Black Holes as 2 Dimensional Objects

  • Thread starter Vastin
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Sorry, new here, and my actual mathematical training is v.limited, so I have to restrict myself mostly to thought experiments, alas. Anyway...

Whenever I study up on black holes, it doesn't take very long before the text or discussion quickly devolves into how impossibly abberant singularities are, and all the mind-bending issues with information loss and time reversal that they raise.

It seems to me (naively perhaps), that black holes would be far more simply represented as purely two dimensional masses with NO interior whatsoever?

Given that the Schwarschild radius is directly proportional to mass, shouldn't we consider whether it IS the mass of the object, accreted onto a 2 dimensional surface of the 'maximum' possible density - as opposed to a 0 dimensional object of ∞ density?

No singularity, no time reversal, because nothing ever falls INTO a black hole - it falls ONTO a black sphere?

Discussed before? Discarded due to mathematical or logical infeasibility? I'm curious. This one has been bugging me for over a year.
 
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It wouldn't agree with the model for Black Holes given by General Relativity, and it still wouldn't agree with some observations of stars orbiting them.
 
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Ok - but why? What are the specific problems this model introduces?
 
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General Relativity predicts the existence of Black Holes as three-dimensional spheres, not two-dimensional circles.

Also, this model wouldn't predict that we'd observe identical orbits for stars with orbits in planes at different angles with respect to the plane of the Black Hole.
 
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Ah, no. You've missed my premise a bit.

I'm not suggesting that black holes are FLAT. That would be very odd indeed. ;)

I'm suggesting that they are two dimensional - ie, a sphere with only one side (the outside) and no volume.

Two dimensional objects have no volume, but they are not required to be flat, they may have curvature so long as they are examined from a three dimensional perspective.
 
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In the model I'm suggesting, the familiar (simplified 2D) space-time graph of a black hole would look slightly different. Rather than the throat of the graph proceeding down to presumed infinity, it would stop with an open ring where the event horizon is drawn.

If one were to watch the graph form during a supernova event, one would observe the star's central mass compressing and stretching the graph downwards to that critical point, at which point the ring would open, rather than stretching indefinitely.

This ring (Domain Wall? Is that the correct term?) would then expand as further mass was compressed into it - but the distortion in space-time would not be infinite. It would instead have a distinct boundary at the event horizon beyond which nothing would pass whatsoever - there being no-where for it to pass to.

Again, I'm not suggesting that it would actually be a flat disc or ring - the phenomena would in fact appear in 3D space as a sphere, but with the critical distinction of it not having an interior. Its mass would be entirely accreted onto the surface of the sphere, forming the domain wall.
 

fzero

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Such a model has in fact been studied and is known as the "membrane paradigm." http://en.wikipedia.org/wiki/Membrane_paradigm However, in this picture, there's nothing to suggest that the nonsingular part of the spacetime inside the horizon is unphysical. It just that an outside observer cannot make measurements inside the horizon, so there must be a way to discuss black hole physics that is independent of the description of the interior. This is related to the notion of "black hole complementarity" http://en.wikipedia.org/wiki/Black_hole_complementarity. An infalling observer would use the interior geometry to describe physics without any problems.
 
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Hmm. Given that it would take me an infinite amount of time (from your point of view) to pass through the event horizon, I'm not sure that I see the difference?

Shouldn't matter accrete to an infinitesimally small (plank scale?) distance from the event horizon, and never succeed in crossing over?

Also given the sort of space warping effects we see from frame dragging, what is the theoretical problem with positing a non-space (a true non-spatial void) 'within' the event horizon?

On another related point, saying that the in-falling person would see 'nothing unusual' as they passed through the event horizon is a bit misleading. They would presumably see the near-instantaneous heat-death of the universe around them, as well as the evaporation of the black hole they were trying to fall into?

In short, I don't see how it would be possible for a singularity to form in the first place. I've always been a bit leary about the whole 'infinite time dilation yet I manage to get in' thing... :confused:
 
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Indeed, the more I think about it, the more likely it seems that if I try to fling myself into a black hole the object will quite literally evaporate before I could 'hit' it, shrinking faster than I can fall (from my point of view).

From your point of view I got sucked into the accretion layer and have been sitting there slowly evaporating for the last several trillion years.
 

fzero

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Hmm. Given that it would take me an infinite amount of time (from your point of view) to pass through the event horizon, I'm not sure that I see the difference?

Shouldn't matter accrete to an infinitesimally small (plank scale?) distance from the event horizon, and never succeed in crossing over?
That is the way that the outside observer perceives what has happened. The infalling observer measures time normally (for him) and perceives that he crosses the horizon.

Also given the sort of space warping effects we see from frame dragging, what is the theoretical problem with positing a non-space (a true non-spatial void) 'within' the event horizon?
It doesn't agree with what an infalling observer would find. Black hole complementarity suggests that there is a description from the point of view of an outside observer where all of the physics of the interior is captured by degrees of freedom on the horizon, but it doesn't invalidate the existence of the interior for the infalling observer. It is saying that there are two equally valid ways of discussing the black hole.

On another related point, saying that the in-falling person would see 'nothing unusual' as they passed through the event horizon is a bit misleading. They would presumably see the near-instantaneous heat-death of the universe around them, as well as the evaporation of the black hole they were trying to fall into?
The infalling observer would continue to see information from the universe around them, as that falls into the black hole with them. If the infalling observer were to try to communicate with an observer outside the horizon, they'd hit the singularity before they realized that they weren't going to get an answer back.

The Hawking radiation isn't a clue about the horizon, since an accelerating observer should see a thermal spectrum of radiation anyway, due to the Unruh effect http://en.wikipedia.org/wiki/Unruh_effect. There's no way for the infalling observer to distinguish between the two. Depending on how the information paradox is resolved, it might be possible for an outside observer to correlate information in Hawking radiation with information that went into the black hole, but that's not part of the issue here.

I have been ignoring tidal forces, which would eventually destroy the observer before they reached the singularity. For a large enough black hole, the tidal forces could be small enough to survive passing the horizon.

In short, I don't see how it would be possible for a singularity to form in the first place. I've always been a bit leary about the whole 'infinite time dilation yet I manage to get in' thing... :confused:
This is a confusion about how time works in relativity, not specifically about black holes.

Indeed, the more I think about it, the more likely it seems that if I try to fling myself into a black hole the object will quite literally evaporate before I could 'hit' it, shrinking faster than I can fall (from my point of view).

From your point of view I got sucked into the accretion layer and have been sitting there slowly evaporating for the last several trillion years.
Again, this is a confusion between how the outside and infalling observers perceive time. For a large enough black hole, the infalling observer can reach the singularity before the black hole completely evaporates.
 
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Yes, lets assume our observer is an imaginary immutable particle so that tidal forces aren't getting in the way.

I'm not overly concerned with the specific properties of hawking radiation in the scope of this discussion - save that it exists and posits a finite lifetime for our black hole through evaporation. That bit is important.


It doesn't agree with what an infalling observer would find. Black hole complementarity suggests that there is a description from the point of view of an outside observer where all of the physics of the interior is captured by degrees of freedom on the horizon, but it doesn't invalidate the existence of the interior for the infalling observer. It is saying that there are two equally valid ways of discussing the black hole.
The interior space time issue is the one I'm least familiar with, and I'm not entirely certain it is directly relevant to this concept. Whether there is a space-time there to traverse, or not, probably doesn't matter. Lets assume that there is.

That is the way that the outside observer perceives what has happened. The infalling observer measures time normally (for him) and perceives that he crosses the horizon.

...

The infalling observer would continue to see information from the universe around them, as that falls into the black hole with them. If the infalling observer were to try to communicate with an observer outside the horizon, they'd hit the singularity before they realized that they weren't going to get an answer back.

Again, this is a confusion between how the outside and infalling observers perceive time. For a large enough black hole, the infalling observer can reach the singularity before the black hole completely evaporates.
Ok, here is where I become very confused with relativity. If I fly to Alpha Centauri at .99c and back, you watch as ~eight years pass while I am gone. (lets ignore acceleration)

I reach the star and return, but due to time dilation my perception of the trip is that it took a relatively shorter period of time - for the sake of argument, lets say 1 year.

But now I'm back and in the same frame as you again. The only way my impression of the trip can be reconciled with yours, is if my perception of the rest of the universe is sped up considerably over the course of the trip. If I could have watched you the whole time, you would have appeared to age eight years in the space of my one year journey - and when I get back, you have.

Now I'm a particle falling towards a black hole, as I approach the event horizon, I am approaching an essentially asymptotic distortion of space time. Exactly 'at' the event horizon, my acceleration would become infinite and I would theoretically reach (exceed?) the speed of light. My time dilation should likewise be rapidly approaching infinity.

At that moment, my perception of the rest of the universe should speed up to near infinite levels - I say near infinite because the black hole should evaporate before I truly reach it (along with the rest of the universe). Like Zeno's Arrow, I never quite reach the wall.

It seems to me that the time dilation effect should be exactly what is preventing an anomaly like a singularity - or a truly infinite gravity well - from forming.
 
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Argh. I feel like I really have no grasp of relativity no matter how many times I read up on it. Is there an actual distortion of time, or just the perception of distortion. It never seems clear.

Reading in detail now, it does appear to be a real distortion, in which case I still fail to see how you could ever crest an asymptotic gravity well.
 
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fzero

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Now I'm a particle falling towards a black hole, as I approach the event horizon, I am approaching an essentially asymptotic distortion of space time. Exactly 'at' the event horizon, my acceleration would become infinite and I would theoretically reach (exceed?) the speed of light. My time dilation should likewise be rapidly approaching infinity.

At that moment, my perception of the rest of the universe should speed up to near infinite levels - I say near infinite because the black hole should evaporate before I truly reach it (along with the rest of the universe). Like Zeno's Arrow, I never quite reach the wall.

It seems to me that the time dilation effect should be exactly what is preventing an anomaly like a singularity - or a truly infinite gravity well - from forming.
The gravitational acceleration for a free-falling observer does not diverge at the horizon. What actually diverges is the proper acceleration of a stationary observer that is approaching the horizon. By stationary, imagine an astronaut at the end of a cable that is being lowered toward the horizon from a spacecraft that is orbiting the black hole. The proper acceleration being measured is the tension in this cable. That the proper acceleration diverges at the horizon is completely consistent with the trapping of objects that pass beyond the horizon.

In contrast, the proper acceleration of a free-falling observer is zero. Free-fall is inertial motion.

Argh. I feel like I really have no grasp of relativity no matter how many times I read up on it. Is there an ACTUAL distortion of time, or just the perception of distortion. It never seems clear.

If a ship takes off from Alpha Centauri and flies towards earth at .9c, from an observer ON earth, I would expect it to appear as if it were approaching at ~10x the speed of light, even though its journey actually took much longer than that, because the photons should arrive at earth in a highly compressed front just a short way in front of the ship itself.
No, if we used radar to track the ship, we'd measure .9c. Remember that there are, in principle, photons which are arriving after being emitted from the ship at arbitrarily large distances from Earth. These photons arrive an arbitrarily long period of time before the ship does.

Likewise, on the ship I would expect time on earth to appear as if it were progressing at x10 normal, while time on alpha centauri would appear to be progressing at 1/10th normal.

In reality, time is progressing perfectly normally for everyone, it's all just perception based on the relative speed of photons we're interacting with.

Is THAT correct? Because that is a lot less unusual. That's just what I'd expect in a purely classical sense.
Time dilation is real and not just perception based. By any possible measurement, clocks do run slower in frames that are moving at high speeds compared to a stationary reference frame. As an example, highly relativistic unstable particles like the muon have a much longer range of travel before decay than they would if there was no time dilation. See http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relrange.html and http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html for some illustration.
 
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Ok good. The second example I gave was not how I thought relativity worked, so I'm glad to hear your confirm that. It is a real distortion.

The gravitational acceleration for a free-falling observer does not diverge at the horizon. What actually diverges is the proper acceleration of a stationary observer that is approaching the horizon. By stationary, imagine an astronaut at the end of a cable that is being lowered toward the horizon from a spacecraft that is orbiting the black hole. The proper acceleration being measured is the tension in this cable. That the proper acceleration diverges at the horizon is completely consistent with the trapping of objects that pass beyond the horizon.

In contrast, the proper acceleration of a free-falling observer is zero. Free-fall is inertial motion.
Ok, this I don't get. I mean, I understand free fall as inertial motion - but I don't understand why acceleration would be considered any differently for the free falling object vs. the tethered/stationary one.

As I approach any mass, the gravitational acceleration that mass is exerting upon me increases - I didn't think my current vector had any bearing on that whatsoever, stationary or otherwise... :frown:

Ok, let me frame my question differently.

Given my (possibly incorrect) assumptions:

A) Time is dilating as an object approaches a black hole due to the curvature of space time.

B) This dilation effect is in some fashion proportional to the space time curvature.

C) The curvature of space time is behaving asymptotically as we approach the event horizon.

Under those circumstances, what is preventing the time dilation effect from likewise behaving asymptotically and causing the universe to age out of existence (from my perspective) before I reach the event horizon?

If I were in a starship somehow undergoing asymptotic acceleration, this is what I expect would happen mathematically, given the examples I have read of how the effect works from the perspective of our intergalactic speedster. I would fly towards the far end of the universe, 13 billion light years distant, and seem to reach it in seconds - but the stars would have gone out by the time I got there.
 
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fzero

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Ok, this I don't get. I mean, I understand free fall as inertial motion - but I don't understand why acceleration would be considered any differently for the free falling object vs. the tethered/stationary one.

As I approach any mass, the gravitational acceleration that mass is exerting upon me increases - I didn't think my current vector had any bearing on that whatsoever, stationary or otherwise... :frown:
What you are considering is the coordinate acceleration. This is the acceleration of an object that is measured by a stationary observer. In your example, this means stationary with respect to the black hole, or, equivalently, at a fixed point in Schwarzschild coordinates. In contrast, the proper acceleration is the measurable acceleration in the frame of the object. It is defined to be zero for a free-falling observer and also corresponds to the coordinate acceleration measured with respect to a free-falling observer that is at rest with respect to the object at the instant of measurement.

Ok, let me frame my question differently.

Given my (possibly incorrect) assumptions:

A) Time is dilating as an object approaches a black hole due to the curvature of space time.

B) This dilation effect is in some fashion proportional to the space time curvature.

C) The curvature of space time is behaving asymptotically as we approach the event horizon.

Your problem here is really tied to assumption C above. The singularity in the curvature at the horizon is what's known as a coordinate singularity. The curvature diverges there only because of the choice of Schwarzschild coordinates. There exist other coordinate systems where the curvature is finite at the horizon and only diverges at the location of the black hole center:

http://en.wikipedia.org/wiki/Lemaitre_coordinates
http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates
http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates

Under those circumstances, what is preventing the time dilation effect from likewise behaving asymptotically and causing the universe to age out of existence (from my perspective) before I reach the event horizon?
The time measured in the inertial frame of the infalling observer is the proper time. If you pick good coordinates like the Kruskal coordinates above, the metric is nonsingular everywhere but at the origin. So it's relatively noncomplicated to compute the proper time that it takes to reach the singularity: the result is finite. Whatever might be happening outside of the black hole, by this time the infalling observer ceases to exist.

It is true that the disparate perspectives of the outside and infalling observers is very peculiar from ordinary, Newtonian reasoning. The consequences for aspects like the information paradox have not been completely understood.

If I were in a starship somehow undergoing asymptotic acceleration, this is what I expect would happen mathematically, given the examples I have read of how the effect works from the perspective of our intergalactic speedster. I would fly towards the far end of the universe, 13 billion light years distant, and seem to reach it in seconds - but the stars would have gone out by the time I got there.
I'm not sure what you are referring to as asymptotic acceleration, but no amount of acceleration can accelerate an object past the speed of light.
 
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Ah, ok. This is interesting, though I think it's going to take a while for me to comprehend the Kurskal-Szekeres coord system. I'm only modestly familiar with the classic Schwarzchild system, which is what I was basing the idea on.

I'm curious, is the Kurskal-Szekeres considered a more accurate representation than Scwarzchild that deals with a wider array of conditions, or should the one you use depend on your frame of reference? (external observer vs. infalling observer)

Any solution that ends in a singularity does bother me, I'll admit. Not so much that the singularity and it's infinite gravity well is a rather nasty mathematical aberration - though that is an issue. I'm not overly fond of magical numbers like infinity in reality.

No, the real problem I'm trying to solve has to do with data. I'm in computer science, and data loss bothers me - and at this time every theory of data representation I am aware of requires surface area to encode it - including black hole theory.

As I understand it, the information that falls into a black hole is of (precisely?) the amount that could be encoded upon the event horizon at plank scale for its given surface area. An interesting coincidence to say the least...

If all that mass is truly being compacted into a singularity at the center, what medium is left to encode this vast amount of data at the event horizon? Is it being 'written' into magnetic fields? Gravitation waves, even at the edge of a black hole, couldn't possibly be granular enough, could they? Data cannot just float in space-time, as far as I know.

Thus why I am looking for solutions that retain the black hole's mass at the event horizon. If it falls through, it appears to me that we are left with no-where feasible to record its existence, which is a very serious problem.
 
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To frame this problem in different terms, a singularity can have very few properties - no more, as far as I can tell, than a single, absurdly massive particle - yet it has supposedly ingested trillions of yottabytes of data.

I can't square that circle, no matter what frame of reference we're talking about. It seems to me that any solution that comes to this conclusion must be highly suspect, unless we have a functional theory for how that data could be retained *within* the singularity.
 

fzero

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Ah, ok. This is interesting, though I think it's going to take a while for me to comprehend the Kurskal-Szekeres coord system. I'm only modestly familiar with the classic Schwarzchild system, which is what I was basing the idea on.

I'm curious, is the Kurskal-Szekeres considered a more accurate representation than Scwarzchild that deals with a wider array of conditions, or should the one you use depend on your frame of reference? (external observer vs. infalling observer)
The Kruskal coordinates are "geodesically complete," which is explained in the last paragraph at http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates#Qualitative_features_of_the_Kruskal-Szekeres_diagram. The Schwarzschild coordinates are fine for the external observer, who only has access to the region outside the horizon. For the infalling observer, the coordinate singularity in the Schwarzschild variables makes them poorly suited, so Kruskal coordinates are more suitable.

Any solution that ends in a singularity does bother me, I'll admit. Not so much that the singularity and it's infinite gravity well is a rather nasty mathematical aberration - though that is an issue. I'm not overly fond of magical numbers like infinity in reality.

No, the real problem I'm trying to solve has to do with data. I'm in computer science, and data loss bothers me - and at this time every theory of data representation I am aware of requires surface area to encode it - including black hole theory.
This is an unusual statement. Most physical systems have data that is most easily described by specifying it with variables that depend on the volume. For example, if we have a box of ideal gas, using the positions and momenta of the gas particles suggests that the entropy depends on the volume of the box, c.f. http://hyperphysics.phy-astr.gsu.edu/hbase/therm/entropgas.html. It is considered unusual that the entropy of a black hole only depends on its surface area. This fact is thought to have deep implications for what properties a theory of quantum gravity must have http://en.wikipedia.org/wiki/Holographic_principle.

As I understand it, the information that falls into a black hole is of (precisely?) the amount that could be encoded upon the event horizon at plank scale for its given surface area. An interesting coincidence to say the least...

If all that mass is truly being compacted into a singularity at the center, what medium is left to encode this vast amount of data at the event horizon? Is it being 'written' into magnetic fields? Gravitation waves, even at the edge of a black hole, couldn't possibly be granular enough, could they? Data cannot just float in space-time, as far as I know.

Thus why I am looking for solutions that retain the black hole's mass at the event horizon. If it falls through, it appears to me that we are left with no-where feasible to record its existence, which is a very serious problem.
To frame this problem in different terms, a singularity can have very few properties - no more, as far as I can tell, than a single, absurdly massive particle - yet it has supposedly ingested trillions of yottabytes of data.

I can't square that circle, no matter what frame of reference we're talking about. It seems to me that any solution that comes to this conclusion must be highly suspect, unless we have a functional theory for how that data could be retained *within* the singularity.
Both the questions of what happens at a gravitational singularity and why the black hole information depends on surface area rather than volume are considered to be in the realm of quantum gravity. There is presently no complete answer to these questions.
 
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Indeed, the Holographic Principle is precisely what got me started on this entire re-examination of the black hole.

Up until I read about GEO 600 results , I hadn't really put much thought into it, but the moment I got what they were getting at, I started thinking about where one would go about finding a two dimensional surface in the universe at large, and a black-hole event horizon was the only ready candidate. Then I began trying to sort out the information storage problems inherent in a singularity, and ended up becoming dissatisfied with the commonly accepted model.

I remain dissatisfied. I will freely admit that my mathematics aren't good enough to properly comprehend the space-time geometries we've been discussing - but I think my grasp of information theory is good enough to tell me that you can't compress data into a singularity by any currently accepted theory - so much so that I feel it must cast suspicion back on any theory that posits such a singularity in the first place.

Given the sheer amount of data and entropy in a black hole, a VAST amount of processing is going on there, over an exorbitant period of time. But how can you process anything in a singularity? I'm fairly certain you can't. It should be essentially inert - thus the search for a solution at the event horizon itself, rather than within.
 
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http://en.wikipedia.org/wiki/Bekenstein_bound

Ah, here we go. It's mainly volume bound, not surface area bound, though in the case of black holes, the surface area does happen to be a perfect bound for the amount of mass/data contained. Given that they are the densest things in the universe, that's close enough, though it doesn't neatly match with my earlier statements. Learning as I go here. ;)

I guess I should simply state that I currently have a higher confidence in this boundary than I do in current gravitational theorem, the topographies we've been discussing, or in any theory that posits infinite informational density or the inscription of data without the corresponding mass/energy to describe it.
 
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fzero, I like your explanations here, but I must point out that the Kruskal-Szekeres coordinate chart is not geodesically complete, but rather maximally extended. There is still the past and future singularities where geodesics can end, though there do exist geodesics which are themselves complete: they extend to infinite proper time in both the positive and negative directions.

And Vastin, have you read Susskind's book "The Black Hole War"? It discusses all of these issues you've brought up and it is aimed at a general audience. Of course, Susskind is biased, so I take him with a grain of salt. But he does say some interesting things.

And have you also read about Hawking's and Penrose's singularity theorems? It basically states that singularities are guaranteed in general relativity inside black holes and at the big bang, given certain requirements. Of course this ignores all quantum effects, which is thought to take over at small scales. It is often hoped that these unknown quantum effects would serve to prevent the formation of a singularity.
 
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It sounds like I have some fun reading to do. :)

While I know the favored existing theorems posit singularities, there is still some real uncertainty about whether they'll hold up through further development of quantum theory - or as you suggest, they may hold up in general, but quantum phenomena may act to block singularity formation regardless.

My bet is on the latter, because the violation of information principles seems to me an intractable problem that is unlikely to be resolved so long as mass is permitted to compact into a singularity. Ergo, there is likely another outcome.

Hawking's admission that black holes likely evaporate their information back out into the universe without destroying it makes the case for this considerably stronger, because again, if it was being compacted into a singularity, it is hard to posit a mechanism that would allow for that information to continue to exist in the interim - unless you want to consider time reversal, and I'm not ready to swallow that one yet either...

I'm not suggesting that the theory that describes all those possible phenomena are wrong, but I'm willing to bet that other factors will inevitably arise to assure they are never realized. Like imaginary numbers, they'll be useful to play with in theoretical terms, but they will never exist.

In any case, we've basically got this super-massive object with both an ideal data storage threshold AND the what appears to be the highest processing power in the universe (as suggested by its entropy) - so some process is going on there that is vastly more complex than any singularity should allow. I'll frankly be rather disappointed if they turn out to be little more than the universe's most thorough wood-chippers.
 

fzero

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I guess I should simply state that I currently have a higher confidence in this boundary than I do in current gravitational theorem, the topographies we've been discussing, or in any theory that posits infinite informational density or the inscription of data without the corresponding mass/energy to describe it.
This is probably not a good attitude to have. General relativity has been well-tested from terrestrial to cosmological scales, and is in excellent agreement with experiment. We should view singularities as something whose correct description will require something beyond general relativity, presumably a theory of quantum gravity. This is not a reason to have lower confidence in GR: we simply must recognize it's limitations.

In constrast, all other theories about black holes, from the Bekenstein-Hawking entropy, to the membrane paradigm and the holographic principle are complete speculation. Sure, some of it (esp. the BH entropy) are extremely well-motivated conjecture, but none of it has been tested. This is not a reason to discount it, as many of the ideas are worthy of further study. But it would be folly to believe in them at the expense of something that's as well-grounded as GR. In fact, almost all of the motivations for the speculative ideas involve agreement with GR in the appropriate limits.
 

fzero

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fzero, I like your explanations here, but I must point out that the Kruskal-Szekeres coordinate chart is not geodesically complete, but rather maximally extended. There is still the past and future singularities where geodesics can end, though there do exist geodesics which are themselves complete: they extend to infinite proper time in both the positive and negative directions.
You are correct, I wasn't being careful enough with the definition of the term. I linked to the (better than my own) discussion at the wiki in the hope that anyone reading what I wrote would get a more accurate summary. Thanks.
 

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