{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Equatorial geodesics in Kerr spacetime" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The spacetime manifold and Boyer-Lindquist coordinates:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)$ " ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M')\n", "X. = M.chart(r\"t r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi\")\n", "X" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The spacetime metric:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g = M.lorentzian_metric('g')\n", "var('m, a', domain='real')\n", "rho2 = r^2 + (a*cos(th))^2\n", "Delta = r^2 -2*m*r + a^2\n", "g[0,0] = -(1-2*m*r/rho2)\n", "g[0,3] = -2*a*m*r*sin(th)^2/rho2\n", "g[1,1], g[2,2] = rho2/Delta, rho2\n", "g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*sin(th)^2\n", "g.display()" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} u_{\\phantom{\\, t}}^{ \\, t } & = & \\frac{{\\left(a^{2} {\\left(\\frac{2 \\, m}{r} + 1\\right)} + r^{2}\\right)} {\\varepsilon} - \\frac{2 \\, a {\\ell} m}{r}}{a^{2} - 2 \\, m r + r^{2}} \\\\ u_{\\phantom{\\, r}}^{ \\, r } & = & \\sqrt{{\\varepsilon}^{2} + \\frac{2 \\, {\\left(a {\\varepsilon} - {\\ell}\\right)}^{2} m}{r^{3}} + \\frac{2 \\, m}{r} + \\frac{{\\left({\\varepsilon}^{2} - 1\\right)} a^{2} - {\\ell}^{2}}{r^{2}} - 1} \\\\ u_{\\phantom{\\, {\\phi}}}^{ \\, {\\phi} } & = & \\frac{\\frac{2 \\, a {\\varepsilon} m}{r} - {\\ell} {\\left(\\frac{2 \\, m}{r} - 1\\right)}}{a^{2} - 2 \\, m r + r^{2}} \\end{array}$ " ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "u = M.vector_field('u')\n", "var('eps', latex_name=r'\\varepsilon')\n", "var('ell', latex_name=r'\\ell')\n", "u[0] = ((r^2 + a^2*(1+2*m/r))*eps - 2*a*m/r*ell)/Delta\n", "u[1] = sqrt(eps^2 - 1 + 2*m/r - (ell^2-a^2*(eps^2-1))/r^2 \n", " + 2*m/r^3*(ell-a*eps)^2)\n", "u[3] = (2*a*m/r*eps + (1-2*m/r)*ell)/Delta\n", "u.display_comp()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a^{2} {\\varepsilon}^{2} - 2 \\, a^{2} - {\\ell}^{2}\\right)} r^{3} \\cos\\left({\\theta}\\right)^{2} - r^{5} + {\\left(2 \\, {\\left(a^{4} {\\varepsilon}^{2} - 2 \\, a^{3} {\\ell} {\\varepsilon} + a^{2} {\\ell}^{2}\\right)} m + {\\left(a^{4} {\\varepsilon}^{2} - a^{4} - a^{2} {\\ell}^{2}\\right)} r\\right)} \\cos\\left({\\theta}\\right)^{4}}{a^{2} r^{3} \\cos\\left({\\theta}\\right)^{2} + r^{5}}$ " ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "norm = g(u,u)\n", "norm.coord_function()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Value of $g(u,u)$ in the equatorial plane ($\\theta=\\frac{\\pi}{2}$):" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1$ " ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "norm.coord_function()(t,r,pi/2,ph)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "nabla = g.connection()\n", "print(nabla)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The 4-acceleration vector $a = \\nabla_{u}\\, u$:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} a_{\\phantom{\\, t}}^{ \\, t } & = & -\\frac{2 \\, {\\left({\\left({\\left(a^{4} {\\varepsilon} - a^{3} {\\ell}\\right)} m r^{2} + {\\left(a^{6} {\\varepsilon} - a^{5} {\\ell}\\right)} m\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left({\\left(a^{2} {\\varepsilon} - a {\\ell}\\right)} m r^{4} + {\\left(a^{4} {\\varepsilon} - a^{3} {\\ell}\\right)} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sqrt{{\\left({\\varepsilon}^{2} - 1\\right)} r^{3} + 2 \\, m r^{2} + 2 \\, {\\left(a^{2} {\\varepsilon}^{2} - 2 \\, a {\\ell} {\\varepsilon} + {\\ell}^{2}\\right)} m + {\\left(a^{2} {\\varepsilon}^{2} - a^{2} - {\\ell}^{2}\\right)} r}}{{\\left(a^{2} r^{7} - 2 \\, m r^{8} + r^{9} + {\\left(a^{6} r^{3} - 2 \\, a^{4} m r^{4} + a^{4} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{5} - 2 \\, a^{2} m r^{6} + a^{2} r^{7}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sqrt{r}} \\\\ a_{\\phantom{\\, r}}^{ \\, r } & = & \\frac{{\\left({\\left(a^{6} {\\varepsilon}^{2} + 4 \\, a^{5} {\\ell} {\\varepsilon} - 4 \\, a^{6} - 5 \\, a^{4} {\\ell}^{2}\\right)} m r^{4} - {\\left(a^{6} {\\varepsilon}^{2} - a^{6} - a^{4} {\\ell}^{2}\\right)} r^{5} - 2 \\, {\\left(a^{8} {\\varepsilon}^{2} - a^{8} - a^{6} {\\ell}^{2} - 2 \\, {\\left(a^{6} {\\varepsilon}^{2} - 3 \\, a^{5} {\\ell} {\\varepsilon} + a^{6} + 2 \\, a^{4} {\\ell}^{2}\\right)} m^{2}\\right)} r^{3} - 2 \\, {\\left(2 \\, {\\left(a^{6} {\\varepsilon}^{2} - 2 \\, a^{5} {\\ell} {\\varepsilon} + a^{4} {\\ell}^{2}\\right)} m^{3} + {\\left(a^{8} {\\varepsilon}^{2} - 5 \\, a^{7} {\\ell} {\\varepsilon} + 2 \\, a^{8} + 4 \\, a^{6} {\\ell}^{2}\\right)} m\\right)} r^{2} - 3 \\, {\\left(a^{10} {\\varepsilon}^{2} - 2 \\, a^{9} {\\ell} {\\varepsilon} + a^{8} {\\ell}^{2}\\right)} m - {\\left(a^{10} {\\varepsilon}^{2} - a^{10} - a^{8} {\\ell}^{2} - 8 \\, {\\left(a^{8} {\\varepsilon}^{2} - 2 \\, a^{7} {\\ell} {\\varepsilon} + a^{6} {\\ell}^{2}\\right)} m^{2}\\right)} r\\right)} \\cos\\left({\\theta}\\right)^{6} + {\\left({\\left(a^{4} {\\varepsilon}^{2} + 6 \\, a^{3} {\\ell} {\\varepsilon} - 8 \\, a^{4} - 7 \\, a^{2} {\\ell}^{2}\\right)} m r^{6} - 2 \\, {\\left(a^{4} {\\varepsilon}^{2} - a^{4} - a^{2} {\\ell}^{2}\\right)} r^{7} - 4 \\, {\\left(a^{6} {\\varepsilon}^{2} - a^{6} - a^{4} {\\ell}^{2} - {\\left(a^{4} {\\varepsilon}^{2} - 2 \\, a^{3} {\\ell} {\\varepsilon} + 2 \\, a^{4} + a^{2} {\\ell}^{2}\\right)} m^{2}\\right)} r^{5} + 2 \\, {\\left(2 \\, {\\left(a^{4} {\\varepsilon}^{2} - 2 \\, a^{3} {\\ell} {\\varepsilon} + a^{2} {\\ell}^{2}\\right)} m^{3} - {\\left(3 \\, a^{6} {\\varepsilon}^{2} - 10 \\, a^{5} {\\ell} {\\varepsilon} + 4 \\, a^{6} + 7 \\, a^{4} {\\ell}^{2}\\right)} m\\right)} r^{4} - 7 \\, {\\left(a^{8} {\\varepsilon}^{2} - 2 \\, a^{7} {\\ell} {\\varepsilon} + a^{6} {\\ell}^{2}\\right)} m r^{2} - 2 \\, {\\left(a^{8} {\\varepsilon}^{2} - a^{8} - a^{6} {\\ell}^{2} - 6 \\, {\\left(a^{6} {\\varepsilon}^{2} - 2 \\, a^{5} {\\ell} {\\varepsilon} + a^{4} {\\ell}^{2}\\right)} m^{2}\\right)} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} - {\\left(2 \\, {\\left(a^{2} {\\varepsilon}^{2} - 3 \\, a {\\ell} {\\varepsilon} + 2 \\, a^{2} + 2 \\, {\\ell}^{2}\\right)} m r^{8} + {\\left(a^{2} {\\varepsilon}^{2} - a^{2} - {\\ell}^{2}\\right)} r^{9} + 2 \\, {\\left(4 \\, a^{4} {\\varepsilon}^{2} - 9 \\, a^{3} {\\ell} {\\varepsilon} + 2 \\, a^{4} + 5 \\, a^{2} {\\ell}^{2}\\right)} m r^{6} + 2 \\, {\\left(a^{4} {\\varepsilon}^{2} - a^{4} - a^{2} {\\ell}^{2} - 2 \\, {\\left(2 \\, a^{2} {\\varepsilon}^{2} - 3 \\, a {\\ell} {\\varepsilon} + a^{2} + {\\ell}^{2}\\right)} m^{2}\\right)} r^{7} + 6 \\, {\\left(a^{6} {\\varepsilon}^{2} - 2 \\, a^{5} {\\ell} {\\varepsilon} + a^{4} {\\ell}^{2}\\right)} m r^{4} + {\\left(a^{6} {\\varepsilon}^{2} - a^{6} - a^{4} {\\ell}^{2} - 12 \\, {\\left(a^{4} {\\varepsilon}^{2} - 2 \\, a^{3} {\\ell} {\\varepsilon} + a^{2} {\\ell}^{2}\\right)} m^{2}\\right)} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{10} - 2 \\, m r^{11} + r^{12} + {\\left(a^{8} r^{4} - 2 \\, a^{6} m r^{5} + a^{6} r^{6}\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{6} r^{6} - 2 \\, a^{4} m r^{7} + a^{4} r^{8}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{4} r^{8} - 2 \\, a^{2} m r^{9} + a^{2} r^{10}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ a_{\\phantom{\\, {\\theta}}}^{ \\, {\\theta} } & = & -\\frac{{\\left({\\left(2 \\, {\\left(2 \\, a^{5} {\\ell} {\\varepsilon} - a^{6} - 2 \\, a^{4} {\\ell}^{2}\\right)} m r^{2} - {\\left(a^{6} {\\varepsilon}^{2} - a^{6} - a^{4} {\\ell}^{2}\\right)} r^{3} - 2 \\, {\\left(a^{8} {\\varepsilon}^{2} - 2 \\, a^{7} {\\ell} {\\varepsilon} + a^{6} {\\ell}^{2}\\right)} m - {\\left(a^{8} {\\varepsilon}^{2} - a^{8} - a^{6} {\\ell}^{2} - 4 \\, {\\left(a^{6} {\\varepsilon}^{2} - 2 \\, a^{5} {\\ell} {\\varepsilon} + a^{4} {\\ell}^{2}\\right)} m^{2}\\right)} r\\right)} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(2 \\, {\\left(2 \\, a^{3} {\\ell} {\\varepsilon} - a^{4} - 2 \\, a^{2} {\\ell}^{2}\\right)} m r^{4} - {\\left(a^{4} {\\varepsilon}^{2} - a^{4} - a^{2} {\\ell}^{2}\\right)} r^{5} - 2 \\, {\\left(a^{6} {\\varepsilon}^{2} - 2 \\, a^{5} {\\ell} {\\varepsilon} + a^{4} {\\ell}^{2}\\right)} m r^{2} - {\\left(a^{6} {\\varepsilon}^{2} - a^{6} - a^{4} {\\ell}^{2} - 4 \\, {\\left(a^{4} {\\varepsilon}^{2} - 2 \\, a^{3} {\\ell} {\\varepsilon} + a^{2} {\\ell}^{2}\\right)} m^{2}\\right)} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(2 \\, {\\left(a^{2} {\\varepsilon}^{2} - a^{2} - {\\ell}^{2}\\right)} m r^{6} - {\\left(a^{2} {\\varepsilon}^{2} - a^{2} - {\\ell}^{2}\\right)} r^{7} - {\\left(a^{4} {\\varepsilon}^{2} - a^{4} - a^{2} {\\ell}^{2}\\right)} r^{5}\\right)} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{2} r^{9} - 2 \\, m r^{10} + r^{11} + {\\left(a^{8} r^{3} - 2 \\, a^{6} m r^{4} + a^{6} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{6} r^{5} - 2 \\, a^{4} m r^{6} + a^{4} r^{7}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{4} r^{7} - 2 \\, a^{2} m r^{8} + a^{2} r^{9}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ a_{\\phantom{\\, {\\phi}}}^{ \\, {\\phi} } & = & -\\frac{2 \\, {\\left(3 \\, {\\left(a^{3} {\\varepsilon} - a^{2} {\\ell}\\right)} m r^{2} \\cos\\left({\\theta}\\right)^{2} + {\\left(a^{5} {\\varepsilon} - a^{4} {\\ell}\\right)} m \\cos\\left({\\theta}\\right)^{4}\\right)} \\sqrt{{\\left({\\varepsilon}^{2} - 1\\right)} r^{3} + 2 \\, m r^{2} + 2 \\, {\\left(a^{2} {\\varepsilon}^{2} - 2 \\, a {\\ell} {\\varepsilon} + {\\ell}^{2}\\right)} m + {\\left(a^{2} {\\varepsilon}^{2} - a^{2} - {\\ell}^{2}\\right)} r}}{{\\left(a^{2} r^{7} - 2 \\, m r^{8} + r^{9} + {\\left(a^{6} r^{3} - 2 \\, a^{4} m r^{4} + a^{4} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{5} - 2 \\, a^{2} m r^{6} + a^{2} r^{7}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sqrt{r}} \\end{array}$ " ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Du = nabla(u)\n", "a = u.contract(0, Du, 1)\n", "a.set_name('a')\n", "a.display_comp()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Values of the 4-acceleration in the equatorial plane ($\\theta=\\frac{\\pi}{2}$):" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ] }, "execution_count": 9, "metadata": { }, "output_type": "execute_result" } ], "source": [ "a[0](t,r,pi/2,ph)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "a[1](t,r,pi/2,ph)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "a[2](t,r,pi/2,ph)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "a[3](t,r,pi/2,ph)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The (non-zero and non-redundant) Christoffel symbols in Boyer-Lindquist coordinates:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & \\frac{a^{2} m r^{2} + m r^{4} - {\\left(a^{4} m + a^{2} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{4} - 2 \\, m r^{5} + r^{6} + {\\left(a^{6} - 2 \\, a^{4} m r + a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} - 2 \\, a^{2} m r^{3} + a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a^{2} m r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, r \\, {\\phi} }^{ \\, t \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{3} m r^{2} + 3 \\, a m r^{4} - {\\left(a^{5} m - a^{3} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} - 2 \\, m r^{5} + r^{6} + {\\left(a^{6} - 2 \\, a^{4} m r + a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} - 2 \\, a^{2} m r^{3} + a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, {\\theta} \\, {\\phi} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{5} m r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{5} - {\\left(a^{5} m r + a^{3} m r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{3}\\right)}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & \\frac{a^{2} m r^{2} - 2 \\, m^{2} r^{3} + m r^{4} - {\\left(a^{4} m - 2 \\, a^{2} m^{2} r + a^{2} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, {\\phi} }^{ \\, r \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{3} m r^{2} - 2 \\, a m^{2} r^{3} + a m r^{4} - {\\left(a^{5} m - 2 \\, a^{3} m^{2} r + a^{3} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{a^{2} r - m r^{2} + {\\left(a^{2} m - a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{2} - 2 \\, m r^{3} + r^{4} + {\\left(a^{4} - 2 \\, a^{2} m r + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} r - 2 \\, m r^{2} + r^{3}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{4} m r^{2} - 2 \\, a^{2} m^{2} r^{3} + a^{2} m r^{4} - {\\left(a^{6} m - 2 \\, a^{4} m^{2} r + a^{4} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} r^{5} - 2 \\, m r^{6} + r^{7} + {\\left(a^{6} r - 2 \\, a^{4} m r^{2} + a^{4} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{3} - 2 \\, a^{2} m r^{4} + a^{2} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, t \\, t }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, a^{2} m r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, t \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, {\\left(a^{3} m r + a m r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, r }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} r^{2} - 2 \\, m r^{3} + r^{4} + {\\left(a^{4} - 2 \\, a^{2} m r + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left({\\left(a^{6} - 2 \\, a^{4} m r + a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{4} r^{2} - 2 \\, a^{2} m r^{3} + a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(2 \\, a^{4} m r + 4 \\, a^{2} m r^{3} + a^{2} r^{4} + r^{6}\\right)} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, r }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{a^{3} m \\cos\\left({\\theta}\\right)^{2} - a m r^{2}}{a^{2} r^{4} - 2 \\, m r^{5} + r^{6} + {\\left(a^{6} - 2 \\, a^{4} m r + a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} - 2 \\, a^{2} m r^{3} + a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a m r \\cos\\left({\\theta}\\right)}{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}\\right)} \\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{a^{2} m r^{2} + 2 \\, m r^{4} - r^{5} + {\\left(a^{4} m - a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} - {\\left(a^{4} m - a^{2} m r^{2} + 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{4} - 2 \\, m r^{5} + r^{6} + {\\left(a^{6} - 2 \\, a^{4} m r + a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} - 2 \\, a^{2} m r^{3} + a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{a^{4} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{4} - 2 \\, {\\left(a^{4} - a^{2} m r + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{2} + {\\left(a^{4} + 2 \\, a^{2} r^{2} + r^{4}\\right)} \\cos\\left({\\theta}\\right)}{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}\\right)} \\sin\\left({\\theta}\\right)} \\end{array}$ " ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath (stable)", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 0 }