Strengths of singularities in spherical symmetry

Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry

It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation (massless scalar field). In the spherically symmetric case it is further sho...

Singularities of centre symmetry sets

We study the singularities of envelopes of families of chords (special straight lines) intrinsically related to a hypersurface M ⊂ R embedded into an affine space. The origin of this investigation is the paper [10] of Janeczko. He described a generalization of central symmetry in which a single point—the centre of symmetry—is replaced by the bifurcation set of a certain family of ratios. In [6]...

Gravitational energy in spherical symmetry.

Various properties of the Misner-Sharp spherically symmetric gravitational energy E are derived, and known properties reviewed. In the Newtonian limit, E yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. In vacuo, E reduces to the Schwarzschild parameter. At null and spatial infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misn...

Emergence, Singularities, and Symmetry Breaking

Topological singularities and symmetry breaking in development

The review presents a topological interpretation of some morphogenetic events through the use of well-known mathematical concepts and theorems. Spatial organization of the biological fields is analyzable in topological terms. Topological singularities inevitably emerging in biological morphogenesis are retained and transformed during pattern formation. It is the topological language that can pr...