TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES

Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.

SINGULARITAS SEMU PADA RUANG-WAKTU REISSNER-NORDSTRÖM

ABSTRAKPenelitian ini mengkaji singularitas semu pada metrik Reissner-Nordström, yang merupakan solusi persamaan medan Einstein untuk model partikel bermuatan. Kajian dilakukan dengan menganalisis titik-titik singular pada metrik, menghitung tensor kelengkungan Riemann, serta menghitung scalar Kretschmann pada titik-titik tersebut. Perhitungan dilakukan dengan bantuan program Maxima. Hasilnya, singularitas nyata hanya terjadi pada r = 0, sedangkan singularitas semu terjadi pada . Singularitas semu tersebut merupakan representasi dari horizon peristiwa. Terdapat tiga kemungkinan situasi pada horizon peristiwa. Hal menarik terdapat pada situasi r = M, dimana terjadi keseimbangan antara massa dan muatan yang memungkinkan tarikan gravitasi dan tolakan elektromagnetik saling meniadakan. Penelitian ini juga menghasilkan persamaan geodesik pada titik-titik yang tidak menghasilkan nilai infinite pada skalar Kretschmaan. Kata Kunci : Kelengkungan Riemann, Metrik Reisner-Nordström, Singularitas, Persamaan Geodesik ABSTRACTThis study examines pseudo singularities on the Reissner-Nordström metric which is a solution to Einstein's field equations for charged particle models. The study was carried out by analyzing the singular points on the metric calculating the Riemann curvature tensor, and calculating Kretschmann's scalar at these points. The results show that real singularities only occur at r = 0, whereas pseudo singularity occurs at . There is a point of pseudo singularity that representing the event horizon. There are two possible situations on the event horizon. Interesting things are in the case r = M, where here is a balance between mass and charge which allows gravitational pull and electromagnetic repulsion to cancel each other. This study also yields the geodesic equation point that not yields infinite value of Kretschmaan scalar. Keywords: Riemann tensor, Reissner-Nordström Metrik, Singularities, Geodesik equation

Magnetic charge and photon mass: Physical string singularities, Dirac condition, and magnetic confinement

We find exact, simple solutions to the Proca version of Maxwell’s equations with magnetic sources. Several properties of these solutions differ from the usual case of magnetic charge with a massless photon: (i) the string singularities of the usual 3-vector potentials become real singularities in the magnetic fields; (ii) the different 3-vector potentials become gauge inequivalent and physically distinct solutions; (iii) the magnetic field depends on r and [Formula: see text] and thus is no longer rotationally symmetric; (iv) a combined system of electric and magnetic charge carries a field angular momentum even when the electric and magnetic charges are located at the same place (i.e. for dyons); (v) for these dyons, one recovers the standard Dirac condition despite the photon being massive. We discuss the reason for this. We conclude by proposing that the string singularity in the magnetic field of an isolated magnetic charge suggests a confinement mechanism for magnetic charge, similar to the flux tube confinement of quarks in QCD.