Space-time extensions II

The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by $\gamma$ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime $(M,g_{ab})$. First, it is shown that it is always possible to select a synchronised family of causal geodesics $\Gamma$ and an open neighbourhood $\mathcal{U}$ of a final segment of $\gamma$ in $M$ such that $\mathcal{U}$ is comprised by members of $\Gamma$, and suitable local coordinates can be defined everywhere on $\mathcal{U}$ provided that $\gamma$ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime, $(M,g_{ab})$, is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order $k-1$ are bounded on $\mathcal{U}$, and also the line integrals of the components of the $k^{th}$-order covariant derivatives are finite along the members of $\Gamma$---where all the components are meant to be registered with respect to a synchronised frame field on $\mathcal{U}$---then there exists a $C^{k-}$ extension $\Phi: (M,g_{ab}) \rightarrow (\widehat{M},\widehat{g}_{ab})$ so that for each $\bar\gamma\in\Gamma$, which is inextendible in $(M,g_{ab})$, the image, $\Phi\circ\bar\gamma$, is extendible in $(\widehat{M},\widehat{g}_{ab})$. Finally, it is also proved that whenever $\gamma$ does terminate on a topological singularity $(M,g_{ab})$ cannot be generic.Comment: 42 pages, no figures, small changes to match the published versio

Trajectories of the S-matrix poles in Salamon-Vertse potential

The trajectories of S-matrix poles are calculated in the finite-range phenomenological potential introduced recently by P. Salamon and T. Vertse (SV). The trajectories of the resonance poles in this SV potential are compared to the corresponding trajectories in a cut-off Woods-Saxon (WS) potential for l>0. The dependence on the cut-off radius is demonstrated. The starting points of the trajectories turn out to be related to the average ranges of the two terms in the SV potential

Space-Times Admitting Isolated Horizons

We characterize a general solution to the vacuum Einstein equations which admits isolated horizons. We show it is a non-linear superposition -- in precise sense -- of the Schwarzschild metric with a certain free data set propagating tangentially to the horizon. This proves Ashtekar's conjecture about the structure of spacetime near the isolated horizon. The same superposition method applied to the Kerr metric gives another class of vacuum solutions admitting isolated horizons. More generally, a vacuum spacetime admitting any null, non expanding, shear free surface is characterized. The results are applied to show that, generically, the non-rotating isolated horizon does not admit a Killing vector field and a spacetime is not spherically symmetric near a symmetric horizon.Comment: 11 pages, no figure

Formation of Liesegang patterns: Simulations using a kinetic Ising model

A kinetic Ising model description of Liesegang phenomena is studied using Monte Carlo simulations. The model takes into account thermal fluctuations, contains noise in the chemical reactions, and its control parameters are experimentally accessible. We find that noisy, irregular precipitation takes place in dimension d=2 while, depending on the values of the control parameters, either irregular patterns or precipitation bands satisfying the regular spacing law emerge in d=3.Comment: 7 pages, 8 ps figures, RevTe