A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the scalar invariant curvatures of spacetime, some of which are a measure of the density of matter.
For the purposes of proving the Penrose–Hawking singularity theorems, a spacetime with 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. This is a different definition, useful for proving theorems.
The two most important types of spacetime singularities are curvature singularities and conical singularities. Singularities can also be divided according to whether they are covered by an event horizon or not (naked singularities). According to general relativity, the initial state of the universe, at the beginning of the Big Bang, was a singularity. Another type of singularity predicted by general relativity is inside a black hole: any star collapsing beyond a certain point would form a black hole, inside which a singularity (covered by an event horizon) would be formed, as all the matter would flow into a certain point (or a circular line, if the black hole is rotating). These singularities are also known as curvature singularities.
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Interpretation
Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the rate of change of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the Ultraviolet Catastrophe and in renormalization.
In supersymmetry, a singularity in the moduli space happens usually when there are additional massless degrees of freedom in that certain point. Similarly, it is thought that singularities in spacetime often mean that there are additional degrees of freedom that exist only within the vicinity of the singularity. The same fields related to the whole spacetime, also exist; for example, the electromagnetic field. In known examples of string theory, the latter degrees of freedom are related to closed strings, while the degrees of freedom are "stuck" to the singularity and related either to open strings or to the twisted sector of an orbifold.
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