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.
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