{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Locally nonrotating frames and NEC\n", "\n", "This Jupyter/SageMath notebook is related to the article [Lamy et al, arXiv:1802.01635](https://arxiv.org/abs/1802.01635)." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.1, Release Date: 2017-12-07'" ] }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Metric" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional differentiable manifold M\n" ] } ], "source": [ "M = Manifold(4, 'M')\n", "print(M)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M_1,(t, r, {\\theta}, {\\phi})\\right)$ " ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "M1 = M.open_subset('M_1')\n", "XBL1. = M1.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "XBL1" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M_2,(t, r, {\\theta}, {\\phi})\\right)$ " ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "M2 = M.open_subset('M_2')\n", "forget(r>0)\n", "XBL2. = M2.chart(r't r:(-oo,0) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "M._top_charts.append(XBL2)\n", "XBL2" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|t|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|th|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\theta} > 0, {\\theta} < \\pi, \\verb|ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi\\right]$ " ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "forget(r<0)\n", "assumptions()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "g = M.lorentzian_metric('g')" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "a, b= var('a b')\n", "Sigma = r^2 + a^2*cos(th)^2" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, r^{4}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{\\frac{2 \\, r^{4}}{r^{3} + 2 \\, b^{2}} - a^{2} - r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + {\\left(\\frac{2 \\, a^{2} r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ] }, "execution_count": 9, "metadata": { }, "output_type": "execute_result" } ], "source": [ "m=r^3/(r^3+2*b^2)\n", "m1 = m\n", "Delta1 = r^2 - 2*m*r + a^2\n", "g1 = g.restrict(M1)\n", "g1[0,0] = -(1 - 2*r*m/Sigma)\n", "g1[0,3] = -2*a*r*sin(th)^2*m/Sigma\n", "g1[1,1] = Sigma/Delta1\n", "g1[2,2] = Sigma\n", "g1[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2*m/Sigma)*sin(th)^2\n", "g.display(XBL1.frame())" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, r^{4}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} - 1 \\\\ g_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{\\frac{2 \\, r^{4}}{r^{3} + 2 \\, b^{2}} - a^{2} - r^{2}} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\\\ g_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & {\\left(\\frac{2 \\, a^{2} r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} + 2 \\, b^{2}\\right)}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\end{array}$ " ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.display_comp(XBL1.frame())" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, r^{4}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{\\frac{2 \\, r^{4}}{r^{3} - 2 \\, b^{2}} - a^{2} - r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + {\\left(\\frac{2 \\, a^{2} r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "m = -r^3/(-r^3 + 2*b^2)\n", "m2 = m\n", "Delta2 = r^2 - 2*m*r + a^2\n", "g2 = g.restrict(M2)\n", "g2[0,0] = -(1 - 2*r*m/Sigma)\n", "g2[0,3] = -2*a*r*sin(th)^2*m/Sigma\n", "g2[1,1] = Sigma/Delta2\n", "g2[2,2] = Sigma\n", "g2[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2*m/Sigma)*sin(th)^2\n", "g.display(XBL2.frame())" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, r^{4}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} - 1 \\\\ g_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{\\frac{2 \\, r^{4}}{r^{3} - 2 \\, b^{2}} - a^{2} - r^{2}} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\\\ g_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & {\\left(\\frac{2 \\, a^{2} r^{4} \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(r^{3} - 2 \\, b^{2}\\right)}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\end{array}$ " ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.display_comp(XBL2.frame())" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Locally nonrotating frames" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "A1=(r^2+a^2)^2-a^2*Delta1*sin(th)^2\n", "A2=(r^2+a^2)^2-a^2*Delta2*sin(th)^2" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M_1, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$ " ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b1=XBL1.frame()\n", "b1" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M_1, \\left(e1_{0},e1_{1},e1_{2},e1_{3}\\right)\\right)$ " ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "change_frame = M1.automorphism_field()\n", "change_frame[:] = [[sqrt(A1/(Sigma*Delta1)),0,0,0],\n", " [0,sqrt(Delta1/Sigma),0,0],[0,0,1/sqrt(Sigma),0],\n", " [2*m1*a*r/sqrt(A1*Sigma*Delta1),0,0,sqrt(Sigma/A1)/sin(th)]]; #old covariant basis in terms of the new covariant basis\n", "e1 = b1.new_frame(change_frame, 'e1') ; e1" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M_2, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$ " ] }, "execution_count": 16, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b2=XBL2.frame();\n", "b2" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M_2, \\left(e2_{0},e2_{1},e2_{2},e2_{3}\\right)\\right)$ " ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "change_frame = M2.automorphism_field()\n", "change_frame[:] = [[sqrt(A2/(Sigma*Delta2)),0,0,0],\n", " [0,sqrt(Delta2/Sigma),0,0],[0,0,1/sqrt(Sigma),0],\n", " [2*m2*a*r/sqrt(A2*Sigma*Delta2),0,0,sqrt(Sigma/A2)/sin(th)]]; #old covariant basis in terms of the new covariant basis\n", "e2 = b2.new_frame(change_frame, 'e2') ; e2" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{\\,0\\,0}^{\\phantom{\\, 0}\\phantom{\\, 0}} & = & -1 \\\\ g_{\\,1\\,1}^{\\phantom{\\, 1}\\phantom{\\, 1}} & = & 1 \\\\ g_{\\,2\\,2}^{\\phantom{\\, 2}\\phantom{\\, 2}} & = & 1 \\\\ g_{\\,3\\,3}^{\\phantom{\\, 3}\\phantom{\\, 3}} & = & 1 \\end{array}$ " ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.display_comp(e1)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{\\,0\\,0}^{\\phantom{\\, 0}\\phantom{\\, 0}} & = & -1 \\\\ g_{\\,1\\,1}^{\\phantom{\\, 1}\\phantom{\\, 1}} & = & 1 \\\\ g_{\\,2\\,2}^{\\phantom{\\, 2}\\phantom{\\, 2}} & = & 1 \\\\ g_{\\,3\\,3}^{\\phantom{\\, 3}\\phantom{\\, 3}} & = & 1 \\end{array}$ " ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.display_comp(e2)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Energy momentum tensor" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g)-unnamed metric on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "Ricci = g.ricci()\n", "Ricci_scal = g.ricci_scalar()\n", "G = Ricci-1/2*g*Ricci_scal; print G" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "G1=G.display(e1)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "G2=G.display(e2)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right)-\\mbox{unnamed metric} = \\left( \\frac{12 \\, {\\left(3 \\, a^{2} b^{2} r^{7} + b^{2} r^{9} + 2 \\, b^{4} r^{6} - 2 \\, b^{2} r^{8} - {\\left(a^{4} b^{2} r^{5} - 4 \\, a^{4} b^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + {\\left(a^{4} b^{2} r^{5} - 2 \\, a^{2} b^{2} r^{7} - 4 \\, a^{4} b^{4} r^{2} + 2 \\, a^{2} b^{4} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)}}{r^{15} + 6 \\, b^{2} r^{12} + 12 \\, b^{4} r^{9} + 8 \\, b^{6} r^{6} + {\\left(a^{6} r^{9} + 6 \\, a^{6} b^{2} r^{6} + 12 \\, a^{6} b^{4} r^{3} + 8 \\, a^{6} b^{6}\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{4} r^{11} + 6 \\, a^{4} b^{2} r^{8} + 12 \\, a^{4} b^{4} r^{5} + 8 \\, a^{4} b^{6} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{2} r^{13} + 6 \\, a^{2} b^{2} r^{10} + 12 \\, a^{2} b^{4} r^{7} + 8 \\, a^{2} b^{6} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{12 \\, {\\left({\\left(a^{5} b^{2} r^{5} + a^{3} b^{2} r^{7} - 4 \\, a^{5} b^{4} r^{2} - 4 \\, a^{3} b^{4} r^{4}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{5} b^{2} r^{5} + 4 \\, a^{3} b^{2} r^{7} + 3 \\, a b^{2} r^{9} - 4 \\, a^{5} b^{4} r^{2} - 4 \\, a^{3} b^{4} r^{4} - 2 \\, a b^{2} r^{8}\\right)} \\sin\\left({\\theta}\\right)^{2}\\right)}}{r^{15} + 6 \\, b^{2} r^{12} + 12 \\, b^{4} r^{9} + 8 \\, b^{6} r^{6} + {\\left(a^{6} r^{9} + 6 \\, a^{6} b^{2} r^{6} + 12 \\, a^{6} b^{4} r^{3} + 8 \\, a^{6} b^{6}\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{4} r^{11} + 6 \\, a^{4} b^{2} r^{8} + 12 \\, a^{4} b^{4} r^{5} + 8 \\, a^{4} b^{6} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{2} r^{13} + 6 \\, a^{2} b^{2} r^{10} + 12 \\, a^{2} b^{4} r^{7} + 8 \\, a^{2} b^{6} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{12 \\, b^{2} r^{4}}{a^{2} r^{8} + r^{10} + 4 \\, a^{2} b^{2} r^{5} + 4 \\, b^{2} r^{7} - 2 \\, r^{9} + 4 \\, a^{2} b^{4} r^{2} + 4 \\, b^{4} r^{4} - 4 \\, b^{2} r^{6} + {\\left(a^{4} r^{6} + a^{2} r^{8} + 4 \\, a^{4} b^{2} r^{3} + 4 \\, a^{2} b^{2} r^{5} - 2 \\, a^{2} r^{7} + 4 \\, a^{4} b^{4} + 4 \\, a^{2} b^{4} r^{2} - 4 \\, a^{2} b^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{12 \\, {\\left(2 \\, b^{2} r^{7} - 2 \\, b^{4} r^{4} + {\\left(a^{2} b^{2} r^{5} - 4 \\, a^{2} b^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)}}{r^{11} + 6 \\, b^{2} r^{8} + 12 \\, b^{4} r^{5} + 8 \\, b^{6} r^{2} + {\\left(a^{2} r^{9} + 6 \\, a^{2} b^{2} r^{6} + 12 \\, a^{2} b^{4} r^{3} + 8 \\, a^{2} b^{6}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{12 \\, {\\left({\\left(a^{5} b^{2} r^{5} + a^{3} b^{2} r^{7} - 4 \\, a^{5} b^{4} r^{2} - 4 \\, a^{3} b^{4} r^{4}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{5} b^{2} r^{5} + 4 \\, a^{3} b^{2} r^{7} + 3 \\, a b^{2} r^{9} - 4 \\, a^{5} b^{4} r^{2} - 4 \\, a^{3} b^{4} r^{4} - 2 \\, a b^{2} r^{8}\\right)} \\sin\\left({\\theta}\\right)^{2}\\right)}}{r^{15} + 6 \\, b^{2} r^{12} + 12 \\, b^{4} r^{9} + 8 \\, b^{6} r^{6} + {\\left(a^{6} r^{9} + 6 \\, a^{6} b^{2} r^{6} + 12 \\, a^{6} b^{4} r^{3} + 8 \\, a^{6} b^{6}\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{4} r^{11} + 6 \\, a^{4} b^{2} r^{8} + 12 \\, a^{4} b^{4} r^{5} + 8 \\, a^{4} b^{6} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{2} r^{13} + 6 \\, a^{2} b^{2} r^{10} + 12 \\, a^{2} b^{4} r^{7} + 8 \\, a^{2} b^{6} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{12 \\, {\\left({\\left(a^{6} b^{2} r^{5} + a^{4} b^{2} r^{7} - 4 \\, a^{6} b^{4} r^{2} - 10 \\, a^{4} b^{4} r^{4} - 6 \\, a^{2} b^{4} r^{6} + 2 \\, a^{2} b^{2} r^{8}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{6} b^{2} r^{5} + 4 \\, a^{4} b^{2} r^{7} + 5 \\, a^{2} b^{2} r^{9} + 2 \\, b^{2} r^{11} - 4 \\, a^{6} b^{4} r^{2} - 10 \\, a^{4} b^{4} r^{4} - 8 \\, a^{2} b^{4} r^{6} - 2 \\, b^{4} r^{8}\\right)} \\sin\\left({\\theta}\\right)^{2}\\right)}}{r^{15} + 6 \\, b^{2} r^{12} + 12 \\, b^{4} r^{9} + 8 \\, b^{6} r^{6} + {\\left(a^{6} r^{9} + 6 \\, a^{6} b^{2} r^{6} + 12 \\, a^{6} b^{4} r^{3} + 8 \\, a^{6} b^{6}\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{4} r^{11} + 6 \\, a^{4} b^{2} r^{8} + 12 \\, a^{4} b^{4} r^{5} + 8 \\, a^{4} b^{6} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{2} r^{13} + 6 \\, a^{2} b^{2} r^{10} + 12 \\, a^{2} b^{4} r^{7} + 8 \\, a^{2} b^{6} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ] }, "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Null Energy Condition" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}k1 = e1_{0}+e1_{1}$ " ] }, "execution_count": 24, "metadata": { }, "output_type": "execute_result" } ], "source": [ "k1 = M1.vector_field('k1')\n", "k1[e1,:] = [1,1,0,0]\n", "k1.display(e1)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M_1 & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$ " ] }, "execution_count": 25, "metadata": { }, "output_type": "execute_result" } ], "source": [ "k1d=k1.down(g)\n", "k1k1=k1d['_a']*k1['^a']\n", "k1k1.display()" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}k2 = e2_{0}+e2_{1}$ " ] }, "execution_count": 26, "metadata": { }, "output_type": "execute_result" } ], "source": [ "k2 = M2.vector_field('k2')\n", "k2[e2,:] = [1,1,0,0]\n", "k2.display(e2)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M_2 & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$ " ] }, "execution_count": 27, "metadata": { }, "output_type": "execute_result" } ], "source": [ "k2d=k2.down(g)\n", "k2k2=k2d['_a']*k2['^a']\n", "k2k2.display()" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M_1 & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{96 \\, {\\left({\\left(\\pi a^{6} b^{2} r^{8} + \\pi a^{4} b^{2} r^{10} - 2 \\, \\pi a^{6} b^{4} r^{5} - 2 \\, \\pi a^{4} b^{4} r^{7} - 2 \\, \\pi a^{4} b^{2} r^{9} - 8 \\, \\pi a^{6} b^{6} r^{2} - 8 \\, \\pi a^{4} b^{6} r^{4} + 8 \\, \\pi a^{4} b^{4} r^{6}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(\\pi a^{6} b^{2} r^{8} + 4 \\, \\pi a^{4} b^{2} r^{10} + 3 \\, \\pi a^{2} b^{2} r^{12} - 2 \\, \\pi a^{6} b^{4} r^{5} + 4 \\, \\pi a^{4} b^{4} r^{7} - 6 \\, \\pi a^{2} b^{2} r^{11} - 8 \\, \\pi a^{6} b^{6} r^{2} - 8 \\, \\pi a^{4} b^{6} r^{4} + 8 \\, \\pi a^{4} b^{4} r^{6} - 2 \\, {\\left(\\pi a^{4} b^{2} - 3 \\, \\pi a^{2} b^{4}\\right)} r^{9}\\right)} \\sin\\left({\\theta}\\right)^{2}\\right)}}{a^{2} r^{18} + r^{20} + 8 \\, a^{2} b^{2} r^{15} + 24 \\, a^{2} b^{4} r^{12} + 2 \\, {\\left(a^{2} + 4 \\, b^{2}\\right)} r^{17} + 32 \\, a^{2} b^{6} r^{9} + 16 \\, a^{2} b^{8} r^{6} + 12 \\, {\\left(a^{2} b^{2} + 2 \\, b^{4}\\right)} r^{14} + 8 \\, {\\left(3 \\, a^{2} b^{4} + 4 \\, b^{6}\\right)} r^{11} + 16 \\, {\\left(a^{2} b^{6} + b^{8}\\right)} r^{8} + {\\left(a^{8} r^{12} + a^{6} r^{14} + 8 \\, a^{8} b^{2} r^{9} + 8 \\, a^{6} b^{2} r^{11} - 2 \\, a^{6} r^{13} + 24 \\, a^{8} b^{4} r^{6} + 24 \\, a^{6} b^{4} r^{8} - 12 \\, a^{6} b^{2} r^{10} + 32 \\, a^{8} b^{6} r^{3} + 32 \\, a^{6} b^{6} r^{5} - 24 \\, a^{6} b^{4} r^{7} + 16 \\, a^{8} b^{8} + 16 \\, a^{6} b^{8} r^{2} - 16 \\, a^{6} b^{6} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{6} + {\\left(3 \\, a^{6} r^{14} + 3 \\, a^{4} r^{16} + 24 \\, a^{6} b^{2} r^{11} - 4 \\, a^{4} r^{15} + 72 \\, a^{6} b^{4} r^{8} - 24 \\, a^{4} b^{2} r^{12} + 96 \\, a^{6} b^{6} r^{5} - 48 \\, a^{4} b^{4} r^{9} + 48 \\, a^{6} b^{8} r^{2} - 32 \\, a^{4} b^{6} r^{6} + 2 \\, {\\left(a^{6} + 12 \\, a^{4} b^{2}\\right)} r^{13} + 12 \\, {\\left(a^{6} b^{2} + 6 \\, a^{4} b^{4}\\right)} r^{10} + 24 \\, {\\left(a^{6} b^{4} + 4 \\, a^{4} b^{6}\\right)} r^{7} + 16 \\, {\\left(a^{6} b^{6} + 3 \\, a^{4} b^{8}\\right)} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{4} + {\\left(3 \\, a^{4} r^{16} + 3 \\, a^{2} r^{18} + 24 \\, a^{4} b^{2} r^{13} - 2 \\, a^{2} r^{17} + 72 \\, a^{4} b^{4} r^{10} - 12 \\, a^{2} b^{2} r^{14} + 96 \\, a^{4} b^{6} r^{7} - 24 \\, a^{2} b^{4} r^{11} + 48 \\, a^{4} b^{8} r^{4} - 16 \\, a^{2} b^{6} r^{8} + 4 \\, {\\left(a^{4} + 6 \\, a^{2} b^{2}\\right)} r^{15} + 24 \\, {\\left(a^{4} b^{2} + 3 \\, a^{2} b^{4}\\right)} r^{12} + 48 \\, {\\left(a^{4} b^{4} + 2 \\, a^{2} b^{6}\\right)} r^{9} + 16 \\, {\\left(2 \\, a^{4} b^{6} + 3 \\, a^{2} b^{8}\\right)} r^{6}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\end{array}$ " ] }, "execution_count": 28, "metadata": { }, "output_type": "execute_result" } ], "source": [ "NN=G['_{ab}']*k1['^a']\n", "NEC1=8*pi*NN['_a']*k1['^a']\n", "NEC1.display()" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M_2 & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{96 \\, {\\left({\\left(\\pi a^{6} b^{2} r^{8} + \\pi a^{4} b^{2} r^{10} + 2 \\, \\pi a^{6} b^{4} r^{5} + 2 \\, \\pi a^{4} b^{4} r^{7} - 2 \\, \\pi a^{4} b^{2} r^{9} - 8 \\, \\pi a^{6} b^{6} r^{2} - 8 \\, \\pi a^{4} b^{6} r^{4} - 8 \\, \\pi a^{4} b^{4} r^{6}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(\\pi a^{6} b^{2} r^{8} + 4 \\, \\pi a^{4} b^{2} r^{10} + 3 \\, \\pi a^{2} b^{2} r^{12} + 2 \\, \\pi a^{6} b^{4} r^{5} - 4 \\, \\pi a^{4} b^{4} r^{7} - 6 \\, \\pi a^{2} b^{2} r^{11} - 8 \\, \\pi a^{6} b^{6} r^{2} - 8 \\, \\pi a^{4} b^{6} r^{4} - 8 \\, \\pi a^{4} b^{4} r^{6} - 2 \\, {\\left(\\pi a^{4} b^{2} + 3 \\, \\pi a^{2} b^{4}\\right)} r^{9}\\right)} \\sin\\left({\\theta}\\right)^{2}\\right)}}{a^{2} r^{18} + r^{20} - 8 \\, a^{2} b^{2} r^{15} + 24 \\, a^{2} b^{4} r^{12} + 2 \\, {\\left(a^{2} - 4 \\, b^{2}\\right)} r^{17} - 32 \\, a^{2} b^{6} r^{9} + 16 \\, a^{2} b^{8} r^{6} - 12 \\, {\\left(a^{2} b^{2} - 2 \\, b^{4}\\right)} r^{14} + 8 \\, {\\left(3 \\, a^{2} b^{4} - 4 \\, b^{6}\\right)} r^{11} - 16 \\, {\\left(a^{2} b^{6} - b^{8}\\right)} r^{8} + {\\left(a^{8} r^{12} + a^{6} r^{14} - 8 \\, a^{8} b^{2} r^{9} - 8 \\, a^{6} b^{2} r^{11} - 2 \\, a^{6} r^{13} + 24 \\, a^{8} b^{4} r^{6} + 24 \\, a^{6} b^{4} r^{8} + 12 \\, a^{6} b^{2} r^{10} - 32 \\, a^{8} b^{6} r^{3} - 32 \\, a^{6} b^{6} r^{5} - 24 \\, a^{6} b^{4} r^{7} + 16 \\, a^{8} b^{8} + 16 \\, a^{6} b^{8} r^{2} + 16 \\, a^{6} b^{6} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{6} + {\\left(3 \\, a^{6} r^{14} + 3 \\, a^{4} r^{16} - 24 \\, a^{6} b^{2} r^{11} - 4 \\, a^{4} r^{15} + 72 \\, a^{6} b^{4} r^{8} + 24 \\, a^{4} b^{2} r^{12} - 96 \\, a^{6} b^{6} r^{5} - 48 \\, a^{4} b^{4} r^{9} + 48 \\, a^{6} b^{8} r^{2} + 32 \\, a^{4} b^{6} r^{6} + 2 \\, {\\left(a^{6} - 12 \\, a^{4} b^{2}\\right)} r^{13} - 12 \\, {\\left(a^{6} b^{2} - 6 \\, a^{4} b^{4}\\right)} r^{10} + 24 \\, {\\left(a^{6} b^{4} - 4 \\, a^{4} b^{6}\\right)} r^{7} - 16 \\, {\\left(a^{6} b^{6} - 3 \\, a^{4} b^{8}\\right)} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{4} + {\\left(3 \\, a^{4} r^{16} + 3 \\, a^{2} r^{18} - 24 \\, a^{4} b^{2} r^{13} - 2 \\, a^{2} r^{17} + 72 \\, a^{4} b^{4} r^{10} + 12 \\, a^{2} b^{2} r^{14} - 96 \\, a^{4} b^{6} r^{7} - 24 \\, a^{2} b^{4} r^{11} + 48 \\, a^{4} b^{8} r^{4} + 16 \\, a^{2} b^{6} r^{8} + 4 \\, {\\left(a^{4} - 6 \\, a^{2} b^{2}\\right)} r^{15} - 24 \\, {\\left(a^{4} b^{2} - 3 \\, a^{2} b^{4}\\right)} r^{12} + 48 \\, {\\left(a^{4} b^{4} - 2 \\, a^{2} b^{6}\\right)} r^{9} - 16 \\, {\\left(2 \\, a^{4} b^{6} - 3 \\, a^{2} b^{8}\\right)} r^{6}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\end{array}$ " ] }, "execution_count": 29, "metadata": { }, "output_type": "execute_result" } ], "source": [ "NNN=G['_{ab}']*k2['^a']\n", "NEC2=8*pi*NNN['_a']*k2['^a']\n", "NEC2.display()" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "nec1(a,b,r,th)=NEC1.expr()\n", "nec2(a,b,r,th)=NEC2.expr()" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "a=0.9\n", "b=1\n", "plot1=plot((nec1(a,b,r,pi/6), nec1(a,b,r,pi/3), nec1(a,b,r,pi/2)), (r,0,4), \n", " ymin=-70,ymax=70,\n", " legend_label=[r'$\\theta=\\frac{\\pi}{6}$',r'$\\theta=\\frac{\\pi}{3}$',\n", " r'$\\theta=\\frac{\\pi}{2}$'],\n", " axes_labels=[r'$r/m$',''], plot_points=400)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "a=0.9\n", "b=1\n", "plot2=plot((nec2(a,b,r,pi/6), nec2(a,b,r,pi/3), nec2(a,b,r,pi/2)),(r,-4,0),\n", " axes_labels=[r'$r/m$',r'$T_{\\hat{\\mu}\\hat{\\nu}}k^{\\hat{\\mu}}k^{\\hat{\\nu}}$'],\n", " ymin=-70, ymax=70, plot_points=400)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "038021c57158cb3aa55731cbf005426cb782a5d7" } } ], "source": [ "graph = plot1 + plot2\n", "show(graph)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "graph.save('NEC_a=09_b=1.pdf')" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "ww1(a,b,r,th)=-12*((a^6*b^2*r^8 + a^4*b^2*r^10 - 2*a^6*b^4*r^5 - 2*a^4*b^4*r^7 - 2*a^4*b^2*r^9 - 8*a^6*b^6*r^2 - 8*a^4*b^6*r^4 + 8*a^4*b^4*r^6)*sin(th)^4 - (a^6*b^2*r^8 + 4*a^4*b^2*r^10 + 3*a^2*b^2*r^12 - 2*a^6*b^4*r^5 + 4*a^4*b^4*r^7 - 6*a^2*b^2*r^11 - 8*a^6*b^6*r^2 - 8*a^4*b^6*r^4 + 8*a^4*b^4*r^6 - 2*(a^4*b^2 - 3*a^2*b^4)*r^9)*sin(th)^2)/(a^2*r^18 + r^20 + 8*a^2*b^2*r^15 + 24*a^2*b^4*r^12 + 2*(a^2 + 4*b^2)*r^17 + 32*a^2*b^6*r^9 + 16*a^2*b^8*r^6 + 12*(a^2*b^2 + 2*b^4)*r^14 + 8*(3*a^2*b^4 + 4*b^6)*r^11 + 16*(a^2*b^6 + b^8)*r^8 + (a^8*r^12 + a^6*r^14 + 8*a^8*b^2*r^9 + 8*a^6*b^2*r^11 - 2*a^6*r^13 + 24*a^8*b^4*r^6 + 24*a^6*b^4*r^8 - 12*a^6*b^2*r^10 + 32*a^8*b^6*r^3 + 32*a^6*b^6*r^5 - 24*a^6*b^4*r^7 + 16*a^8*b^8 + 16*a^6*b^8*r^2 - 16*a^6*b^6*r^4)*cos(th)^6 + (3*a^6*r^14 + 3*a^4*r^16 + 24*a^6*b^2*r^11 - 4*a^4*r^15 + 72*a^6*b^4*r^8 - 24*a^4*b^2*r^12 + 96*a^6*b^6*r^5 - 48*a^4*b^4*r^9 + 48*a^6*b^8*r^2 - 32*a^4*b^6*r^6 + 2*(a^6 + 12*a^4*b^2)*r^13 + 12*(a^6*b^2 + 6*a^4*b^4)*r^10 + 24*(a^6*b^4 + 4*a^4*b^6)*r^7 + 16*(a^6*b^6 + 3*a^4*b^8)*r^4)*cos(th)^4 + (3*a^4*r^16 + 3*a^2*r^18 + 24*a^4*b^2*r^13 - 2*a^2*r^17 + 72*a^4*b^4*r^10 - 12*a^2*b^2*r^14 + 96*a^4*b^6*r^7 - 24*a^2*b^4*r^11 + 48*a^4*b^8*r^4 - 16*a^2*b^6*r^8 + 4*(a^4 + 6*a^2*b^2)*r^15 + 24*(a^4*b^2 + 3*a^2*b^4)*r^12 + 48*(a^4*b^4 + 2*a^2*b^6)*r^9 + 16*(2*a^4*b^6 + 3*a^2*b^8)*r^6)*cos(th)^2)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "ww2(a,b,r,th)=12*((a^6*b^2*r^8 + a^4*b^2*r^10 + 2*a^6*b^4*r^5 + 2*a^4*b^4*r^7 - 2*a^4*b^2*r^9 - 8*a^6*b^6*r^2 - 8*a^4*b^6*r^4 - 8*a^4*b^4*r^6)*sin(th)^4 - (a^6*b^2*r^8 + 4*a^4*b^2*r^10 + 3*a^2*b^2*r^12 + 2*a^6*b^4*r^5 - 4*a^4*b^4*r^7 - 6*a^2*b^2*r^11 - 8*a^6*b^6*r^2 - 8*a^4*b^6*r^4 - 8*a^4*b^4*r^6 - 2*(a^4*b^2 + 3*a^2*b^4)*r^9)*sin(th)^2)/(a^2*r^18 + r^20 - 8*a^2*b^2*r^15 + 24*a^2*b^4*r^12 + 2*(a^2 - 4*b^2)*r^17 - 32*a^2*b^6*r^9 + 16*a^2*b^8*r^6 - 12*(a^2*b^2 - 2*b^4)*r^14 + 8*(3*a^2*b^4 - 4*b^6)*r^11 - 16*(a^2*b^6 - b^8)*r^8 + (a^8*r^12 + a^6*r^14 - 8*a^8*b^2*r^9 - 8*a^6*b^2*r^11 - 2*a^6*r^13 + 24*a^8*b^4*r^6 + 24*a^6*b^4*r^8 + 12*a^6*b^2*r^10 - 32*a^8*b^6*r^3 - 32*a^6*b^6*r^5 - 24*a^6*b^4*r^7 + 16*a^8*b^8 + 16*a^6*b^8*r^2 + 16*a^6*b^6*r^4)*cos(th)^6 + (3*a^6*r^14 + 3*a^4*r^16 - 24*a^6*b^2*r^11 - 4*a^4*r^15 + 72*a^6*b^4*r^8 + 24*a^4*b^2*r^12 - 96*a^6*b^6*r^5 - 48*a^4*b^4*r^9 + 48*a^6*b^8*r^2 + 32*a^4*b^6*r^6 + 2*(a^6 - 12*a^4*b^2)*r^13 - 12*(a^6*b^2 - 6*a^4*b^4)*r^10 + 24*(a^6*b^4 - 4*a^4*b^6)*r^7 - 16*(a^6*b^6 - 3*a^4*b^8)*r^4)*cos(th)^4 + (3*a^4*r^16 + 3*a^2*r^18 - 24*a^4*b^2*r^13 - 2*a^2*r^17 + 72*a^4*b^4*r^10 + 12*a^2*b^2*r^14 - 96*a^4*b^6*r^7 - 24*a^2*b^4*r^11 + 48*a^4*b^8*r^4 + 16*a^2*b^6*r^8 + 4*(a^4 - 6*a^2*b^2)*r^15 - 24*(a^4*b^2 - 3*a^2*b^4)*r^12 + 48*(a^4*b^4 - 2*a^2*b^6)*r^9 - 16*(2*a^4*b^6 - 3*a^2*b^8)*r^6)*cos(th)^2)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.1", "name": "sage-8.1" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 0 }