Implications of Spontaneous Glitches in the Mass and Angular Momentum in Kerr Space-Time

The outward-pointing principal null direction of the Schwarzschild Riemann tensor is null hypersurface-forming. If the Schwarzschild mass spontaneously jumps across one such hypersurface then the hypersurface is the history of an outgoing light-like shell. The outward-- pointing principal null direction of the Kerr Riemann tensor is asymptotically (in the neighbourhood of future null infinity) null hypersurface-forming. If the Kerr parameters of mass and angular momentum spontaneously jump across one such asymptotic hypersurface then the asymptotic hypersurface is shown to be the history of an outgoing light-like shell and a wire singularity-free spherical impulsive gravitational wave.Comment: 16 pages, TeX, no figures, accepted for publication in Phys. Rev.

M\"obius and Laguerre geometry of Dupin Hypersurfaces

In this paper we show that a Dupin hypersurface with constant M\"{o}bius curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin hypersurfaces.Comment: 45 pages. arXiv admin note: text overlap with arXiv:math/0512090 by other author

Matching LTB and FRW spacetimes through a null hypersurface

Matching of a LTB metric representing dust matter to a background FRW universe across a null hypersurface is studied. In general, an unrestricted matching is possible only if the background FRW is flat or open. There is in general no gravitational impulsive wave present on the null hypersurface which is shear-free and expanding. Special cases of the vanishing pressure or energy density on the hypersurface is discussed. In the case of vanishing energy momentum tensor of the null hypersurface, i.e. in the case of a null boundary, it turns out that all possible definitions of the Hubble parameter on the null hypersurface, being those of LTB or that of FRW, are equivalent, and that a flat FRW can only be joined smoothly to a flat LTB.Comment: 9 page

Convergence of formal embeddings between real-analytic hypersurfaces in codimension one

We show that every formal embedding sending a real-analytic strongly pseudoconvex hypersurface in M\subset \C^N into another such hypersurface in M'\subset \C^{N+1} is convergent. More generally, if $M$ and $M'$ are merely Levi-nondegenerate, the same conclusion holds for any formal embedding provided either that the embedding is CR transversal or the target hypersurface does not contain any complex curves.Comment: 8 page

Mapped Null Hypersurfaces and Legendrian Maps

For an $(m+1)$-dimensional space-time $(X^{m+1}, g),$ define a mapped null hypersurface to be a smooth map $\nu:N^{m}\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $\nu$ that are both tangent and $g$-orthogonal to $\nu.$ We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle $ST^*M$ of an immersed spacelike hypersurface $\mu:M^m\to X^{m+1}.$ We show that a Legendrian map \wt \lambda: L^{m-1}\to (ST^*M)^{2m-1} defines a mapped null hypersurface in $X.$ On the other hand, the intersection of a mapped null hypersurface $\nu:N^m\to X^{m+1}$ with an immersed spacelike hypersurface $\mu':M'^m\to X^{m+1}$ defines a Legendrian map to the spherical cotangent bundle $ST^*M'.$ This map is a Legendrian immersion if $\nu$ came from a Legendrian immersion to $ST^*M$ for some immersed spacelike hypersurface $\mu:M^m\to X^{m+1}.$Comment: 13 pages, 1 figur