that is without consequence, as it's easily verified (I can give a simple example). It appears that the same problem occurred in GR, but without the possibility for a direct "reality check".

On a side note there is a somewhat similar case in SR, with tachyons. Are tachyons really SR? Must they exist if one can "fix" the math to contain their mathematical possibility?

 

[PAllen]

However, that brings us immediately to the real sticking point that has all the time been lurking over the discussions of the last weeks:

If you use a valid coordinate system, then there is nothing against it. The issue is about what kind of coordinate systems are valid in GR, and if perhaps contradictory mapping models can be made that match the mathematics of GR (but perhaps not all equally well matching the foundations), thus resulting in contradictory predictions.

We know that this can happen with Earth maps; however that is without consequence, as it's easily verified (I can give a simple example). It appears that the same problem occurred in GR, but without the possibility for a direct "reality check".

On a side note there is a somewhat similar case in SR, with tachyons. Are tachyons really SR? Must they exist if one can "fix" the math to contain their mathematical possibility?

For the case of Earth maps, do you claim there is case of conflicting prediction for maps as used in differential geometry:

- associated with each map is a metric expression, such that each map expressed the same geometry
- if one is talking about the same sphere using different maps, and one map doesn't cover all of the sphere, you use other maps to cover the rest, such that you are always describing the same complete sphere.

Who decides what is a valid coordinate system? Differential geometry has a well defined, precise, answer to this (see any definition of topological manifold, refined further to become a pseudo-riemannian manifold).

If you think this is wrong, then what is your precise criteria for a valid coordinate system? If it is different from above, you have a new theory, not GR as understood by everyone else. And in this new theory, general covariance is rejected, because that requires that any coordinates allowed by the criteria in the prior paragraph or good.

One possible analogy for our disagreement is:

- Imagine a 2-sphere in polar coordinates. Bob doesn't like what happens at or near the poles. So Bob decides to analyze only different object: a sphere missing a little disk around each pole. This is a valid, different geometric object. It is easy to demonstrate that you have holes using only polar coordinates with metric.

Now in the case of O-S collapse, the hole you are proposing 'must' be accepted as the correct prediction of GR is rather strange. A clock in the middle of collapsing dust ball stops for no reason. It stops in a strange sense - locally everything proceeds at a normal rate until it is declared to stop.

Note that for Krauss, et. all, assuming their quantum simulation is correct, they have good physical justification for this - this central clock is not acatually stopping; it evaporates in finite local time. Then it makes sense to talk about chopping a classical model at similar point.

If, instead, you accept the the interior clock proceeds normally, there is no escaping (using any coordinates), that the clock is proceeding for some time after an event horizon has formed around it. Any signals it sends will not escape, but it can readily receive signals from an external observer.

 

[PeterDonis]

With SC type time slicing there is no EH and no interior region for collapsing mass. All you can get is "frozen star". EH appears at infinite future i.e. never.

No, this is not correct. The correct statement is: SC type time slicing cannot *cover* the EH and interior region.

In order to have EH and interior region with SC type time slicing you have to have eternal BH.

This is not correct either. SC type time slicing cannot cover the EH and interior region for *any* black hole spacetime; it only covers the exterior. But in both cases you cite (collapsing mass and eternal BH), the EH and interior region are part of the spacetime; they are just not covered by the SC type time slicing.

 

[PAllen]

So basically you are saying that simultaneity is a convention, right?
But it does not mean that you can use different conventions at the same time.

Who says? It is no different than saying: I have two problems in analytic plane geometry. One is easier to compute in cartesian coordinates, one in polar coordinates. So I do one calculation one way, the other a different way.

With SC type time slicing there is no EH and no interior region for collapsing mass. All you can get is "frozen star". EH appears at infinite future i.e. never.

In order to have EH and interior region with SC type time slicing you have to have eternal BH.

This is just false. For an O-S collapse, there is always an interior because it is a collapsing ball of dust. If you restrict yourself to what a distant observer sees, what they see is a ball that freezes throughout, at a radius just larger than the SC radius.

Outside the frozen ball, you can apply SC coordinates and metric (or others). Inside (where matter is), you must do something else.

Now, SC time slicing is a specific choice of simultaneity (of which there are infinite choices, including simple, physically defined ones: https://www.physicsforums.com/showpost.php?p=4165220&postcount=23 ). If I pick a different choice, and use it throughout, then the external observer assigns well defined time coordinates to the dust ball collapsing inside the horizon, and even a well defined time to the formation of a singularity.

 

[PeterDonis]

The issue is about what kind of coordinate systems are valid in GR, and if perhaps contradictory mapping models can be made that match the mathematics of GR (but perhaps not all equally well matching the foundations), thus resulting in contradictory predictions.

I don't think this can happen, because any valid coordinate system in GR has to preserve geometric invariants, and all of the physical predictions of GR depend only on geometric invariants. So any valid coordinate system in GR must lead to the same physical predictions as any other valid coordinate system.

We know that this can happen with Earth maps; however that is without consequence, as it's easily verified (I can give a simple example).

Please do; I don't understand what you're referring to here. See my comments above about what a "valid" coordinate system is. Any valid map of the Earth would also have to preserve geometric invariants; that is, you would have to be able to calculate, say, the correct great-circle distance between New York and Sydney using any valid map (though the calculation might be easier in some maps than in others). Note that this is *not* the same as how the distance "looks" on the map: the NY-Sydney great circle looks very different on a Mercator projection than it does on a stereographic projection, but both allow you to calculate that the physical distance is the same; it's just represented differently in terms of the coordinates.

On a side note there is a somewhat similar case in SR, with tachyons. Are tachyons really SR? Must they exist if one can "fix" the math to contain their mathematical possibility?

I'm not aware of any requirement that everything mathematically possible according to any theory must exist. The maximally extended Schwarzschild spacetime, including a white hole and a second exterior region, is mathematically possible in GR, but nobody, AFAIK, thinks it's physically possible.

 

[zonde]

Who says? It is no different than saying: I have two problems in analytic plane geometry. One is easier to compute in cartesian coordinates, one in polar coordinates. So I do one calculation one way, the other a different way.

Hmm, I meant it differently. You can't use different conventions within single calculation/reasoning. Because there can be convention dependent statements.

This is just false. For an O-S collapse, there is always an interior because it is a collapsing ball of dust. If you restrict yourself to what a distant observer sees, what they see is a ball that freezes throughout, at a radius just larger than the SC radius.

Outside the frozen ball, you can apply SC coordinates and metric (or others). Inside (where matter is), you must do something else.

With "inside" I mean "inside EH" not "inside gravitating mass".
And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates.

 

[PAllen]

With "inside" I mean "inside EH" not "inside gravitating mass".
And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates.

OK, if you assume a transparent pressure less dust, and do this radially from infinity, you get this SC time slicing outside and something similar inside. The whole point of this thread is that neither SR nor GR say this is the only allowed way to slice spacetime. And it is provable that you have hole in spacetime (world lines of particles that just end for no reason, at finite proper time along them), or a region not covered by these coordinates.

The point of this thread, is that you can choose many other intuitive simultaneity conventions for a distant observe (and simultaneity convention identically equals time sicing), which don't have a hole, and assign finite time coordinates to events inside the EH, which itself labeled with a finite time coordinate.

 

[PeterDonis]

And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates.

You can do this, as PAllen said, but it's important to note that if you do, this time slicing still won't cover the horizon or the region of spacetime inside it. That includes the portion of the collapsing matter that is inside the horizon.

 

[pervect]

I've got several comments:

On the map issue. There ARE coordinates that are mildly "special" at any given point. These coordinates are the ones where your map is drawn to scale. The mathematical feature of such maps is that the metric is diag(-c, 1, 1, 1).

Because these maps are to scale, one can freely interchange coordinate distances (i.e. changes in coordinate) with physical distances. They're easy to work with. Discussions don't go off in as many strange tangents when you use maps that are to scale, and people don't get "lost" so much, I find.

Some recent discussion reads to me something like this:

"Your'e reading the map wrong - here , use this one, It's almost to scale - well, not really to scale, actually, but the distortions are at least finiite."

"No - I like the one that's infinitely distorted better, because - nevermind why, I just like it better."

People for some reason seem to have a really hard time dealing with distances in GR but if done correctly it's not that complex - just make sure your map is to scale (and if its not, learn enough math to covert your maps so they are to scale, this can reasonably be motivated with as little math as algebra).

The only remaining issue with maps is the SR issue. This is understanding that time is not absolute, that simultaneity is relative, that as you change your notion of simultaneity your notion of distance also changes correspondingly because space-time is a continuum, and space and time are fundamentally linked.

Mathematically: the maps of SR and GR preserve the Lorentz interval.

This is where I keep feeling the communication is lacking - but many people who say they "get" this point obviously don't :-(.

The notion of "now" is relative. The local notion of now is determined by backwards compatibility with Newton's laws, but said notion of now (necessary for this backwards compatibility) is purely local, and not universal. This relativity means that this, or any other , notion of "now" is not used to determine cause and effect, but rather one uses light cones. To put it succinctly, "now" is relative, light cones are absolute.

The last issue. I think there are some people who believe in "white holes", but not in the context of classical GR. See the thread on Nikodem Poplawski I started

 

[harrylin]

For the case of Earth maps, do you claim there is case of conflicting prediction for maps as used in differential geometry:

- associated with each map is a metric expression, such that each map expressed the same geometry
- if one is talking about the same sphere using different maps, and one map doesn't cover all of the sphere, you use other maps to cover the rest, such that you are always describing the same complete sphere.[..]

No, I meant nothing like that. Instead my illustration is about different mapping systems, which are related by conformal transformations. It's a very simple example that I thought of when I awoke one morning at the time that we were discussing Hamilton's model, which as we all agreed, in the way he literally pictures it isn't exactly GR. My example is a bit silly, please don't laugh - but you may smile.

There is a flatland country near the equator, and sailors are setting out on voyages to distant destinations. Now comes a cartographer, who decides to make a map that can serve as a travel guide. Based on intuition (inspiration?) he then develops a map of the world with the strange property that it has medians that converge when going away from the equator towards the North. As a result everything ends in a singularity, which he gives the name North Pole. Thus it appears that one cannot get further away than the North pole, which is intellectually unsatisfactory.

Then comes along a different cartographer who thinks up a conformal transformation that results in a very similar map, but now with the medians running parallel. The funny thing of that map is that the singularity is gone; on that map people can continue beyond the North pole. Of course, the two maps can be transformed from one into the other without problems up to that singularity, but predictions definitely differ from that point onward.

Now they have a problem; at best one of the two will match reality. Either Flatlander Earth physics law can tell them which one to choose as the most correct one, or Earth science is incomplete, so that the flatlanders don't know yet which mapping would best, even just in theory.

 

[harrylin]

I don't think this can happen, because any valid coordinate system in GR has to preserve geometric invariants, and all of the physical predictions of GR depend only on geometric invariants. So any valid coordinate system in GR must lead to the same physical predictions as any other valid coordinate system.

I agree with you about valid coordinate systems. However, it appears that the disagreements in the physics community that I noticed relate to the issue of what is valid. If so, then this is a "hot potato".

Note: regretfully I can merely choose like a voter about a political issue: Yes, I do have an opinion, and No, I probably don't have enough GR expertise to make a case. So, please regard me as a science reporter who meddles in expert discussions and asks annoying questions.

[..] I'm not aware of any requirement that everything mathematically possible according to any theory must exist. The maximally extended Schwarzschild spacetime, including a white hole and a second exterior region, is mathematically possible in GR, but nobody, AFAIK, thinks it's physically possible.

Yes, that is what I think too. I got the impression that PAllen was heading for the contrary opinion, although within certain limits.

 

[PAllen]

Yes, that is what I think too. I got the impression that PAllen was heading for the contrary opinion, although within certain limits.

Not at all. I believe (physically intuit?) that tachyons are not likely to exist in our universe; nor white holes; nor closed time like curves; nor super-extremal kerr black holes; nor alcubierre drive; nor actual singularities. I also admit that all of these have mathematically consistent treatment.

But in each of these cases, I see a clear reason to 'draw a line'. Tachyons have to be added to SR in violation of the causal interpretation of SR. White holes have no process by which they can form. Similarly, for most of the others, there is no known process they can form out of plausible initial conditions. Singularities I interpret as a clear sign that GR has broken in this domain.

With black holes, treated classically, we have, instead, the singularity theorems: With almost any reasonable starting point, once collapse has gotten close to a critical radius, it must proceed all the way to a singularity. Further, all the approaches, classically, to try to avoid an event horizon formation amount to my example chopping the poles off a sphere because I don't like what they do to my coordinate preference.

 

[PeterDonis]

I agree with you about valid coordinate systems. However, it appears that the disagreements in the physics community that I noticed relate to the issue of what is valid.

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.

For example, the black hole information loss problem basically comes down to: which do you want to keep when push comes to shove, general relativity or quantum mechanics? GR predicts that gravitational collapse leads to the formation of regions of spacetime from which information can't get back out; that means a quantum state that falls in gets lost, so unitarity, a central principle of QM, is violated. Hawking's position (at least until about 2004-ish; he seems to have switched sides in what Susskind calls the "Black Hole War" about then) was, so much the worse for QM. Susskind's and t'Hooft's position (there were others, but they seem to have been the primary ones who held to this position throughout) was, so much the worse for GR. They can't both be right. But nobody, as far as I can see, was arguing one way or the other based on which coordinates were valid and which weren't.

Note: regretfully I can merely choose like a voter about a political issue: Yes, I do have an opinion, and No, I probably don't have enough GR expertise to make a case. So, please regard me as a science reporter who meddles in expert discussions and asks annoying questions.

Fair enough. You ask better questions than a lot of science reporters do.

 

[zonde]

OK, if you assume a transparent pressure less dust, and do this radially from infinity, you get this SC time slicing outside and something similar inside. The whole point of this thread is that neither SR nor GR say this is the only allowed way to slice spacetime. And it is provable that you have hole in spacetime (world lines of particles that just end for no reason, at finite proper time along them), or a region not covered by these coordinates.

There is no hole in spacetime with SC type time slicing for collapsing mass. EH forms at infinite future i.e. never. So there is no hole in spacetime.

And wordlines of particles do not end. They just extend toward infinite future.
You would not claim that there is some problem with this expression, right?
\lim_{t\to\infty}f(t)=a
Let's say that t is coordinate time and f(t) is proper time of infalling particle.

The point of this thread, is that you can choose many other intuitive simultaneity conventions for a distant observe (and simultaneity convention identically equals time sicing), which don't have a hole, and assign finite time coordinates to events inside the EH, which itself labeled with a finite time coordinate.

You can choose different simultaneity conventions but then we need good understanding of the things that depend on simultaneity convention as they would change along with it.

And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime.

 

[PAllen]

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. This one is pressure-less, and only the final stages (where the matter is almost light like) is shown (the r=0 line and the shell line need to slanted closer to vertical to consider a collapse from rest not too far from the SC radius):

http://casa.colorado.edu/~ajsh/collapse.html

This paper discusses collapse of a null dust shell, including Kruskal coordinates (section 4):

http://arxiv.org/abs/gr-qc/0502040

While this is not quite the case of interest, the techniques shown for matching interior and exterior are general.

I think the interesting case to consider is a clock sitting inertially at the center of a collapsing shell, under the further assumption that the shell is transparent. There are a number of interesting features:

- The spacetime inside the shell is pure Minkowski flat spacetime; there are no tidal forces at all. There is no matter density at all. In the whole region inside the shell, SR physics applies. The analog of this is obviously true for Newtonian physics; it is also well known that this is exactly true for GR as well (that inside a spherical shell, physics is indistinguishable from SR with no gravity).

- The clock inside operates according to SR physics, time locally flowing normally, until shell collapse reaches singularity. The clock at the center cannot detect the shell passing the event horizon. Nothing about physics within the shell changes on approach to or passing the event horizion - the interior is pure SR until the singularity.

- The clock at the center sees the external 'universe' proceeding normally except highly (but finitely) blue shifted until the moment of singularity. Here, the clock world line ends for a very physical reason: the shell has collapsed to a singularity, bringing the clock with it. One of the things I wanted to get right here was the blueshift relation. Unlike an O-S collapse, where an infaller (with right trajectory) can have very mild or even no blue shift relative to the outside universe, the boundary matching conditions for a shell collapse require that the whole interior (for observers stationary with respect to shell center) experiences (in the limit of zero thickness shell) the same blueshift as a clock riding exactly on the shell.

- A distant observer, of course, never sees the shell reach the event horizon. Similarly, they see the inside clock stop at a time before the clock hits the singularity.

- Later free fall clocks will be seen, from a distance, to stop on reaching the event horizon. Such a clock, itself, will experience no such thing, and its time will progress further until it hits the singularity (the shell is 'long gone' into the singularity).

Classically, one may ask by what possible rationale should one declare that a clock operating according to pure SR physics, be declared to stop for no local physical reason? What does what a distant observer sees have anything to do with their local, pure SR physics?

All of the above is indisputable for classical GR. The Krauss paper, by choice, only covered what the outside observer sees, for the classical case, because of what they discovered about the quantum case. The quantum analysis (if true) showed that, a lot of the above does not happen in our universe. It shows, instead, that (no matter what coordinates are used), the interior clock actually evaporates into not quite thermal radiation well before its classical end. The Kruskal diagram changes in significant ways as a result. Only with this justification, does it make any sense to 'chop' the classical analysis based on Schwarzschild time coordinate.

Note, that in the classical analysis without the quantum chop, perfectly reasonable, physically based simultaneity conventions, establish simultaneity between the interior clock and exterior clocks well after the shell has crossed the event horizon. One simple example is:

https://www.physicsforums.com/showpost.php?p=4165220&postcount=23

 

[PeterDonis]

There is no hole in spacetime with SC type time slicing for collapsing mass. EH forms at infinite future i.e. never. So there is no hole in spacetime.

You think that this is a logical proof, but it isn't. What you have actually proven is only that there is no hole in the region of spacetime covered by the SC type time slicing. You have *not* proven that that region of spacetime is the entire spacetime, and in fact it's easy to prove that it's not. See below.

And wordlines of particles do not end. They just extend toward infinite future.

No, they don't, at least not the way you mean. 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.

You would not claim that there is some problem with this expression, right?
\lim_{t\to\infty}f(t)=a
Let's say that t is coordinate time and f(t) is proper time of infalling particle.

This isn't quite the right expression. The right expression is:

\lim_{t\to\infty} \int_{t_0}^t dt' f(t')

where t_0 is the time by the distant observer's clock that the infalling object starts falling. Substitute the correct function f(t'), for the proper time of the infalling object, do the integral, and take the limit. You will find that it gives a finite answer. This shows that "extending the worldline to infinity" by the distant observer's clock only extends it by a finite length. The SC time coordinate is so distorted at the horizon that it makes finite lengths look like infinite lengths.

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.

You can choose different simultaneity conventions but then we need good understanding of the things that depend on simultaneity convention as they would change along with it.

And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime.

This is a good idea; do you have any suggestions? I would suggest comparing the description of flat spacetime in Rindler coordinates to its description in Minkowski coordinates.

 

[PAllen]

There is no hole in spacetime with SC type time slicing for collapsing mass. EH forms at infinite future i.e. never. So there is no hole in spacetime.

And wordlines of particles do not end. They just extend toward infinite future.
You would not claim that there is some problem with this expression, right?
\lim_{t\to\infty}f(t)=a
Let's say that t is coordinate time and f(t) is proper time of infalling particle.

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? 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 can choose different simultaneity conventions but then we need good understanding of the things that depend on simultaneity convention as they would change along with it.

And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime.

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.

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.

 

[zonde]

Mathematically: the maps of SR and GR preserve the Lorentz interval.

This is where I keep feeling the communication is lacking - but many people who say they "get" this point obviously don't :-(.

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?

 

[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?