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I had this doubt bugging my mind for a long time about singularities in Physics. I heard that R.Penrose and S.Hawking have proposed that there could be singularities at Blackholes and at the time of the origin of universe (at the time of Bigbang). The concept of singularity is essentially a mathematical one. (unless you call anything that you cannot explain physically as a singularity).

My question is (which i want to get it resolved once for all so that i stop bugging people with it :-))....We know that the gravitational field has a singularity at points where there are point masses, so is the case with point charges and electric field. So what is it they have proved about existence of singularities. I assume that they most probably have started with a universe which doesn't have point masses. If they did'nt start with point masses, then what did they actually started with. I want to know what they have started with in mathematical terms and physical terms, once for all. I am open to read some references as well to resolve this doubt.

EDIT :

I am more puzzled as to why I am not getting an answer to the question I've asked which doesn't involve any analysis but a plain request for information. Did they (R.P and S.H) assume existence of point masses in their model of universe (or objects or whatever it is) or they have assumed smooth matter distributions ? This is a fairly straight forward request for information....I do not understand why I don't get the answer for the question I've asked...rather I get something else.

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Have you looked at the following wikipedia article? en.wikipedia.org/wiki/Gravitational_singularity "a singularity is defined to be one that contains geodesics that cannot be extended in a smooth manner. The end of such a geodesic is considered to be the singularity." So a singularity might be interpreted as a straight line that ends suddenly. I would almost think of it as point where transformations can not be described using the GL group of matrices, but I would have to defer to others to verify the truth of such a statement. –  Unassuminglymeek May 22 '11 at 23:22

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What they proved is that in general relativity time-like world-lines (i.e. real observer paths) terminate in finite proper time i.e. the time experienced by the observer on the object is finite. This occurs forward in time if a trapped surface exists in space time. Such a surface has all light rays converging from it whatever direction they point and is a sign of a black hole. Backwards in time it occurs cosmologically because of the expanding universe. Terminating world-lines usually mean singularities on the space time manifold.

Singularities like this occur in regions of space-time where curvature is enormous (and ultimately infinite) and as a result one might suspect therefore that classical general relativity is inapplicable there. Quantum gravity theories can avoid the singularities although I am unaware of a general proof they always do.

The singularity proof relies on so-called positive energy conditions of various kinds which classical matter is assumed to satisfy.

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I don't think there is any doubt on point masses being a singularity --their very definition asserts that. I will be leaving Hawkings out of my post as has retracted from his original position on the big bang being a singularity (general relativity breaks at times < planck times).

Both types of singularities that are under investigation are for real objects. As your question specifically asks about the assumptions Penrose made, so I think that perhaps the best answer can be found in a paper he authored.

Penrose has not proven the existence of singularities. What he has shown is that assuming Einstein's equations hold inside a black hole and if the observed matter densities > 0 for future pointing time like vectors, then we will end up with some sort of a singularity.

There can of course be other, yet undiscovered, interactions that act at closer distances and prevent singularities but that's outside the scope of his theorem (theorems prove something based on some theories and they act to demonstrate what those theories predict).

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