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In a gravitational field, should the mass distributions always behave well?

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No. There are numerous well-known mass distributions that start out non-singular, and which collapse to form a black-hole after a finite amount of time.

Black hole solutions, however, have a region of infinite density and infinite spacetime curvature, so they are not at all well-behaved in any ordinary sense.

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@Jerry Schirmer: Is the converse true ? If there is a point where the distribution is not well behaved then there should be a black hole. –  Rajesh D Nov 21 '10 at 5:36
    
No. All sorts of different singularities are possible. –  Jerry Schirmer Nov 21 '10 at 5:37
    
@Jerry Schirmer: please give some examples –  Rajesh D Nov 21 '10 at 5:47
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In another thread, you're hostile to a discussion that uses technical math. Discussing Taub-NUT singularities and cosmological singularities without technical math is somewhat pointless. But there are definitely singular spacetimes that aren't black hole solutions, irrepsective of anything else. –  Jerry Schirmer Nov 21 '10 at 6:18
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I would suggest picking up Bernard Schutz's "a first course in general relativity". Any answer I give will just generate more questions. To have any satisfactory answer to this broad of a question will require more background than is practical here. –  Jerry Schirmer Nov 22 '10 at 4:28

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