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According to general relativity, an observer not in the close proximity of a black hole, observing a mass fall into that black hole, will never see that mass cross event horizon(it will gradually fade but never actually pass through the event horizon) as it requires an infinite amount of time to do so. So, effectively, for the observer, the mass of the object falling in will never reach the middle of the event horizon, or the so called "singularity", as it will require more-than-infinite amount of time to do so.

Then why do we even define such a thing as a singularity?

I am alright with the singularities in space-time curvature, but not with a finite amount of mass being compressed into zero volume, not because it is silly, because it would never happen in a finite time for a casual observer outside of black hole.

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I am not sure what the actual question is. We don't define any singularity but rather just notice that it is a consequence of the GTR. As for what happens under the horizon: no external observer will ever observe it, so you can ignore singularities if you want (most people would ignore black holes completely anyway, if it wasn't for Star Trek). Nevertheless, it's interesting to ask what your theory tells about the situations you can't yet observe (i.e. you extrapolate the theory). –  Marek Nov 17 '10 at 0:47
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BTW, the singularity is not in the "middle" of the event horizon. An observer who passes through the horizon does not find the singularity at some particular spatial location, rather it is in their future. That's why they can't avoid it. It's like death and taxes. –  pho Dec 16 '10 at 19:20
    
Yes, yes, I have grown much more knowledgeable about the issue of singularity since I asked the question. Thank you for your concern Jeff. –  Cem Dec 16 '10 at 19:44
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While the distant observer will never observe the infalling observer crossing the event horizon, the infalling observer will intersect the horizon in a finite, calculable amount of local time. So, understanding the singularity is important for understanding the fate of that observer, certainly.

Second, if you believe that Hawking radiation is real, then the picture isn't quite so stark--the radius of the black hole will decrease, and it will be possible to see some of the collapsing matter at late times. Understanding the singularity is critical to understanding what exactly happens when the radius of the black hole becomes very small (say, a radius of one planck length).

Third, in addition to black hole singularities, you have big bang singularities, which are also a consequence of a general theorem in relativity. Understanding what happens near the big bang singularity (or what quantum effects may cause the singularity not to happen, and perhaps trigger something like inflation) is one of the central questions in cosmology.

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"While the distant observer will never observe the infalling observer crossing the event horizon, the infalling observer will intersect the horizon in a finite, calculable amount of local time." - yes. "So, understanding the singularity is important for understanding the fate of that observer, certainly." - no. Because the time of life of any BH is finite also, and shorter than the time untill the infalling observer will intersect the horizon. –  Anixx Feb 7 '11 at 0:15
    
@Anixx: No. The lifetime of a black hole cannot even be accurately calculated by current theory, but all estimates for realistic astrophysical black holes say that the lifetime of the hole is very long (millions of years), while the GR calculation for the lifetime of the observer before they intersect the singularity is very, very short (less than a second for radial infall). –  Jerry Schirmer Feb 7 '11 at 1:24
    
In different reference frames, millions of years for distant observer=seconds for the free falling. You just confuse them. For free falling observer the BH also will evaporate in seconds. –  Anixx Feb 7 '11 at 1:28
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@Anixx: NO. PLEASE READ ANY REFERENCE. REALLY, ANY REFERENCE. I REFERRED YOU TO TWO VERY STANDARD ONES. Don't talk like an expert if you don't understand basic terminology. –  Jerry Schirmer Feb 7 '11 at 3:47
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@Anixx: I'm done talking to you until you cite some sort of reference. You repeatedly say wrong things, then cite popular level crap at me, contradicting basic definitions for no apparent reason. I cite back basic, standard texts on General Relativity, and you refuse to make any conterarguments. Cite some actual physics or just say that you don't know what you're talking about. –  Jerry Schirmer Feb 7 '11 at 4:53
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