LRS Bianchi Type II Massive String Cosmological Models with Magnetic Field in Lyra's Geometry

Raj Bali,Lokesh Kumar Gupta,Mahesh Kumar Yadav

doi:10.1155/2013/892361

Abstract

Bianchi type II massive string cosmological models with magnetic field and time dependent gauge function (<svg style="vertical-align:-4.15506pt;width:13.6px;" id="M1" height="17.049999" version="1.1" viewBox="0 0 13.6 17.049999" width="13.6" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.813)"><path id="x1D719" d="M542 242q0 -106 -80 -176.5t-192 -72.5l-55 -251l-6 -3q-18 7 -28 38t1 91l23 124q-85 16 -133.5 72.5t-48.5 137.5q0 95 62.5 159.5t166.5 97.5l14 -31q-77 -32 -115 -82.5t-38 -128.5q0 -63 31.5 -107t74.5 -59l125 597l56 43l16 -10l-40 -168q-9 -40 3 -45
q77 -33 120 -85.5t43 -140.5zM469 232q0 67 -37 115t-77 62l-76 -370q89 -4 139.5 52.5t50.5 140.5z" /></g> <g transform="matrix(.012,-0,0,-.012,9.663,16.838)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" /></g> </svg>) in the frame work of Lyra's geometry are investigated. The magnetic field is in <svg style="vertical-align:-0.0pt;width:23.924999px;" id="M2" height="11.6" version="1.1" viewBox="0 0 23.924999 11.6" width="23.924999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.537)"><path id="x1D44C" d="M667 650l-9 -28q-53 -5 -76 -17t-64 -59q-51 -61 -175 -225q-21 -29 -27 -55l-27 -136q-13 -65 -0.5 -80t83.5 -22l-7 -28h-280l8 28q64 4 81 19t30 83l25 128q6 35 -7 65l-98 231q-17 41 -32.5 52t-68.5 16l8 28h252l-6 -28l-40 -4q-27 -3 -33 -12.5t2 -31.5
q8 -26 43 -107.5t61 -134.5q114 145 174 240q14 24 8 33t-37 13l-34 4l8 28h238z" /></g><g transform="matrix(.017,-0,0,-.017,11.605,11.537)"><path id="x1D44D" d="M698 636l-541 -596q60 -5 176 -5q91 0 138.5 5t69.5 24q44 36 85 124l29 -15q-38 -125 -64 -173h-559l-9 16l545 598h-182q-81 0 -109 -8t-48 -31q-26 -29 -55 -103l-29 3q23 86 42 200h22q11 -16 21 -20.5t34 -4.5h428z" /></g> </svg>-plane. To get the deterministic solution, we have assumed that the shear (<svg style="vertical-align:-0.1638pt;width:9.7250004px;" id="M3" height="8.3000002" version="1.1" viewBox="0 0 9.7250004 8.3000002" width="9.7250004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,8.038)"><path id="x1D70E" d="M548 455q-32 -77 -74 -77q-37 0 -103 10l-3 -2q36 -30 48.5 -62t12.5 -80q0 -103 -75 -179.5t-175 -76.5q-70 0 -113 45t-43 128q0 115 83.5 200.5t209.5 85.5q31 0 77.5 -4t67.5 -4q36 0 62 30zM350 274q0 54 -17 86q-19 35 -57 35q-50 0 -90 -37.5t-59 -91.5t-19 -109
q0 -61 24.5 -96.5t65.5 -35.5q51 0 87.5 46.5t50.5 100.5t14 102z" /></g> </svg>) is proportional to the expansion (<svg style="vertical-align:-0.1638pt;width:8.5874996px;" id="M4" height="12.4375" version="1.1" viewBox="0 0 8.5874996 12.4375" width="8.5874996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D703" d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123
t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" /></g> </svg>). This leads to <svg style="vertical-align:-0.23206pt;width:45.25px;" id="M5" height="13.9" version="1.1" viewBox="0 0 45.25 13.9" width="45.25" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g><g transform="matrix(.017,-0,0,-.017,15.718,13.55)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,30.422,13.55)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68
l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g> <g transform="matrix(.012,-0,0,-.012,38.588,5.388)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> </svg>, where <svg style="vertical-align:-0.1092pt;width:11.075px;" id="M6" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D445"/></g> </svg> and <svg style="vertical-align:-0.23206pt;width:8.2875004px;" id="M7" height="11.75" version="1.1" viewBox="0 0 8.2875004 11.75" width="8.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"/></g> </svg> are metric potentials and <svg style="vertical-align:-0.1638pt;width:8.6625004px;" id="M8" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"/></g> </svg> is a constant. We find that the models start with a big bang at initial singularity and expansion decreases due to lapse of time. The anisotropy is maintained throughout but the model isotropizes when <svg style="vertical-align:-0.1638pt;width:36.237499px;" id="M9" height="11.125" version="1.1" viewBox="0 0 36.237499 11.125" width="36.237499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,13.305,10.862)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,28.008,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg>. The physical and geometrical aspects of the model in the presence and absence of magnetic field are also discussed.

Highlights

Bianchi type II space time successfully explains the initial stage of evolution of universe
The pioneer work in the formation of energy momentum tensor for classical massive strings is due to Letelier [5] who explained that the massive strings are formed by geometric strings (Stachel [6]) with particle attached along its extension
It is interesting to discuss whether it is possible to construct an analogue of cosmic string in the presence of magnetic field in the frame work of Lyra’s geometry

Summary

Introduction

Bianchi type II space time successfully explains the initial stage of evolution of universe. The presence of string in the early universe has been explained by Kibble [2], Vilenkin [3], and Zel’dovich [4] using grand unified theories These strings have stress energy and are classified as massive and geometric strings. It is interesting to discuss whether it is possible to construct an analogue of cosmic string in the presence of magnetic field in the frame work of Lyra’s geometry. Bali et al [28] investigated Bianchi type I string dust magnetized cosmological model in the frame work of Lyra’s geometry. We have investigated LRS Bianchi type II massive string cosmological models with magnetic field in Lyra’s geometry. We find that it is possible to construct an analogue of cosmic string solution in presence of magnetic field in the frame work of Lyra geometry. The physical and geometrical aspects of the model together with behavior of the model in the presence and absence of magnetic field are discussed

The Metric and Field Equations

Solution of Field Equations

Model in Absence of Magnetic Field

Discussion