Singularities
Coordinate systems, in general only cover a portion of the entire manifold of points that are relevant to a gravitational field description. The Schwarzschild coordinates, for example, do not cover the polar axis described by . The line element becomes degenerate at these points. The covariant metric ceases to be of dimension 4. Introducing new coordinates could clear up this problem. For instance introducing the usual Cartesian coordinates under the transformation law
(21.1)
,
would remove the aforementioned Schwarzschild coordinate degeneracy. Points which are problematic like this in one coordinate system but which become nonproblematic in other coordinate systems are called Coordinate Singularities. This type of singularity can be fixed since it is removable and therefore the physical description of the gravitational field does not break down at this troublesome point. The other points that cause problems for the Schwarzschild solution are the points:
(called the Schwarzschild Radius) and .
To test for nonremovable singularities in general relativity on calculates the squared Riemann tensor scalar called the Kretshmann Scalar given as follows:
(21.2) 


If this scalar blows up at any point in the manifold then you have a nonremovable singularity at that point. In the case of the Schwarzschild solution this scalar gives the following result
(21.3) 


We see that the hypersurface and the problem at must be removable since these problem values do not make the Kretshmann Scalar infinite. These are therefore coordinate singularities. They can be transformed away. However, there is no such luck with the problem. The singularity at the origin is not removable. This type of singularity is given one of the following labels in the General Relativity literature.