The God has created a man in order that he creates that the God fails to do

Saturday, 23 July 2016

Foundations of Modern Physics 12: What are gravitational singularities

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.


WikipediA Gravitational singularity

G.Sardanashvily, Gravitational singularities of the caustic typearXiv:gr-qc/9404024

G.Sardanashvily, V.Yanchevsky, Caustics of space-time foliations in General Relativity, Acta Phys. Polon. B 17 (1986) 1017 – 1028. #

Wednesday, 20 July 2016

57th International Mathematical Olympiad 2016. Results

Hong Kong is proud to be hosting the brightest secondary school mathematics talents from over 100 countries and regions at the 57th International Mathematical Olympiad, 6 - 16 July 2016. 

The Results (#)

1. USA, 2. Korea, 3. China ..., 7. Russia, ...

Saturday, 9 July 2016

Our recent article: Partially superintegrable systems on Poisson manifolds

Our recent article: A.Kurov and G.Sardanashvily, “Partially superintegrable systems on Poisson manifolds” in arXiv: 1606.03868

Abstract. Superintegrable systems on a symplectic manifold conventionally are considered. However, their definition implies a rather restrictive condition 2n=k+m where 2n is a dimension of a symplectic manifold, k is a dimension of a pointwise Lie algebra of a superintegrable system, and m is its corank. To solve this problem, we aim to consider partially superintegrable systems on Poisson manifolds where k+m is the rank of a compatible Poisson structure. The according extensions of the Mishchenko-Fomenko theorem on generalized action-angle coordinates is formulated.