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Black Holes as 2 Dimensional Objects 
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#1
May2312, 07:35 PM

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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 mindbending 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. 


#2
May2412, 10:05 AM

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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.



#3
May2412, 10:47 AM

P: 17

Ok  but why? What are the specific problems this model introduces?



#4
May2412, 10:50 AM

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Black Holes as 2 Dimensional Objects
General Relativity predicts the existence of Black Holes as threedimensional spheres, not twodimensional 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. 


#5
May2412, 12:58 PM

P: 17

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. 


#6
May2412, 03:28 PM

P: 17

In the model I'm suggesting, the familiar (simplified 2D) spacetime 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 spacetime would not be infinite. It would instead have a distinct boundary at the event horizon beyond which nothing would pass whatsoever  there being nowhere 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. 


#7
May2412, 04:40 PM

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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.



#8
May2412, 05:09 PM

P: 17

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 nonspace (a true nonspatial void) 'within' the event horizon? On another related point, saying that the infalling person would see 'nothing unusual' as they passed through the event horizon is a bit misleading. They would presumably see the nearinstantaneous heatdeath 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... 


#9
May2412, 05:24 PM

P: 17

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. 


#10
May2412, 06:17 PM

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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. 


#11
May2412, 07:43 PM

P: 17

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. 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. 


#12
May2412, 08:02 PM

P: 17

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. 


#13
May2412, 09:04 PM

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In contrast, the proper acceleration of a freefalling observer is zero. Freefall is inertial motion. 


#14
May2412, 10:32 PM

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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.
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... 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. 


#15
May2512, 12:51 AM

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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/Eddingt...in_coordinates http://en.wikipedia.org/wiki/Kruskal...es_coordinates 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. 


#16
May2512, 09:15 AM

P: 17

Ah, ok. This is interesting, though I think it's going to take a while for me to comprehend the KurskalSzekeres 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 KurskalSzekeres 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 spacetime, 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 nowhere feasible to record its existence, which is a very serious problem. 


#17
May2512, 09:55 AM

P: 17

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. 


#18
May2512, 02:12 PM

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