{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Asymptotic 5D Kerr-AdS metric with b=0 in light-cone coordinates\n", "\n", "This [SageMath](https://www.sagemath.org/) notebook is relative to the article *Heavy quarks in rotating plasma via holography* by Anastasia A. Golubtsova, Eric Gourgoulhon and Marina K. Usova, [arXiv:2107.11672](https://arxiv.org/abs/2107.11672).\n", "\n", "The involved differential geometry computations are based on tools developed through the [SageManifolds](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.3, Release Date: 2021-05-09'" ] }, "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": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "M = Manifold(5, 'M', r'\\mathcal{M}', structure='Lorentzian', metric_name='G')\n", "print(M)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Asymptotically AdS coordinates" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(T, y, {\\Theta}, {\\Phi}, {\\Psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(T, y, {\\Theta}, {\\Phi}, {\\Psi})\\right)$$" ], "text/plain": [ "Chart (M, (T, y, Th, Ph, Ps))" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "AdSc. = M.chart(r'T y:(0,+oo) Th:(0,pi/2):\\Theta Ph:(0,2*pi):\\Phi Ps:(0,2*pi):\\Psi')\n", "AdSc" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$$" ], "text/plain": [ "(m, a, b)" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('m a b', domain='real')" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "b = 0 # assumed in Sec. 5" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "keep_Delta = True # change to False to provide explicit expression for Delta" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "if keep_Delta:\n", " Delta = var('Delta', latex_name=r'\\Delta', domain='real')\n", "else:\n", " Delta = 1 - a^2*sin(Th)^2 - b^2*cos(Th)^2" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "G = M.metric()\n", "G[0,0] = - (1 + y^2) + 2*m/(Delta^3*y^2)\n", "G[0,3] = -2*a*m*sin(Th)^2/(Delta^3*y^2)\n", "G[0,4] = -2*b*m*cos(Th)^2/(Delta^3*y^2)\n", "G[1,1] = 1/(1 + y^2 - 2*m/(Delta^2*y^2))\n", "G[2,2] = y^2\n", "G[3,3] = y^2*sin(Th)^2 + 2*a^2*m*sin(Th)^4/(Delta^3*y^2)\n", "G[3,4] = 2*a*b*m*sin(Th)^2*cos(Th)^2/(Delta^3*y^2)\n", "G[4,4] = y^2*cos(Th)^2 + 2*b^2*m*cos(Th)^4/(Delta^3*y^2)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check of Eq. (5.38):" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T -\\frac{2 \\, a m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Phi} + \\left( \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{2} y^{2}} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + y^{2} \\mathrm{d} {\\Theta}\\otimes \\mathrm{d} {\\Theta} -\\frac{2 \\, a m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} T + \\left( y^{2} \\sin\\left({\\Theta}\\right)^{2} + \\frac{2 \\, a^{2} m \\sin\\left({\\Theta}\\right)^{4}}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Phi} + y^{2} \\cos\\left({\\Theta}\\right)^{2} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T -\\frac{2 \\, a m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Phi} + \\left( \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{2} y^{2}} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + y^{2} \\mathrm{d} {\\Theta}\\otimes \\mathrm{d} {\\Theta} -\\frac{2 \\, a m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} T + \\left( y^{2} \\sin\\left({\\Theta}\\right)^{2} + \\frac{2 \\, a^{2} m \\sin\\left({\\Theta}\\right)^{4}}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Phi} + y^{2} \\cos\\left({\\Theta}\\right)^{2} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$$" ], "text/plain": [ "G = (-y^2 + 2*m/(Delta^3*y^2) - 1) dT*dT - 2*a*m*sin(Th)^2/(Delta^3*y^2) dT*dPh + 1/(y^2 - 2*m/(Delta^2*y^2) + 1) dy*dy + y^2 dTh*dTh - 2*a*m*sin(Th)^2/(Delta^3*y^2) dPh*dT + (y^2*sin(Th)^2 + 2*a^2*m*sin(Th)^4/(Delta^3*y^2)) dPh*dPh + y^2*cos(Th)^2 dPs*dPs" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, T \\, T }^{ \\phantom{\\, T}\\phantom{\\, T} } & = & -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\\\ G_{ \\, T \\, {\\Phi} }^{ \\phantom{\\, T}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{2} y^{2}} + 1} \\\\ G_{ \\, {\\Theta} \\, {\\Theta} }^{ \\phantom{\\, {\\Theta}}\\phantom{\\, {\\Theta}} } & = & y^{2} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & y^{2} \\sin\\left({\\Theta}\\right)^{2} + \\frac{2 \\, a^{2} m \\sin\\left({\\Theta}\\right)^{4}}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & y^{2} \\cos\\left({\\Theta}\\right)^{2} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, T \\, T }^{ \\phantom{\\, T}\\phantom{\\, T} } & = & -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\\\ G_{ \\, T \\, {\\Phi} }^{ \\phantom{\\, T}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{2} y^{2}} + 1} \\\\ G_{ \\, {\\Theta} \\, {\\Theta} }^{ \\phantom{\\, {\\Theta}}\\phantom{\\, {\\Theta}} } & = & y^{2} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & y^{2} \\sin\\left({\\Theta}\\right)^{2} + \\frac{2 \\, a^{2} m \\sin\\left({\\Theta}\\right)^{4}}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & y^{2} \\cos\\left({\\Theta}\\right)^{2} \\end{array}$$" ], "text/plain": [ "G_T,T = -y^2 + 2*m/(Delta^3*y^2) - 1 \n", "G_T,Ph = -2*a*m*sin(Th)^2/(Delta^3*y^2) \n", "G_y,y = 1/(y^2 - 2*m/(Delta^2*y^2) + 1) \n", "G_Th,Th = y^2 \n", "G_Ph,Ph = y^2*sin(Th)^2 + 2*a^2*m*sin(Th)^4/(Delta^3*y^2) \n", "G_Ps,Ps = y^2*cos(Th)^2 " ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Light cone coordinates" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},({x^+}, {x^-}, y, {\\Theta}, {\\Psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},({x^+}, {x^-}, y, {\\Theta}, {\\Psi})\\right)$$" ], "text/plain": [ "Chart (M, (xp, xm, y, Th, Ps))" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "LC. = M.chart(r'xp:x^+ xm:x^- y:(0,+oo) Th:(0,pi/2):\\Theta Ps:(0,2*pi):\\Psi')\n", "LC" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The transformation from AdS coordinates to light cone coordinates is defined by Eq. (5.40) of the paper:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {x^+} & = & -{\\Phi} a + T \\\\ {x^-} & = & {\\Phi} a + T \\\\ y & = & y \\\\ {\\Theta} & = & {\\Theta} \\\\ {\\Psi} & = & {\\Psi} \\end{array}\\right.$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {x^+} & = & -{\\Phi} a + T \\\\ {x^-} & = & {\\Phi} a + T \\\\ y & = & y \\\\ {\\Theta} & = & {\\Theta} \\\\ {\\Psi} & = & {\\Psi} \\end{array}\\right.$$" ], "text/plain": [ "xp = -Ph*a + T\n", "xm = Ph*a + T\n", "y = y\n", "Th = Th\n", "Ps = Ps" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "AdSc_to_LC = AdSc.transition_map(LC, [T - a*Ph, T + a*Ph, y, Th, Ps])\n", "AdSc_to_LC.display()" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} T & = & \\frac{1}{2} \\, {x^-} + \\frac{1}{2} \\, {x^+} \\\\ y & = & y \\\\ {\\Theta} & = & {\\Theta} \\\\ {\\Phi} & = & \\frac{{x^-} - {x^+}}{2 \\, a} \\\\ {\\Psi} & = & {\\Psi} \\end{array}\\right.$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} T & = & \\frac{1}{2} \\, {x^-} + \\frac{1}{2} \\, {x^+} \\\\ y & = & y \\\\ {\\Theta} & = & {\\Theta} \\\\ {\\Phi} & = & \\frac{{x^-} - {x^+}}{2 \\, a} \\\\ {\\Psi} & = & {\\Psi} \\end{array}\\right.$$" ], "text/plain": [ "T = 1/2*xm + 1/2*xp\n", "y = y\n", "Th = Th\n", "Ph = 1/2*(xm - xp)/a\n", "Ps = Ps" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "AdSc_to_LC.inverse().display()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} - 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} + 2 \\, \\sin\\left({\\Theta}\\right)^{2} + 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} + 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} - 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^-} + \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} + 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} - 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} - 2 \\, a^{2} m \\cos\\left({\\Theta}\\right)^{4} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\cos\\left({\\Theta}\\right)^{2} - {\\Delta}^{3}\\right)} y^{4}}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^-} + \\left( \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + y^{2} \\mathrm{d} {\\Theta}\\otimes \\mathrm{d} {\\Theta} + y^{2} \\cos\\left({\\Theta}\\right)^{2} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} - 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} + 2 \\, \\sin\\left({\\Theta}\\right)^{2} + 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} + 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} - 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^-} + \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} + 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} - 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{{\\Delta}^{3} a^{2} y^{2} - 2 \\, a^{2} m \\cos\\left({\\Theta}\\right)^{4} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\cos\\left({\\Theta}\\right)^{2} - {\\Delta}^{3}\\right)} y^{4}}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^-} + \\left( \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + y^{2} \\mathrm{d} {\\Theta}\\otimes \\mathrm{d} {\\Theta} + y^{2} \\cos\\left({\\Theta}\\right)^{2} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$$" ], "text/plain": [ "G = -1/4*(Delta^3*a^2*y^2 + (Delta^3*a^2 - Delta^3*sin(Th)^2)*y^4 - 2*(sin(Th)^4 + 2*sin(Th)^2 + 1)*a^2*m)/(Delta^3*a^2*y^2) dxp*dxp - 1/4*(Delta^3*a^2*y^2 + (Delta^3*a^2 + Delta^3*sin(Th)^2)*y^4 + 2*(sin(Th)^4 - 1)*a^2*m)/(Delta^3*a^2*y^2) dxp*dxm - 1/4*(Delta^3*a^2*y^2 + (Delta^3*a^2 + Delta^3*sin(Th)^2)*y^4 + 2*(sin(Th)^4 - 1)*a^2*m)/(Delta^3*a^2*y^2) dxm*dxp - 1/4*(Delta^3*a^2*y^2 - 2*a^2*m*cos(Th)^4 + (Delta^3*a^2 + Delta^3*cos(Th)^2 - Delta^3)*y^4)/(Delta^3*a^2*y^2) dxm*dxm + Delta^2*y^2/(Delta^2*y^4 + Delta^2*y^2 - 2*m) dy*dy + y^2 dTh*dTh + y^2*cos(Th)^2 dPs*dPs" ] }, "execution_count": 16, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display(LC)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, {x^+} \\, {x^+} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^+}} } & = & -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} - 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} + 2 \\, \\sin\\left({\\Theta}\\right)^{2} + 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\\\ G_{ \\, {x^+} \\, {x^-} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^-}} } & = & -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} + 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} - 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\\\ G_{ \\, {x^-} \\, {x^-} }^{ \\phantom{\\, {x^-}}\\phantom{\\, {x^-}} } & = & -\\frac{{\\Delta}^{3} a^{2} y^{2} - 2 \\, a^{2} m \\cos\\left({\\Theta}\\right)^{4} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\cos\\left({\\Theta}\\right)^{2} - {\\Delta}^{3}\\right)} y^{4}}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\\\ G_{ \\, {\\Theta} \\, {\\Theta} }^{ \\phantom{\\, {\\Theta}}\\phantom{\\, {\\Theta}} } & = & y^{2} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & y^{2} \\cos\\left({\\Theta}\\right)^{2} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, {x^+} \\, {x^+} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^+}} } & = & -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} - 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} + 2 \\, \\sin\\left({\\Theta}\\right)^{2} + 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\\\ G_{ \\, {x^+} \\, {x^-} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^-}} } & = & -\\frac{{\\Delta}^{3} a^{2} y^{2} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\sin\\left({\\Theta}\\right)^{2}\\right)} y^{4} + 2 \\, {\\left(\\sin\\left({\\Theta}\\right)^{4} - 1\\right)} a^{2} m}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\\\ G_{ \\, {x^-} \\, {x^-} }^{ \\phantom{\\, {x^-}}\\phantom{\\, {x^-}} } & = & -\\frac{{\\Delta}^{3} a^{2} y^{2} - 2 \\, a^{2} m \\cos\\left({\\Theta}\\right)^{4} + {\\left({\\Delta}^{3} a^{2} + {\\Delta}^{3} \\cos\\left({\\Theta}\\right)^{2} - {\\Delta}^{3}\\right)} y^{4}}{4 \\, {\\Delta}^{3} a^{2} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\\\ G_{ \\, {\\Theta} \\, {\\Theta} }^{ \\phantom{\\, {\\Theta}}\\phantom{\\, {\\Theta}} } & = & y^{2} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & y^{2} \\cos\\left({\\Theta}\\right)^{2} \\end{array}$$" ], "text/plain": [ "G_xp,xp = -1/4*(Delta^3*a^2*y^2 + (Delta^3*a^2 - Delta^3*sin(Th)^2)*y^4 - 2*(sin(Th)^4 + 2*sin(Th)^2 + 1)*a^2*m)/(Delta^3*a^2*y^2) \n", "G_xp,xm = -1/4*(Delta^3*a^2*y^2 + (Delta^3*a^2 + Delta^3*sin(Th)^2)*y^4 + 2*(sin(Th)^4 - 1)*a^2*m)/(Delta^3*a^2*y^2) \n", "G_xm,xm = -1/4*(Delta^3*a^2*y^2 - 2*a^2*m*cos(Th)^4 + (Delta^3*a^2 + Delta^3*cos(Th)^2 - Delta^3)*y^4)/(Delta^3*a^2*y^2) \n", "G_y,y = Delta^2*y^2/(Delta^2*y^4 + Delta^2*y^2 - 2*m) \n", "G_Th,Th = y^2 \n", "G_Ps,Ps = y^2*cos(Th)^2 " ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(chart=LC, only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "M.set_default_chart(LC)\n", "M.set_default_frame(LC.frame())" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (5.41)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{1}{4} \\, y^{2} + \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^-} + \\left( -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\cos\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{y^{2}}{4 \\, a^{2}} + \\frac{m \\cos\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^-} + \\left( \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + y^{2} \\mathrm{d} {\\Theta}\\otimes \\mathrm{d} {\\Theta} + y^{2} \\cos\\left({\\Theta}\\right)^{2} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{1}{4} \\, y^{2} + \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^+}\\otimes \\mathrm{d} {x^-} + \\left( -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^+} + \\left( -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\cos\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{y^{2}}{4 \\, a^{2}} + \\frac{m \\cos\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\right) \\mathrm{d} {x^-}\\otimes \\mathrm{d} {x^-} + \\left( \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + y^{2} \\mathrm{d} {\\Theta}\\otimes \\mathrm{d} {\\Theta} + y^{2} \\cos\\left({\\Theta}\\right)^{2} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$$" ], "text/plain": [ "G = (-1/4*y^2 + 1/4*y^2*sin(Th)^2/a^2 + 1/2*m*sin(Th)^4/(Delta^3*y^2) + m*sin(Th)^2/(Delta^3*y^2) + 1/2*m/(Delta^3*y^2) - 1/4) dxp*dxp + (-1/4*y^2 - 1/4*y^2*sin(Th)^2/a^2 - 1/2*m*sin(Th)^4/(Delta^3*y^2) + 1/2*m/(Delta^3*y^2) - 1/4) dxp*dxm + (-1/4*y^2 - 1/4*y^2*sin(Th)^2/a^2 - 1/2*m*sin(Th)^4/(Delta^3*y^2) + 1/2*m/(Delta^3*y^2) - 1/4) dxm*dxp + (-1/4*y^2 - 1/4*y^2*cos(Th)^2/a^2 + 1/4*y^2/a^2 + 1/2*m*cos(Th)^4/(Delta^3*y^2) - 1/4) dxm*dxm + Delta^2*y^2/(Delta^2*y^4 + Delta^2*y^2 - 2*m) dy*dy + y^2 dTh*dTh + y^2*cos(Th)^2 dPs*dPs" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.apply_map(expand, keep_other_components=True)\n", "G.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, {x^+} \\, {x^+} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^+}} } & = & -\\frac{1}{4} \\, y^{2} + \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, {x^+} \\, {x^-} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^-}} } & = & -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, {x^-} \\, {x^+} }^{ \\phantom{\\, {x^-}}\\phantom{\\, {x^+}} } & = & -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, {x^-} \\, {x^-} }^{ \\phantom{\\, {x^-}}\\phantom{\\, {x^-}} } & = & -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\cos\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{y^{2}}{4 \\, a^{2}} + \\frac{m \\cos\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\\\ G_{ \\, {\\Theta} \\, {\\Theta} }^{ \\phantom{\\, {\\Theta}}\\phantom{\\, {\\Theta}} } & = & y^{2} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & y^{2} \\cos\\left({\\Theta}\\right)^{2} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, {x^+} \\, {x^+} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^+}} } & = & -\\frac{1}{4} \\, y^{2} + \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m \\sin\\left({\\Theta}\\right)^{2}}{{\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, {x^+} \\, {x^-} }^{ \\phantom{\\, {x^+}}\\phantom{\\, {x^-}} } & = & -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, {x^-} \\, {x^+} }^{ \\phantom{\\, {x^-}}\\phantom{\\, {x^+}} } & = & -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\sin\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} - \\frac{m \\sin\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} + \\frac{m}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, {x^-} \\, {x^-} }^{ \\phantom{\\, {x^-}}\\phantom{\\, {x^-}} } & = & -\\frac{1}{4} \\, y^{2} - \\frac{y^{2} \\cos\\left({\\Theta}\\right)^{2}}{4 \\, a^{2}} + \\frac{y^{2}}{4 \\, a^{2}} + \\frac{m \\cos\\left({\\Theta}\\right)^{4}}{2 \\, {\\Delta}^{3} y^{2}} - \\frac{1}{4} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\Delta}^{2} y^{2}}{{\\Delta}^{2} y^{4} + {\\Delta}^{2} y^{2} - 2 \\, m} \\\\ G_{ \\, {\\Theta} \\, {\\Theta} }^{ \\phantom{\\, {\\Theta}}\\phantom{\\, {\\Theta}} } & = & y^{2} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & y^{2} \\cos\\left({\\Theta}\\right)^{2} \\end{array}$$" ], "text/plain": [ "G_xp,xp = -1/4*y^2 + 1/4*y^2*sin(Th)^2/a^2 + 1/2*m*sin(Th)^4/(Delta^3*y^2) + m*sin(Th)^2/(Delta^3*y^2) + 1/2*m/(Delta^3*y^2) - 1/4 \n", "G_xp,xm = -1/4*y^2 - 1/4*y^2*sin(Th)^2/a^2 - 1/2*m*sin(Th)^4/(Delta^3*y^2) + 1/2*m/(Delta^3*y^2) - 1/4 \n", "G_xm,xp = -1/4*y^2 - 1/4*y^2*sin(Th)^2/a^2 - 1/2*m*sin(Th)^4/(Delta^3*y^2) + 1/2*m/(Delta^3*y^2) - 1/4 \n", "G_xm,xm = -1/4*y^2 - 1/4*y^2*cos(Th)^2/a^2 + 1/4*y^2/a^2 + 1/2*m*cos(Th)^4/(Delta^3*y^2) - 1/4 \n", "G_y,y = Delta^2*y^2/(Delta^2*y^4 + Delta^2*y^2 - 2*m) \n", "G_Th,Th = y^2 \n", "G_Ps,Ps = y^2*cos(Th)^2 " ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The above components fully agree with Eq. (5.41)." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.3", "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 10, "url": "https://www.sagemath.org/" } }, "name": "sage-9.3", "resource_dir": "/ext/jupyter/kernels/sage-9.3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.2" } }, "nbformat": 4, "nbformat_minor": 4 }