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What is actual meaning of singularity can we use this term for conclusion in any research paper( related to cosmological models ).what r the types .

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We should narrow the question to a concrete theory, like general relativity, to get a definite answer. In general relativity a singularity in a spacetime is characterized by timelike and/or lightlike paths that are not extendable to infinite time in the past and the future: That is, there are paths that end before "time itself comes to an end" (speaking metaphorically) or that begin after "time itself has begun" (again, metaphorically). The former describe observers that fall into a singularity, the latter describe observers escaping from a singularity.

Naively, one would have expected that "singularity" in general relativity is simply defined as a point of a spacetime = Lorentzian manifold where the curvature diverges. This definition runs into problems, which are very nicely explained in the classic book

  • Hawking, Ellis: "The large scale structure in space-time".

You may have heard that both Hawking and Ellis are famous experts and that their work is hard to understand: While this may be true, their book is very well and clearly written and should be understandable to anyone with some background in GR and differential geometry. Try it.

Anyway, I'd recommend that you try to become a little bit more specific with your question and narrow it down to a specific physical setting or theory, like: what is a singularity in viscuous flows described by the Navier-Stokes equation, what is the description of a black hole like singularity in a spacetime according to the XY theory of quantum gravity etc.

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