The most of physically significant solutions of Einstein’s equations possess gravitational singularities. A problem however is that there is no generally accepted criterion of gravitational singularities.
It seems natural to identify gravitational singularities with singular values of a pseudo-Riemannian metric g, or a curvature tensor R, or scalar polynomials of a curvature tensor and its derivatives. However, this criterion is not quite satisfactory.
Firstly, the regularity of all these quantities fails to prevent us from such singular situations as incomplete geodesics, a breakdown of causality etc.
Secondly, it may happen that non-scalar gravitational quantities are singular relative to a certain reference frame while scalar polynomials are regular. One can think of such a singularity as being fictitious, which one can avoid by reference frame transformations. However, these transformations are singular.
Thirdly, if some scalar curvature polynomial takes a singular value, one can exclude a point of this singularity from a space-time. Although the remainder is singular too, the criterion under discussion fails to indicate its singularity.
For instance, a gravitational field g of a black hole is singular on its gravitational radius, whereas all scalar curvature polynomials remain regular. Consequently, this singularity is fictitious, while a real singularity lies in the center of a black hole.
At present, the most recognized criterion of gravitational singularities is based on the notion of so called b-incompleteness. By virtue of this criterion, there is a gravitational singularity if some smooth curve in a space-time X can not be prolonged up to any finite value of its generalized affine parameter. In the case of time-like geodesics, this parameter is a usual proper time.
In order to describe such a b-singularity, singular points are replaced with a set of points, called the b-boundary, which a curve is prolonged to. Then one study the behavior of gravitational quantities with respect to a particular frame, propagated in parallel, as one approaches the b-boundary. In particular, one separated the regular (removable) singularities, scalar and non-scalar curvature singularities, and quasi-regular (locally-extendible) singularities. Unfortunately, the b-criterion also is not quite satisfactory as follows.
(i) It is impossible to examine the b-completeness of all curves in a space-time.
(ii) The construction of a b-boundary is very complicated, and one can define it only in a few particular cases. For instance, if X is a regular space-time and we exclude its regular point, the b-boundary need not coincide with this point.
(iii) The definition of a generalized affine parameter depends on a connection on X, but not a pseudo-Riemannian metric g.
(iv) The b-criterion of gravitational singularities can not indicate a breakdown of space-time causality, e.g., the existence of time-like cycles.
In a different way, gravitation singularities can be described as singularities of an associated space-time structure which is characterized by a time-like differential one-form h. In particular, no gravitational singularity is present if there exists a nowhere vanishing time-like exact form h=df. Then the equations f=const. define a foliation of X in space-like hypersurfaces and t=f is a global time. Space-time singularities are exemplified by a breakdown of causality, when h is not exact, topological transitions at points, where df=0, and the caustics of space-like hypersurfaces at points where a time function f becomes multivalued. However, this description of gravitation singularities also meets problems. For instance, the Minkowski space admits space-time caustics.