{ "cells": [ { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|SageMath|\\phantom{\\verb!x!}\\verb|version|\\phantom{\\verb!x!}\\verb|9.0,|\\phantom{\\verb!x!}\\verb|Release|\\phantom{\\verb!x!}\\verb|Date:|\\phantom{\\verb!x!}\\verb|2020-01-01|$ " ], "text/plain": [ "'SageMath version 9.0, Release Date: 2020-01-01'" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}gT = e^{\\left(x y\\right)} \\mathrm{d} u\\otimes \\mathrm{d} u -\\frac{1}{2} \\mathrm{d} u\\otimes \\mathrm{d} v -\\frac{1}{2} \\mathrm{d} v\\otimes \\mathrm{d} u +\\mathrm{d} x\\otimes \\mathrm{d} x +\\mathrm{d} y\\otimes \\mathrm{d} y$ " ], "text/plain": [ "gT = e^(x*y) du*du - 1/2 du*dv - 1/2 dv*du + dx*dx + dy*dy" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "CPU times: user 4.98 s, sys: 148 ms, total: 5.12 s\n", "Wall time: 4.91 s\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, {\\left(x^{4} + 2 \\, x^{2} y^{2} + y^{4} + 8 \\, x y + 4\\right)} e^{\\left(x y\\right)} \\mathrm{d} u\\otimes \\mathrm{d} u$ " ], "text/plain": [ "1/2*(x^4 + 2*x^2*y^2 + y^4 + 8*x*y + 4)*e^(x*y) du*du" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "CPU times: user 22.2 s, sys: 643 ms, total: 22.9 s\n", "Wall time: 21.6 s\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, {\\left(x^{4} + 2 \\, x^{2} y^{2} + y^{4} + 8 \\, x y + 4\\right)} e^{\\left(x y\\right)} \\mathrm{d} u\\otimes \\mathrm{d} u$ " ], "text/plain": [ "-1/4*(x^4 + 2*x^2*y^2 + y^4 + 8*x*y + 4)*e^(x*y) du*du" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "CPU times: user 5.01 s, sys: 130 ms, total: 5.14 s\n", "Wall time: 4.79 s\n", "CPU times: user 1.59 s, sys: 37.4 ms, total: 1.62 s\n", "Wall time: 1.48 s\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, {\\left(x^{4} + 2 \\, x^{2} y^{2} + y^{4} + 8 \\, x y + 4\\right)} e^{\\left(x y\\right)} \\mathrm{d} u\\otimes \\mathrm{d} u$ " ], "text/plain": [ "-1/4*(x^4 + 2*x^2*y^2 + y^4 + 8*x*y + 4)*e^(x*y) du*du" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(version())\n", "#MacBook Pro dualcore i5, hyperthreaded. Should support 4 ply multiprocessing.\n", "\n", "%display latex # LaTeX rendering turned on\n", "\n", "Parallelism().set('tensor',1) #Only works if set to one.\n", "Parallelism().set('linbox',4) \n", "Parallelism()\n", "\n", "M = Manifold(4, 'M')\n", "MChart = M.open_subset('MChart')\n", "Chart. = MChart.chart(r'u:(-oo,+oo) v:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo)') \n", "\n", "gT= MChart.riemannian_metric('gT')\n", "var('du','dv','dx','dy')\n", "\n", "dsds= -du*dv+dx*dx+dy*dy+e^(x*y)*du*du\n", "\n", "\n", "dsds=dsds.expand()\n", "g00=dsds.coefficient(du,2)\n", "g11=dsds.coefficient(dv,2)\n", "g22=dsds.coefficient(dx,2)\n", "g33=dsds.coefficient(dy,2)\n", "g01=dsds.coefficient(du*dv,1)\n", "g01=g01/2\n", "g10=g01\n", "\n", "gT[0,0] = g00.factor() #du du\n", "gT[1,1] = g11.factor() #dv dv\n", "gT[2,2] = g22.factor() #dx dx\n", "gT[3,3] = g33.factor() #dy dy\n", "gT[0,1] = g01.factor() #du dv \n", "%display latex\n", "show(gT.display())\n", "\n", "Metric=gT\n", "\n", "Nabla = Metric.connection()\n", "\n", "#https://arxiv.org/pdf/gr-qc/0309008.pdf equation 54\n", "%time Bach=(Nabla(Metric.cotton()).up(Metric,3)['^u_aub'])+((Metric.schouten().up(Metric))*(Metric.weyl().down(Metric)))['^uv_aubv']\n", "show(Bach.display())\n", "%time Bach=Nabla(Nabla(Metric.weyl().down(Metric))).up(Metric,4).up(Metric,5)['^bd_abcd']-(1/2)*((Metric.ricci().up(Metric))*(Metric.weyl().down(Metric)))['^bd_abcd']\n", "show(Bach.display())\n", "%time Bach=-(Metric.schouten()*(Metric.weyl().down(Metric))).up(Metric,1).up(Metric,2)['^cd_cdab'] - Nabla(Nabla(Metric.schouten())).up(Metric,3)['^c_cab'] + Nabla(Nabla(Metric.schouten())).up(Metric,3)['^c_abc']\n", "\n", "Cotton=-Nabla(Metric.ricci())['_i[jk]']-1/(2*(4-1))*Nabla(Metric.ricci_scalar()*Metric)['_i[kj]']\n", "\n", "%time Bach=(Nabla(Cotton).up(Metric,3)['^u_aub'])+((Metric.schouten().up(Metric))*(Metric.weyl().down(Metric)))['^uv_aubv']\n", "show(Bach.display())" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "Nabla = Metric.connection()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, v} \\, u \\, x }^{ \\, v \\phantom{\\, u} \\phantom{\\, x} } & = & -y e^{\\left(x y\\right)} \\\\ \\Gamma_{ \\phantom{\\, v} \\, u \\, y }^{ \\, v \\phantom{\\, u} \\phantom{\\, y} } & = & -x e^{\\left(x y\\right)} \\\\ \\Gamma_{ \\phantom{\\, v} \\, x \\, u }^{ \\, v \\phantom{\\, x} \\phantom{\\, u} } & = & -y e^{\\left(x y\\right)} \\\\ \\Gamma_{ \\phantom{\\, v} \\, y \\, u }^{ \\, v \\phantom{\\, y} \\phantom{\\, u} } & = & -x e^{\\left(x y\\right)} \\\\ \\Gamma_{ \\phantom{\\, x} \\, u \\, u }^{ \\, x \\phantom{\\, u} \\phantom{\\, u} } & = & -\\frac{1}{2} \\, y e^{\\left(x y\\right)} \\\\ \\Gamma_{ \\phantom{\\, y} \\, u \\, u }^{ \\, y \\phantom{\\, u} \\phantom{\\, u} } & = & -\\frac{1}{2} \\, x e^{\\left(x y\\right)} \\end{array}$ " ], "text/plain": [ "Gam^v_ux = -y*e^(x*y) \n", "Gam^v_uy = -x*e^(x*y) \n", "Gam^v_xu = -y*e^(x*y) \n", "Gam^v_yu = -x*e^(x*y) \n", "Gam^x_uu = -1/2*y*e^(x*y) \n", "Gam^y_uu = -1/2*x*e^(x*y) " ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Nabla.display()" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}gMann = \\left( -\\frac{a^{4} k \\cos\\left(\\mathit{th}\\right)^{4} - k r^{4} + a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2} + r u}{a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{a k r^{4} \\cos\\left(\\mathit{th}\\right)^{2} - a k r^{4} + {\\left(a^{5} k + a^{3} k r^{2}\\right)} \\cos\\left(\\mathit{th}\\right)^{4} - {\\left(a r \\cos\\left(\\mathit{th}\\right)^{2} - a r\\right)} u}{a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} \\phi + \\left( -\\frac{a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2}}{k r^{4} - a^{2} - r^{2} - r u} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( -\\frac{{\\left(a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2}\\right)} \\sin\\left(\\mathit{th}\\right)^{2}}{a^{2} k \\cos\\left(\\mathit{th}\\right)^{4} + \\cos\\left(\\mathit{th}\\right)^{2} - 1} \\right) \\mathrm{d} \\mathit{th}\\otimes \\mathrm{d} \\mathit{th} + \\left( \\frac{a k r^{4} \\cos\\left(\\mathit{th}\\right)^{2} - a k r^{4} + {\\left(a^{5} k + a^{3} k r^{2}\\right)} \\cos\\left(\\mathit{th}\\right)^{4} - {\\left(a r \\cos\\left(\\mathit{th}\\right)^{2} - a r\\right)} u}{a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2}} \\right) \\mathrm{d} \\phi\\otimes \\mathrm{d} t + \\left( \\frac{{\\left(a^{2} k + 1\\right)} r^{4} - {\\left(a^{6} k + a^{4} + {\\left(2 \\, a^{4} k + a^{2}\\right)} r^{2}\\right)} \\cos\\left(\\mathit{th}\\right)^{4} + a^{2} r^{2} - {\\left({\\left(2 \\, a^{2} k + 1\\right)} r^{4} - a^{4}\\right)} \\cos\\left(\\mathit{th}\\right)^{2} - {\\left(a^{2} r \\cos\\left(\\mathit{th}\\right)^{4} - 2 \\, a^{2} r \\cos\\left(\\mathit{th}\\right)^{2} + a^{2} r\\right)} u}{a^{2} \\cos\\left(\\mathit{th}\\right)^{2} + r^{2}} \\right) \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi$ " ], "text/plain": [ "gMann = -(a^4*k*cos(th)^4 - k*r^4 + a^2*cos(th)^2 + r^2 + r*u)/(a^2*cos(th)^2 + r^2) dt*dt + (a*k*r^4*cos(th)^2 - a*k*r^4 + (a^5*k + a^3*k*r^2)*cos(th)^4 - (a*r*cos(th)^2 - a*r)*u)/(a^2*cos(th)^2 + r^2) dt*dphi - (a^2*cos(th)^2 + r^2)/(k*r^4 - a^2 - r^2 - r*u) dr*dr - (a^2*cos(th)^2 + r^2)*sin(th)^2/(a^2*k*cos(th)^4 + cos(th)^2 - 1) dth*dth + (a*k*r^4*cos(th)^2 - a*k*r^4 + (a^5*k + a^3*k*r^2)*cos(th)^4 - (a*r*cos(th)^2 - a*r)*u)/(a^2*cos(th)^2 + r^2) dphi*dt + ((a^2*k + 1)*r^4 - (a^6*k + a^4 + (2*a^4*k + a^2)*r^2)*cos(th)^4 + a^2*r^2 - ((2*a^2*k + 1)*r^4 - a^4)*cos(th)^2 - (a^2*r*cos(th)^4 - 2*a^2*r*cos(th)^2 + a^2*r)*u)/(a^2*cos(th)^2 + r^2) dphi*dphi" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ ".\n" ] } ], "source": [ "M = Manifold(4, 'M')\n", "MChart = M.open_subset('MChart')\n", "Chart. = MChart.chart(r't:(-oo,+oo) r:(-oo,+oo) th:(-oo,+oo) phi:(-oo,+oo)') \n", "var('a','A','B','C','D','E','F')\n", "var('k','u')\n", "gMann= MChart.lorentzian_metric('gMann')\n", "var('dt','dr','dth','dphi')\n", "#https://arxiv.org/pdf/1401.6503.pdf\n", "y=cos(th)\n", "dy=diff(y)*dth\n", "x=r\n", "dx=diff(x)*dr\n", "A=a^2+u*x+x^2-k*x^4\n", "B=a^2+x^2\n", "C=a\n", "D=1-y^2-a^2*k*y^4\n", "E=a*(1-y^2)\n", "F=1\n", "\n", "dsds=((B*F-C*E)*(dx^2/A+dy^2/D))+(1/(B*F-C*E))*((D*(B*dphi-C*dt)^2-A*(E*dphi-F*dt)^2)).expand()\n", "\n", "\n", "dsds=dsds.expand()\n", "g00=dsds.coefficient(dt,2)\n", "g11=dsds.coefficient(dr,2)\n", "g22=dsds.coefficient(dth,2)\n", "g33=dsds.coefficient(dphi,2)\n", "g03=dsds.coefficient(dt*dphi,1)\n", "g03=g03/2\n", "g30=g03\n", "\n", "gMann[0,0] = g00.full_simplify() #du du\n", "gMann[1,1] = g11.full_simplify() #dv dv\n", "gMann[2,2] = g22.full_simplify() #dx dx\n", "gMann[3,3] = g33.full_simplify() #dy dy\n", "gMann[0,3] = g03.full_simplify() #du dv \n", "%display latex\n", "show(gMann.display())\n", "\n", "Metric=gMann\n", "\n", "Nabla = Metric.connection()\n", "\n", "#https://arxiv.org/pdf/gr-qc/0309008.pdf equation 54\n", "#%time Bach=(Nabla(Metric.cotton()).up(Metric,3)['^u_aub'])+((Metric.schouten().up(Metric))*(Metric.weyl().down(Metric)))['^uv_aubv']\n", "#%time Bach=Nabla(Nabla(Metric.weyl().down(Metric))).up(Metric,4).up(Metric,5)['^bd_abcd']-(1/2)*((Metric.ricci().up(Metric))*(Metric.weyl().down(Metric)))['^bd_abcd']\n", "print('.')\n", "#Cotton=-Nabla(Metric.ricci())['_i[jk]']-1/(2*(4-1))*Nabla(Metric.ricci_scalar()*Metric)['_i[kj]']\n", "#show(Cotton.display())\n", "#%time Bach=(Nabla(Cotton).up(Metric,3)['^u_aub'])+((Metric.schouten().up(Metric))*(Metric.weyl().down(Metric)))['^uv_aubv']\n", "#%time Bach=-(Metric.schouten()*(Metric.weyl().down(Metric))).up(Metric,1).up(Metric,2)['^cd_cdab'] - Nabla(Nabla(Metric.schouten())).up(Metric,3)['^c_cab'] + Nabla(Nabla(Metric.schouten())).up(Metric,3)['^c_abc']\n", "#Bach.display()\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "\n", "%display latex\n", "dsds=((B*F-C*E)*(dx^2/A+dy^2/D))+(1/(B*F-C*E))*((D*(B*dphi-C*dt)^2-A*(E*dphi-F*dt)^2)).expand()\n", "\n", "dsthth=dsds.coefficient(dth,2)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]$ " ], "text/plain": [ "[]" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve((dsthth).denominator()==0,th)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a^{2} k \\cos\\left(\\mathit{th}\\right)^{4} + \\cos\\left(\\mathit{th}\\right)^{2} - 1$ " ], "text/plain": [ "a^2*k*cos(th)^4 + cos(th)^2 - 1" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dsthth.denominator().full_simplify()" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(dsthth.denominator().subs(a=9/10,k=1),th,-pi/2,pi/2,color='blue')+plot(dsthth.denominator().subs(a=9/10,k=1/100),th,-pi/2,pi/2,color='red')+plot(dsthth.denominator().subs(a=9/10,k=0),th,-pi/2,pi/2,color='green')" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left\\{\\mathit{th} : \\pi - \\arccos\\left(\\frac{1}{6} \\, \\sqrt{10} \\sqrt{\\left(\\frac{1}{9}\\right)^{\\frac{1}{3}} {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}} - \\frac{12 \\, \\left(\\frac{1}{9}\\right)^{\\frac{2}{3}}}{{\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}}}} + \\frac{1}{2} \\, \\sqrt{-\\frac{10}{9} \\, \\left(\\frac{1}{9}\\right)^{\\frac{1}{3}} {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}} + \\frac{40 \\, \\left(\\frac{1}{9}\\right)^{\\frac{2}{3}}}{3 \\, {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}}} + \\frac{20 \\, \\sqrt{10}}{27 \\, \\sqrt{\\left(\\frac{1}{9}\\right)^{\\frac{1}{3}} {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}} - \\frac{12 \\, \\left(\\frac{1}{9}\\right)^{\\frac{2}{3}}}{{\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}}}}}}\\right)\\right\\}, \\left\\{\\mathit{th} : \\pi - \\arccos\\left(\\frac{1}{6} \\, \\sqrt{10} \\sqrt{\\left(\\frac{1}{9}\\right)^{\\frac{1}{3}} {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}} - \\frac{12 \\, \\left(\\frac{1}{9}\\right)^{\\frac{2}{3}}}{{\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}}}} - \\frac{1}{2} \\, \\sqrt{-\\frac{10}{9} \\, \\left(\\frac{1}{9}\\right)^{\\frac{1}{3}} {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}} + \\frac{40 \\, \\left(\\frac{1}{9}\\right)^{\\frac{2}{3}}}{3 \\, {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}}} + \\frac{20 \\, \\sqrt{10}}{27 \\, \\sqrt{\\left(\\frac{1}{9}\\right)^{\\frac{1}{3}} {\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}} - \\frac{12 \\, \\left(\\frac{1}{9}\\right)^{\\frac{2}{3}}}{{\\left(\\sqrt{217} + 5\\right)}^{\\frac{1}{3}}}}}}\\right)\\right\\}\\right]$ " ], "text/plain": [ "[{th: pi - arccos(1/6*sqrt(10)*sqrt((1/9)^(1/3)*(sqrt(217) + 5)^(1/3) - 12*(1/9)^(2/3)/(sqrt(217) + 5)^(1/3)) + 1/2*sqrt(-10/9*(1/9)^(1/3)*(sqrt(217) + 5)^(1/3) + 40/3*(1/9)^(2/3)/(sqrt(217) + 5)^(1/3) + 20/27*sqrt(10)/sqrt((1/9)^(1/3)*(sqrt(217) + 5)^(1/3) - 12*(1/9)^(2/3)/(sqrt(217) + 5)^(1/3))))},\n", " {th: pi - arccos(1/6*sqrt(10)*sqrt((1/9)^(1/3)*(sqrt(217) + 5)^(1/3) - 12*(1/9)^(2/3)/(sqrt(217) + 5)^(1/3)) - 1/2*sqrt(-10/9*(1/9)^(1/3)*(sqrt(217) + 5)^(1/3) + 40/3*(1/9)^(2/3)/(sqrt(217) + 5)^(1/3) + 20/27*sqrt(10)/sqrt((1/9)^(1/3)*(sqrt(217) + 5)^(1/3) - 12*(1/9)^(2/3)/(sqrt(217) + 5)^(1/3))))}]" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "solve((a^2*k*cos(th)^4+cos(th)^1-1==0).subs(a=9/10,k=1/1000000),th, solution_dict=True)\n", "[[s[th].n(5)] for s in solns]\n", "solns" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.0", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }