{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', latex_name=r'\\mathcal{M}', structure='Lorentzian'); M" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "var('a', domain='real')\n", "assume(a > 0)\n", "assume(a < 1)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL. = M.chart(r't r:(1+sqrt(1-a^2),+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi'); BL" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "g = M.metric()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + {\\left(\\frac{2 \\, a^{2} r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = (2*r/(a^2*cos(th)^2 + r^2) - 1) dt*dt - 2*a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt*dph + (a^2*cos(th)^2 + r^2)/(a^2 + r^2 - 2*r) dr*dr + (a^2*cos(th)^2 + r^2) dth*dth - 2*a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph*dt + (2*a^2*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph*dph" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Sigma = r^2 + (a*cos(th))^2\n", "Delta = r^2 - 2*r + a^2\n", "g[0,0] = -(1-2*r/Sigma)\n", "g[0,3] = -2*a*r*sin(th)^2/Sigma\n", "g[1,1] = Sigma/Delta\n", "g[2,2] = Sigma\n", "g[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2/Sigma)*sin(th)^2\n", "g.display()" ] }, { "cell_type": "code", "execution_count": 7, "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^{4} - r^{4} - {\\left(a^{4} + a^{2} r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a^{2} 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} r^{2} + 3 \\, a r^{4} - {\\left(a^{5} - a^{3} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, {\\theta} \\, {\\phi} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{5} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{5} - {\\left(a^{5} r + a^{3} 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} r^{2} + r^{4} - 2 \\, r^{3} - {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\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} r^{2} + a r^{4} - 2 \\, a r^{3} - {\\left(a^{5} + a^{3} r^{2} - 2 \\, a^{3} r\\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{{\\left(a^{2} r - a^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} + a^{2} - r^{2}}{a^{2} r^{2} + r^{4} - 2 \\, r^{3} + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\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 + r^{3} - 2 \\, r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3} - {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} r^{5} + r^{7} - 2 \\, r^{6} + {\\left(a^{6} r + a^{4} r^{3} - 2 \\, a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{3} + a^{2} r^{5} - 2 \\, a^{2} r^{4}\\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} 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} r + a 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} + r^{4} - 2 \\, r^{3} + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\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} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{4} r + 4 \\, a^{2} r^{3}\\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} \\cos\\left({\\theta}\\right)^{2} - a r^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a 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{r^{5} + {\\left(a^{4} r - a^{4}\\right)} \\cos\\left({\\theta}\\right)^{4} - a^{2} r^{2} - 2 \\, r^{4} + {\\left(2 \\, a^{2} r^{3} + a^{4} - a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{a^{4} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{2} r^{2} - a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(r^{4} + 2 \\, a^{2} r\\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}$ " ], "text/plain": [ "Gam^t_t,r = -(a^4 - r^4 - (a^4 + a^2*r^2)*sin(th)^2)/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^t_t,th = -2*a^2*r*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) \n", "Gam^t_r,ph = -(a^3*r^2 + 3*a*r^4 - (a^5 - a^3*r^2)*cos(th)^2)*sin(th)^2/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^t_th,ph = -2*(a^5*r*cos(th)*sin(th)^5 - (a^5*r + a^3*r^3)*cos(th)*sin(th)^3)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^r_t,t = (a^2*r^2 + r^4 - 2*r^3 - (a^4 + a^2*r^2 - 2*a^2*r)*cos(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^r_t,ph = -(a^3*r^2 + a*r^4 - 2*a*r^3 - (a^5 + a^3*r^2 - 2*a^3*r)*cos(th)^2)*sin(th)^2/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^r_r,r = ((a^2*r - a^2)*sin(th)^2 + a^2 - r^2)/(a^2*r^2 + r^4 - 2*r^3 + (a^4 + a^2*r^2 - 2*a^2*r)*cos(th)^2) \n", "Gam^r_r,th = -a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) \n", "Gam^r_th,th = -(a^2*r + r^3 - 2*r^2)/(a^2*cos(th)^2 + r^2) \n", "Gam^r_ph,ph = ((a^4*r^2 + a^2*r^4 - 2*a^2*r^3 - (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^2)*sin(th)^4 - (a^2*r^5 + r^7 - 2*r^6 + (a^6*r + a^4*r^3 - 2*a^4*r^2)*cos(th)^4 + 2*(a^4*r^3 + a^2*r^5 - 2*a^2*r^4)*cos(th)^2)*sin(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^th_t,t = -2*a^2*r*cos(th)*sin(th)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^th_t,ph = 2*(a^3*r + a*r^3)*cos(th)*sin(th)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^th_r,r = a^2*cos(th)*sin(th)/(a^2*r^2 + r^4 - 2*r^3 + (a^4 + a^2*r^2 - 2*a^2*r)*cos(th)^2) \n", "Gam^th_r,th = r/(a^2*cos(th)^2 + r^2) \n", "Gam^th_th,th = -a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) \n", "Gam^th_ph,ph = -((a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^5 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^3 + (a^2*r^4 + r^6 + 2*a^4*r + 4*a^2*r^3)*cos(th))*sin(th)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^ph_t,r = -(a^3*cos(th)^2 - a*r^2)/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^ph_t,th = -2*a*r*cos(th)/((a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)*sin(th)) \n", "Gam^ph_r,ph = (r^5 + (a^4*r - a^4)*cos(th)^4 - a^2*r^2 - 2*r^4 + (2*a^2*r^3 + a^4 - a^2*r^2)*cos(th)^2)/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^ph_th,ph = (a^4*cos(th)^5 + 2*(a^2*r^2 - a^2*r)*cos(th)^3 + (r^4 + 2*a^2*r)*cos(th))/((a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)*sin(th)) " ] }, "execution_count": 7, "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 8.2", "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 1, "url": "https://www.sagemath.org/" } }, "name": "sage-8.2" }, "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 }