{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "# 5D Kerr-AdS spacetime with a Nambu-Goto string\n", "\n", "## Case b = n a" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "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": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "*NB:* a version of SageMath at least equal to 9.1 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "size": 4 }, "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": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Since some computations are quite long, we ask for running them in parallel on 8 cores:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Spacetime manifold\n", "\n", "We declare the Kerr-AdS spacetime as a 5-dimensional Lorentzian manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "size": 4 }, "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, "size": 4 }, "source": [ "Let us define **Boyer-Lindquist-type coordinates (rational polynomial version)** on $\\mathcal{M}$, via the method `chart()`, the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\\infty,+\\infty)$) and their LaTeX symbols:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$$" ], "text/plain": [ "Chart (M, (t, r, mu, ph, ps))" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL. = M.chart(r't r:(0,+oo) mu:(0,1):\\mu ph:(0,2*pi):\\phi ps:(0,2*pi):\\psi')\n", "BL" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The coordinate $\\mu$ is related to the standard Boyer-Lindquist coordinate $\\theta$ by\n", "$$ \\mu = \\cos\\theta$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coordinate ranges are" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "t: (-oo, +oo); r: (0, +oo); mu: (0, 1); ph: (0, 2*pi); ps: (0, 2*pi)" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL.coord_range()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Note that contrary to the 4-dimensional case, the range of $\\mu$ is $(0,1)$ only (cf. Sec. 1.2 of [R.C. Myers, arXiv:1111.1903](https://arxiv.org/abs/1111.1903) or Sec. 2 of [G.W. Gibbons, H. Lüb, Don N. Page, C.N. Pope, J. Geom. Phys. **53**, 49 (2005)](https://doi.org/10.1016/j.geomphys.2004.05.001)). In other words, the range of $\\theta$ is $\\left(0, \\frac{\\pi}{2}\\right)$ only. " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Metric tensor\n", "\n", "The 4 parameters $m$, $a$, $b$ and $\\ell$ of the Kerr-AdS spacetime are declared as symbolic variables, $a$ and $b$ being the two angular momentum parameters and $\\ell$ being related to the cosmological constant by $\\Lambda = - 6 \\ell^2$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "size": 4 }, "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": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('m a b', domain='real')" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ell}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ell}$$" ], "text/plain": [ "l" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('l', domain='real', latex_name=r'\\ell')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We assume that $b = n a$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "n = var('n', domain='real')\n", "b = n*a" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Some auxiliary functions:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "keep_Delta = True # change to False to provide explicit expression for Delta_r, Xi_a, etc..." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "sig = (1 + r^2*l^2)/r^2\n", "costh2 = mu^2\n", "sinth2 = 1 - mu^2\n", "rho2 = r^2 + a^2*mu^2 + b^2*sinth2\n", "# Explicit expressions:\n", "Delta_r_expr = (r^2+a^2)*(r^2+b^2)*sig - 2*m\n", "Delta_th_expr = 1 - a^2*l^2*costh2 - b^2*l^2*sinth2\n", "Xi_a_expr = 1 - a^2*l^2\n", "Xi_b_expr = 1 - b^2*l^2\n", "if keep_Delta:\n", " Delta_r = var('Delta_r', latex_name=r'\\Delta_r', domain='real')\n", " Delta_th = var('Delta_th', latex_name=r'\\Delta_\\theta', domain='real')\n", " if a == b:\n", " Xi_a = var('Xi', latex_name=r'\\Xi', domain='real')\n", " Xi_b = Xi_a\n", " else:\n", " Xi_a = var('Xi_a', latex_name=r'\\Xi_a', domain='real')\n", " Xi_b = var('Xi_b', latex_name=r'\\Xi_b', domain='real')\n", "else:\n", " Delta_r = Delta_r_expr\n", " Delta_th = Delta_th_expr\n", " Xi_a = Xi_a_expr\n", " Xi_b = Xi_b_expr" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components can be deduced from\n", "Eq. (5.22) of the article [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005) (the check of agreement with this equation is performed below):" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "G = M.metric()\n", "tmp = 1/rho2*( -Delta_r + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig )\n", "G[0,0] = tmp.simplify_full()\n", "tmp = a*sinth2/(rho2*Xi_a)*( Delta_r - (r^2+a^2)*(Delta_th + b^2*sig) )\n", "G[0,3] = tmp.simplify_full()\n", "tmp = b*mu^2/(rho2*Xi_b)*( Delta_r - (r^2+b^2)*(Delta_th + a^2*sig) )\n", "G[0,4] = tmp.simplify_full()\n", "G[1,1] = (rho2/Delta_r).simplify_full()\n", "G[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full()\n", "tmp = sinth2/(rho2*Xi_a^2)*( - Delta_r*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) ) \n", "G[3,3] = tmp.simplify_full()\n", "tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta_r + sig*(r^2+a^2)*(r^2+b^2) )\n", "G[3,4] = tmp.simplify_full()\n", "tmp = mu^2/(rho2*Xi_b^2)*( - Delta_r*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) )\n", "G[4,4] = tmp.simplify_full()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{a^{4} n^{2} - {\\left({\\Delta_\\theta} a^{2} {\\mu}^{2} - {\\Delta_\\theta} a^{2} - {\\left(a^{4} {\\ell}^{2} + {\\Delta_\\theta} a^{2} {\\mu}^{2}\\right)} n^{2} + {\\Delta_r}\\right)} r^{2}}{r^{4} + {\\left(a^{2} {\\mu}^{2} - {\\left(a^{2} {\\mu}^{2} - a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & \\frac{{\\left({\\Delta_\\theta} a {\\mu}^{2} + {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} n^{2} - {\\Delta_\\theta} a\\right)} r^{4} + {\\left(a^{5} {\\mu}^{2} - a^{5}\\right)} n^{2} - {\\left({\\Delta_\\theta} a^{3} - {\\left({\\Delta_\\theta} a^{3} - {\\Delta_r} a\\right)} {\\mu}^{2} + {\\left(a^{5} {\\ell}^{2} + a^{3} - {\\left(a^{5} {\\ell}^{2} + a^{3}\\right)} {\\mu}^{2}\\right)} n^{2} - {\\Delta_r} a\\right)} r^{2}}{{\\Xi_a} r^{4} + {\\left({\\Xi_a} a^{2} {\\mu}^{2} - {\\left({\\Xi_a} a^{2} {\\mu}^{2} - {\\Xi_a} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{a^{5} {\\mu}^{2} n^{3} + {\\left(a^{3} {\\ell}^{2} + {\\Delta_\\theta} a\\right)} {\\mu}^{2} n r^{4} + {\\left({\\left(a^{5} {\\ell}^{2} + {\\Delta_\\theta} a^{3}\\right)} {\\mu}^{2} n^{3} + {\\left(a^{3} - {\\Delta_r} a\\right)} {\\mu}^{2} n\\right)} r^{2}}{{\\Xi_b} r^{4} + {\\left({\\Xi_b} a^{2} {\\mu}^{2} - {\\left({\\Xi_b} a^{2} {\\mu}^{2} - {\\Xi_b} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} {\\mu}^{2} - {\\left(a^{2} {\\mu}^{2} - a^{2}\\right)} n^{2} + r^{2}}{{\\Delta_r}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} {\\mu}^{2} - {\\left(a^{2} {\\mu}^{2} - a^{2}\\right)} n^{2} + r^{2}}{{\\Delta_\\theta} {\\mu}^{2} - {\\Delta_\\theta}} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{{\\left({\\Delta_\\theta} {\\mu}^{2} - {\\left(a^{2} {\\ell}^{2} {\\mu}^{4} - 2 \\, a^{2} {\\ell}^{2} {\\mu}^{2} + a^{2} {\\ell}^{2}\\right)} n^{2} - {\\Delta_\\theta}\\right)} r^{6} + {\\left(2 \\, {\\Delta_\\theta} a^{2} {\\mu}^{2} - 2 \\, {\\Delta_\\theta} a^{2} - {\\left(2 \\, a^{4} {\\ell}^{2} + {\\left(2 \\, a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{4} - 2 \\, {\\left(2 \\, a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{2} + a^{2}\\right)} n^{2}\\right)} r^{4} - {\\left(a^{6} {\\mu}^{4} - 2 \\, a^{6} {\\mu}^{2} + a^{6}\\right)} n^{2} + {\\left({\\Delta_r} a^{2} {\\mu}^{4} - {\\Delta_\\theta} a^{4} + {\\Delta_r} a^{2} + {\\left({\\Delta_\\theta} a^{4} - 2 \\, {\\Delta_r} a^{2}\\right)} {\\mu}^{2} - {\\left(a^{6} {\\ell}^{2} + {\\left(a^{6} {\\ell}^{2} + 2 \\, a^{4}\\right)} {\\mu}^{4} + 2 \\, a^{4} - 2 \\, {\\left(a^{6} {\\ell}^{2} + 2 \\, a^{4}\\right)} {\\mu}^{2}\\right)} n^{2}\\right)} r^{2}}{{\\Xi_a}^{2} r^{4} + {\\left({\\Xi_a}^{2} a^{2} {\\mu}^{2} - {\\left({\\Xi_a}^{2} a^{2} {\\mu}^{2} - {\\Xi_a}^{2} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{{\\left(a^{2} {\\ell}^{2} {\\mu}^{4} - a^{2} {\\ell}^{2} {\\mu}^{2}\\right)} n r^{6} + {\\left({\\left(a^{4} {\\ell}^{2} {\\mu}^{4} - a^{4} {\\ell}^{2} {\\mu}^{2}\\right)} n^{3} + {\\left({\\left(a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{4} - {\\left(a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{2}\\right)} n\\right)} r^{4} + {\\left(a^{6} {\\mu}^{4} - a^{6} {\\mu}^{2}\\right)} n^{3} + {\\left({\\left({\\left(a^{6} {\\ell}^{2} + a^{4}\\right)} {\\mu}^{4} - {\\left(a^{6} {\\ell}^{2} + a^{4}\\right)} {\\mu}^{2}\\right)} n^{3} + {\\left({\\left(a^{4} - {\\Delta_r} a^{2}\\right)} {\\mu}^{4} - {\\left(a^{4} - {\\Delta_r} a^{2}\\right)} {\\mu}^{2}\\right)} n\\right)} r^{2}}{{\\Xi_a} {\\Xi_b} r^{4} + {\\left({\\Xi_a} {\\Xi_b} a^{2} {\\mu}^{2} - {\\left({\\Xi_a} {\\Xi_b} a^{2} {\\mu}^{2} - {\\Xi_a} {\\Xi_b} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{a^{6} {\\mu}^{4} n^{4} + {\\left(a^{2} {\\ell}^{2} {\\mu}^{4} + {\\Delta_\\theta} {\\mu}^{2}\\right)} r^{6} + {\\left(a^{2} {\\mu}^{4} + 2 \\, {\\left(a^{4} {\\ell}^{2} {\\mu}^{4} + {\\Delta_\\theta} a^{2} {\\mu}^{2}\\right)} n^{2}\\right)} r^{4} + {\\left({\\left(2 \\, a^{4} - {\\Delta_r} a^{2}\\right)} {\\mu}^{4} n^{2} + {\\left(a^{6} {\\ell}^{2} {\\mu}^{4} + {\\Delta_\\theta} a^{4} {\\mu}^{2}\\right)} n^{4}\\right)} r^{2}}{{\\Xi_b}^{2} r^{4} + {\\left({\\Xi_b}^{2} a^{2} {\\mu}^{2} - {\\left({\\Xi_b}^{2} a^{2} {\\mu}^{2} - {\\Xi_b}^{2} a^{2}\\right)} n^{2}\\right)} r^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{a^{4} n^{2} - {\\left({\\Delta_\\theta} a^{2} {\\mu}^{2} - {\\Delta_\\theta} a^{2} - {\\left(a^{4} {\\ell}^{2} + {\\Delta_\\theta} a^{2} {\\mu}^{2}\\right)} n^{2} + {\\Delta_r}\\right)} r^{2}}{r^{4} + {\\left(a^{2} {\\mu}^{2} - {\\left(a^{2} {\\mu}^{2} - a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & \\frac{{\\left({\\Delta_\\theta} a {\\mu}^{2} + {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} n^{2} - {\\Delta_\\theta} a\\right)} r^{4} + {\\left(a^{5} {\\mu}^{2} - a^{5}\\right)} n^{2} - {\\left({\\Delta_\\theta} a^{3} - {\\left({\\Delta_\\theta} a^{3} - {\\Delta_r} a\\right)} {\\mu}^{2} + {\\left(a^{5} {\\ell}^{2} + a^{3} - {\\left(a^{5} {\\ell}^{2} + a^{3}\\right)} {\\mu}^{2}\\right)} n^{2} - {\\Delta_r} a\\right)} r^{2}}{{\\Xi_a} r^{4} + {\\left({\\Xi_a} a^{2} {\\mu}^{2} - {\\left({\\Xi_a} a^{2} {\\mu}^{2} - {\\Xi_a} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{a^{5} {\\mu}^{2} n^{3} + {\\left(a^{3} {\\ell}^{2} + {\\Delta_\\theta} a\\right)} {\\mu}^{2} n r^{4} + {\\left({\\left(a^{5} {\\ell}^{2} + {\\Delta_\\theta} a^{3}\\right)} {\\mu}^{2} n^{3} + {\\left(a^{3} - {\\Delta_r} a\\right)} {\\mu}^{2} n\\right)} r^{2}}{{\\Xi_b} r^{4} + {\\left({\\Xi_b} a^{2} {\\mu}^{2} - {\\left({\\Xi_b} a^{2} {\\mu}^{2} - {\\Xi_b} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} {\\mu}^{2} - {\\left(a^{2} {\\mu}^{2} - a^{2}\\right)} n^{2} + r^{2}}{{\\Delta_r}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} {\\mu}^{2} - {\\left(a^{2} {\\mu}^{2} - a^{2}\\right)} n^{2} + r^{2}}{{\\Delta_\\theta} {\\mu}^{2} - {\\Delta_\\theta}} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{{\\left({\\Delta_\\theta} {\\mu}^{2} - {\\left(a^{2} {\\ell}^{2} {\\mu}^{4} - 2 \\, a^{2} {\\ell}^{2} {\\mu}^{2} + a^{2} {\\ell}^{2}\\right)} n^{2} - {\\Delta_\\theta}\\right)} r^{6} + {\\left(2 \\, {\\Delta_\\theta} a^{2} {\\mu}^{2} - 2 \\, {\\Delta_\\theta} a^{2} - {\\left(2 \\, a^{4} {\\ell}^{2} + {\\left(2 \\, a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{4} - 2 \\, {\\left(2 \\, a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{2} + a^{2}\\right)} n^{2}\\right)} r^{4} - {\\left(a^{6} {\\mu}^{4} - 2 \\, a^{6} {\\mu}^{2} + a^{6}\\right)} n^{2} + {\\left({\\Delta_r} a^{2} {\\mu}^{4} - {\\Delta_\\theta} a^{4} + {\\Delta_r} a^{2} + {\\left({\\Delta_\\theta} a^{4} - 2 \\, {\\Delta_r} a^{2}\\right)} {\\mu}^{2} - {\\left(a^{6} {\\ell}^{2} + {\\left(a^{6} {\\ell}^{2} + 2 \\, a^{4}\\right)} {\\mu}^{4} + 2 \\, a^{4} - 2 \\, {\\left(a^{6} {\\ell}^{2} + 2 \\, a^{4}\\right)} {\\mu}^{2}\\right)} n^{2}\\right)} r^{2}}{{\\Xi_a}^{2} r^{4} + {\\left({\\Xi_a}^{2} a^{2} {\\mu}^{2} - {\\left({\\Xi_a}^{2} a^{2} {\\mu}^{2} - {\\Xi_a}^{2} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{{\\left(a^{2} {\\ell}^{2} {\\mu}^{4} - a^{2} {\\ell}^{2} {\\mu}^{2}\\right)} n r^{6} + {\\left({\\left(a^{4} {\\ell}^{2} {\\mu}^{4} - a^{4} {\\ell}^{2} {\\mu}^{2}\\right)} n^{3} + {\\left({\\left(a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{4} - {\\left(a^{4} {\\ell}^{2} + a^{2}\\right)} {\\mu}^{2}\\right)} n\\right)} r^{4} + {\\left(a^{6} {\\mu}^{4} - a^{6} {\\mu}^{2}\\right)} n^{3} + {\\left({\\left({\\left(a^{6} {\\ell}^{2} + a^{4}\\right)} {\\mu}^{4} - {\\left(a^{6} {\\ell}^{2} + a^{4}\\right)} {\\mu}^{2}\\right)} n^{3} + {\\left({\\left(a^{4} - {\\Delta_r} a^{2}\\right)} {\\mu}^{4} - {\\left(a^{4} - {\\Delta_r} a^{2}\\right)} {\\mu}^{2}\\right)} n\\right)} r^{2}}{{\\Xi_a} {\\Xi_b} r^{4} + {\\left({\\Xi_a} {\\Xi_b} a^{2} {\\mu}^{2} - {\\left({\\Xi_a} {\\Xi_b} a^{2} {\\mu}^{2} - {\\Xi_a} {\\Xi_b} a^{2}\\right)} n^{2}\\right)} r^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{a^{6} {\\mu}^{4} n^{4} + {\\left(a^{2} {\\ell}^{2} {\\mu}^{4} + {\\Delta_\\theta} {\\mu}^{2}\\right)} r^{6} + {\\left(a^{2} {\\mu}^{4} + 2 \\, {\\left(a^{4} {\\ell}^{2} {\\mu}^{4} + {\\Delta_\\theta} a^{2} {\\mu}^{2}\\right)} n^{2}\\right)} r^{4} + {\\left({\\left(2 \\, a^{4} - {\\Delta_r} a^{2}\\right)} {\\mu}^{4} n^{2} + {\\left(a^{6} {\\ell}^{2} {\\mu}^{4} + {\\Delta_\\theta} a^{4} {\\mu}^{2}\\right)} n^{4}\\right)} r^{2}}{{\\Xi_b}^{2} r^{4} + {\\left({\\Xi_b}^{2} a^{2} {\\mu}^{2} - {\\left({\\Xi_b}^{2} a^{2} {\\mu}^{2} - {\\Xi_b}^{2} a^{2}\\right)} n^{2}\\right)} r^{2}} \\end{array}$$" ], "text/plain": [ "G_t,t = (a^4*n^2 - (Delta_th*a^2*mu^2 - Delta_th*a^2 - (a^4*l^2 + Delta_th*a^2*mu^2)*n^2 + Delta_r)*r^2)/(r^4 + (a^2*mu^2 - (a^2*mu^2 - a^2)*n^2)*r^2) \n", "G_t,ph = ((Delta_th*a*mu^2 + (a^3*l^2*mu^2 - a^3*l^2)*n^2 - Delta_th*a)*r^4 + (a^5*mu^2 - a^5)*n^2 - (Delta_th*a^3 - (Delta_th*a^3 - Delta_r*a)*mu^2 + (a^5*l^2 + a^3 - (a^5*l^2 + a^3)*mu^2)*n^2 - Delta_r*a)*r^2)/(Xi_a*r^4 + (Xi_a*a^2*mu^2 - (Xi_a*a^2*mu^2 - Xi_a*a^2)*n^2)*r^2) \n", "G_t,ps = -(a^5*mu^2*n^3 + (a^3*l^2 + Delta_th*a)*mu^2*n*r^4 + ((a^5*l^2 + Delta_th*a^3)*mu^2*n^3 + (a^3 - Delta_r*a)*mu^2*n)*r^2)/(Xi_b*r^4 + (Xi_b*a^2*mu^2 - (Xi_b*a^2*mu^2 - Xi_b*a^2)*n^2)*r^2) \n", "G_r,r = (a^2*mu^2 - (a^2*mu^2 - a^2)*n^2 + r^2)/Delta_r \n", "G_mu,mu = -(a^2*mu^2 - (a^2*mu^2 - a^2)*n^2 + r^2)/(Delta_th*mu^2 - Delta_th) \n", "G_ph,ph = -((Delta_th*mu^2 - (a^2*l^2*mu^4 - 2*a^2*l^2*mu^2 + a^2*l^2)*n^2 - Delta_th)*r^6 + (2*Delta_th*a^2*mu^2 - 2*Delta_th*a^2 - (2*a^4*l^2 + (2*a^4*l^2 + a^2)*mu^4 - 2*(2*a^4*l^2 + a^2)*mu^2 + a^2)*n^2)*r^4 - (a^6*mu^4 - 2*a^6*mu^2 + a^6)*n^2 + (Delta_r*a^2*mu^4 - Delta_th*a^4 + Delta_r*a^2 + (Delta_th*a^4 - 2*Delta_r*a^2)*mu^2 - (a^6*l^2 + (a^6*l^2 + 2*a^4)*mu^4 + 2*a^4 - 2*(a^6*l^2 + 2*a^4)*mu^2)*n^2)*r^2)/(Xi_a^2*r^4 + (Xi_a^2*a^2*mu^2 - (Xi_a^2*a^2*mu^2 - Xi_a^2*a^2)*n^2)*r^2) \n", "G_ph,ps = -((a^2*l^2*mu^4 - a^2*l^2*mu^2)*n*r^6 + ((a^4*l^2*mu^4 - a^4*l^2*mu^2)*n^3 + ((a^4*l^2 + a^2)*mu^4 - (a^4*l^2 + a^2)*mu^2)*n)*r^4 + (a^6*mu^4 - a^6*mu^2)*n^3 + (((a^6*l^2 + a^4)*mu^4 - (a^6*l^2 + a^4)*mu^2)*n^3 + ((a^4 - Delta_r*a^2)*mu^4 - (a^4 - Delta_r*a^2)*mu^2)*n)*r^2)/(Xi_a*Xi_b*r^4 + (Xi_a*Xi_b*a^2*mu^2 - (Xi_a*Xi_b*a^2*mu^2 - Xi_a*Xi_b*a^2)*n^2)*r^2) \n", "G_ps,ps = (a^6*mu^4*n^4 + (a^2*l^2*mu^4 + Delta_th*mu^2)*r^6 + (a^2*mu^4 + 2*(a^4*l^2*mu^4 + Delta_th*a^2*mu^2)*n^2)*r^4 + ((2*a^4 - Delta_r*a^2)*mu^4*n^2 + (a^6*l^2*mu^4 + Delta_th*a^4*mu^2)*n^4)*r^2)/(Xi_b^2*r^4 + (Xi_b^2*a^2*mu^2 - (Xi_b^2*a^2*mu^2 - Xi_b^2*a^2)*n^2)*r^2) " ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (2.9)\n", "\n", "We need the 1-forms $\\mathrm{d}t$, $\\mathrm{d}r$, $\\mathrm{d}\\mu$, $\\mathrm{d}\\phi$ and $\\mathrm{d}\\psi$:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$$" ], "text/plain": [ "(1-form dt on the 5-dimensional Lorentzian manifold M,\n", " 1-form dr on the 5-dimensional Lorentzian manifold M,\n", " 1-form dmu on the 5-dimensional Lorentzian manifold M,\n", " 1-form dph on the 5-dimensional Lorentzian manifold M,\n", " 1-form dps on the 5-dimensional Lorentzian manifold M)" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange())\n", "dt, dr, dmu, dph, dps" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form dt on the 5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(dt)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In agreement with $\\mu = \\cos\\theta$, we introduce the 1-form\n", "$$\\mathrm{d}\\theta = - \\mathrm{d}\\mu /\\sin\\theta ,$$\n", "with\n", "$\\sin\\theta = \\sqrt{1-\\mu^2}$ since $\\theta\\in\\left(0, \\frac{\\pi}{2}\\right)$ :" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "dth = - 1/sqrt(1 - mu^2)*dmu" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( \\frac{a {\\mu}^{2} - a}{{\\Xi_a}} \\right) \\mathrm{d} {\\phi} -\\frac{a {\\mu}^{2} n}{{\\Xi_b}} \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( \\frac{a {\\mu}^{2} - a}{{\\Xi_a}} \\right) \\mathrm{d} {\\phi} -\\frac{a {\\mu}^{2} n}{{\\Xi_b}} \\mathrm{d} {\\psi}$$" ], "text/plain": [ "dt + (a*mu^2 - a)/Xi_a dph - a*mu^2*n/Xi_b dps" ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps\n", "s1.display()" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( -\\frac{a^{2} + r^{2}}{{\\Xi_a}} \\right) \\mathrm{d} {\\phi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( -\\frac{a^{2} + r^{2}}{{\\Xi_a}} \\right) \\mathrm{d} {\\phi}$$" ], "text/plain": [ "a dt - (a^2 + r^2)/Xi_a dph" ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = a*dt - (r^2 + a^2)/Xi_a*dph\n", "s2.display()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a n \\mathrm{d} t + \\left( -\\frac{a^{2} n^{2} + r^{2}}{{\\Xi_b}} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a n \\mathrm{d} t + \\left( -\\frac{a^{2} n^{2} + r^{2}}{{\\Xi_b}} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "a*n dt - (a^2*n^2 + r^2)/Xi_b dps" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s3 = b*dt - (r^2 + b^2)/Xi_b*dps\n", "s3.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a^{2} n \\mathrm{d} t + \\left( \\frac{{\\left(a {\\mu}^{2} - a\\right)} n r^{2} + {\\left(a^{3} {\\mu}^{2} - a^{3}\\right)} n}{{\\Xi_a}} \\right) \\mathrm{d} {\\phi} + \\left( -\\frac{a^{3} {\\mu}^{2} n^{2} + a {\\mu}^{2} r^{2}}{{\\Xi_b}} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a^{2} n \\mathrm{d} t + \\left( \\frac{{\\left(a {\\mu}^{2} - a\\right)} n r^{2} + {\\left(a^{3} {\\mu}^{2} - a^{3}\\right)} n}{{\\Xi_a}} \\right) \\mathrm{d} {\\phi} + \\left( -\\frac{a^{3} {\\mu}^{2} n^{2} + a {\\mu}^{2} r^{2}}{{\\Xi_b}} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "a^2*n dt + ((a*mu^2 - a)*n*r^2 + (a^3*mu^2 - a^3)*n)/Xi_a dph - (a^3*mu^2*n^2 + a*mu^2*r^2)/Xi_b dps" ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps\n", "s4.display()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "G0 = - Delta_r/rho2 * s1*s1 + Delta_th*sinth2/rho2 * s2*s2 \\\n", " + Delta_th*costh2/rho2 * s3*s3 + rho2/Delta_r * dr*dr \\\n", " + rho2/Delta_th * dth*dth + sig/rho2 * s4*s4" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check of Eq. (2.9):" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 22, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G0 == G" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Einstein equation\n", "\n", "The Ricci tensor of $g$ is" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "if not keep_Delta:\n", " # Ric = G.ricci()\n", " # print(Ric)\n", " pass" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "if not keep_Delta:\n", " # show(Ric.display_comp(only_nonredundant=True))\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us check that $g$ is a solution of the vacuum Einstein equation with the cosmological constant $\\Lambda = - 6 \\ell^2$:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "Lambda = -6*l^2\n", "if not keep_Delta:\n", " # print(Ric == 2/3*Lambda*G)\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## String worldsheet" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The string worldsheet as a 2-dimensional Lorentzian submanifold of $\\mathcal{M}$:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional Lorentzian submanifold W immersed in the 5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "W = Manifold(2, 'W', ambient=M, structure='Lorentzian', \n", " latex_name=r'\\mathcal{W}')\n", "print(W)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us assume that the string worldsheet is parametrized by $(t,r)$:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{W},(t, r)\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{W},(t, r)\\right)$$" ], "text/plain": [ "Chart (W, (t, r))" ] }, "execution_count": 27, "metadata": { }, "output_type": "execute_result" } ], "source": [ "XW. = W.chart(r't r:(0,+oo)')\n", "XW" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The string embedding in Kerr-AdS spacetime, as an expansion about a \n", "straight string solution in AdS (Eq. (4.5) of the paper):" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& \\mathcal{W} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, r\\right) & \\longmapsto & \\left(t, r, {\\mu}, {\\phi}, {\\psi}\\right) = \\left(t, r, {\\left(a n + a\\right)}^{2} \\mu_{1}\\left(r\\right) + {\\mu_0}, a {\\ell}^{2} t + a \\phi_{1}\\left(r\\right) + {\\Phi_0}, a {\\ell}^{2} n t + a n \\psi_{1}\\left(r\\right) + {\\Psi_0}\\right) \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& \\mathcal{W} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, r\\right) & \\longmapsto & \\left(t, r, {\\mu}, {\\phi}, {\\psi}\\right) = \\left(t, r, {\\left(a n + a\\right)}^{2} \\mu_{1}\\left(r\\right) + {\\mu_0}, a {\\ell}^{2} t + a \\phi_{1}\\left(r\\right) + {\\Phi_0}, a {\\ell}^{2} n t + a n \\psi_{1}\\left(r\\right) + {\\Psi_0}\\right) \\end{array}$$" ], "text/plain": [ "F: W --> M\n", " (t, r) |--> (t, r, mu, ph, ps) = (t, r, (a*n + a)^2*mu_1(r) + Mu0, a*l^2*t + a*phi_1(r) + Phi0, a*l^2*n*t + a*n*psi_1(r) + Psi0)" ] }, "execution_count": 28, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Mu0 = var('Mu0', latex_name=r'\\mu_0', domain='real')\n", "Phi0 = var('Phi0', latex_name=r'\\Phi_0', domain='real')\n", "Psi0 = var('Psi0', latex_name=r'\\Psi_0', domain='real')\n", "\n", "cosTh0 = Mu0\n", "sinTh0 = sqrt(1 - Mu0^2)\n", "\n", "mu_s = Mu0 + (a+b)^2*function('mu_1')(r)\n", "ph_s = Phi0 + a*l^2*t + a*function('phi_1')(r)\n", "ps_s = Psi0 + b*l^2*t + b*function('psi_1')(r)\n", "\n", "F = W.diff_map(M, {(XW, BL): [t, r, mu_s, ph_s, ps_s]}, name='F') \n", "\n", "W.set_embedding(F)\n", "\n", "F.display()" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "0 & {\\left(a^{2} n^{2} + 2 \\, a^{2} n + a^{2}\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) \\\\\n", "a {\\ell}^{2} & a \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) \\\\\n", "a {\\ell}^{2} n & a n \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "0 & {\\left(a^{2} n^{2} + 2 \\, a^{2} n + a^{2}\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) \\\\\n", "a {\\ell}^{2} & a \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) \\\\\n", "a {\\ell}^{2} n & a n \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 1 0]\n", "[ 0 1]\n", "[ 0 (a^2*n^2 + 2*a^2*n + a^2)*diff(mu_1(r), r)]\n", "[ a*l^2 a*diff(phi_1(r), r)]\n", "[ a*l^2*n a*n*diff(psi_1(r), r)]" ] }, "execution_count": 29, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F.jacobian_matrix()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "### Induced metric on the string worldsheet" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Because of the bug [#27492](https://trac.sagemath.org/ticket/27492), which impedes parallel computations involving symbolic functions, such as $\\mu_1$, we switch back to serial computations:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "Parallelism().set(nproc=1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The metric on the string worldsheet $\\mathcal{W}$ is the metric $g$ induced by the spacetime metric $G$, i.e. the pullback of $G$ by the embedding $F$:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "g = W.induced_metric()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Nambu-Goto action" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The determinant of $g$ is" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg = g.determinant().expr()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Expanding at fourth order in $a$ (will be required latter):" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a4 = detg.series(a, 5).truncate().simplify_full()" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a40 = detg_a4" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a4 = detg_a40" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a4 = detg_a4.subs({Xi_a: Xi_a_expr, Xi_b: Xi_b_expr, \n", " Delta_r: Delta_r_expr, Delta_th: Delta_th_expr})\n", "detg_a4 = detg_a4.subs({mu: mu_s})" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a4 = detg_a4.series(a, 5).truncate().simplify_full()" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\mu_0}, a, {\\ell}, m, n, r\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\mu_0}, a, {\\ell}, m, n, r\\right)$$" ], "text/plain": [ "(Mu0, a, l, m, n, r)" ] }, "execution_count": 38, "metadata": { }, "output_type": "execute_result" } ], "source": [ "detg_a4.variables()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "For the time being, only the expansion at second order in $a$ is required:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a2 = detg_a4.series(a, 3).truncate().simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The Nambu-Goto Lagrangian at second order in $a$:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(3 \\, {\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} - 3 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{4} - 2 \\, {\\ell}^{2}\\right)} r^{4} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2}\\right)} n^{2} + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{4} r^{8} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{6} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m r^{2} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2}\\right)} r^{4}\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} r^{8} + 2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{6} - 4 \\, {\\mu_0}^{2} a^{2} m n^{2} r^{2} + 4 \\, {\\mu_0}^{2} a^{2} m^{2} n^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2}\\right)} n^{2} r^{4}\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} + 4 \\, {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} + 1\\right)} m - 2 \\, r^{2}}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(3 \\, {\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} - 3 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{4} - 2 \\, {\\ell}^{2}\\right)} r^{4} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2}\\right)} n^{2} + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{4} r^{8} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{6} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m r^{2} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2}\\right)} r^{4}\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} r^{8} + 2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{6} - 4 \\, {\\mu_0}^{2} a^{2} m n^{2} r^{2} + 4 \\, {\\mu_0}^{2} a^{2} m^{2} n^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2}\\right)} n^{2} r^{4}\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} + 4 \\, {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} + 1\\right)} m - 2 \\, r^{2}}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}}$$" ], "text/plain": [ "-1/2*((3*Mu0^2*a^2*l^4*n^2 - 3*(Mu0^2 - 1)*a^2*l^4 - 2*l^2)*r^4 - (Mu0^2 - 1)*a^2 - (4*Mu0^2*a^2*l^2*m - Mu0^2*a^2)*n^2 + ((Mu0^2 - 1)*a^2*l^4*r^8 + 2*(Mu0^2 - 1)*a^2*l^2*r^6 - 4*(Mu0^2 - 1)*a^2*m*r^2 + 4*(Mu0^2 - 1)*a^2*m^2 - (4*(Mu0^2 - 1)*a^2*l^2*m - (Mu0^2 - 1)*a^2)*r^4)*diff(phi_1(r), r)^2 - (Mu0^2*a^2*l^4*n^2*r^8 + 2*Mu0^2*a^2*l^2*n^2*r^6 - 4*Mu0^2*a^2*m*n^2*r^2 + 4*Mu0^2*a^2*m^2*n^2 - (4*Mu0^2*a^2*l^2*m - Mu0^2*a^2)*n^2*r^4)*diff(psi_1(r), r)^2 + 4*((Mu0^2 - 1)*a^2*l^2 + 1)*m - 2*r^2)/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 40, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_a2 = (sqrt(-detg_a2)).series(a, 3).truncate().simplify_full()\n", "L_a2" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} r^{8} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - {\\mu_0}^{2} a^{2} {\\ell}^{4} r^{8} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + a^{2} {\\ell}^{4} r^{8} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{6} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m n^{2} r^{4} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 3 \\, {\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} r^{4} - 2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} r^{6} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 3 \\, {\\mu_0}^{2} a^{2} {\\ell}^{4} r^{4} + 2 \\, a^{2} {\\ell}^{2} r^{6} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + {\\mu_0}^{2} a^{2} n^{2} r^{4} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 4 \\, a^{2} {\\ell}^{2} m r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} m n^{2} r^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 3 \\, a^{2} {\\ell}^{4} r^{4} - {\\mu_0}^{2} a^{2} r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} m^{2} n^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m n^{2} + 4 \\, {\\mu_0}^{2} a^{2} m r^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} m^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + a^{2} r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m - 4 \\, a^{2} m r^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - {\\mu_0}^{2} a^{2} n^{2} + 2 \\, {\\ell}^{2} r^{4} + 4 \\, a^{2} m^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 4 \\, a^{2} {\\ell}^{2} m + {\\mu_0}^{2} a^{2} - a^{2} + 2 \\, r^{2} - 4 \\, m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} r^{8} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - {\\mu_0}^{2} a^{2} {\\ell}^{4} r^{8} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + a^{2} {\\ell}^{4} r^{8} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{6} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m n^{2} r^{4} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 3 \\, {\\mu_0}^{2} a^{2} {\\ell}^{4} n^{2} r^{4} - 2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} r^{6} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 3 \\, {\\mu_0}^{2} a^{2} {\\ell}^{4} r^{4} + 2 \\, a^{2} {\\ell}^{2} r^{6} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + {\\mu_0}^{2} a^{2} n^{2} r^{4} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 4 \\, a^{2} {\\ell}^{2} m r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} m n^{2} r^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 3 \\, a^{2} {\\ell}^{4} r^{4} - {\\mu_0}^{2} a^{2} r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} m^{2} n^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m n^{2} + 4 \\, {\\mu_0}^{2} a^{2} m r^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} m^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + a^{2} r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} m - 4 \\, a^{2} m r^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - {\\mu_0}^{2} a^{2} n^{2} + 2 \\, {\\ell}^{2} r^{4} + 4 \\, a^{2} m^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + 4 \\, a^{2} {\\ell}^{2} m + {\\mu_0}^{2} a^{2} - a^{2} + 2 \\, r^{2} - 4 \\, m$$" ], "text/plain": [ "Mu0^2*a^2*l^4*n^2*r^8*diff(psi_1(r), r)^2 - Mu0^2*a^2*l^4*r^8*diff(phi_1(r), r)^2 + a^2*l^4*r^8*diff(phi_1(r), r)^2 + 2*Mu0^2*a^2*l^2*n^2*r^6*diff(psi_1(r), r)^2 - 4*Mu0^2*a^2*l^2*m*n^2*r^4*diff(psi_1(r), r)^2 - 3*Mu0^2*a^2*l^4*n^2*r^4 - 2*Mu0^2*a^2*l^2*r^6*diff(phi_1(r), r)^2 + 4*Mu0^2*a^2*l^2*m*r^4*diff(phi_1(r), r)^2 + 3*Mu0^2*a^2*l^4*r^4 + 2*a^2*l^2*r^6*diff(phi_1(r), r)^2 + Mu0^2*a^2*n^2*r^4*diff(psi_1(r), r)^2 - 4*a^2*l^2*m*r^4*diff(phi_1(r), r)^2 - 4*Mu0^2*a^2*m*n^2*r^2*diff(psi_1(r), r)^2 - 3*a^2*l^4*r^4 - Mu0^2*a^2*r^4*diff(phi_1(r), r)^2 + 4*Mu0^2*a^2*m^2*n^2*diff(psi_1(r), r)^2 + 4*Mu0^2*a^2*l^2*m*n^2 + 4*Mu0^2*a^2*m*r^2*diff(phi_1(r), r)^2 - 4*Mu0^2*a^2*m^2*diff(phi_1(r), r)^2 + a^2*r^4*diff(phi_1(r), r)^2 - 4*Mu0^2*a^2*l^2*m - 4*a^2*m*r^2*diff(phi_1(r), r)^2 - Mu0^2*a^2*n^2 + 2*l^2*r^4 + 4*a^2*m^2*diff(phi_1(r), r)^2 + 4*a^2*l^2*m + Mu0^2*a^2 - a^2 + 2*r^2 - 4*m" ] }, "execution_count": 41, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_a2.numerator()" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\ell}^{2} r^{4} + 2 \\, r^{2} - 4 \\, m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\ell}^{2} r^{4} + 2 \\, r^{2} - 4 \\, m$$" ], "text/plain": [ "2*l^2*r^4 + 2*r^2 - 4*m" ] }, "execution_count": 42, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_a2.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Euler-Lagrange equations" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def euler_lagrange(lagr, qs, var):\n", " r\"\"\"\n", " Derive the Euler-Lagrange equations from a given Lagrangian.\n", "\n", " INPUT:\n", "\n", " - ``lagr`` -- symbolic expression representing the Lagrangian density\n", " - ``qs`` -- either a single symbolic function or a list/tuple of\n", " symbolic functions, representing the `q`'s; these functions must\n", " appear in ``lagr`` up to at most their first derivatives\n", " - ``var`` -- either a single variable, typically `t` (1-dimensional\n", " problem) or a list/tuple of symbolic variables\n", "\n", " OUTPUT:\n", "\n", " - list of Euler-Lagrange equations; if only one function is involved, the\n", " single Euler-Lagrannge equation is returned instead.\n", "\n", " \"\"\"\n", " if not isinstance(qs, (list, tuple)):\n", " qs = [qs]\n", " if not isinstance(var, (list, tuple)):\n", " var = [var]\n", " n = len(qs)\n", " d = len(var)\n", " qv = [SR.var('qxxxx{}'.format(q)) for q in qs]\n", " dqv = [[SR.var('qxxxx{}_{}'.format(q, v)) for v in var] for q in qs]\n", " subs = {qs[i](*var): qv[i] for i in range(n)}\n", " subs_inv = {qv[i]: qs[i](*var) for i in range(n)}\n", " for i in range(n):\n", " subs.update({diff(qs[i](*var), var[j]): dqv[i][j]\n", " for j in range(d)})\n", " subs_inv.update({dqv[i][j]: diff(qs[i](*var), var[j])\n", " for j in range(d)})\n", " lg = lagr.substitute(subs)\n", " eqs = []\n", " for i in range(n):\n", " dLdq = diff(lg, qv[i]).simplify_full()\n", " dLdq = dLdq.substitute(subs_inv)\n", " ddL = 0\n", " for j in range(d):\n", " h = diff(lg, dqv[i][j]).simplify_full()\n", " h = h.substitute(subs_inv)\n", " ddL += diff(h, var[j])\n", " eqs.append((dLdq - ddL).simplify_full() == 0)\n", " if n == 1:\n", " return eqs[0]\n", " return eqs" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We compute the Euler-Lagrange equations at order $a^2$ for $\\phi_1$ and $\\psi_1$:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\phi_{1}\\left(r\\right) = 0, -2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{3} + {\\mu_0}^{2} a^{2} n^{2} r\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - {\\left({\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{4} + {\\mu_0}^{2} a^{2} n^{2} r^{2} - 2 \\, {\\mu_0}^{2} a^{2} m n^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\psi_{1}\\left(r\\right) = 0\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\phi_{1}\\left(r\\right) = 0, -2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{3} + {\\mu_0}^{2} a^{2} n^{2} r\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - {\\left({\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{4} + {\\mu_0}^{2} a^{2} n^{2} r^{2} - 2 \\, {\\mu_0}^{2} a^{2} m n^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\psi_{1}\\left(r\\right) = 0\\right]$$" ], "text/plain": [ "[2*(2*(Mu0^2 - 1)*a^2*l^2*r^3 + (Mu0^2 - 1)*a^2*r)*diff(phi_1(r), r) + ((Mu0^2 - 1)*a^2*l^2*r^4 + (Mu0^2 - 1)*a^2*r^2 - 2*(Mu0^2 - 1)*a^2*m)*diff(phi_1(r), r, r) == 0,\n", " -2*(2*Mu0^2*a^2*l^2*n^2*r^3 + Mu0^2*a^2*n^2*r)*diff(psi_1(r), r) - (Mu0^2*a^2*l^2*n^2*r^4 + Mu0^2*a^2*n^2*r^2 - 2*Mu0^2*a^2*m*n^2)*diff(psi_1(r), r, r) == 0]" ] }, "execution_count": 44, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eqs = euler_lagrange(L_a2, [phi_1, psi_1], r)\n", "eqs" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### Solving the equation for $\\phi_1$ (Eq. (4.8))" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\phi_{1}\\left(r\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\phi_{1}\\left(r\\right) = 0$$" ], "text/plain": [ "2*(2*(Mu0^2 - 1)*a^2*l^2*r^3 + (Mu0^2 - 1)*a^2*r)*diff(phi_1(r), r) + ((Mu0^2 - 1)*a^2*l^2*r^4 + (Mu0^2 - 1)*a^2*r^2 - 2*(Mu0^2 - 1)*a^2*m)*diff(phi_1(r), r, r) == 0" ] }, "execution_count": 45, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_phi1 = eqs[0]\n", "eq_phi1" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} r^{3} + r\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\phi_{1}\\left(r\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} r^{3} + r\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\phi_{1}\\left(r\\right) = 0$$" ], "text/plain": [ "2*(2*l^2*r^3 + r)*diff(phi_1(r), r) + (l^2*r^4 + r^2 - 2*m)*diff(phi_1(r), r, r) == 0" ] }, "execution_count": 46, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_phi1 = (eq_phi1/(a^2*(Mu0^2-1))).simplify_full()\n", "eq_phi1" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r} + K_{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r} + K_{2}$$" ], "text/plain": [ "_K1*integrate(1/(l^2*r^4 + r^2 - 2*m), r) + _K2" ] }, "execution_count": 47, "metadata": { }, "output_type": "execute_result" } ], "source": [ "phi1_sol(r) = desolve(eq_phi1, phi_1(r), ivar=r)\n", "phi1_sol(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We recover Eqs. (4.8) with $K_1 = \\mathfrak{p}$ and $K_2=0$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The symbolic constants $K_1$ and $K_2$ are actually denoted `_K1` and `_K2` by SageMath, as the `print` reveals:" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "_K1*integrate(1/(l^2*r^4 + r^2 - 2*m), r) + _K2\n" ] } ], "source": [ "print(phi1_sol(r))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Hence we perform the substitutions with `SR.var('_K1')` and `SR.var('_K2')`:" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{p}} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{p}} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r}$$" ], "text/plain": [ "pf*integrate(1/(l^2*r^4 + r^2 - 2*m), r)" ] }, "execution_count": 49, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pf = var(\"pf\", latex_name=r\"\\mathfrak{p}\")\n", "phi1_sol(r) = phi1_sol(r).subs({SR.var('_K1'): pf, SR.var('_K2'): 0})\n", "phi1_sol(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### Solving the equation for $\\psi_1$ (Eq. (4.8))" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{3} + {\\mu_0}^{2} a^{2} n^{2} r\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - {\\left({\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{4} + {\\mu_0}^{2} a^{2} n^{2} r^{2} - 2 \\, {\\mu_0}^{2} a^{2} m n^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\psi_{1}\\left(r\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{3} + {\\mu_0}^{2} a^{2} n^{2} r\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - {\\left({\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{4} + {\\mu_0}^{2} a^{2} n^{2} r^{2} - 2 \\, {\\mu_0}^{2} a^{2} m n^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\psi_{1}\\left(r\\right) = 0$$" ], "text/plain": [ "-2*(2*Mu0^2*a^2*l^2*n^2*r^3 + Mu0^2*a^2*n^2*r)*diff(psi_1(r), r) - (Mu0^2*a^2*l^2*n^2*r^4 + Mu0^2*a^2*n^2*r^2 - 2*Mu0^2*a^2*m*n^2)*diff(psi_1(r), r, r) == 0" ] }, "execution_count": 50, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_psi1 = eqs[1]\n", "eq_psi1" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\ell}^{2} n^{2} r^{3} + n^{2} r\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - {\\left({\\ell}^{2} n^{2} r^{4} + n^{2} r^{2} - 2 \\, m n^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\psi_{1}\\left(r\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\ell}^{2} n^{2} r^{3} + n^{2} r\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - {\\left({\\ell}^{2} n^{2} r^{4} + n^{2} r^{2} - 2 \\, m n^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\psi_{1}\\left(r\\right) = 0$$" ], "text/plain": [ "-2*(2*l^2*n^2*r^3 + n^2*r)*diff(psi_1(r), r) - (l^2*n^2*r^4 + n^2*r^2 - 2*m*n^2)*diff(psi_1(r), r, r) == 0" ] }, "execution_count": 51, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_psi1 = (eq_psi1/(a^2*Mu0^2)).simplify_full()\n", "eq_psi1" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r} + K_{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r} + K_{2}$$" ], "text/plain": [ "_K1*integrate(1/(l^2*r^4 + r^2 - 2*m), r) + _K2" ] }, "execution_count": 52, "metadata": { }, "output_type": "execute_result" } ], "source": [ "psi1_sol(r) = desolve(eq_psi1, psi_1(r), ivar=r)\n", "psi1_sol(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We recover Eq. (4.8) with $K_1 = \\mathfrak{q}$ and $K_2=0$." ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{q}} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{q}} \\int \\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}\\,{d r}$$" ], "text/plain": [ "qf*integrate(1/(l^2*r^4 + r^2 - 2*m), r)" ] }, "execution_count": 53, "metadata": { }, "output_type": "execute_result" } ], "source": [ "qf = var('qf', latex_name=r\"\\mathfrak{q}\")\n", "psi1_sol(r) = psi1_sol(r).subs({SR.var('_K1'): qf, SR.var('_K2'): 0})\n", "psi1_sol(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Nambu-Goto Lagrangian at fourth order in $a$" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "L_a4 = (sqrt(-detg_a4)).series(a, 5).truncate().simplify_full()" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "eqs = euler_lagrange(L_a4, [phi_1, psi_1, mu_1], r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### The equation for $\\mu_1$" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{4} + 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4}\\right)} r^{4} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4}\\right)} r^{8} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2}\\right)} r^{6} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n\\right)} r^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{4} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{3} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{2}\\right)} r^{8} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{2}\\right)} r^{6} - {\\left({\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{4} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{2}\\right)} r^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{2}\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n - 2 \\, {\\left(2 \\, {\\left(a^{4} {\\ell}^{4} n^{4} + 4 \\, a^{4} {\\ell}^{4} n^{3} + 6 \\, a^{4} {\\ell}^{4} n^{2} + 4 \\, a^{4} {\\ell}^{4} n + a^{4} {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left(a^{4} {\\ell}^{2} n^{4} + 4 \\, a^{4} {\\ell}^{2} n^{3} + 6 \\, a^{4} {\\ell}^{2} n^{2} + 4 \\, a^{4} {\\ell}^{2} n + a^{4} {\\ell}^{2}\\right)} r^{5} - {\\left(4 \\, a^{4} {\\ell}^{2} m + {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{4} - a^{4} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{3} + 6 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{2} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n\\right)} r^{3} - 2 \\, {\\left(a^{4} m n^{4} + 4 \\, a^{4} m n^{3} + 6 \\, a^{4} m n^{2} + 4 \\, a^{4} m n + a^{4} m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) - {\\left(4 \\, a^{4} m^{2} n^{4} + 16 \\, a^{4} m^{2} n^{3} + {\\left(a^{4} {\\ell}^{4} n^{4} + 4 \\, a^{4} {\\ell}^{4} n^{3} + 6 \\, a^{4} {\\ell}^{4} n^{2} + 4 \\, a^{4} {\\ell}^{4} n + a^{4} {\\ell}^{4}\\right)} r^{8} + 24 \\, a^{4} m^{2} n^{2} + 16 \\, a^{4} m^{2} n + 2 \\, {\\left(a^{4} {\\ell}^{2} n^{4} + 4 \\, a^{4} {\\ell}^{2} n^{3} + 6 \\, a^{4} {\\ell}^{2} n^{2} + 4 \\, a^{4} {\\ell}^{2} n + a^{4} {\\ell}^{2}\\right)} r^{6} + 4 \\, a^{4} m^{2} - {\\left(4 \\, a^{4} {\\ell}^{2} m + {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{4} - a^{4} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{3} + 6 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{2} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n\\right)} r^{4} - 4 \\, {\\left(a^{4} m n^{4} + 4 \\, a^{4} m n^{3} + 6 \\, a^{4} m n^{2} + 4 \\, a^{4} m n + a^{4} m\\right)} r^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right)}{{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m} = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{4} + 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4}\\right)} r^{4} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4}\\right)} r^{8} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2}\\right)} r^{6} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n\\right)} r^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{4} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{3} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m^{2} n^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{4} n^{2}\\right)} r^{8} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} n^{2}\\right)} r^{6} - {\\left({\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{4} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n^{2}\\right)} r^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} m n^{2}\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4}\\right)} n - 2 \\, {\\left(2 \\, {\\left(a^{4} {\\ell}^{4} n^{4} + 4 \\, a^{4} {\\ell}^{4} n^{3} + 6 \\, a^{4} {\\ell}^{4} n^{2} + 4 \\, a^{4} {\\ell}^{4} n + a^{4} {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left(a^{4} {\\ell}^{2} n^{4} + 4 \\, a^{4} {\\ell}^{2} n^{3} + 6 \\, a^{4} {\\ell}^{2} n^{2} + 4 \\, a^{4} {\\ell}^{2} n + a^{4} {\\ell}^{2}\\right)} r^{5} - {\\left(4 \\, a^{4} {\\ell}^{2} m + {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{4} - a^{4} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{3} + 6 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{2} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n\\right)} r^{3} - 2 \\, {\\left(a^{4} m n^{4} + 4 \\, a^{4} m n^{3} + 6 \\, a^{4} m n^{2} + 4 \\, a^{4} m n + a^{4} m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) - {\\left(4 \\, a^{4} m^{2} n^{4} + 16 \\, a^{4} m^{2} n^{3} + {\\left(a^{4} {\\ell}^{4} n^{4} + 4 \\, a^{4} {\\ell}^{4} n^{3} + 6 \\, a^{4} {\\ell}^{4} n^{2} + 4 \\, a^{4} {\\ell}^{4} n + a^{4} {\\ell}^{4}\\right)} r^{8} + 24 \\, a^{4} m^{2} n^{2} + 16 \\, a^{4} m^{2} n + 2 \\, {\\left(a^{4} {\\ell}^{2} n^{4} + 4 \\, a^{4} {\\ell}^{2} n^{3} + 6 \\, a^{4} {\\ell}^{2} n^{2} + 4 \\, a^{4} {\\ell}^{2} n + a^{4} {\\ell}^{2}\\right)} r^{6} + 4 \\, a^{4} m^{2} - {\\left(4 \\, a^{4} {\\ell}^{2} m + {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{4} - a^{4} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{3} + 6 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n^{2} + 4 \\, {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} n\\right)} r^{4} - 4 \\, {\\left(a^{4} m n^{4} + 4 \\, a^{4} m n^{3} + 6 \\, a^{4} m n^{2} + 4 \\, a^{4} m n + a^{4} m\\right)} r^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right)}{{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m} = 0$$" ], "text/plain": [ "-(4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4 - (4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n^4 + 3*((Mu0^3 - Mu0)*a^4*l^4*n^4 + 2*(Mu0^3 - Mu0)*a^4*l^4*n^3 - 2*(Mu0^3 - Mu0)*a^4*l^4*n - (Mu0^3 - Mu0)*a^4*l^4)*r^4 - 2*(4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n^3 + (4*(Mu0^3 - Mu0)*a^4*m^2*n^2 + ((Mu0^3 - Mu0)*a^4*l^4*n^2 + 2*(Mu0^3 - Mu0)*a^4*l^4*n + (Mu0^3 - Mu0)*a^4*l^4)*r^8 + 8*(Mu0^3 - Mu0)*a^4*m^2*n + 4*(Mu0^3 - Mu0)*a^4*m^2 + 2*((Mu0^3 - Mu0)*a^4*l^2*n^2 + 2*(Mu0^3 - Mu0)*a^4*l^2*n + (Mu0^3 - Mu0)*a^4*l^2)*r^6 - (4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4 + (4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n^2 + 2*(4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n)*r^4 - 4*((Mu0^3 - Mu0)*a^4*m*n^2 + 2*(Mu0^3 - Mu0)*a^4*m*n + (Mu0^3 - Mu0)*a^4*m)*r^2)*diff(phi_1(r), r)^2 - (4*(Mu0^3 - Mu0)*a^4*m^2*n^4 + 8*(Mu0^3 - Mu0)*a^4*m^2*n^3 + 4*(Mu0^3 - Mu0)*a^4*m^2*n^2 + ((Mu0^3 - Mu0)*a^4*l^4*n^4 + 2*(Mu0^3 - Mu0)*a^4*l^4*n^3 + (Mu0^3 - Mu0)*a^4*l^4*n^2)*r^8 + 2*((Mu0^3 - Mu0)*a^4*l^2*n^4 + 2*(Mu0^3 - Mu0)*a^4*l^2*n^3 + (Mu0^3 - Mu0)*a^4*l^2*n^2)*r^6 - ((4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n^4 + 2*(4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n^3 + (4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n^2)*r^4 - 4*((Mu0^3 - Mu0)*a^4*m*n^4 + 2*(Mu0^3 - Mu0)*a^4*m*n^3 + (Mu0^3 - Mu0)*a^4*m*n^2)*r^2)*diff(psi_1(r), r)^2 + 2*(4*(Mu0^3 - Mu0)*a^4*l^2*m - (Mu0^3 - Mu0)*a^4)*n - 2*(2*(a^4*l^4*n^4 + 4*a^4*l^4*n^3 + 6*a^4*l^4*n^2 + 4*a^4*l^4*n + a^4*l^4)*r^7 + 3*(a^4*l^2*n^4 + 4*a^4*l^2*n^3 + 6*a^4*l^2*n^2 + 4*a^4*l^2*n + a^4*l^2)*r^5 - (4*a^4*l^2*m + (4*a^4*l^2*m - a^4)*n^4 - a^4 + 4*(4*a^4*l^2*m - a^4)*n^3 + 6*(4*a^4*l^2*m - a^4)*n^2 + 4*(4*a^4*l^2*m - a^4)*n)*r^3 - 2*(a^4*m*n^4 + 4*a^4*m*n^3 + 6*a^4*m*n^2 + 4*a^4*m*n + a^4*m)*r)*diff(mu_1(r), r) - (4*a^4*m^2*n^4 + 16*a^4*m^2*n^3 + (a^4*l^4*n^4 + 4*a^4*l^4*n^3 + 6*a^4*l^4*n^2 + 4*a^4*l^4*n + a^4*l^4)*r^8 + 24*a^4*m^2*n^2 + 16*a^4*m^2*n + 2*(a^4*l^2*n^4 + 4*a^4*l^2*n^3 + 6*a^4*l^2*n^2 + 4*a^4*l^2*n + a^4*l^2)*r^6 + 4*a^4*m^2 - (4*a^4*l^2*m + (4*a^4*l^2*m - a^4)*n^4 - a^4 + 4*(4*a^4*l^2*m - a^4)*n^3 + 6*(4*a^4*l^2*m - a^4)*n^2 + 4*(4*a^4*l^2*m - a^4)*n)*r^4 - 4*(a^4*m*n^4 + 4*a^4*m*n^3 + 6*a^4*m*n^2 + 4*a^4*m*n + a^4*m)*r^2)*diff(mu_1(r), r, r))/((Mu0^2 - 1)*l^2*r^4 + (Mu0^2 - 1)*r^2 - 2*(Mu0^2 - 1)*m) == 0" ] }, "execution_count": 56, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = eqs[2]\n", "eq_mu1" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} r^{4} + {\\left({\\mu_0}^{2} - 1\\right)} r^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m$$" ], "text/plain": [ "(Mu0^2 - 1)*l^2*r^4 + (Mu0^2 - 1)*r^2 - 2*(Mu0^2 - 1)*m" ] }, "execution_count": 57, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1.lhs().denominator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "eq_mu1 = eq_mu1.lhs().numerator().simplify_full() == 0\n", "#eq_mu1" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\mu_0}^{3} - {\\left({\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2}\\right)} r^{6} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + {\\mu_0}\\right)} r^{4} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + {\\left({\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{2}\\right)} r^{8} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{4} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{2}\\right)} r^{6} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{3} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{2} - {\\left({\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{2}\\right)} r^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{2}\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) + {\\left({\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} + 4 \\, m^{2} n^{4} + 16 \\, m^{2} n^{3} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 24 \\, m^{2} n^{2} + 16 \\, m^{2} n - 4 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) - {\\mu_0} = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\mu_0}^{3} - {\\left({\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2}\\right)} r^{6} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + {\\mu_0}\\right)} r^{4} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right)^{2} + {\\left({\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{2}\\right)} r^{8} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{4} + 2 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} n^{2}\\right)} r^{6} + 8 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{3} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m^{2} n^{2} - {\\left({\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{2}\\right)} r^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} m n^{2}\\right)} r^{2}\\right)} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)^{2} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) + {\\left({\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} + 4 \\, m^{2} n^{4} + 16 \\, m^{2} n^{3} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 24 \\, m^{2} n^{2} + 16 \\, m^{2} n - 4 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) - {\\mu_0} = 0$$" ], "text/plain": [ "(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^4 - 3*((Mu0^3 - Mu0)*l^4*n^4 + 2*(Mu0^3 - Mu0)*l^4*n^3 - 2*(Mu0^3 - Mu0)*l^4*n - (Mu0^3 - Mu0)*l^4)*r^4 - 4*(Mu0^3 - Mu0)*l^2*m + 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^3 + Mu0^3 - (((Mu0^3 - Mu0)*l^4*n^2 + 2*(Mu0^3 - Mu0)*l^4*n + (Mu0^3 - Mu0)*l^4)*r^8 + 2*((Mu0^3 - Mu0)*l^2*n^2 + 2*(Mu0^3 - Mu0)*l^2*n + (Mu0^3 - Mu0)*l^2)*r^6 + 4*(Mu0^3 - Mu0)*m^2*n^2 - (4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + (4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^2 + 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n + Mu0)*r^4 + 8*(Mu0^3 - Mu0)*m^2*n + 4*(Mu0^3 - Mu0)*m^2 - 4*((Mu0^3 - Mu0)*m*n^2 + 2*(Mu0^3 - Mu0)*m*n + (Mu0^3 - Mu0)*m)*r^2)*diff(phi_1(r), r)^2 + (((Mu0^3 - Mu0)*l^4*n^4 + 2*(Mu0^3 - Mu0)*l^4*n^3 + (Mu0^3 - Mu0)*l^4*n^2)*r^8 + 4*(Mu0^3 - Mu0)*m^2*n^4 + 2*((Mu0^3 - Mu0)*l^2*n^4 + 2*(Mu0^3 - Mu0)*l^2*n^3 + (Mu0^3 - Mu0)*l^2*n^2)*r^6 + 8*(Mu0^3 - Mu0)*m^2*n^3 + 4*(Mu0^3 - Mu0)*m^2*n^2 - ((4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^4 + 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^3 + (4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^2)*r^4 - 4*((Mu0^3 - Mu0)*m*n^4 + 2*(Mu0^3 - Mu0)*m*n^3 + (Mu0^3 - Mu0)*m*n^2)*r^2)*diff(psi_1(r), r)^2 - 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n + 2*(2*(l^4*n^4 + 4*l^4*n^3 + 6*l^4*n^2 + 4*l^4*n + l^4)*r^7 + 3*(l^2*n^4 + 4*l^2*n^3 + 6*l^2*n^2 + 4*l^2*n + l^2)*r^5 - ((4*l^2*m - 1)*n^4 + 4*(4*l^2*m - 1)*n^3 + 4*l^2*m + 6*(4*l^2*m - 1)*n^2 + 4*(4*l^2*m - 1)*n - 1)*r^3 - 2*(m*n^4 + 4*m*n^3 + 6*m*n^2 + 4*m*n + m)*r)*diff(mu_1(r), r) + ((l^4*n^4 + 4*l^4*n^3 + 6*l^4*n^2 + 4*l^4*n + l^4)*r^8 + 2*(l^2*n^4 + 4*l^2*n^3 + 6*l^2*n^2 + 4*l^2*n + l^2)*r^6 + 4*m^2*n^4 + 16*m^2*n^3 - ((4*l^2*m - 1)*n^4 + 4*(4*l^2*m - 1)*n^3 + 4*l^2*m + 6*(4*l^2*m - 1)*n^2 + 4*(4*l^2*m - 1)*n - 1)*r^4 + 24*m^2*n^2 + 16*m^2*n - 4*(m*n^4 + 4*m*n^3 + 6*m*n^2 + 4*m*n + m)*r^2 + 4*m^2)*diff(mu_1(r), r, r) - Mu0 == 0" ] }, "execution_count": 59, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = (eq_mu1/a^4).simplify_full()\n", "eq_mu1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We plug the solutions obtained previously for $\\phi_1(r)$ and $\\psi_1(r)$ in this equation:" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\mu_0}^{3} - {\\left({\\mu_0}^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n - {\\mu_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2}\\right)} {\\mathfrak{q}}^{2} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) + {\\left({\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} + 4 \\, m^{2} n^{4} + 16 \\, m^{2} n^{3} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 24 \\, m^{2} n^{2} + 16 \\, m^{2} n - 4 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) - {\\mu_0} = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\mu_0}^{3} - {\\left({\\mu_0}^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n - {\\mu_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2}\\right)} {\\mathfrak{q}}^{2} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) + {\\left({\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} + 4 \\, m^{2} n^{4} + 16 \\, m^{2} n^{3} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 24 \\, m^{2} n^{2} + 16 \\, m^{2} n - 4 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) - {\\mu_0} = 0$$" ], "text/plain": [ "(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^4 - 3*((Mu0^3 - Mu0)*l^4*n^4 + 2*(Mu0^3 - Mu0)*l^4*n^3 - 2*(Mu0^3 - Mu0)*l^4*n - (Mu0^3 - Mu0)*l^4)*r^4 - 4*(Mu0^3 - Mu0)*l^2*m + 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^3 + Mu0^3 - (Mu0^3 + (Mu0^3 - Mu0)*n^2 + 2*(Mu0^3 - Mu0)*n - Mu0)*pf^2 + ((Mu0^3 - Mu0)*n^4 + 2*(Mu0^3 - Mu0)*n^3 + (Mu0^3 - Mu0)*n^2)*qf^2 - 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n + 2*(2*(l^4*n^4 + 4*l^4*n^3 + 6*l^4*n^2 + 4*l^4*n + l^4)*r^7 + 3*(l^2*n^4 + 4*l^2*n^3 + 6*l^2*n^2 + 4*l^2*n + l^2)*r^5 - ((4*l^2*m - 1)*n^4 + 4*(4*l^2*m - 1)*n^3 + 4*l^2*m + 6*(4*l^2*m - 1)*n^2 + 4*(4*l^2*m - 1)*n - 1)*r^3 - 2*(m*n^4 + 4*m*n^3 + 6*m*n^2 + 4*m*n + m)*r)*diff(mu_1(r), r) + ((l^4*n^4 + 4*l^4*n^3 + 6*l^4*n^2 + 4*l^4*n + l^4)*r^8 + 2*(l^2*n^4 + 4*l^2*n^3 + 6*l^2*n^2 + 4*l^2*n + l^2)*r^6 + 4*m^2*n^4 + 16*m^2*n^3 - ((4*l^2*m - 1)*n^4 + 4*(4*l^2*m - 1)*n^3 + 4*l^2*m + 6*(4*l^2*m - 1)*n^2 + 4*(4*l^2*m - 1)*n - 1)*r^4 + 24*m^2*n^2 + 16*m^2*n - 4*(m*n^4 + 4*m*n^3 + 6*m*n^2 + 4*m*n + m)*r^2 + 4*m^2)*diff(mu_1(r), r, r) - Mu0 == 0" ] }, "execution_count": 60, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = eq_mu1.substitute_function(phi_1, phi1_sol).substitute_function(psi_1, psi1_sol)\n", "eq_mu1 = eq_mu1.simplify_full()\n", "eq_mu1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.9)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\mu_0}^{3} - {\\left({\\mu_0}^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n - {\\mu_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2}\\right)} {\\mathfrak{q}}^{2} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) + {\\left({\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} + 4 \\, m^{2} n^{4} + 16 \\, m^{2} n^{3} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 24 \\, m^{2} n^{2} + 16 \\, m^{2} n - 4 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) - {\\mu_0}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{4} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{3} - 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{3} + {\\mu_0}^{3} - {\\left({\\mu_0}^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n - {\\mu_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{4} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{3} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2}\\right)} {\\mathfrak{q}}^{2} - 2 \\, {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) + {\\left({\\left({\\ell}^{4} n^{4} + 4 \\, {\\ell}^{4} n^{3} + 6 \\, {\\ell}^{4} n^{2} + 4 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{4} + 4 \\, {\\ell}^{2} n^{3} + 6 \\, {\\ell}^{2} n^{2} + 4 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} + 4 \\, m^{2} n^{4} + 16 \\, m^{2} n^{3} - {\\left({\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{4} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{3} + 4 \\, {\\ell}^{2} m + 6 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 4 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 24 \\, m^{2} n^{2} + 16 \\, m^{2} n - 4 \\, {\\left(m n^{4} + 4 \\, m n^{3} + 6 \\, m n^{2} + 4 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) - {\\mu_0}$$" ], "text/plain": [ "(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^4 - 3*((Mu0^3 - Mu0)*l^4*n^4 + 2*(Mu0^3 - Mu0)*l^4*n^3 - 2*(Mu0^3 - Mu0)*l^4*n - (Mu0^3 - Mu0)*l^4)*r^4 - 4*(Mu0^3 - Mu0)*l^2*m + 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^3 + Mu0^3 - (Mu0^3 + (Mu0^3 - Mu0)*n^2 + 2*(Mu0^3 - Mu0)*n - Mu0)*pf^2 + ((Mu0^3 - Mu0)*n^4 + 2*(Mu0^3 - Mu0)*n^3 + (Mu0^3 - Mu0)*n^2)*qf^2 - 2*(4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n + 2*(2*(l^4*n^4 + 4*l^4*n^3 + 6*l^4*n^2 + 4*l^4*n + l^4)*r^7 + 3*(l^2*n^4 + 4*l^2*n^3 + 6*l^2*n^2 + 4*l^2*n + l^2)*r^5 - ((4*l^2*m - 1)*n^4 + 4*(4*l^2*m - 1)*n^3 + 4*l^2*m + 6*(4*l^2*m - 1)*n^2 + 4*(4*l^2*m - 1)*n - 1)*r^3 - 2*(m*n^4 + 4*m*n^3 + 6*m*n^2 + 4*m*n + m)*r)*diff(mu_1(r), r) + ((l^4*n^4 + 4*l^4*n^3 + 6*l^4*n^2 + 4*l^4*n + l^4)*r^8 + 2*(l^2*n^4 + 4*l^2*n^3 + 6*l^2*n^2 + 4*l^2*n + l^2)*r^6 + 4*m^2*n^4 + 16*m^2*n^3 - ((4*l^2*m - 1)*n^4 + 4*(4*l^2*m - 1)*n^3 + 4*l^2*m + 6*(4*l^2*m - 1)*n^2 + 4*(4*l^2*m - 1)*n - 1)*r^4 + 24*m^2*n^2 + 16*m^2*n - 4*(m*n^4 + 4*m*n^3 + 6*m*n^2 + 4*m*n + m)*r^2 + 4*m^2)*diff(mu_1(r), r, r) - Mu0" ] }, "execution_count": 61, "metadata": { }, "output_type": "execute_result" } ], "source": [ "lhs = eq_mu1.lhs()\n", "lhs = lhs.simplify_full()\n", "lhs" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{4}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{4}$$" ], "text/plain": [ "(l^2*r^4 + r^2 - 2*m)^2*(n + 1)^4" ] }, "execution_count": 62, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = lhs.coefficient(diff(mu_1(r), r, 2)) # coefficient of mu_1''\n", "s.factor()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + {\\mu_0}^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathfrak{p}}^{2} + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left(4 \\, {\\ell}^{2} m + {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{2} + 2 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) - {\\mu_0}}{{\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} - {\\left(4 \\, {\\ell}^{2} m + {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 4 \\, m^{2} n^{2} + 8 \\, m^{2} n - 4 \\, {\\left(m n^{2} + 2 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\mu_0}^{3} - {\\mu_0}\\right)} n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4} n^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{4}\\right)} r^{4} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m + {\\mu_0}^{3} + {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\ell}^{2} m - {\\mu_0}^{3} + {\\mu_0}\\right)} n^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathfrak{p}}^{2} + 2 \\, {\\left(2 \\, {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{7} + 3 \\, {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{5} - {\\left(4 \\, {\\ell}^{2} m + {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{3} - 2 \\, {\\left(m n^{2} + 2 \\, m n + m\\right)} r\\right)} \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right) - {\\mu_0}}{{\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{8} + 2 \\, {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{6} - {\\left(4 \\, {\\ell}^{2} m + {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n^{2} + 2 \\, {\\left(4 \\, {\\ell}^{2} m - 1\\right)} n - 1\\right)} r^{4} + 4 \\, m^{2} n^{2} + 8 \\, m^{2} n - 4 \\, {\\left(m n^{2} + 2 \\, m n + m\\right)} r^{2} + 4 \\, m^{2}}$$" ], "text/plain": [ "((Mu0^3 - Mu0)*n^2*qf^2 - 3*((Mu0^3 - Mu0)*l^4*n^2 - (Mu0^3 - Mu0)*l^4)*r^4 - 4*(Mu0^3 - Mu0)*l^2*m + Mu0^3 + (4*(Mu0^3 - Mu0)*l^2*m - Mu0^3 + Mu0)*n^2 - (Mu0^3 - Mu0)*pf^2 + 2*(2*(l^4*n^2 + 2*l^4*n + l^4)*r^7 + 3*(l^2*n^2 + 2*l^2*n + l^2)*r^5 - (4*l^2*m + (4*l^2*m - 1)*n^2 + 2*(4*l^2*m - 1)*n - 1)*r^3 - 2*(m*n^2 + 2*m*n + m)*r)*diff(mu_1(r), r) - Mu0)/((l^4*n^2 + 2*l^4*n + l^4)*r^8 + 2*(l^2*n^2 + 2*l^2*n + l^2)*r^6 - (4*l^2*m + (4*l^2*m - 1)*n^2 + 2*(4*l^2*m - 1)*n - 1)*r^4 + 4*m^2*n^2 + 8*m^2*n - 4*(m*n^2 + 2*m*n + m)*r^2 + 4*m^2)" ] }, "execution_count": 63, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = (lhs/s - diff(mu_1(r), r, 2)).simplify_full()\n", "s1" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} r^{2} + 1\\right)} r}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} r^{2} + 1\\right)} r}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "2*(2*l^2*r^2 + 1)*r/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 64, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b1 = s1.coefficient(diff(mu_1(r), r)).factor()\n", "b1" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(3 \\, {\\ell}^{4} n^{2} r^{4} - 3 \\, {\\ell}^{4} r^{4} - 4 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} + 4 \\, {\\ell}^{2} m + n^{2} + {\\mathfrak{p}}^{2} - 1\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(3 \\, {\\ell}^{4} n^{2} r^{4} - 3 \\, {\\ell}^{4} r^{4} - 4 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} + 4 \\, {\\ell}^{2} m + n^{2} + {\\mathfrak{p}}^{2} - 1\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{2}}$$" ], "text/plain": [ "-(3*l^4*n^2*r^4 - 3*l^4*r^4 - 4*l^2*m*n^2 - n^2*qf^2 + 4*l^2*m + n^2 + pf^2 - 1)*(Mu0 + 1)*(Mu0 - 1)*Mu0/((l^2*r^4 + r^2 - 2*m)^2*(n + 1)^2)" ] }, "execution_count": 65, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b2 = (s1 - b1*diff(mu_1(r), r)).simplify_full().factor()\n", "b2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The equation for $\\mu_1$ is thus:" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} r^{2} + 1\\right)} r \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right)}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m} - \\frac{{\\left(3 \\, {\\ell}^{4} n^{2} r^{4} - 3 \\, {\\ell}^{4} r^{4} - 4 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} + 4 \\, {\\ell}^{2} m + n^{2} + {\\mathfrak{p}}^{2} - 1\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{2}} + \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} r^{2} + 1\\right)} r \\frac{\\partial}{\\partial r}\\mu_{1}\\left(r\\right)}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m} - \\frac{{\\left(3 \\, {\\ell}^{4} n^{2} r^{4} - 3 \\, {\\ell}^{4} r^{4} - 4 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} + 4 \\, {\\ell}^{2} m + n^{2} + {\\mathfrak{p}}^{2} - 1\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{2}} + \\frac{\\partial^{2}}{(\\partial r)^{2}}\\mu_{1}\\left(r\\right) = 0$$" ], "text/plain": [ "2*(2*l^2*r^2 + 1)*r*diff(mu_1(r), r)/(l^2*r^4 + r^2 - 2*m) - (3*l^4*n^2*r^4 - 3*l^4*r^4 - 4*l^2*m*n^2 - n^2*qf^2 + 4*l^2*m + n^2 + pf^2 - 1)*(Mu0 + 1)*(Mu0 - 1)*Mu0/((l^2*r^4 + r^2 - 2*m)^2*(n + 1)^2) + diff(mu_1(r), r, r) == 0" ] }, "execution_count": 66, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = diff(mu_1(r), r, 2) + b1*diff(mu_1(r), r) + b2 == 0\n", "eq_mu1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us define\n", "$$ \\Theta_2 := 2 \\Theta_0$$" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "Th2 = var('Th2', latex_name=r'\\Theta_2', \n", " domain='real')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Given that \n", "$$ \\mu_1(r) = - \\sin\\Theta_0 \\; \\theta_1(r) = - \\sqrt{1-\\mu_0^2} \\; \\theta_1(r)$$\n", "and\n", "$$\\sin2\\Theta_0 = 2\\mu_0\\sqrt{1-\\mu_0^2},$$\n", "we get the equation for $\\Upsilon := \\theta_1' = - \\frac{\\mu_1'}{\\sqrt{1 - \\mu_0}}$:" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} r^{2} + 1\\right)} r \\Upsilon\\left(r\\right)}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m} - \\frac{{\\left(3 \\, {\\ell}^{4} n^{2} r^{4} - 3 \\, {\\ell}^{4} r^{4} - 4 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} + 4 \\, {\\ell}^{2} m + n^{2} + {\\mathfrak{p}}^{2} - 1\\right)} \\sin\\left({\\Theta_2}\\right)}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{2}} + \\frac{\\partial}{\\partial r}\\Upsilon\\left(r\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} r^{2} + 1\\right)} r \\Upsilon\\left(r\\right)}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m} - \\frac{{\\left(3 \\, {\\ell}^{4} n^{2} r^{4} - 3 \\, {\\ell}^{4} r^{4} - 4 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} + 4 \\, {\\ell}^{2} m + n^{2} + {\\mathfrak{p}}^{2} - 1\\right)} \\sin\\left({\\Theta_2}\\right)}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}^{2} {\\left(n + 1\\right)}^{2}} + \\frac{\\partial}{\\partial r}\\Upsilon\\left(r\\right) = 0$$" ], "text/plain": [ "2*(2*l^2*r^2 + 1)*r*Y(r)/(l^2*r^4 + r^2 - 2*m) - 1/2*(3*l^4*n^2*r^4 - 3*l^4*r^4 - 4*l^2*m*n^2 - n^2*qf^2 + 4*l^2*m + n^2 + pf^2 - 1)*sin(Th2)/((l^2*r^4 + r^2 - 2*m)^2*(n + 1)^2) + diff(Y(r), r) == 0" ] }, "execution_count": 68, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y = function('Y', latex_name=r'\\Upsilon')\n", "eq_Y = diff(Y(r), r) + b1*Y(r) \\\n", " - (b2/(2*(1-Mu0^2)*Mu0)*sin(Th2)).factor() == 0\n", "eq_Y" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This agrees with Eq. (4.9) of the paper." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Solving the equation for $\\Upsilon := \\theta_1'$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We use the function `desolve` to solve the differential equation for $\\Upsilon$:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "Y_sol(r) = desolve(eq_Y, Y(r), ivar=r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The solution involves an integral that SageMath is not capable to evaluate with the default integrator. Trying to display `Y_sol` would make SageMath hang. Instead, we print `Y_sol` to get the unvaluated form of the integral, in order to compute it by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1/2*(2*_C + 3*(l^2*n*sin(Th2) - l^2*sin(Th2))*r/(n + 1) - integrate((n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*r^2 - pf^2 + 1)/(l^2*r^4 + r^2 - 2*m), r)*sin(Th2)/(n^2 + 2*n + 1))/(l^2*r^4 + r^2 - 2*m)\n" ] } ], "source": [ "print(Y_sol(r))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The solution involves some constant, denoted `_C` by SageMath. We rename it `C_1` and \n", "rewrite the above solution as" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, C_{1} + \\frac{3 \\, {\\left({\\ell}^{2} n \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r}{n + 1} - \\frac{{\\rm Integ}\\left(r\\right) \\sin\\left({\\Theta_2}\\right)}{n^{2} + 2 \\, n + 1}}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, C_{1} + \\frac{3 \\, {\\left({\\ell}^{2} n \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r}{n + 1} - \\frac{{\\rm Integ}\\left(r\\right) \\sin\\left({\\Theta_2}\\right)}{n^{2} + 2 \\, n + 1}}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)}}$$" ], "text/plain": [ "1/2*(2*C_1 + 3*(l^2*n*sin(Th2) - l^2*sin(Th2))*r/(n + 1) - Integ(r)*sin(Th2)/(n^2 + 2*n + 1))/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 71, "metadata": { }, "output_type": "execute_result" } ], "source": [ "C_1 = var('C_1')\n", "Integ(r) = function('Integ')(r)\n", "Y_sol0(r) = 1/2*(2*C_1 + 3*(l^2*n*sin(Th2) - l^2*sin(Th2))*r/(n + 1) \\\n", " - Integ(r)*sin(Th2)/(n^2 + 2*n + 1))/(l^2*r^4 + r^2 - 2*m)\n", "Y_sol0(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "`Integ(r)` represents the integral $I(r)$, whose integrand, $F(r)$ say, is read from the\n", "output of `print(Y_sol(r))`:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r^{2} - {\\mathfrak{p}}^{2} + 1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r^{2} - {\\mathfrak{p}}^{2} + 1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "(n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*r^2 - pf^2 + 1)/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 72, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F(r) = (n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*r^2 - pf^2 + 1) \\\n", " /(l^2*r^4 + r^2 - 2*m)\n", "F(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We split the integral in two parts:\n", "$$ I(r) = F_1 \\; s_1(r) + F_2 \\; s_2(r)$$\n", "with \n", "$$ s_1(r) := \\int^r \\frac{\\bar{r}^2}{\\ell^2 \\bar{r}^4 + \\bar{r}^2 - 2m} \\, \\mathrm{d}\\bar{r}, \\qquad s_2(r) := \\int^r \\frac{\\mathrm{d}\\bar{r}}{\\ell^2 \\bar{r}^4 + \\bar{r}^2 - 2m} $$\n", "and" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\ell}^{2} n^{2} - 3 \\, {\\ell}^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\ell}^{2} n^{2} - 3 \\, {\\ell}^{2}$$" ], "text/plain": [ "3*l^2*n^2 - 3*l^2" ] }, "execution_count": 73, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F1 = 3*(l^2*n^2 - l^2)\n", "F1" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} - {\\mathfrak{p}}^{2} + 1$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} - {\\mathfrak{p}}^{2} + 1$$" ], "text/plain": [ "n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 - pf^2 + 1" ] }, "execution_count": 74, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F2 = n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 - pf^2 + 1\n", "F2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 75, "metadata": { }, "output_type": "execute_result" } ], "source": [ "bool(F(r) == F1*r^2/(l^2*r^4 + r^2 - 2*m) + F2/(l^2*r^4 + r^2 - 2*m))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us evaluate $s_1(r)$ by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + r\\right)$$" ], "text/plain": [ "1/2*sqrt(1/2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))*log(sqrt(1/2)*(8*l^4*m + l^2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + r) - 1/2*sqrt(1/2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))*log(-sqrt(1/2)*(8*l^4*m + l^2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + r) - 1/2*sqrt(1/2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))*log(sqrt(1/2)*(8*l^4*m + l^2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + r) + 1/2*sqrt(1/2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))*log(-sqrt(1/2)*(8*l^4*m + l^2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + r)" ] }, "execution_count": 76, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = integrate(r^2/(l^2*r^4 + r^2 - 2*m), r, algorithm='fricas')\n", "s1" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r - \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r + \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + \\sqrt{2} \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r + \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r - \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} {\\ell}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r - \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r + \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + \\sqrt{2} \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r + \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} r - \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} {\\ell}}$$" ], "text/plain": [ "1/4*(sqrt(2)*sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1)*log((sqrt(2)*(8*l^2*m + 1)^(1/4)*l*r - sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1))/(sqrt(2)*(8*l^2*m + 1)^(1/4)*l*r + sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1))) + sqrt(2)*sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1)*log((sqrt(2)*(8*l^2*m + 1)^(1/4)*l*r + sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1))/(sqrt(2)*(8*l^2*m + 1)^(1/4)*l*r - sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1))))/((8*l^2*m + 1)^(3/4)*l)" ] }, "execution_count": 77, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.canonicalize_radical().simplify_log()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{2}}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{2}}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "r^2/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 78, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(s1, r).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Similarly, we evaluate $s_2(r)$ by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} r - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right)$$" ], "text/plain": [ "-1/4*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))*log(2*l^2*r + 1/2*(8*l^2*m - (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))) + 1/4*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))*log(2*l^2*r - 1/2*(8*l^2*m - (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))) - 1/4*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))*log(2*l^2*r + 1/2*(8*l^2*m + (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))) + 1/4*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))*log(2*l^2*r - 1/2*(8*l^2*m + (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m)))" ] }, "execution_count": 79, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = integrate(1/(l^2*r^4 + r^2 - 2*m), r, algorithm='fricas')\n", "s2" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r - \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r + \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r - \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r - \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r + \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r - \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} r + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$$" ], "text/plain": [ "1/4*(sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*log((4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*r - sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*(sqrt(8*l^2*m + 1) + 1))/(4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*r + sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*(sqrt(8*l^2*m + 1) + 1))) + sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*log((4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*r - sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*(sqrt(8*l^2*m + 1) - 1))/(4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*r + sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*(sqrt(8*l^2*m + 1) - 1))))/((8*l^2*m + 1)^(3/4)*sqrt(m))" ] }, "execution_count": 80, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.canonicalize_radical().simplify_log()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 81, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(s2, r).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the above expressions for $s_1(r)$ and $s_2(r)$ there appears $\\sqrt{1 + 8 \\ell^2 m}$,\n", "which can be rewritten\n", "$$\n", " \\sqrt{1 + 8 \\ell^2 m} = 2 \\ell^2 r_H^2 + 1 \n", "$$\n", "where $r_H$ is the positive root of $\\ell^2 r_H^4 + r_H^2 - 2m = 0$. More precisely, we perform the following substitution:\n", "$$\n", " m = \\frac{1}{2} r_H^2 (\\ell^2 r_H^2 + 1)\n", "$$" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\ell} {r_H} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right) + i \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{{\\ell} r - i \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}{{\\ell} r + i \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{3} {r_H}^{2} + {\\ell}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\ell} {r_H} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right) + i \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{{\\ell} r - i \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}{{\\ell} r + i \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{3} {r_H}^{2} + {\\ell}\\right)}}$$" ], "text/plain": [ "-1/2*(l*rH*log((r + rH)/(r - rH)) + I*sqrt(l^2*rH^2 + 1)*log((l*r - I*sqrt(l^2*rH^2 + 1))/(l*r + I*sqrt(l^2*rH^2 + 1))))/(2*l^3*rH^2 + l)" ] }, "execution_count": 82, "metadata": { }, "output_type": "execute_result" } ], "source": [ "rH = var('rH', latex_name=r'r_H', domain='real')\n", "assume(rH > 0)\n", "m_rH = rH^2*(l^2*rH^2 + 1)/2\n", "s1 = s1.subs({m: m_rH}).canonicalize_radical().simplify_log()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the second $\\log$, we recognize the $\\mathrm{arccot}$ function, via the identity\n", "$$\n", " \\mathrm{arccot}\\, x = \\frac{i}{2} \\ln\\left( \\frac{x - i}{x + i} \\right) . \n", "$$\n", "Given that $\\mathrm{arccot}\\, x = \\pi/2 - \\mathrm{arctan}\\, x$, we use this identity as\n", "$$\n", "i \\ln\\left( \\frac{x - i}{x + i} \\right) = - 2 \\, \\mathrm{arctan}(x) + \\pi\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Thus, we perform the following substitution, disregarding the additive constant $\\pi$:" ] }, { "cell_type": "code", "execution_count": 83, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\ell} {r_H} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right) - 2 \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{3} {r_H}^{2} + {\\ell}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\ell} {r_H} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right) - 2 \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{3} {r_H}^{2} + {\\ell}\\right)}}$$" ], "text/plain": [ "-1/2*(l*rH*log((r + rH)/(r - rH)) - 2*sqrt(l^2*rH^2 + 1)*arctan(l*r/sqrt(l^2*rH^2 + 1)))/(2*l^3*rH^2 + l)" ] }, "execution_count": 83, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.subs({I*sqrt(l^2*rH^2 + 1)*log((l*r - I*sqrt(l^2*rH^2 + 1))/(l*r + I*sqrt(l^2*rH^2 + 1))):\n", " -2*sqrt(l^2*rH^2 + 1)*atan(l*r/sqrt(l^2*rH^2 + 1))})\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we have indeed a primitive of $r\\mapsto \\frac{r^2}{\\ell^2 r^4 + r^2 - 2m}$:" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{2}}{{\\ell}^{2} r^{4} - {\\ell}^{2} {r_H}^{4} + r^{2} - {r_H}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{2}}{{\\ell}^{2} r^{4} - {\\ell}^{2} {r_H}^{4} + r^{2} - {r_H}^{2}}$$" ], "text/plain": [ "r^2/(l^2*r^4 - l^2*rH^4 + r^2 - rH^2)" ] }, "execution_count": 84, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds1 = diff(s1, r).simplify_full()\n", "Ds1" ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{2}}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{2}}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "r^2/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 85, "metadata": { }, "output_type": "execute_result" } ], "source": [ "rH_m = sqrt(sqrt(1 + 8*l^2*m) - 1)/(sqrt(2)*l)\n", "Ds1.subs({rH: rH_m}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Similarly, let us express $s_2$ in terms of $r_H$:" ] }, { "cell_type": "code", "execution_count": 86, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, {\\ell} {r_H} \\log\\left(\\frac{-i \\, {\\ell}^{2} {r_H}^{2} + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} r - i}{i \\, {\\ell}^{2} {r_H}^{2} + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} r + i}\\right) + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r - {r_H}}{r + {r_H}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, {\\ell} {r_H} \\log\\left(\\frac{-i \\, {\\ell}^{2} {r_H}^{2} + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} r - i}{i \\, {\\ell}^{2} {r_H}^{2} + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} r + i}\\right) + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r - {r_H}}{r + {r_H}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "1/2*(I*l*rH*log((-I*l^2*rH^2 + sqrt(l^2*rH^2 + 1)*l*r - I)/(I*l^2*rH^2 + sqrt(l^2*rH^2 + 1)*l*r + I)) + sqrt(l^2*rH^2 + 1)*log((r - rH)/(r + rH)))/((2*l^2*rH^3 + rH)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 86, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({m: m_rH}).canonicalize_radical().simplify_log()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Again, we use the identity\n", "$$\n", "i \\ln\\left( \\frac{x - i}{x + i} \\right) = - 2 \\, \\mathrm{arctan}(x) + \\pi\n", "$$\n", "to rewrite $s_2$ as" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\ell} {r_H} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r - {r_H}}{r + {r_H}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\ell} {r_H} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r - {r_H}}{r + {r_H}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*(2*l*rH*arctan(l*r/sqrt(l^2*rH^2 + 1)) - sqrt(l^2*rH^2 + 1)*log((r - rH)/(r + rH)))/((2*l^2*rH^3 + rH)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 87, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({I*l*rH*log((-I*l^2*rH^2 + sqrt(l^2*rH^2 + 1)*l*r - I)/(I*l^2*rH^2 + sqrt(l^2*rH^2 + 1)*l*r + I)):\n", " -2*l*rH*atan(l*r/sqrt(l^2*rH^2 + 1))})\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us also replace $\\ln\\left(\\frac{r - r_H}{r + r_H}\\right)$ by $-\\ln\\left(\\frac{r + r_H}{r - r_H}\\right)$\n", "in order to have the same log term as in $s_1(r)$:" ] }, { "cell_type": "code", "execution_count": 88, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\ell} {r_H} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\ell} {r_H} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*(2*l*rH*arctan(l*r/sqrt(l^2*rH^2 + 1)) + sqrt(l^2*rH^2 + 1)*log((r + rH)/(r - rH)))/((2*l^2*rH^3 + rH)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 88, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({log((r - rH)/(r + rH)): - log((r + rH)/(r - rH))})\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we have indeed a primitive of $r\\mapsto \\frac{1}{\\ell^2 r^4 + r^2 - 2m}$:" ] }, { "cell_type": "code", "execution_count": 89, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} - {\\ell}^{2} {r_H}^{4} + r^{2} - {r_H}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} - {\\ell}^{2} {r_H}^{4} + r^{2} - {r_H}^{2}}$$" ], "text/plain": [ "1/(l^2*r^4 - l^2*rH^4 + r^2 - rH^2)" ] }, "execution_count": 89, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds2 = diff(s2, r).simplify_full()\n", "Ds2" ] }, { "cell_type": "code", "execution_count": 90, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 90, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds2.subs({rH: rH_m}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The full integral is thus" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right) - 2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} - {\\ell}^{3}\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{3} m - 2 \\, {\\left({\\ell}^{3} m + 2 \\, {\\ell}\\right)} n^{2} - {\\ell} {\\mathfrak{p}}^{2} + 4 \\, {\\ell}\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right) - 2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} - {\\ell}^{3}\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{3} m - 2 \\, {\\left({\\ell}^{3} m + 2 \\, {\\ell}\\right)} n^{2} - {\\ell} {\\mathfrak{p}}^{2} + 4 \\, {\\ell}\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*((n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*rH^2 - pf^2 + 1)*sqrt(l^2*rH^2 + 1)*log((r + rH)/(r - rH)) - 2*(3*(l^3*n^2 - l^3)*rH^3 - (l*n^2*qf^2 + 2*l^3*m - 2*(l^3*m + 2*l)*n^2 - l*pf^2 + 4*l)*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)))/((2*l^2*rH^3 + rH)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 91, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Integ0 = (F1*s1 + F2*s2).simplify_full()\n", "Integ0" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "so that the solution is" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left({\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 4 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(4 \\, {\\left(2 \\, C_{1} {\\ell}^{2} n^{2} + 4 \\, C_{1} {\\ell}^{2} n + 2 \\, C_{1} {\\ell}^{2} + 3 \\, {\\left({\\ell}^{4} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{4} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H}^{3} + 2 \\, {\\left(2 \\, C_{1} n^{2} + 4 \\, C_{1} n + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r + 2 \\, C_{1}\\right)} {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) - {\\left(2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)}}{4 \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(2 \\, {\\left(2 \\, {\\ell}^{2} m n^{2} - {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{4} + 4 \\, {\\ell}^{2} m n + 2 \\, {\\ell}^{2} m - {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{2}\\right)} {r_H}^{3} - {\\left({\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{4} - 2 \\, m n^{2} + {\\left(n^{2} + 2 \\, n + 1\\right)} r^{2} - 4 \\, m n - 2 \\, m\\right)} {r_H}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left({\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 4 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(4 \\, {\\left(2 \\, C_{1} {\\ell}^{2} n^{2} + 4 \\, C_{1} {\\ell}^{2} n + 2 \\, C_{1} {\\ell}^{2} + 3 \\, {\\left({\\ell}^{4} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{4} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H}^{3} + 2 \\, {\\left(2 \\, C_{1} n^{2} + 4 \\, C_{1} n + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r + 2 \\, C_{1}\\right)} {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) - {\\left(2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)}}{4 \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(2 \\, {\\left(2 \\, {\\ell}^{2} m n^{2} - {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{4} + 4 \\, {\\ell}^{2} m n + 2 \\, {\\ell}^{2} m - {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{2}\\right)} {r_H}^{3} - {\\left({\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{4} - 2 \\, m n^{2} + {\\left(n^{2} + 2 \\, n + 1\\right)} r^{2} - 4 \\, m n - 2 \\, m\\right)} {r_H}\\right)}}$$" ], "text/plain": [ "1/4*(2*(3*(l^3*n^2*sin(Th2) - l^3*sin(Th2))*rH^3 - (l*n^2*qf^2*sin(Th2) + 2*l^3*m*sin(Th2) - l*pf^2*sin(Th2) - 2*(l^3*m*sin(Th2) + 2*l*sin(Th2))*n^2 + 4*l*sin(Th2))*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)) - sqrt(l^2*rH^2 + 1)*(4*(2*C_1*l^2*n^2 + 4*C_1*l^2*n + 2*C_1*l^2 + 3*(l^4*n^2*sin(Th2) - l^4*sin(Th2))*r)*rH^3 + 2*(2*C_1*n^2 + 4*C_1*n + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*r + 2*C_1)*rH + (n^2*qf^2*sin(Th2) + 2*l^2*m*sin(Th2) - (2*l^2*m*sin(Th2) + sin(Th2))*n^2 + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*rH^2 - pf^2*sin(Th2) + sin(Th2))*log((r + rH)/(r - rH))))/(sqrt(l^2*rH^2 + 1)*(2*(2*l^2*m*n^2 - (l^4*n^2 + 2*l^4*n + l^4)*r^4 + 4*l^2*m*n + 2*l^2*m - (l^2*n^2 + 2*l^2*n + l^2)*r^2)*rH^3 - ((l^2*n^2 + 2*l^2*n + l^2)*r^4 - 2*m*n^2 + (n^2 + 2*n + 1)*r^2 - 4*m*n - 2*m)*rH))" ] }, "execution_count": 92, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y_sol(r) = Y_sol0(r).subs({Integ(r): Integ0}).simplify_full()\n", "Y_sol(r)" ] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left({\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 4 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(4 \\, {\\left(2 \\, C_{1} {\\ell}^{2} n^{2} + 4 \\, C_{1} {\\ell}^{2} n + 2 \\, C_{1} {\\ell}^{2} + 3 \\, {\\left({\\ell}^{4} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{4} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H}^{3} + 2 \\, {\\left(2 \\, C_{1} n^{2} + 4 \\, C_{1} n + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r + 2 \\, C_{1}\\right)} {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) - {\\left(2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left({\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 4 \\, {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) + \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(4 \\, {\\left(2 \\, C_{1} {\\ell}^{2} n^{2} + 4 \\, C_{1} {\\ell}^{2} n + 2 \\, C_{1} {\\ell}^{2} + 3 \\, {\\left({\\ell}^{4} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{4} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H}^{3} + 2 \\, {\\left(2 \\, C_{1} n^{2} + 4 \\, C_{1} n + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r + 2 \\, C_{1}\\right)} {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) - {\\left(2 \\, {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + \\sin\\left({\\Theta_2}\\right)\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)}$$" ], "text/plain": [ "-2*(3*(l^3*n^2*sin(Th2) - l^3*sin(Th2))*rH^3 - (l*n^2*qf^2*sin(Th2) + 2*l^3*m*sin(Th2) - l*pf^2*sin(Th2) - 2*(l^3*m*sin(Th2) + 2*l*sin(Th2))*n^2 + 4*l*sin(Th2))*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)) + sqrt(l^2*rH^2 + 1)*(4*(2*C_1*l^2*n^2 + 4*C_1*l^2*n + 2*C_1*l^2 + 3*(l^4*n^2*sin(Th2) - l^4*sin(Th2))*r)*rH^3 + 2*(2*C_1*n^2 + 4*C_1*n + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*r + 2*C_1)*rH + (n^2*qf^2*sin(Th2) + 2*l^2*m*sin(Th2) - (2*l^2*m*sin(Th2) + sin(Th2))*n^2 + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*rH^2 - pf^2*sin(Th2) + sin(Th2))*log((r + rH)/(r - rH)))" ] }, "execution_count": 93, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y_sol(r).numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 94, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(n + 1\\right)}^{2} {r_H}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(n + 1\\right)}^{2} {r_H}$$" ], "text/plain": [ "4*(l^2*r^4 + r^2 - 2*m)*(2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)*(n + 1)^2*rH" ] }, "execution_count": 94, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y_sol(r).denominator().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that `Y_sol` is indeed a solution of the differential equation for $\\Upsilon$:" ] }, { "cell_type": "code", "execution_count": 95, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$" ], "text/plain": [ "0 == 0" ] }, "execution_count": 95, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_Y.substitute_function(Y, Y_sol).subs({rH: rH_m}).simplify_full()" ] }, { "cell_type": "code", "execution_count": 96, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1/4*(2*(3*(l^3*n^2*sin(Th2) - l^3*sin(Th2))*rH^3 - (l*n^2*qf^2*sin(Th2) + 2*l^3*m*sin(Th2) - l*pf^2*sin(Th2) - 2*(l^3*m*sin(Th2) + 2*l*sin(Th2))*n^2 + 4*l*sin(Th2))*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)) - sqrt(l^2*rH^2 + 1)*(4*(2*C_1*l^2*n^2 + 4*C_1*l^2*n + 2*C_1*l^2 + 3*(l^4*n^2*sin(Th2) - l^4*sin(Th2))*r)*rH^3 + 2*(2*C_1*n^2 + 4*C_1*n + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*r + 2*C_1)*rH + (n^2*qf^2*sin(Th2) + 2*l^2*m*sin(Th2) - (2*l^2*m*sin(Th2) + sin(Th2))*n^2 + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*rH^2 - pf^2*sin(Th2) + sin(Th2))*log((r + rH)/(r - rH))))/(sqrt(l^2*rH^2 + 1)*(2*(2*l^2*m*n^2 - (l^4*n^2 + 2*l^4*n + l^4)*r^4 + 4*l^2*m*n + 2*l^2*m - (l^2*n^2 + 2*l^2*n + l^2)*r^2)*rH^3 - ((l^2*n^2 + 2*l^2*n + l^2)*r^4 - 2*m*n^2 + (n^2 + 2*n + 1)*r^2 - 4*m*n - 2*m)*rH))\n" ] } ], "source": [ "print(Y_sol(r))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.10) (expression of $\\theta'_1 = \\Upsilon$)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The term involving the constant $C_1$ agrees with that of Eq. (4.10):" ] }, { "cell_type": "code", "execution_count": 97, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} r^{4} + r^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(l^2*r^4 + r^2 - 2*m)" ] }, "execution_count": 97, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = Y_sol(r).coefficient(C_1).simplify_full()\n", "s" ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 98, "metadata": { }, "output_type": "execute_result" } ], "source": [ "fr4 = l^2*r^4 + r^2 - 2*m\n", "bool(s == 1/fr4)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us remove it from $\\Upsilon$ and divide the result by $\\sin(2\\Theta_0)$:" ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} - {\\ell}^{3}\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{3} m - 2 \\, {\\left({\\ell}^{3} m + 2 \\, {\\ell}\\right)} n^{2} - {\\ell} {\\mathfrak{p}}^{2} + 4 \\, {\\ell}\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - {\\left(12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r {r_H}^{3} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}{4 \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(2 \\, {\\left(2 \\, {\\ell}^{2} m n^{2} - {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{4} + 4 \\, {\\ell}^{2} m n + 2 \\, {\\ell}^{2} m - {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{2}\\right)} {r_H}^{3} - {\\left({\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{4} - 2 \\, m n^{2} + {\\left(n^{2} + 2 \\, n + 1\\right)} r^{2} - 4 \\, m n - 2 \\, m\\right)} {r_H}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(3 \\, {\\left({\\ell}^{3} n^{2} - {\\ell}^{3}\\right)} {r_H}^{3} - {\\left({\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{3} m - 2 \\, {\\left({\\ell}^{3} m + 2 \\, {\\ell}\\right)} n^{2} - {\\ell} {\\mathfrak{p}}^{2} + 4 \\, {\\ell}\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - {\\left(12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r {r_H}^{3} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}{4 \\, \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(2 \\, {\\left(2 \\, {\\ell}^{2} m n^{2} - {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{4} + 4 \\, {\\ell}^{2} m n + 2 \\, {\\ell}^{2} m - {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{2}\\right)} {r_H}^{3} - {\\left({\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{4} - 2 \\, m n^{2} + {\\left(n^{2} + 2 \\, n + 1\\right)} r^{2} - 4 \\, m n - 2 \\, m\\right)} {r_H}\\right)}}$$" ], "text/plain": [ "1/4*(2*(3*(l^3*n^2 - l^3)*rH^3 - (l*n^2*qf^2 + 2*l^3*m - 2*(l^3*m + 2*l)*n^2 - l*pf^2 + 4*l)*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)) - (12*(l^4*n^2 - l^4)*r*rH^3 + 6*(l^2*n^2 - l^2)*r*rH + (n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*rH^2 - pf^2 + 1)*log((r + rH)/(r - rH)))*sqrt(l^2*rH^2 + 1))/(sqrt(l^2*rH^2 + 1)*(2*(2*l^2*m*n^2 - (l^4*n^2 + 2*l^4*n + l^4)*r^4 + 4*l^2*m*n + 2*l^2*m - (l^2*n^2 + 2*l^2*n + l^2)*r^2)*rH^3 - ((l^2*n^2 + 2*l^2*n + l^2)*r^4 - 2*m*n^2 + (n^2 + 2*n + 1)*r^2 - 4*m*n - 2*m)*rH))" ] }, "execution_count": 99, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y1 = ((Y_sol(r) - s*C_1)/sin(Th2)).simplify_full()\n", "Y1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the arctan term is" ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(3 \\, {\\ell}^{2} n^{2} {r_H}^{2} + 2 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\ell}^{2} {r_H}^{2} - 2 \\, {\\ell}^{2} m + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} {\\ell}}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(n + 1\\right)}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(3 \\, {\\ell}^{2} n^{2} {r_H}^{2} + 2 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\ell}^{2} {r_H}^{2} - 2 \\, {\\ell}^{2} m + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} {\\ell}}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(n + 1\\right)}^{2}}$$" ], "text/plain": [ "-1/2*(3*l^2*n^2*rH^2 + 2*l^2*m*n^2 - n^2*qf^2 - 3*l^2*rH^2 - 2*l^2*m + 4*n^2 + pf^2 - 4)*l/((l^2*r^4 + r^2 - 2*m)*(2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)*(n + 1)^2)" ] }, "execution_count": 100, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = Y1.coefficient(arctan(l*r/sqrt(l^2*rH^2 + 1))).simplify_full().factor()\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The numerator of this term agrees with Eq. (4.10):" ] }, { "cell_type": "code", "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\ell}^{4} n^{2} {r_H}^{4} - {\\ell}^{4} {r_H}^{4} + 4 \\, {\\ell}^{2} n^{2} {r_H}^{2} - n^{2} {\\mathfrak{q}}^{2} - 4 \\, {\\ell}^{2} {r_H}^{2} + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} {\\ell}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\ell}^{4} n^{2} {r_H}^{4} - {\\ell}^{4} {r_H}^{4} + 4 \\, {\\ell}^{2} n^{2} {r_H}^{2} - n^{2} {\\mathfrak{q}}^{2} - 4 \\, {\\ell}^{2} {r_H}^{2} + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} {\\ell}$$" ], "text/plain": [ "-(l^4*n^2*rH^4 - l^4*rH^4 + 4*l^2*n^2*rH^2 - n^2*qf^2 - 4*l^2*rH^2 + 4*n^2 + pf^2 - 4)*l" ] }, "execution_count": 101, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.numerator().subs({m: m_rH}).factor()" ] }, { "cell_type": "code", "execution_count": 102, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 102, "metadata": { }, "output_type": "execute_result" } ], "source": [ "bool(s.numerator().subs({m: m_rH}) \n", " == l*((1 - n^2)*(l^2*rH^2 + 2)^2 - pf^2 + n^2*qf^2))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The denominator agrees with Eq. (4.10) as well:" ] }, { "cell_type": "code", "execution_count": 103, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(n + 1\\right)}^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(n + 1\\right)}^{2}$$" ], "text/plain": [ "2*(l^2*r^4 + r^2 - 2*m)*(2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)*(n + 1)^2" ] }, "execution_count": 103, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us remove the arctan term from $\\Upsilon$:" ] }, { "cell_type": "code", "execution_count": 104, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r {r_H}^{3} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)}{4 \\, {\\left(2 \\, {\\left(2 \\, {\\ell}^{2} m n^{2} - {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{4} + 4 \\, {\\ell}^{2} m n + 2 \\, {\\ell}^{2} m - {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{2}\\right)} {r_H}^{3} - {\\left({\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{4} - 2 \\, m n^{2} + {\\left(n^{2} + 2 \\, n + 1\\right)} r^{2} - 4 \\, m n - 2 \\, m\\right)} {r_H}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r {r_H}^{3} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)}{4 \\, {\\left(2 \\, {\\left(2 \\, {\\ell}^{2} m n^{2} - {\\left({\\ell}^{4} n^{2} + 2 \\, {\\ell}^{4} n + {\\ell}^{4}\\right)} r^{4} + 4 \\, {\\ell}^{2} m n + 2 \\, {\\ell}^{2} m - {\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{2}\\right)} {r_H}^{3} - {\\left({\\left({\\ell}^{2} n^{2} + 2 \\, {\\ell}^{2} n + {\\ell}^{2}\\right)} r^{4} - 2 \\, m n^{2} + {\\left(n^{2} + 2 \\, n + 1\\right)} r^{2} - 4 \\, m n - 2 \\, m\\right)} {r_H}\\right)}}$$" ], "text/plain": [ "-1/4*(12*(l^4*n^2 - l^4)*r*rH^3 + 6*(l^2*n^2 - l^2)*r*rH + (n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*rH^2 - pf^2 + 1)*log((r + rH)/(r - rH)))/(2*(2*l^2*m*n^2 - (l^4*n^2 + 2*l^4*n + l^4)*r^4 + 4*l^2*m*n + 2*l^2*m - (l^2*n^2 + 2*l^2*n + l^2)*r^2)*rH^3 - ((l^2*n^2 + 2*l^2*n + l^2)*r^4 - 2*m*n^2 + (n^2 + 2*n + 1)*r^2 - 4*m*n - 2*m)*rH)" ] }, "execution_count": 104, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y2 = (Y1 - s*arctan(l*r/sqrt(l^2*rH^2 + 1))).simplify_full()\n", "Y2" ] }, { "cell_type": "code", "execution_count": 105, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r {r_H}^{3} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r {r_H}^{3} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r {r_H} + {\\left(n^{2} {\\mathfrak{q}}^{2} + 2 \\, {\\ell}^{2} m - {\\left(2 \\, {\\ell}^{2} m + 1\\right)} n^{2} + 3 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} {r_H}^{2} - {\\mathfrak{p}}^{2} + 1\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)$$" ], "text/plain": [ "12*(l^4*n^2 - l^4)*r*rH^3 + 6*(l^2*n^2 - l^2)*r*rH + (n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*rH^2 - pf^2 + 1)*log((r + rH)/(r - rH))" ] }, "execution_count": 105, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y2.numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 106, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left(n + 1\\right)}^{2} {r_H}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left(n + 1\\right)}^{2} {r_H}$$" ], "text/plain": [ "4*(l^2*r^4 + r^2 - 2*m)*(2*l^2*rH^2 + 1)*(n + 1)^2*rH" ] }, "execution_count": 106, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y2.denominator().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the log term is" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3 \\, {\\ell}^{2} n^{2} {r_H}^{2} - 2 \\, {\\ell}^{2} m n^{2} + n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\ell}^{2} {r_H}^{2} + 2 \\, {\\ell}^{2} m - n^{2} - {\\mathfrak{p}}^{2} + 1}{4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left(n + 1\\right)}^{2} {r_H}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3 \\, {\\ell}^{2} n^{2} {r_H}^{2} - 2 \\, {\\ell}^{2} m n^{2} + n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\ell}^{2} {r_H}^{2} + 2 \\, {\\ell}^{2} m - n^{2} - {\\mathfrak{p}}^{2} + 1}{4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left(n + 1\\right)}^{2} {r_H}}$$" ], "text/plain": [ "1/4*(3*l^2*n^2*rH^2 - 2*l^2*m*n^2 + n^2*qf^2 - 3*l^2*rH^2 + 2*l^2*m - n^2 - pf^2 + 1)/((l^2*r^4 + r^2 - 2*m)*(2*l^2*rH^2 + 1)*(n + 1)^2*rH)" ] }, "execution_count": 107, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = Y2.coefficient(log((r + rH)/(r - rH))).simplify_full().factor()\n", "s" ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\ell}^{4} n^{2} {r_H}^{4} + {\\ell}^{4} {r_H}^{4} + 2 \\, {\\ell}^{2} n^{2} {r_H}^{2} + n^{2} {\\mathfrak{q}}^{2} - 2 \\, {\\ell}^{2} {r_H}^{2} - n^{2} - {\\mathfrak{p}}^{2} + 1$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\ell}^{4} n^{2} {r_H}^{4} + {\\ell}^{4} {r_H}^{4} + 2 \\, {\\ell}^{2} n^{2} {r_H}^{2} + n^{2} {\\mathfrak{q}}^{2} - 2 \\, {\\ell}^{2} {r_H}^{2} - n^{2} - {\\mathfrak{p}}^{2} + 1$$" ], "text/plain": [ "-l^4*n^2*rH^4 + l^4*rH^4 + 2*l^2*n^2*rH^2 + n^2*qf^2 - 2*l^2*rH^2 - n^2 - pf^2 + 1" ] }, "execution_count": 108, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.numerator().subs({m: m_rH}).factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check against Eq. (4.10):" ] }, { "cell_type": "code", "execution_count": 109, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 109, "metadata": { }, "output_type": "execute_result" } ], "source": [ "bool(s.numerator().subs({m: m_rH}) == (1 - n^2)*(l^2*rH^2 - 1)^2 - pf^2 + n^2*qf^2)" ] }, { "cell_type": "code", "execution_count": 110, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left(n + 1\\right)}^{2} {r_H}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left(n + 1\\right)}^{2} {r_H}$$" ], "text/plain": [ "4*(l^2*r^4 + r^2 - 2*m)*(2*l^2*rH^2 + 1)*(n + 1)^2*rH" ] }, "execution_count": 110, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Given that \n", "$$ \\mathrm{artanh}\\, x = \\frac{1}{2} \\ln\\left( \\frac{1 + x}{1 - x} \\right) $$\n", "we have\n", "$$\n", " \\ln \\left( \\frac{x + 1}{x - 1} \\right) = 2\\, \\mathrm{artanh}\\left(\\frac{1}{x}\\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Hence the term in $\\ln\\left(\\frac{r + r_H}{r - r_H}\\right)$ agrees with the corresponding term in Eq. (4.10)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Finally, the last term in $\\Upsilon$ is" ] }, { "cell_type": "code", "execution_count": 111, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3 \\, {\\ell}^{2} {\\left(n - 1\\right)} r}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(n + 1\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3 \\, {\\ell}^{2} {\\left(n - 1\\right)} r}{2 \\, {\\left({\\ell}^{2} r^{4} + r^{2} - 2 \\, m\\right)} {\\left(n + 1\\right)}}$$" ], "text/plain": [ "3/2*l^2*(n - 1)*r/((l^2*r^4 + r^2 - 2*m)*(n + 1))" ] }, "execution_count": 111, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y3 = (Y2 - s*log((r + rH)/(r - rH))).simplify_full()\n", "Y3.factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This term agrees with Eq. (4.10), given the simplification \n", "$\\frac{1 - n^2}{(1 + n)^2} = -\\frac{n - 1}{n + 1}$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Conclusion:** we have full agreement with Eq. (4.10)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Conjugate momenta" ] }, { "cell_type": "code", "execution_count": 112, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def conjugate_momenta(lagr, qs, var):\n", " r\"\"\"\n", " Compute the conjugate momenta from a given Lagrangian.\n", "\n", " INPUT:\n", "\n", " - ``lagr`` -- symbolic expression representing the Lagrangian density\n", " - ``qs`` -- either a single symbolic function or a list/tuple of\n", " symbolic functions, representing the `q`'s; these functions must\n", " appear in ``lagr`` up to at most their first derivatives\n", " - ``var`` -- either a single variable, typically `t` (1-dimensional\n", " problem) or a list/tuple of symbolic variables; in the latter case the\n", " time coordinate must the first one\n", "\n", " OUTPUT:\n", "\n", " - list of conjugate momenta; if only one function is involved, the\n", " single conjugate momentum is returned instead.\n", "\n", " \"\"\"\n", " if not isinstance(qs, (list, tuple)):\n", " qs = [qs]\n", " if not isinstance(var, (list, tuple)):\n", " var = [var]\n", " n = len(qs)\n", " d = len(var)\n", " dqvt = [SR.var('qxxxx{}_t'.format(q)) for q in qs]\n", " subs = {diff(qs[i](*var), var[0]): dqvt[i] for i in range(n)}\n", " subs_inv = {dqvt[i]: diff(qs[i](*var), var[0]) for i in range(n)}\n", " lg = lagr.substitute(subs)\n", " ps = [diff(lg, dotq).simplify_full().substitute(subs_inv) for dotq in dqvt]\n", " if n == 1:\n", " return ps[0]\n", " return ps" ] }, { "cell_type": "code", "execution_count": 113, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right), {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{4} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) + {\\mu_0}^{2} a^{2} n^{2} r^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - 2 \\, {\\mu_0}^{2} a^{2} m n^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\ell}^{2} r^{4} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} r^{2} \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right) + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} m \\frac{\\partial}{\\partial r}\\phi_{1}\\left(r\\right), {\\mu_0}^{2} a^{2} {\\ell}^{2} n^{2} r^{4} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) + {\\mu_0}^{2} a^{2} n^{2} r^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right) - 2 \\, {\\mu_0}^{2} a^{2} m n^{2} \\frac{\\partial}{\\partial r}\\psi_{1}\\left(r\\right)\\right]$$" ], "text/plain": [ "[-(Mu0^2 - 1)*a^2*l^2*r^4*diff(phi_1(r), r) - (Mu0^2 - 1)*a^2*r^2*diff(phi_1(r), r) + 2*(Mu0^2 - 1)*a^2*m*diff(phi_1(r), r),\n", " Mu0^2*a^2*l^2*n^2*r^4*diff(psi_1(r), r) + Mu0^2*a^2*n^2*r^2*diff(psi_1(r), r) - 2*Mu0^2*a^2*m*n^2*diff(psi_1(r), r)]" ] }, "execution_count": 113, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pis = conjugate_momenta(L_a2, [phi_1, psi_1], r)\n", "pis" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.15):" ] }, { "cell_type": "code", "execution_count": 114, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\mu_0}^{2} - 1\\right)} a {\\mathfrak{p}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\mu_0}^{2} - 1\\right)} a {\\mathfrak{p}}$$" ], "text/plain": [ "-(Mu0^2 - 1)*a*pf" ] }, "execution_count": 114, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_phi_r = (pis[0]/a).substitute_function(phi_1, phi1_sol).simplify_full()\n", "pi_phi_r" ] }, { "cell_type": "code", "execution_count": 115, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} a n^{2} {\\mathfrak{q}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} a n^{2} {\\mathfrak{q}}$$" ], "text/plain": [ "Mu0^2*a*n^2*qf" ] }, "execution_count": 115, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_psi_r = (pis[1]/a).substitute_function(psi_1, psi1_sol).simplify_full()\n", "pi_psi_r" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.14):" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We start from $\\pi^r_\\theta$ as given by Eq. (4.13):" ] }, { "cell_type": "code", "execution_count": 116, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(3 \\, {\\left(a^{2} {\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{3} - {\\left(a^{2} {\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + 4 \\, a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left(a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2}\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} {\\ell}^{2} n^{2} + 4 \\, C_{1} a^{2} {\\ell}^{2} n + 2 \\, C_{1} a^{2} {\\ell}^{2} + 3 \\, {\\left(a^{2} {\\ell}^{4} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{4} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} n^{2} + 4 \\, C_{1} a^{2} n + 2 \\, C_{1} a^{2} + 3 \\, {\\left(a^{2} {\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H} + {\\left(a^{2} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, a^{2} {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) - a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - {\\left(2 \\, a^{2} {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) + a^{2} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 3 \\, {\\left(a^{2} {\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} + a^{2} \\sin\\left({\\Theta_2}\\right)\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)}}{4 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(3 \\, {\\left(a^{2} {\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{3} - {\\left(a^{2} {\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + 4 \\, a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left(a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2}\\right)} {r_H}\\right)} \\arctan\\left(\\frac{{\\ell} r}{\\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}\\right) - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} {\\ell}^{2} n^{2} + 4 \\, C_{1} a^{2} {\\ell}^{2} n + 2 \\, C_{1} a^{2} {\\ell}^{2} + 3 \\, {\\left(a^{2} {\\ell}^{4} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{4} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} n^{2} + 4 \\, C_{1} a^{2} n + 2 \\, C_{1} a^{2} + 3 \\, {\\left(a^{2} {\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r\\right)} {r_H} + {\\left(a^{2} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, a^{2} {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) - a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - {\\left(2 \\, a^{2} {\\ell}^{2} m \\sin\\left({\\Theta_2}\\right) + a^{2} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} + 3 \\, {\\left(a^{2} {\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} + a^{2} \\sin\\left({\\Theta_2}\\right)\\right)} \\log\\left(\\frac{r + {r_H}}{r - {r_H}}\\right)\\right)}}{4 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{3} + {r_H}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "-1/4*(2*(3*(a^2*l^3*n^2*sin(Th2) - a^2*l^3*sin(Th2))*rH^3 - (a^2*l*n^2*qf^2*sin(Th2) + 2*a^2*l^3*m*sin(Th2) - a^2*l*pf^2*sin(Th2) + 4*a^2*l*sin(Th2) - 2*(a^2*l^3*m*sin(Th2) + 2*a^2*l*sin(Th2))*n^2)*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)) - sqrt(l^2*rH^2 + 1)*(4*(2*C_1*a^2*l^2*n^2 + 4*C_1*a^2*l^2*n + 2*C_1*a^2*l^2 + 3*(a^2*l^4*n^2*sin(Th2) - a^2*l^4*sin(Th2))*r)*rH^3 + 2*(2*C_1*a^2*n^2 + 4*C_1*a^2*n + 2*C_1*a^2 + 3*(a^2*l^2*n^2*sin(Th2) - a^2*l^2*sin(Th2))*r)*rH + (a^2*n^2*qf^2*sin(Th2) + 2*a^2*l^2*m*sin(Th2) - a^2*pf^2*sin(Th2) - (2*a^2*l^2*m*sin(Th2) + a^2*sin(Th2))*n^2 + 3*(a^2*l^2*n^2*sin(Th2) - a^2*l^2*sin(Th2))*rH^2 + a^2*sin(Th2))*log((r + rH)/(r - rH))))/((2*l^2*rH^3 + rH)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 116, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_theta = (fr4*(a + b)^2*Y_sol(r)).simplify_full()\n", "pi_theta " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us perform an expansion in $1/r$ for $r\\rightarrow +\\infty$:" ] }, { "cell_type": "code", "execution_count": 117, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3}{2} \\, {\\left(a^{2} {\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r + \\frac{3 \\, {\\left(a^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right)\\right)}}{2 \\, r} + \\frac{\\pi a^{2} {\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, \\pi a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - \\pi a^{2} {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + 4 \\, \\pi a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left(\\pi a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, \\pi a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} - 3 \\, {\\left(\\pi a^{2} {\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - \\pi a^{2} {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} + 4 \\, {\\left(C_{1} a^{2} n^{2} + 2 \\, C_{1} a^{2} n + C_{1} a^{2} + 2 \\, {\\left(C_{1} a^{2} {\\ell}^{2} n^{2} + 2 \\, C_{1} a^{2} {\\ell}^{2} n + C_{1} a^{2} {\\ell}^{2}\\right)} {r_H}^{2}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}{4 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3}{2} \\, {\\left(a^{2} {\\ell}^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} {\\ell}^{2} \\sin\\left({\\Theta_2}\\right)\\right)} r + \\frac{3 \\, {\\left(a^{2} n^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right)\\right)}}{2 \\, r} + \\frac{\\pi a^{2} {\\ell} n^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, \\pi a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) - \\pi a^{2} {\\ell} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) + 4 \\, \\pi a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right) - 2 \\, {\\left(\\pi a^{2} {\\ell}^{3} m \\sin\\left({\\Theta_2}\\right) + 2 \\, \\pi a^{2} {\\ell} \\sin\\left({\\Theta_2}\\right)\\right)} n^{2} - 3 \\, {\\left(\\pi a^{2} {\\ell}^{3} n^{2} \\sin\\left({\\Theta_2}\\right) - \\pi a^{2} {\\ell}^{3} \\sin\\left({\\Theta_2}\\right)\\right)} {r_H}^{2} + 4 \\, {\\left(C_{1} a^{2} n^{2} + 2 \\, C_{1} a^{2} n + C_{1} a^{2} + 2 \\, {\\left(C_{1} a^{2} {\\ell}^{2} n^{2} + 2 \\, C_{1} a^{2} {\\ell}^{2} n + C_{1} a^{2} {\\ell}^{2}\\right)} {r_H}^{2}\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}{4 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "3/2*(a^2*l^2*n^2*sin(Th2) - a^2*l^2*sin(Th2))*r + 3/2*(a^2*n^2*sin(Th2) - a^2*sin(Th2))/r + 1/4*(pi*a^2*l*n^2*qf^2*sin(Th2) + 2*pi*a^2*l^3*m*sin(Th2) - pi*a^2*l*pf^2*sin(Th2) + 4*pi*a^2*l*sin(Th2) - 2*(pi*a^2*l^3*m*sin(Th2) + 2*pi*a^2*l*sin(Th2))*n^2 - 3*(pi*a^2*l^3*n^2*sin(Th2) - pi*a^2*l^3*sin(Th2))*rH^2 + 4*(C_1*a^2*n^2 + 2*C_1*a^2*n + C_1*a^2 + 2*(C_1*a^2*l^2*n^2 + 2*C_1*a^2*l^2*n + C_1*a^2*l^2)*rH^2)*sqrt(l^2*rH^2 + 1))/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 117, "metadata": { }, "output_type": "execute_result" } ], "source": [ "assume(l > 0)\n", "\n", "u = var('u')\n", "assume(u > 0)\n", "s = pi_theta.subs({r: 1/u}).simplify_log()\n", "s = s.taylor(u, 0, 2)\n", "s = s.subs({u: 1/r})\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We consider $\\frac{\\pi^r_\\theta}{(a^2/2)\\sin(2\\Theta_0)}$:" ] }, { "cell_type": "code", "execution_count": 118, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{3 \\, \\pi {\\ell}^{3} n^{2} {r_H}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} - \\frac{\\pi {\\ell}^{3} m n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + 3 \\, {\\ell}^{2} n^{2} r + \\frac{4 \\, C_{1} {\\ell}^{2} n^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{\\pi {\\ell} n^{2} {\\mathfrak{q}}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{3 \\, \\pi {\\ell}^{3} {r_H}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{8 \\, C_{1} {\\ell}^{2} n {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{\\pi {\\ell}^{3} m}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} - 3 \\, {\\ell}^{2} r + \\frac{4 \\, C_{1} {\\ell}^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} - \\frac{2 \\, \\pi {\\ell} n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} - \\frac{\\pi {\\ell} {\\mathfrak{p}}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{3 \\, n^{2}}{r} + \\frac{2 \\, C_{1} n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, \\pi {\\ell}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{4 \\, C_{1} n}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} - \\frac{3}{r} + \\frac{2 \\, C_{1}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{3 \\, \\pi {\\ell}^{3} n^{2} {r_H}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} - \\frac{\\pi {\\ell}^{3} m n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + 3 \\, {\\ell}^{2} n^{2} r + \\frac{4 \\, C_{1} {\\ell}^{2} n^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{\\pi {\\ell} n^{2} {\\mathfrak{q}}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{3 \\, \\pi {\\ell}^{3} {r_H}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{8 \\, C_{1} {\\ell}^{2} n {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{\\pi {\\ell}^{3} m}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} - 3 \\, {\\ell}^{2} r + \\frac{4 \\, C_{1} {\\ell}^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} - \\frac{2 \\, \\pi {\\ell} n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} - \\frac{\\pi {\\ell} {\\mathfrak{p}}^{2}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{3 \\, n^{2}}{r} + \\frac{2 \\, C_{1} n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, \\pi {\\ell}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}} + \\frac{4 \\, C_{1} n}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} - \\frac{3}{r} + \\frac{2 \\, C_{1}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)}$$" ], "text/plain": [ "-3/2*pi*l^3*n^2*rH^2/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) - pi*l^3*m*n^2/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) + 3*l^2*n^2*r + 4*C_1*l^2*n^2*rH^2/((2*l^2*rH^2 + 1)*sin(Th2)) + 1/2*pi*l*n^2*qf^2/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) + 3/2*pi*l^3*rH^2/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) + 8*C_1*l^2*n*rH^2/((2*l^2*rH^2 + 1)*sin(Th2)) + pi*l^3*m/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) - 3*l^2*r + 4*C_1*l^2*rH^2/((2*l^2*rH^2 + 1)*sin(Th2)) - 2*pi*l*n^2/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) - 1/2*pi*l*pf^2/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) + 3*n^2/r + 2*C_1*n^2/((2*l^2*rH^2 + 1)*sin(Th2)) + 2*pi*l/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1)) + 4*C_1*n/((2*l^2*rH^2 + 1)*sin(Th2)) - 3/r + 2*C_1/((2*l^2*rH^2 + 1)*sin(Th2))" ] }, "execution_count": 118, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = (s/(a^2/2*sin(Th2))).expand()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The term in factor of $C_1$ is" ] }, { "cell_type": "code", "execution_count": 119, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, {\\ell}^{2} n^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{8 \\, {\\ell}^{2} n {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{4 \\, {\\ell}^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{4 \\, n}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, {\\ell}^{2} n^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{8 \\, {\\ell}^{2} n {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{4 \\, {\\ell}^{2} {r_H}^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, n^{2}}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{4 \\, n}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2}{{\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)}$$" ], "text/plain": [ "4*l^2*n^2*rH^2/((2*l^2*rH^2 + 1)*sin(Th2)) + 8*l^2*n*rH^2/((2*l^2*rH^2 + 1)*sin(Th2)) + 4*l^2*rH^2/((2*l^2*rH^2 + 1)*sin(Th2)) + 2*n^2/((2*l^2*rH^2 + 1)*sin(Th2)) + 4*n/((2*l^2*rH^2 + 1)*sin(Th2)) + 2/((2*l^2*rH^2 + 1)*sin(Th2))" ] }, "execution_count": 119, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1.coefficient(C_1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "It is a constant term, of the type $\\tilde{C}_1/\\sin(2\\Theta_0)$, in agreement with Eq. (4.14).\n", "We remove it from the main term:" ] }, { "cell_type": "code", "execution_count": 120, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4} + {\\left({\\ell}^{6} n^{2} - {\\ell}^{6}\\right)} r^{2}\\right)} {r_H}^{4} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r^{2} + 18 \\, {\\left({\\ell}^{2} n^{2} + {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r^{2} - {\\ell}^{2}\\right)} {r_H}^{2} + 6 \\, n^{2} - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(3 \\, {\\left(\\pi {\\ell}^{3} n^{2} - \\pi {\\ell}^{3}\\right)} r {r_H}^{2} - {\\left(\\pi {\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, \\pi {\\ell}^{3} m - \\pi {\\ell} {\\mathfrak{p}}^{2} - 2 \\, {\\left(\\pi {\\ell}^{3} m + 2 \\, \\pi {\\ell}\\right)} n^{2} + 4 \\, \\pi {\\ell}\\right)} r\\right)} - 6}{2 \\, {\\left(2 \\, {\\ell}^{4} r {r_H}^{4} + 3 \\, {\\ell}^{2} r {r_H}^{2} + r\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4} + {\\left({\\ell}^{6} n^{2} - {\\ell}^{6}\\right)} r^{2}\\right)} {r_H}^{4} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r^{2} + 18 \\, {\\left({\\ell}^{2} n^{2} + {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r^{2} - {\\ell}^{2}\\right)} {r_H}^{2} + 6 \\, n^{2} - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(3 \\, {\\left(\\pi {\\ell}^{3} n^{2} - \\pi {\\ell}^{3}\\right)} r {r_H}^{2} - {\\left(\\pi {\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, \\pi {\\ell}^{3} m - \\pi {\\ell} {\\mathfrak{p}}^{2} - 2 \\, {\\left(\\pi {\\ell}^{3} m + 2 \\, \\pi {\\ell}\\right)} n^{2} + 4 \\, \\pi {\\ell}\\right)} r\\right)} - 6}{2 \\, {\\left(2 \\, {\\ell}^{4} r {r_H}^{4} + 3 \\, {\\ell}^{2} r {r_H}^{2} + r\\right)}}$$" ], "text/plain": [ "1/2*(12*(l^4*n^2 - l^4 + (l^6*n^2 - l^6)*r^2)*rH^4 + 6*(l^2*n^2 - l^2)*r^2 + 18*(l^2*n^2 + (l^4*n^2 - l^4)*r^2 - l^2)*rH^2 + 6*n^2 - sqrt(l^2*rH^2 + 1)*(3*(pi*l^3*n^2 - pi*l^3)*r*rH^2 - (pi*l*n^2*qf^2 + 2*pi*l^3*m - pi*l*pf^2 - 2*(pi*l^3*m + 2*pi*l)*n^2 + 4*pi*l)*r) - 6)/(2*l^4*r*rH^4 + 3*l^2*r*rH^2 + r)" ] }, "execution_count": 120, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = (s1 - s1.coefficient(C_1)*C_1).simplify_full()\n", "s2" ] }, { "cell_type": "code", "execution_count": 121, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4} + {\\left({\\ell}^{6} n^{2} - {\\ell}^{6}\\right)} r^{2}\\right)} {r_H}^{4} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r^{2} + 18 \\, {\\left({\\ell}^{2} n^{2} + {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r^{2} - {\\ell}^{2}\\right)} {r_H}^{2} + 6 \\, n^{2} - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(3 \\, {\\left(\\pi {\\ell}^{3} n^{2} - \\pi {\\ell}^{3}\\right)} r {r_H}^{2} - {\\left(\\pi {\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, \\pi {\\ell}^{3} m - \\pi {\\ell} {\\mathfrak{p}}^{2} - 2 \\, {\\left(\\pi {\\ell}^{3} m + 2 \\, \\pi {\\ell}\\right)} n^{2} + 4 \\, \\pi {\\ell}\\right)} r\\right)} - 6$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}12 \\, {\\left({\\ell}^{4} n^{2} - {\\ell}^{4} + {\\left({\\ell}^{6} n^{2} - {\\ell}^{6}\\right)} r^{2}\\right)} {r_H}^{4} + 6 \\, {\\left({\\ell}^{2} n^{2} - {\\ell}^{2}\\right)} r^{2} + 18 \\, {\\left({\\ell}^{2} n^{2} + {\\left({\\ell}^{4} n^{2} - {\\ell}^{4}\\right)} r^{2} - {\\ell}^{2}\\right)} {r_H}^{2} + 6 \\, n^{2} - \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\left(3 \\, {\\left(\\pi {\\ell}^{3} n^{2} - \\pi {\\ell}^{3}\\right)} r {r_H}^{2} - {\\left(\\pi {\\ell} n^{2} {\\mathfrak{q}}^{2} + 2 \\, \\pi {\\ell}^{3} m - \\pi {\\ell} {\\mathfrak{p}}^{2} - 2 \\, {\\left(\\pi {\\ell}^{3} m + 2 \\, \\pi {\\ell}\\right)} n^{2} + 4 \\, \\pi {\\ell}\\right)} r\\right)} - 6$$" ], "text/plain": [ "12*(l^4*n^2 - l^4 + (l^6*n^2 - l^6)*r^2)*rH^4 + 6*(l^2*n^2 - l^2)*r^2 + 18*(l^2*n^2 + (l^4*n^2 - l^4)*r^2 - l^2)*rH^2 + 6*n^2 - sqrt(l^2*rH^2 + 1)*(3*(pi*l^3*n^2 - pi*l^3)*r*rH^2 - (pi*l*n^2*qf^2 + 2*pi*l^3*m - pi*l*pf^2 - 2*(pi*l^3*m + 2*pi*l)*n^2 + 4*pi*l)*r) - 6" ] }, "execution_count": 121, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2.numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 122, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left({\\ell}^{2} {r_H}^{2} + 1\\right)} r$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left({\\ell}^{2} {r_H}^{2} + 1\\right)} r$$" ], "text/plain": [ "2*(2*l^2*rH^2 + 1)*(l^2*rH^2 + 1)*r" ] }, "execution_count": 122, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2.denominator().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let divide both the numerator and denominator by $r$:" ] }, { "cell_type": "code", "execution_count": 123, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}12 \\, {\\ell}^{6} n^{2} r {r_H}^{4} - 12 \\, {\\ell}^{6} r {r_H}^{4} + 18 \\, {\\ell}^{4} n^{2} r {r_H}^{2} + \\frac{12 \\, {\\ell}^{4} n^{2} {r_H}^{4}}{r} - 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} n^{2} {r_H}^{2} - 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m n^{2} - 18 \\, {\\ell}^{4} r {r_H}^{2} - \\frac{12 \\, {\\ell}^{4} {r_H}^{4}}{r} + \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} {\\mathfrak{q}}^{2} + 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} {r_H}^{2} + 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m + 6 \\, {\\ell}^{2} n^{2} r + \\frac{18 \\, {\\ell}^{2} n^{2} {r_H}^{2}}{r} - 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} - \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} {\\mathfrak{p}}^{2} - 6 \\, {\\ell}^{2} r - \\frac{18 \\, {\\ell}^{2} {r_H}^{2}}{r} + 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} + \\frac{6 \\, n^{2}}{r} - \\frac{6}{r}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}12 \\, {\\ell}^{6} n^{2} r {r_H}^{4} - 12 \\, {\\ell}^{6} r {r_H}^{4} + 18 \\, {\\ell}^{4} n^{2} r {r_H}^{2} + \\frac{12 \\, {\\ell}^{4} n^{2} {r_H}^{4}}{r} - 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} n^{2} {r_H}^{2} - 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m n^{2} - 18 \\, {\\ell}^{4} r {r_H}^{2} - \\frac{12 \\, {\\ell}^{4} {r_H}^{4}}{r} + \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} {\\mathfrak{q}}^{2} + 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} {r_H}^{2} + 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m + 6 \\, {\\ell}^{2} n^{2} r + \\frac{18 \\, {\\ell}^{2} n^{2} {r_H}^{2}}{r} - 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} - \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} {\\mathfrak{p}}^{2} - 6 \\, {\\ell}^{2} r - \\frac{18 \\, {\\ell}^{2} {r_H}^{2}}{r} + 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} + \\frac{6 \\, n^{2}}{r} - \\frac{6}{r}$$" ], "text/plain": [ "12*l^6*n^2*r*rH^4 - 12*l^6*r*rH^4 + 18*l^4*n^2*r*rH^2 + 12*l^4*n^2*rH^4/r - 3*pi*sqrt(l^2*rH^2 + 1)*l^3*n^2*rH^2 - 2*pi*sqrt(l^2*rH^2 + 1)*l^3*m*n^2 - 18*l^4*r*rH^2 - 12*l^4*rH^4/r + pi*sqrt(l^2*rH^2 + 1)*l*n^2*qf^2 + 3*pi*sqrt(l^2*rH^2 + 1)*l^3*rH^2 + 2*pi*sqrt(l^2*rH^2 + 1)*l^3*m + 6*l^2*n^2*r + 18*l^2*n^2*rH^2/r - 4*pi*sqrt(l^2*rH^2 + 1)*l*n^2 - pi*sqrt(l^2*rH^2 + 1)*l*pf^2 - 6*l^2*r - 18*l^2*rH^2/r + 4*pi*sqrt(l^2*rH^2 + 1)*l + 6*n^2/r - 6/r" ] }, "execution_count": 123, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2n = (s2.numerator()/r).expand()\n", "s2n" ] }, { "cell_type": "code", "execution_count": 124, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left({\\ell}^{2} {r_H}^{2} + 1\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} {\\left({\\ell}^{2} {r_H}^{2} + 1\\right)}$$" ], "text/plain": [ "2*(2*l^2*rH^2 + 1)*(l^2*rH^2 + 1)" ] }, "execution_count": 124, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2d = (s2.denominator()/r).factor()\n", "s2d" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the term in $r$ is" ] }, { "cell_type": "code", "execution_count": 125, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\ell}^{2} {\\left(n + 1\\right)} {\\left(n - 1\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\ell}^{2} {\\left(n + 1\\right)} {\\left(n - 1\\right)}$$" ], "text/plain": [ "3*l^2*(n + 1)*(n - 1)" ] }, "execution_count": 125, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = s2n.coefficient(r).factor()\n", "s/s2d" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This is in agreement with Eq. (4.14)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We remove it:" ] }, { "cell_type": "code", "execution_count": 126, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{12 \\, {\\ell}^{4} n^{2} {r_H}^{4}}{r} - 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} n^{2} {r_H}^{2} - 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m n^{2} - \\frac{12 \\, {\\ell}^{4} {r_H}^{4}}{r} + \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} {\\mathfrak{q}}^{2} + 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} {r_H}^{2} + 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m + \\frac{18 \\, {\\ell}^{2} n^{2} {r_H}^{2}}{r} - 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} - \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} {\\mathfrak{p}}^{2} - \\frac{18 \\, {\\ell}^{2} {r_H}^{2}}{r} + 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} + \\frac{6 \\, n^{2}}{r} - \\frac{6}{r}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{12 \\, {\\ell}^{4} n^{2} {r_H}^{4}}{r} - 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} n^{2} {r_H}^{2} - 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m n^{2} - \\frac{12 \\, {\\ell}^{4} {r_H}^{4}}{r} + \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} {\\mathfrak{q}}^{2} + 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} {r_H}^{2} + 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m + \\frac{18 \\, {\\ell}^{2} n^{2} {r_H}^{2}}{r} - 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} - \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} {\\mathfrak{p}}^{2} - \\frac{18 \\, {\\ell}^{2} {r_H}^{2}}{r} + 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} + \\frac{6 \\, n^{2}}{r} - \\frac{6}{r}$$" ], "text/plain": [ "12*l^4*n^2*rH^4/r - 3*pi*sqrt(l^2*rH^2 + 1)*l^3*n^2*rH^2 - 2*pi*sqrt(l^2*rH^2 + 1)*l^3*m*n^2 - 12*l^4*rH^4/r + pi*sqrt(l^2*rH^2 + 1)*l*n^2*qf^2 + 3*pi*sqrt(l^2*rH^2 + 1)*l^3*rH^2 + 2*pi*sqrt(l^2*rH^2 + 1)*l^3*m + 18*l^2*n^2*rH^2/r - 4*pi*sqrt(l^2*rH^2 + 1)*l*n^2 - pi*sqrt(l^2*rH^2 + 1)*l*pf^2 - 18*l^2*rH^2/r + 4*pi*sqrt(l^2*rH^2 + 1)*l + 6*n^2/r - 6/r" ] }, "execution_count": 126, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s3n = (s2n - s*r).simplify_full().expand()\n", "s3n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the term in $1/r$ is" ] }, { "cell_type": "code", "execution_count": 127, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\left(n + 1\\right)} {\\left(n - 1\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\left(n + 1\\right)} {\\left(n - 1\\right)}$$" ], "text/plain": [ "3*(n + 1)*(n - 1)" ] }, "execution_count": 127, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = s3n.coefficient(r^(-1)).factor()\n", "s/s2d" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This is in agreement with Eq. (4.14)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Finally the remaining term is" ] }, { "cell_type": "code", "execution_count": 128, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} n^{2} {r_H}^{2} - 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m n^{2} + \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} {\\mathfrak{q}}^{2} + 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} {r_H}^{2} + 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m - 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} - \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} {\\mathfrak{p}}^{2} + 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} n^{2} {r_H}^{2} - 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m n^{2} + \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} {\\mathfrak{q}}^{2} + 3 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} {r_H}^{2} + 2 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}^{3} m - 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} n^{2} - \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell} {\\mathfrak{p}}^{2} + 4 \\, \\pi \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}$$" ], "text/plain": [ "-3*pi*sqrt(l^2*rH^2 + 1)*l^3*n^2*rH^2 - 2*pi*sqrt(l^2*rH^2 + 1)*l^3*m*n^2 + pi*sqrt(l^2*rH^2 + 1)*l*n^2*qf^2 + 3*pi*sqrt(l^2*rH^2 + 1)*l^3*rH^2 + 2*pi*sqrt(l^2*rH^2 + 1)*l^3*m - 4*pi*sqrt(l^2*rH^2 + 1)*l*n^2 - pi*sqrt(l^2*rH^2 + 1)*l*pf^2 + 4*pi*sqrt(l^2*rH^2 + 1)*l" ] }, "execution_count": 128, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4n = (s3n - s/r).simplify_full().expand()\n", "s4n" ] }, { "cell_type": "code", "execution_count": 129, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi {\\left(3 \\, {\\ell}^{2} n^{2} {r_H}^{2} + 2 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\ell}^{2} {r_H}^{2} - 2 \\, {\\ell}^{2} m + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi {\\left(3 \\, {\\ell}^{2} n^{2} {r_H}^{2} + 2 \\, {\\ell}^{2} m n^{2} - n^{2} {\\mathfrak{q}}^{2} - 3 \\, {\\ell}^{2} {r_H}^{2} - 2 \\, {\\ell}^{2} m + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1} {\\ell}$$" ], "text/plain": [ "-pi*(3*l^2*n^2*rH^2 + 2*l^2*m*n^2 - n^2*qf^2 - 3*l^2*rH^2 - 2*l^2*m + 4*n^2 + pf^2 - 4)*sqrt(l^2*rH^2 + 1)*l" ] }, "execution_count": 129, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4n.factor()" ] }, { "cell_type": "code", "execution_count": 130, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi {\\left({\\ell}^{4} n^{2} {r_H}^{4} - {\\ell}^{4} {r_H}^{4} + 4 \\, {\\ell}^{2} n^{2} {r_H}^{2} - n^{2} {\\mathfrak{q}}^{2} - 4 \\, {\\ell}^{2} {r_H}^{2} + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} {\\ell}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi {\\left({\\ell}^{4} n^{2} {r_H}^{4} - {\\ell}^{4} {r_H}^{4} + 4 \\, {\\ell}^{2} n^{2} {r_H}^{2} - n^{2} {\\mathfrak{q}}^{2} - 4 \\, {\\ell}^{2} {r_H}^{2} + 4 \\, n^{2} + {\\mathfrak{p}}^{2} - 4\\right)} {\\ell}}{2 \\, {\\left(2 \\, {\\ell}^{2} {r_H}^{2} + 1\\right)} \\sqrt{{\\ell}^{2} {r_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*pi*(l^4*n^2*rH^4 - l^4*rH^4 + 4*l^2*n^2*rH^2 - n^2*qf^2 - 4*l^2*rH^2 + 4*n^2 + pf^2 - 4)*l/((2*l^2*rH^2 + 1)*sqrt(l^2*rH^2 + 1))" ] }, "execution_count": 130, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = (s4n.factor()/s2d).subs({m: m_rH}).factor()\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The denominator clearly agrees with Eq. (4.14); the numerator agrees as well:" ] }, { "cell_type": "code", "execution_count": 131, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 131, "metadata": { }, "output_type": "execute_result" } ], "source": [ "bool(s.numerator()/(pi*l) == (1 - n^2)*(l^2*rH^2 + 2)^2 - pf^2 + n^2*qf^2)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Conclusion:** we have full agreement with Eq. (4.14)." ] }, { "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 }