{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Kerr spacetime in Kerr coordinates\n", "\n", "This Jupyter/SageMath worksheet is relative to the lectures\n", "[Geometry and physics of black holes](https://luth.obspm.fr/~luthier/gourgoulhon/bh16/)\n", "\n", "These computations are based on tools developed through the [SageManifolds](http://sagemanifolds.obspm.fr) project.\n", "\n", "*NB:* a version of SageMath at least equal to 8.2 is required to run this worksheet: " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.9.beta4, Release Date: 2019-07-28'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To speed up computations, we ask for running them in parallel on 8 threads:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime\n", "\n", "We declare the spacetime manifold $M$:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "M = Manifold(4, 'M', structure='Lorentzian')\n", "print(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and the Kerr coordinates $(v,r,\\theta,\\phi)$ as a chart on $M$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(v, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (M, (v, r, th, ph))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X. = M.chart(r'v r th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "X" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( -\\infty, +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/plain": [ "v: (-oo, +oo); r: (-oo, +oo); th: (0, pi); ph: (0, 2*pi)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Kerr parameters $m$ and $a$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "m = var('m', domain='real')\n", "assume(m>0)\n", "a = var('a', domain='real')\n", "assume(a>=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Kerr metric\n", "\n", "We define the metric $g$ by its components w.r.t. the Kerr coordinates:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} v\\otimes \\mathrm{d} {\\phi} +\\mathrm{d} r\\otimes \\mathrm{d} v -a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} v -a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dv*dv + dv*dr - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dv*dph + dr*dv - a*sin(th)^2 dr*dph + (a^2*cos(th)^2 + r^2) dth*dth - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph*dv - a*sin(th)^2 dph*dr + (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph*dph" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.metric()\n", "rho2 = r^2 + (a*cos(th))^2\n", "g[0,0] = -(1 - 2*m*r/rho2)\n", "g[0,1] = 1\n", "g[0,3] = -2*a*m*r*sin(th)^2/rho2\n", "g[1,3] = -a*sin(th)^2\n", "g[2,2] = 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": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, v \\, v }^{ \\phantom{\\, v}\\phantom{\\, v} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\\\ g_{ \\, v \\, r }^{ \\phantom{\\, v}\\phantom{\\, r} } & = & 1 \\\\ g_{ \\, v \\, {\\phi} }^{ \\phantom{\\, v}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, r \\, v }^{ \\phantom{\\, r}\\phantom{\\, v} } & = & 1 \\\\ g_{ \\, r \\, {\\phi} }^{ \\phantom{\\, r}\\phantom{\\, {\\phi}} } & = & -a \\sin\\left({\\theta}\\right)^{2} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\\\ g_{ \\, {\\phi} \\, v }^{ \\phantom{\\, {\\phi}}\\phantom{\\, v} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, {\\phi} \\, r }^{ \\phantom{\\, {\\phi}}\\phantom{\\, r} } & = & -a \\sin\\left({\\theta}\\right)^{2} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\end{array}$ " ], "text/plain": [ "g_v,v = 2*m*r/(a^2*cos(th)^2 + r^2) - 1 \n", "g_v,r = 1 \n", "g_v,ph = -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n", "g_r,v = 1 \n", "g_r,ph = -a*sin(th)^2 \n", "g_th,th = a^2*cos(th)^2 + r^2 \n", "g_ph,v = -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n", "g_ph,r = -a*sin(th)^2 \n", "g_ph,ph = (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 " ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The inverse metric is pretty simple:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{a^{2} \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & \\frac{a^{2} + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & \\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\\n", "\\frac{a^{2} + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & \\frac{a^{2} - 2 \\, m r + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & \\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\\n", "0 & 0 & \\frac{1}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 \\\\\n", "\\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & \\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & -\\frac{1}{a^{2} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[ a^2*sin(th)^2/(a^2*cos(th)^2 + r^2) (a^2 + r^2)/(a^2*cos(th)^2 + r^2) 0 a/(a^2*cos(th)^2 + r^2)]\n", "[ (a^2 + r^2)/(a^2*cos(th)^2 + r^2) (a^2 - 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) 0 a/(a^2*cos(th)^2 + r^2)]\n", "[ 0 0 1/(a^2*cos(th)^2 + r^2) 0]\n", "[ a/(a^2*cos(th)^2 + r^2) a/(a^2*cos(th)^2 + r^2) 0 -1/(a^2*sin(th)^4 - (a^2 + r^2)*sin(th)^2)]" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.inverse()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "as well as the determinant w.r.t. to the Kerr coordinates:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & a^{4} \\cos\\left({\\theta}\\right)^{6} - {\\left(a^{4} - 2 \\, a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} - r^{4} - {\\left(2 \\, a^{2} r^{2} - r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2} \\end{array}$ " ], "text/plain": [ "M --> R\n", "(v, r, th, ph) |--> a^4*cos(th)^6 - (a^4 - 2*a^2*r^2)*cos(th)^4 - r^4 - (2*a^2*r^2 - r^4)*cos(th)^2" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.determinant().display()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.determinant() == - (rho2*sin(th))^2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that we are dealing with a solution of the Einstein equation in vacuum:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0$ " ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Christoffel symbols w.r.t. the Kerr coordinates:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/plain": [ "0" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci()[0,0]" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, v} \\, v \\, v }^{ \\, v \\phantom{\\, v} \\phantom{\\, v} } & = & -\\frac{a^{4} m - m r^{4} - {\\left(a^{4} m + a^{2} m r^{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{\\, v} \\, v \\, {\\theta} }^{ \\, v \\phantom{\\, v} \\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{\\, v} \\, v \\, {\\phi} }^{ \\, v \\phantom{\\, v} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{5} m + a^{3} m r^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{5} m - a m r^{4}\\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{\\, v} \\, r \\, {\\theta} }^{ \\, v \\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{\\, v} \\, r \\, {\\phi} }^{ \\, v \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, v} \\, {\\theta} \\, {\\theta} }^{ \\, v \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} r + r^{3}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, v} \\, {\\theta} \\, {\\phi} }^{ \\, v \\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{\\, v} \\, {\\phi} \\, {\\phi} }^{ \\, v \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{4} m r^{2} + a^{2} m r^{4} - {\\left(a^{6} m + a^{4} m r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} r^{5} + r^{7} + {\\left(a^{6} r + a^{4} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{3} + 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{\\, r} \\, v \\, v }^{ \\, r \\phantom{\\, v} \\phantom{\\, v} } & = & \\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} \\, v \\, r }^{ \\, r \\phantom{\\, v} \\phantom{\\, r} } & = & \\frac{a^{2} m \\cos\\left({\\theta}\\right)^{2} - m r^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, v \\, {\\phi} }^{ \\, r \\phantom{\\, v} \\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 \\, {\\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} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{5} m \\cos\\left({\\theta}\\right)^{2} + a^{5} m\\right)} \\sin\\left({\\theta}\\right)^{4} + {\\left(a^{5} r \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{3} r^{3} \\cos\\left({\\theta}\\right)^{2} - a^{5} m + a m r^{4} + a r^{5}\\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} \\, {\\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}} \\, v \\, v }^{ \\, {\\theta} \\phantom{\\, v} \\phantom{\\, v} } & = & -\\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}} \\, v \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, v} \\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 \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{a \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{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}} \\, v \\, v }^{ \\, {\\phi} \\phantom{\\, v} \\phantom{\\, v} } & = & -\\frac{a^{3} m \\cos\\left({\\theta}\\right)^{2} - a m r^{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{\\, {\\phi}} \\, v \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, v} \\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}} \\, v \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, v} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{4} m \\cos\\left({\\theta}\\right)^{2} - a^{2} m r^{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{\\, {\\phi}} \\, r \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{a^{4} \\cos\\left({\\theta}\\right)^{5} - 2 \\, {\\left(a^{2} m r - a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(2 \\, a^{2} m r + 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)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{5} m \\cos\\left({\\theta}\\right)^{2} - a^{3} m r^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} + {\\left(a^{5} r \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{3} r^{3} \\cos\\left({\\theta}\\right)^{2} + a r^{5}\\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}} \\end{array}$ " ], "text/plain": [ "Gam^v_v,v = -(a^4*m - m*r^4 - (a^4*m + a^2*m*r^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^v_v,th = -2*a^2*m*r*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) \n", "Gam^v_v,ph = -((a^5*m + a^3*m*r^2)*sin(th)^4 - (a^5*m - a*m*r^4)*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^v_r,th = -a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) \n", "Gam^v_r,ph = a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n", "Gam^v_th,th = -(a^2*r + r^3)/(a^2*cos(th)^2 + r^2) \n", "Gam^v_th,ph = -2*(a^5*m*r*cos(th)*sin(th)^5 - (a^5*m*r + a^3*m*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^v_ph,ph = ((a^4*m*r^2 + a^2*m*r^4 - (a^6*m + a^4*m*r^2)*cos(th)^2)*sin(th)^4 - (a^2*r^5 + r^7 + (a^6*r + a^4*r^3)*cos(th)^4 + 2*(a^4*r^3 + a^2*r^5)*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_v,v = (a^2*m*r^2 - 2*m^2*r^3 + m*r^4 - (a^4*m - 2*a^2*m^2*r + a^2*m*r^2)*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_v,r = (a^2*m*cos(th)^2 - m*r^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) \n", "Gam^r_v,ph = -(a^3*m*r^2 - 2*a*m^2*r^3 + a*m*r^4 - (a^5*m - 2*a^3*m^2*r + a^3*m*r^2)*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,th = -a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) \n", "Gam^r_r,ph = ((a^5*m*cos(th)^2 + a^5*m)*sin(th)^4 + (a^5*r*cos(th)^4 + 2*a^3*r^3*cos(th)^2 - a^5*m + a*m*r^4 + a*r^5)*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_th,th = -(a^2*r - 2*m*r^2 + r^3)/(a^2*cos(th)^2 + r^2) \n", "Gam^r_ph,ph = ((a^4*m*r^2 - 2*a^2*m^2*r^3 + a^2*m*r^4 - (a^6*m - 2*a^4*m^2*r + a^4*m*r^2)*cos(th)^2)*sin(th)^4 - (a^2*r^5 - 2*m*r^6 + r^7 + (a^6*r - 2*a^4*m*r^2 + a^4*r^3)*cos(th)^4 + 2*(a^4*r^3 - 2*a^2*m*r^4 + a^2*r^5)*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_v,v = -2*a^2*m*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_v,ph = 2*(a^3*m*r + a*m*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,th = r/(a^2*cos(th)^2 + r^2) \n", "Gam^th_r,ph = a*cos(th)*sin(th)/(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 - 2*a^4*m*r + a^4*r^2)*cos(th)^5 + 2*(a^4*r^2 - 2*a^2*m*r^3 + a^2*r^4)*cos(th)^3 + (2*a^4*m*r + 4*a^2*m*r^3 + a^2*r^4 + r^6)*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_v,v = -(a^3*m*cos(th)^2 - a*m*r^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^ph_v,th = -2*a*m*r*cos(th)/((a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)*sin(th)) \n", "Gam^ph_v,ph = (a^4*m*cos(th)^2 - a^2*m*r^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^ph_r,th = -a*cos(th)/((a^2*cos(th)^2 + r^2)*sin(th)) \n", "Gam^ph_r,ph = r/(a^2*cos(th)^2 + r^2) \n", "Gam^ph_th,th = -a*r/(a^2*cos(th)^2 + r^2) \n", "Gam^ph_th,ph = (a^4*cos(th)^5 - 2*(a^2*m*r - a^2*r^2)*cos(th)^3 + (2*a^2*m*r + r^4)*cos(th))/((a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)*sin(th)) \n", "Gam^ph_ph,ph = -((a^5*m*cos(th)^2 - a^3*m*r^2)*sin(th)^4 + (a^5*r*cos(th)^4 + 2*a^3*r^3*cos(th)^2 + a*r^5)*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) " ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vector normal to the hypersurfaces $r=\\mathrm{const}$" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form dr on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} r = \\mathrm{d} r$ " ], "text/plain": [ "dr = dr" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dr = X.coframe()[1]\n", "print(dr)\n", "dr.display()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{a^{2} + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{a^{2} - 2 \\, m r + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\frac{\\partial}{\\partial r } + \\left( \\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "(a^2 + r^2)/(a^2*cos(th)^2 + r^2) d/dv + (a^2 - 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) d/dr + a/(a^2*cos(th)^2 + r^2) d/dph" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nr = dr.up(g)\n", "print(nr)\n", "nr.display()" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}m + \\sqrt{-a^{2} + m^{2}}$ " ], "text/plain": [ "m + sqrt(-a^2 + m^2)" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assume(a^2 $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(v, m + \\sqrt{-a^{2} + m^{2}}, {\\theta}, {\\phi}\\right)$ " ], "text/plain": [ "(v, m + sqrt(-a^2 + m^2), th, ph)" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X(p)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector at Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "nrH = nr.at(p)\n", "print(nrH)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent space at Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Tp = M.tangent_space(p)\n", "print(Tp)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)$ " ], "text/plain": [ "Basis (d/dv,d/dr,d/dth,d/dph) on the Tangent space at Point p on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Tp.default_basis()" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-\\frac{2 \\, {\\left(\\sqrt{a + m} \\sqrt{-a + m} m + m^{2}\\right)}}{a^{2} \\sin\\left({\\theta}\\right)^{2} - 2 \\, \\sqrt{a + m} \\sqrt{-a + m} m - 2 \\, m^{2}}, 0, 0, -\\frac{a}{a^{2} \\sin\\left({\\theta}\\right)^{2} - 2 \\, \\sqrt{a + m} \\sqrt{-a + m} m - 2 \\, m^{2}}\\right]$ " ], "text/plain": [ "[-2*(sqrt(a + m)*sqrt(-a + m)*m + m^2)/(a^2*sin(th)^2 - 2*sqrt(a + m)*sqrt(-a + m)*m - 2*m^2),\n", " 0,\n", " 0,\n", " -a/(a^2*sin(th)^2 - 2*sqrt(a + m)*sqrt(-a + m)*m - 2*m^2)]" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nrH[:]" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a}{2 \\, {\\left(m + \\sqrt{-a^{2} + m^{2}}\\right)} m}$ " ], "text/plain": [ "1/2*a/((m + sqrt(-a^2 + m^2))*m)" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "OmegaH = a/(2*m*rp)\n", "OmegaH" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial v }$ " ], "text/plain": [ "Vector field d/dv on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi = X.frame()[0]\n", "xi" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "Vector field d/dph on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eta = X.frame()[3]\n", "eta" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial v } + \\frac{a}{2 \\, {\\left(\\sqrt{a + m} \\sqrt{-a + m} m + m^{2}\\right)}} \\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "d/dv + 1/2*a/(sqrt(a + m)*sqrt(-a + m)*m + m^2) d/dph" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "chi = xi + OmegaH*eta\n", "chi.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Ingoing principal null geodesics" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = -\\frac{\\partial}{\\partial r }$ " ], "text/plain": [ "k = -d/dr" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k = M.vector_field(name='k')\n", "k[1] = -1\n", "k.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $k$ is a null vector:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(k,k\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$ " ], "text/plain": [ "g(k,k): M --> R\n", " (v, r, th, ph) |--> 0" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(k,k).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Computation of $\\nabla_k k$:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/plain": [ "0" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab = g.connection()\n", "acc = nab(k).contract(k)\n", "acc.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Outgoing principal null geodesics" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\ell = \\frac{\\partial}{\\partial v } + \\left( -\\frac{m r}{a^{2} + r^{2}} + \\frac{1}{2} \\right) \\frac{\\partial}{\\partial r } + \\left( \\frac{a}{a^{2} + r^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "el = d/dv + (-m*r/(a^2 + r^2) + 1/2) d/dr + a/(a^2 + r^2) d/dph" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "el = M.vector_field(name='el', latex_name=r'\\ell')\n", "el[0] = 1\n", "el[1] = 1/2 - m*r/(r^2+a^2)\n", "el[3] = a/(r^2+a^2)\n", "el.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $\\ell$ is a null vector:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(\\ell,\\ell\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$ " ], "text/plain": [ "g(el,el): M --> R\n", " (v, r, th, ph) |--> 0" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(el,el).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Computation of $\\nabla_\\ell \\ell$:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{a^{2} m - m r^{2}}{a^{4} + 2 \\, a^{2} r^{2} + r^{4}} \\right) \\frac{\\partial}{\\partial v } + \\left( -\\frac{a^{4} m - 2 \\, a^{2} m^{2} r + 2 \\, m^{2} r^{3} - m r^{4}}{2 \\, {\\left(a^{6} + 3 \\, a^{4} r^{2} + 3 \\, a^{2} r^{4} + r^{6}\\right)}} \\right) \\frac{\\partial}{\\partial r } + \\left( -\\frac{a^{3} m - a m r^{2}}{a^{6} + 3 \\, a^{4} r^{2} + 3 \\, a^{2} r^{4} + r^{6}} \\right) \\frac{\\partial}{\\partial {\\phi} }$ " ], "text/plain": [ "-(a^2*m - m*r^2)/(a^4 + 2*a^2*r^2 + r^4) d/dv - 1/2*(a^4*m - 2*a^2*m^2*r + 2*m^2*r^3 - m*r^4)/(a^6 + 3*a^4*r^2 + 3*a^2*r^4 + r^6) d/dr - (a^3*m - a*m*r^2)/(a^6 + 3*a^4*r^2 + 3*a^2*r^4 + r^6) d/dph" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "acc = nab(el).contract(el)\n", "acc.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We check that $\\nabla_\\ell \\ell \\propto \\ell$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} m - m r^{2}}{a^{4} + 2 \\, a^{2} r^{2} + r^{4}}$ " ], "text/plain": [ "-(a^2*m - m*r^2)/(a^4 + 2*a^2*r^2 + r^4)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} m - m r^{2}}{a^{4} + 2 \\, a^{2} r^{2} + r^{4}}$ " ], "text/plain": [ "-(a^2*m - m*r^2)/(a^4 + 2*a^2*r^2 + r^4)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} m - m r^{2}}{a^{4} + 2 \\, a^{2} r^{2} + r^{4}}$ " ], "text/plain": [ "-(a^2*m - m*r^2)/(a^4 + 2*a^2*r^2 + r^4)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for i in [0,1,3]:\n", " show(acc[i] / el[i])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence we may write $\\nabla_\\ell\\ell = \\kappa \\ell$:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} m - m r^{2}}{a^{4} + 2 \\, a^{2} r^{2} + r^{4}}$ " ], "text/plain": [ "-(a^2*m - m*r^2)/(a^4 + 2*a^2*r^2 + r^4)" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kappa = (acc[0] / el[0]).expr()\n", "kappa" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "acc == kappa * el" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Surface gravity\n", "\n", "On $H$, $\\ell$ coincides with the Killing vector $\\chi$:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "el.at(p) == chi.at(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Therefore the surface gravity of the Kerr black hole is nothing but the value of the non-affinity coefficient of $\\ell$ on $H$:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} - m^{2} - \\sqrt{-a^{2} + m^{2}} m}{2 \\, {\\left(a^{2} m - 2 \\, m^{3} - 2 \\, \\sqrt{-a^{2} + m^{2}} m^{2}\\right)}}$ " ], "text/plain": [ "1/2*(a^2 - m^2 - sqrt(-a^2 + m^2)*m)/(a^2*m - 2*m^3 - 2*sqrt(-a^2 + m^2)*m^2)" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kappaH = kappa.subs(r=rp).simplify_full()\n", "kappaH" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bool(kappaH == sqrt(m^2-a^2)/(2*m*(m+sqrt(m^2-a^2))))" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.9.beta4", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 1 }