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.
References:
G.Sardanashvily, Gravitational singularities of the caustic type, arXiv:gr-qc/9404024
G.Sardanashvily,
V.Yanchevsky, Caustics of space-time foliations in General Relativity, Acta Phys. Polon.
B 17 (1986) 1017 – 1028. #
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