Physics Research Centre

Two flavours Skyrmion will be analyzed analytically, in case of static and rotational Skyrme equations. Numerical solution of a nonlinear scalar field equation, i.e. the Skyrme equation, will be worked with finite difference method. This article is a more comprehensive version of \textit{SU(2) Skyrme Model for Hadron} which have been published at Journal of Theoretical and Computational Studies, Volume \textbf{3} (2004) 0407.

Page 1. Model Skyrme SU(2) untuk Hadron Miftachul Hadi 7 September 2004 Page 2. Ringkasan Ditinjau model Skyrme SU(2) yang memperlakukan hadron sebagai soliton (skyrmion) dari suatu persamaan medan skalar non linier, yakni persamaan Skyrme. Massa nukleon (mN ) dan massa delta (m∆) dihitung dari penjumlahan energi statik dan energi rotasi skyrmion yang dapat diturunkan dari tensor rapat energi-momentum model.

We study nonlinear sigma model, especially Skyrme model without twist and Skyrme model with twist: twisted Skyrmion string. Twist term, mkz, is indicated in vortex
solution. Necessary condition for stability of vortex solution has consequence that energy of vortex is minimum and scale-free (vortex solution is neutrally stable to changes in scale). We
nd numerically that the value of vortex minimum energy per unit length for twisted Skyrmion string is 20.37 x 10^60 eV/m.

We study nonlinear sigma model, especially Skyrme model without twist and Skyrme model with twist: twisted Skyrmion string. Twist term, mkz, is indicated in vortex solution. Necessary condition for stability of vortex solution has consequence that energy of vortex is minimum and scale-free (vortex solution is neutrally stable to changes in scale). We find numerically that the value of vortex minimum energy per unit length for twisted Skyrmion string is 20.37 × 10 60 eV/m.

In this paper we study the gravitational field of a straight string generated from a class of nonlinear sigma models, specifically the Skyrme model without a twist and the Skyrme model with a twist (the twisted Skyrmion string). The twist term, $mkz$, is included to stabilize the vortex solution. To model the effects of gravity, we replace the Minkowski tensor, $\eta^{\mu\nu}$, in the standard Skyrme Lagrangian density with a space-time metric tensor, $g^{\mu\nu}$, assumed to be static and cylindrically symmetric. The Einstein equations for the metric and field components are then derived. This work is still in progress.

We study nonlinear sigma model, especially Skyrme model with twist: twisted Skyrmion string where twist term, mkz , is indicated in vortex solution. To add gravity, we replace g  in Lagrangian system with a space-time metric tensor, g  which in view of the time-independence and cylindrical symmetry of the assumed vortex solution is taken to be a function of r alone. We use ode45 for numerical calculation, i.e. a tool box in Matlab to solve coupled Einstein field equations which have ordinary differential equations (ODE) form.

We explore deeper and analyse in more detail Fermat’s and Hamilton’s principles. We try to address some questions: Is it possible to have δS negative? Is Hamilton’s principle always valid for entire path of the system? Is there a relation between Fermat’s principle and Hamilton’s principle? We assume analogy with Hamilton’s principle, is Fermat’s principle always valid for entire path of the system? Does a least action take a least time for happening?

The refractive index in the bent fibre optics and in curved space of the gravitational field are studied. We obtain the results that refractive index in the bent fibre optics is similar with refractive index in curved space of the gravitational field. Both results show that refractive index increases when light traverses through the bent fibre optics and curved space of the gravitational field. Based on analogy reasoning taken from fibre optics bending loss phenomena, we propose that if light traverses through curved space of the gravitational field, light will loss its energy. We calculate this energy loss numerically using bending loss model of Faustini-Martini. We find that light losses its energy around 467.7073 dB/m, when it traverses through curved space with radius of curvature R = 0.01 m due to massive object around 8.7 × 10 20 kg.

We study a refractive index and its relation with curvature. The space dimension of curvature is extended to more than two dimensions as Riemann-Christoffel curvature tensor. We formulate the refractive index in relations with the Riemann-Christoffel curvature tensor. We find that the refractive index should be a tensor rank-2 as a consequence of the fourth rank tensor of the Riemann-Christoffel curvature tensor. We also formulate the relations between the refractive index and mass of a kink (anti-kink) where mass of the kink (anti-kink) is expressed in relations with a topological charge. Deflection of light is discussed in relations with the mass of the kink (anti-kink).

The refractive index and curved space relation is formulated using the Riemann-Christoffel curvature tensor. As a consequence of the fourth rank tensor of the Riemann-Christoffel curvature tensor, we found that the refractive index should be a second rank tensor. The second rank tensor of the refractive index describes a linear optics. It implies naturally that the Riemann-Christoffel curvature tensor is related to the linear optics. In case of a non-linear optics, the refractive index is a sixth rank tensor, if susceptibility is a fourth rank tensor. The Riemann-Christoffel curvature tensor can be formulated in the non-linear optics but it is reduced by a gradient of susceptibility term.

Two flavours Skyrmion will be analyzed analytically, in case of static and rotational Skyrme equations. Numerical solution of a nonlinear scalar field equation, i.e. the Skyrme equation, will be worked with finite difference method. This article is a more comprehensive version of SU(2) Skyrme Model for Hadron which have been published at Journal of Theoretical and Computational Studies, Volume 3 (2004) 0407.

Two flavours Skyrmion will be analyzed analytically, in case of static and rotational Skyrme equations. Numerical solution of a nonlinear scalar field equation, i.e. the Skyrme equation, will be worked with finite difference method. This article is a more comprehensive version of SU(2) Skyrme Model for Hadron which have been published at Journal of Theoretical and Computational Studies, Volume 3 (2004) 0407.

We study nonlinear sigma model, especially Skyrme model without twist and Skyrme model with twist: twisted Skyrmion string. Twist term, mkz, is indicated in vortex solution. Necessary condition for stability of vortex solution has consequence that energy of vortex is minimum and scale-free (vortex solution is neutrally stable to changes in scale). We find numerically that the value of vortex minimum energy per unit length for twisted Skyrmion string is 20.37 × 10 60 eV/m.

In this paper we study the gravitational field of a straight string generated from a class of nonlinear sigma models, specifically the Skyrme model without a twist and the Skyrme model with a twist (the twisted Skyrmion string). The twist term, $mkz$, is included to stabilize the vortex solution. To model the effects of gravity, we replace the Minkowski tensor, $\eta^{\mu\nu}$, in the standard Skyrme Lagrangian density with a space-time metric tensor, $g^{\mu\nu}$, assumed to be static and cylindrically symmetric. The Einstein equations for the metric and field components are then derived. This work is still in progress.