I am not aware of GR, but due to curiosity i have a question in my mind. Please

Question

I am not aware of GR, but due to curiosity i have a question in my mind. Please let me know if it is inappropriate to ask here.

My question is about singularity. I am under the assumption that singularity means in mathematical terms equivalent to a discontinuity in a function. My question is what type of discontinuity are the ones corresponding to Penrose-Hawking singularity theorems and also to the the naked singularity ? by type i mean

removable (left and right limits exist and are equal but not equal to the function value at that point).

a jump type discontinuity (left and right limits exists and are not equal to each other).

Oscillating discontinuity (as described as described in the above given link)

Explanation / Answer

Note: Depending on one's definition of the discontinuity (whether one allows talking about the continuity of the function at a point where the function is undefined), pole might also be considered an infinite discontinuity.

Singularity is certainly not equivalent to a discontinuity (not even mathematically). It can also mean pole and this is the usual meaning it has in GR. I.e. the curvature of the space-time blows up.

For example, Schwarzschild metric's curvature diverges at the origin as may be checked by looking at Kretschmann invariant. There might also exist coordinate singularities but these are unphysical and can be removed by a better choice of coordinates.

Note: What does one mean by curvature?

The basic object associated with the metric tensor is a connection and for that one can define a curvature form. Now this is quite complicated object and it is not quite clear how to detect singularities in it. One easy way is to define invariants (such as the above-mentioned Kretschmann invariant) associated with this form

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