{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Kerr-Schild coordinates on Kerr spacetime\n", "\n", "This Jupyter/SageMath notebook is relative to the lectures\n", "[Geometry and physics of black holes](https://relativite.obspm.fr/blackholes/).\n", "\n", "The involved 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 8.8 is required to run this notebook: " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.1.beta8, Release Date: 2020-03-18'" ] }, "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": [ "## 3+1 Kerr coordinates $(t,r,\\theta,\\phi)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We restrict the 3+1 Kerr patch to $r>0$, in order to introduce latter the Kerr-Schild coordinates:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X. = M.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "X" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/plain": [ "t: (-oo, +oo); r: (0, +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 3+1 Kerr coordinates:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} r + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + 1 \\right) \\mathrm{d} r\\otimes \\mathrm{d} r -a {\\left(\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + 1\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t -a {\\left(\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + 1\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt*dt + 2*m*r/(a^2*cos(th)^2 + r^2) dt*dr - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt*dph + 2*m*r/(a^2*cos(th)^2 + r^2) dr*dt + (2*m*r/(a^2*cos(th)^2 + r^2) + 1) dr*dr - a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*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*dt - a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*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] = 2*m*r/rho2\n", "g[0,3] = -2*a*m*r*sin(th)^2/rho2\n", "g[1,1] = 1 + 2*m*r/rho2\n", "g[1,3] = -a*(1 + 2*m*r/rho2)*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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\\\ g_{ \\, t \\, r }^{ \\phantom{\\, t}\\phantom{\\, r} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, r \\, t }^{ \\phantom{\\, r}\\phantom{\\, t} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + 1 \\\\ g_{ \\, r \\, {\\phi} }^{ \\phantom{\\, r}\\phantom{\\, {\\phi}} } & = & -a {\\left(\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + 1\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\\\ g_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, {\\phi} \\, r }^{ \\phantom{\\, {\\phi}}\\phantom{\\, r} } & = & -a {\\left(\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + 1\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\end{array}$ " ], "text/plain": [ "g_t,t = 2*m*r/(a^2*cos(th)^2 + r^2) - 1 \n", "g_t,r = 2*m*r/(a^2*cos(th)^2 + r^2) \n", "g_t,ph = -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n", "g_r,t = 2*m*r/(a^2*cos(th)^2 + r^2) \n", "g_r,r = 2*m*r/(a^2*cos(th)^2 + r^2) + 1 \n", "g_r,ph = -a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*sin(th)^2 \n", "g_th,th = a^2*cos(th)^2 + r^2 \n", "g_ph,t = -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n", "g_ph,r = -a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + 2 \\, m r + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & 0 \\\\\n", "\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & \\frac{a^{2} - 2 \\, m r + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & \\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\\n", "0 & 0 & \\frac{1}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 \\\\\n", "0 & \\frac{a}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & -\\frac{1}{a^{2} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[-(a^2*cos(th)^2 + 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) 2*m*r/(a^2*cos(th)^2 + r^2) 0 0]\n", "[ 2*m*r/(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", "[ 0 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 3+1 Kerr coordinates:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & a^{4} \\cos\\left({\\theta}\\right)^{6} - {\\left(a^{4} - 2 \\, a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} - r^{4} - {\\left(2 \\, a^{2} r^{2} - r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2} \\end{array}$ " ], "text/plain": [ "M --> R\n", "(t, 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": [ "## Ingoing principal null geodesics" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\frac{\\partial}{\\partial t }-\\frac{\\partial}{\\partial r }$ " ], "text/plain": [ "k = d/dt - d/dr" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k = M.vector_field(1, -1, 0, 0, name='k')\n", "k.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $k$ is a null vector:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(k,k\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$ " ], "text/plain": [ "g(k,k): M --> R\n", " (t, r, th, ph) |--> 0" ] }, "execution_count": 14, "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": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/plain": [ "0" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla = g.connection()\n", "acc = nabla(k).contract(k)\n", "acc.display()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\mathrm{d} t-\\mathrm{d} r + a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}$ " ], "text/plain": [ "-dt - dr + a*sin(th)^2 dph" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k_form = k.down(g)\n", "k_form.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Kerr-Schild form of the Kerr metric" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce the metric $f$ such that\n", "$$ g = f + 2 H \\underline{k} \\otimes \\underline{k}$$\n", "where $H = m r / \\rho^2$:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} H:& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\end{array}$ " ], "text/plain": [ "H: M --> R\n", " (t, r, th, ph) |--> m*r/(a^2*cos(th)^2 + r^2)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H = M.scalar_field({X: m*r/rho2}, name='H')\n", "H.display()" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}f = -\\mathrm{d} t\\otimes \\mathrm{d} t+\\mathrm{d} r\\otimes \\mathrm{d} r -a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + {\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "f = -dt*dt + dr*dr - a*sin(th)^2 dr*dph + (a^2*cos(th)^2 + r^2) dth*dth - a*sin(th)^2 dph*dr + (a^2 + r^2)*sin(th)^2 dph*dph" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = M.lorentzian_metric('f')\n", "f.set(g - 2*H*(k_form*k_form))\n", "f.display()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & -a \\sin\\left({\\theta}\\right)^{2} \\\\\n", "0 & 0 & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} & 0 \\\\\n", "0 & -a \\sin\\left({\\theta}\\right)^{2} & 0 & {\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[ -1 0 0 0]\n", "[ 0 1 0 -a*sin(th)^2]\n", "[ 0 0 a^2*cos(th)^2 + r^2 0]\n", "[ 0 -a*sin(th)^2 0 (a^2 + r^2)*sin(th)^2]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f$ is a flat metric:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(f\\right) = 0$ " ], "text/plain": [ "Riem(f) = 0" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.riemann().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which proves that $g$ is a Kerr-Schild metric." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $k$ is a null vector for $f$ as well:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/plain": [ "0" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(k,k).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Kerr-Schild coordinates $(t, x, y, z)$" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, x, y, z)\\right)$ " ], "text/plain": [ "Chart (M, (t, x, y, z))" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "KS. = M.chart()\n", "KS" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ x & = & {\\left(r \\cos\\left({\\phi}\\right) - a \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\\\ y & = & {\\left(a \\cos\\left({\\phi}\\right) + r \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\\\ z & = & r \\cos\\left({\\theta}\\right) \\end{array}\\right.$ " ], "text/plain": [ "t = t\n", "x = (r*cos(ph) - a*sin(ph))*sin(th)\n", "y = (a*cos(ph) + r*sin(ph))*sin(th)\n", "z = r*cos(th)" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_to_KS = X.transition_map(KS, [t, \n", " (r*cos(ph) - a*sin(ph))*sin(th),\n", " (r*sin(ph) + a*cos(ph))*sin(th),\n", " r*cos(th)])\n", "X_to_KS.display()" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{-\\frac{1}{2} \\, a^{2} + \\frac{1}{2} \\, x^{2} + \\frac{1}{2} \\, y^{2} + \\frac{1}{2} \\, z^{2} + \\frac{1}{2} \\, \\sqrt{4 \\, a^{2} z^{2} + {\\left(a^{2} - x^{2} - y^{2} - z^{2}\\right)}^{2}}}$ " ], "text/plain": [ "sqrt(-1/2*a^2 + 1/2*x^2 + 1/2*y^2 + 1/2*z^2 + 1/2*sqrt(4*a^2*z^2 + (a^2 - x^2 - y^2 - z^2)^2))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R = sqrt((x^2 + y^2 + z^2 - a^2 \n", " + sqrt((x^2 + y^2 + z^2 - a^2)^2 + 4*a^2*z^2))/2)\n", "R" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [], "source": [ "#X_to_KS.set_inverse(t, R, acos(z/R), \n", "# atan2(R*y - a*x, R*x + a*y))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Check of the identity\n", "$$\\frac{x^2 + y^2}{r^2 + a^2} + \\frac{z^2}{r^2} = 1$$" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}1$ " ], "text/plain": [ "1" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "((x^2 + y^2)/(R^2 + a^2) + z^2/R^2).simplify_full()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Minkowskian expression of $f$ in terms of Kerr-Schild coordinates:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}f = -\\mathrm{d} t\\otimes \\mathrm{d} t+\\mathrm{d} x\\otimes \\mathrm{d} x+\\mathrm{d} y\\otimes \\mathrm{d} y+\\mathrm{d} z\\otimes \\mathrm{d} z$ " ], "text/plain": [ "f = -dt*dt + dx*dx + dy*dy + dz*dz" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f.display(KS.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Equivalently, we may check the following identity:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dt, dx, dy, dz = KS.coframe()[:]\n", "f == - dt*dt + dx*dx + dy*dy + dz*dz" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} x = \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\mathrm{d} r + {\\left(r \\cos\\left({\\phi}\\right) - a \\sin\\left({\\phi}\\right)\\right)} \\cos\\left({\\theta}\\right) \\mathrm{d} {\\theta} -{\\left(a \\cos\\left({\\phi}\\right) + r \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\mathrm{d} {\\phi}$ " ], "text/plain": [ "dx = cos(ph)*sin(th) dr + (r*cos(ph) - a*sin(ph))*cos(th) dth - (a*cos(ph) + r*sin(ph))*sin(th) dph" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dx.display()" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} x \\otimes \\mathrm{d} x + \\mathrm{d} y \\otimes \\mathrm{d} y + \\mathrm{d} z \\otimes \\mathrm{d} z = \\mathrm{d} r\\otimes \\mathrm{d} r -a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -a \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + {\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "dx*dx+dy*dy+dz*dz = dr*dr - a*sin(th)^2 dr*dph + (a^2*cos(th)^2 + r^2) dth*dth - a*sin(th)^2 dph*dr + (a^2 + r^2)*sin(th)^2 dph*dph" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(dx*dx + dy*dy + dz*dz).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Expression of $k$ and $g$ in the Kerr-Schild frame:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\frac{\\partial}{\\partial t } -\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\frac{\\partial}{\\partial x } -\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\frac{\\partial}{\\partial y } -\\cos\\left({\\theta}\\right) \\frac{\\partial}{\\partial z }$ " ], "text/plain": [ "k = d/dt - cos(ph)*sin(th) d/dx - sin(ph)*sin(th) d/dy - cos(th) d/dz" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.display(KS.frame())" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\mathrm{d} t -\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\mathrm{d} x -\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\mathrm{d} y -\\cos\\left({\\theta}\\right) \\mathrm{d} z$ " ], "text/plain": [ "-dt - cos(ph)*sin(th) dx - sin(ph)*sin(th) dy - cos(th) dz" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k_form.display(KS.frame())" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} - 2 \\, m r + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, m r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} x + \\left( \\frac{2 \\, m r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} y + \\left( \\frac{2 \\, m r \\cos\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} z + \\left( \\frac{2 \\, m r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} x\\otimes \\mathrm{d} t + \\left( \\frac{{\\left(2 \\, m r \\cos\\left({\\phi}\\right)^{2} - a^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} + a^{2} + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( -\\frac{2 \\, {\\left(m r \\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right)^{2} \\sin\\left({\\phi}\\right) - m r \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right)\\right)}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( \\frac{2 \\, m r \\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} x\\otimes \\mathrm{d} z + \\left( \\frac{2 \\, m r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} y\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, {\\left(m r \\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right)^{2} \\sin\\left({\\phi}\\right) - m r \\cos\\left({\\phi}\\right) \\sin\\left({\\phi}\\right)\\right)}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( \\frac{{\\left(2 \\, m r \\sin\\left({\\phi}\\right)^{2} - a^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} + a^{2} + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( \\frac{2 \\, m r \\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} y\\otimes \\mathrm{d} z + \\left( \\frac{2 \\, m r \\cos\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} z\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, m r \\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} z\\otimes \\mathrm{d} x + \\left( \\frac{2 \\, m r \\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} z\\otimes \\mathrm{d} y + \\left( \\frac{{\\left(a^{2} + 2 \\, m r\\right)} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} z\\otimes \\mathrm{d} z$ " ], "text/plain": [ "g = -(a^2*cos(th)^2 - 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) dt*dt + 2*m*r*cos(ph)*sin(th)/(a^2*cos(th)^2 + r^2) dt*dx + 2*m*r*sin(ph)*sin(th)/(a^2*cos(th)^2 + r^2) dt*dy + 2*m*r*cos(th)/(a^2*cos(th)^2 + r^2) dt*dz + 2*m*r*cos(ph)*sin(th)/(a^2*cos(th)^2 + r^2) dx*dt + ((2*m*r*cos(ph)^2 - a^2)*sin(th)^2 + a^2 + r^2)/(a^2*cos(th)^2 + r^2) dx*dx - 2*(m*r*cos(ph)*cos(th)^2*sin(ph) - m*r*cos(ph)*sin(ph))/(a^2*cos(th)^2 + r^2) dx*dy + 2*m*r*cos(ph)*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) dx*dz + 2*m*r*sin(ph)*sin(th)/(a^2*cos(th)^2 + r^2) dy*dt - 2*(m*r*cos(ph)*cos(th)^2*sin(ph) - m*r*cos(ph)*sin(ph))/(a^2*cos(th)^2 + r^2) dy*dx + ((2*m*r*sin(ph)^2 - a^2)*sin(th)^2 + a^2 + r^2)/(a^2*cos(th)^2 + r^2) dy*dy + 2*m*r*cos(th)*sin(ph)*sin(th)/(a^2*cos(th)^2 + r^2) dy*dz + 2*m*r*cos(th)/(a^2*cos(th)^2 + r^2) dz*dt + 2*m*r*cos(ph)*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) dz*dx + 2*m*r*cos(th)*sin(ph)*sin(th)/(a^2*cos(th)^2 + r^2) dz*dy + ((a^2 + 2*m*r)*cos(th)^2 + r^2)/(a^2*cos(th)^2 + r^2) dz*dz" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(KS.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Expression of the Killing vector $\\partial/\\partial\\phi$ in terms of the Kerr-Schild frame:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial {\\phi} } = -{\\left(a \\cos\\left({\\phi}\\right) + r \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\frac{\\partial}{\\partial x } + {\\left(r \\cos\\left({\\phi}\\right) - a \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\frac{\\partial}{\\partial y }$ " ], "text/plain": [ "d/dph = -(a*cos(ph) + r*sin(ph))*sin(th) d/dx + (r*cos(ph) - a*sin(ph))*sin(th) d/dy" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.frame()[3].display(KS.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Plots" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [], "source": [ "ap = 0.9 # value of a for the plots\n", "rmax = 3" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "rcol = 'green' # color of the curves (th,ph) = const\n", "thcol = 'red' # color of the curves (r,ph) = const\n", "phcol = 'goldenrod' # color of the curves (r,th) = const\n", "coordcol = {r: rcol, th: thcol, ph: phcol}" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [], "source": [ "opacity = 1\n", "surf_shift = 0.03" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Numerical values of the event and Cauchy horizons:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1.43588989435407, 0.564110105645933\\right)$ " ], "text/plain": [ "(1.43588989435407, 0.564110105645933)" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rHp = 1 + sqrt(1 - ap^2)\n", "rCp = 1 - sqrt(1 - ap^2)\n", "rHp, rCp" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [], "source": [ "X_plot = X.plot(KS, fixed_coords={t: 0, ph: 0}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11,\n", " color=coordcol, thickness=2, max_range=rmax, \n", " label_axes=False)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, {\\left(r \\cos\\left({\\phi}\\right) - a \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right), {\\left(a \\cos\\left({\\phi}\\right) + r \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right)$ " ], "text/plain": [ "(t,\n", " (r*cos(ph) - a*sin(ph))*sin(th),\n", " (a*cos(ph) + r*sin(ph))*sin(th),\n", " r*cos(th))" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_to_KS(t, r, th, ph)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[r \\sin\\left({\\theta}\\right), 0.900000000000000 \\, \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right]$ " ], "text/plain": [ "[r*sin(th), 0.900000000000000*sin(th), r*cos(th)]" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xyz_n = [s.subs({a: ap, ph: 0}) for s in X_to_KS(t, r, th, ph)[1:]]\n", "xyz_n" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.43588989435407 \\, \\sin\\left({\\theta}\\right), 0.900000000000000 \\, \\sin\\left({\\theta}\\right), 1.43588989435407 \\, \\cos\\left({\\theta}\\right)\\right]$ " ], "text/plain": [ "[1.43588989435407*sin(th), 0.900000000000000*sin(th), 1.43588989435407*cos(th)]" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xyz_H = [s.subs({r: rHp}) for s in xyz_n]\n", "xyz_H" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.564110105645933 \\, \\sin\\left({\\theta}\\right), 0.900000000000000 \\, \\sin\\left({\\theta}\\right), 0.564110105645933 \\, \\cos\\left({\\theta}\\right)\\right]$ " ], "text/plain": [ "[0.564110105645933*sin(th),\n", " 0.900000000000000*sin(th),\n", " 0.564110105645933*cos(th)]" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xyz_C = [s.subs({r: rCp}) for s in xyz_n]\n", "xyz_C" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "xyz_n[1] += surf_shift # small adjustment to ensure that the surface does not cover\n", " # the coordinate grid lines" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = parametric_plot3d(xyz_n, (r, 0, rmax), (th, 0, pi), color='ivory',\n", " opacity=opacity)\n", "graph += X_plot\n", "graph += parametric_plot3d(xyz_H, (th, 0, pi), color='black', thickness=6)\n", "graph += parametric_plot3d(xyz_C, (th, 0, pi), color='blue', thickness=6)\n", "graph += line([(0.03,0,0), (0.03,ap,0)], color='red', thickness=6)\n", "graph" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "xyz_n = [s.subs({a: ap, ph: pi}) for s in X_to_KS(t, r, th, ph)[1:]]\n", "xyz_H = [s.subs({r: rHp}) for s in xyz_n]\n", "xyz_C = [s.subs({r: rCp}) for s in xyz_n]\n", "X_plot_pi = X.plot(KS, fixed_coords={t: 0, ph: pi}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11,\n", " color=coordcol, thickness=2, max_range=rmax, \n", " label_axes=False)" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xyz_n[1] += surf_shift # small adjustment to ensure that the surface does not cover\n", " # the coordinate grid lines\n", "graph_pi = parametric_plot3d(xyz_n, (r, 0, rmax), (th, 0, pi), color='ivory',\n", " opacity=opacity)\n", "graph_pi += X_plot_pi\n", "graph_pi += parametric_plot3d(xyz_H, (th, 0, pi), color='black', thickness=6)\n", "graph_pi += parametric_plot3d(xyz_C, (th, 0, pi), color='blue', thickness=6)\n", "graph_pi += line([(0.03,0,0), (0.03,-ap,0)], color='red', thickness=6)\n", "graph_pi += line([(0,0,-1.1*rmax), (0,0,1.1*rmax)], color='green', thickness=4)\n", "ph_slice = graph + graph_pi\n", "show(ph_slice)" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.plot(KS, fixed_coords={t: 0, r: 1}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11,\n", " color=coordcol, label_axes=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The BH event horizon:" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H_plot = X.plot(KS, fixed_coords={t: 0, r: rHp}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11, color='grey', \n", " label_axes=False)\n", "H_plot" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.plot(KS, fixed_coords={t: 0, r: 0}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11, color=coordcol, \n", " label_axes=False)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 31 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "rzero = X.plot(KS, fixed_coords={t: 0, r: 0}, ambient_coords=(x,y), \n", " parameters={a: ap}, number_values={th: 17, ph: 13}, color=coordcol)\n", "rzero += circle((0,0), ap, color='brown', thickness=3) \n", "show(rzero, xmin=-1, xmax=1, ymin=-1, ymax=1, axes=False, frame=True,\n", " axes_labels=[r'$x/m$', r'$y/m$'])" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [], "source": [ "rzero.save('ksm_rzero_disk.pdf', xmin=-1, xmax=1, ymin=-1, ymax=1, \n", " axes=False, frame=True, axes_labels=[r'$x/m$', r'$y/m$'])" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Xth_pi2 = X.plot(KS, fixed_coords={t: 0, th: pi/2}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11, color=coordcol,\n", " max_range=rmax, thickness=1.5, label_axes=False)\n", "Xth_pi2" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X.plot(KS, fixed_coords={t: 0, th: pi/3}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11, color=coordcol,\n", " max_range=rmax, thickness=1.5, label_axes=False)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph = X.plot(KS, fixed_coords={t: 0, th: pi/6}, ambient_coords=(x,y,z), \n", " parameters={a: ap}, number_values=11, color=coordcol,\n", " max_range=rmax, thickness=1.5, label_axes=False) \\\n", " + Xth_pi2 \\\n", " + circle((0,0,0), ap, color='pink', fill=True) \\\n", " + circle((0,0,0), ap, color='brown', thickness=6, linestyle='dashed')\n", "show(graph)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Plot of the $r<0$ domain" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, {x'}, {y'}, {z'})\\right)$ " ], "text/plain": [ "Chart (M, (t, xp, yp, zp))" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "KS2. = M.chart(r\"t xp:x' yp:y' zp:z'\")\n", "KS2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We use replace $r$ by $-r$ in the transformation formulas, because in what follows, we consider that\n", "$r$ is positive:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ {x'} & = & -{\\left(r \\cos\\left({\\phi}\\right) + a \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\\\ {y'} & = & {\\left(a \\cos\\left({\\phi}\\right) - r \\sin\\left({\\phi}\\right)\\right)} \\sin\\left({\\theta}\\right) \\\\ {z'} & = & -r \\cos\\left({\\theta}\\right) \\end{array}\\right.$ " ], "text/plain": [ "t = t\n", "xp = -(r*cos(ph) + a*sin(ph))*sin(th)\n", "yp = (a*cos(ph) - r*sin(ph))*sin(th)\n", "zp = -r*cos(th)" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_to_KS2 = X.transition_map(KS2, [t, \n", " (-r*cos(ph) - a*sin(ph))*sin(th),\n", " (-r*sin(ph) + a*cos(ph))*sin(th),\n", " -r*cos(th)])\n", "X_to_KS2.display()" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [], "source": [ "X_plot2 = X.plot(KS2, fixed_coords={t: 0, ph: 0}, ambient_coords=(xp,yp,zp), \n", " parameters={a: ap}, number_values=11,\n", " color=coordcol, thickness=2, max_range=rmax, \n", " label_axes=False)" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [], "source": [ "surf_shift = 0\n", "xyz_n2 = [s.subs({a: ap, ph: 0}) for s in X_to_KS2(t, r, th, ph)[1:]]\n", "xyz_n2[1] -= surf_shift " ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [], "source": [ "graph2 = parametric_plot3d(xyz_n2, (r, 0, rmax), (th, 0, pi), color='pink',\n", " opacity=opacity)\n", "graph2 += X_plot2" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [], "source": [ "X_plot2_pi = X.plot(KS2, fixed_coords={t: 0, ph: pi}, ambient_coords=(xp,yp,zp), \n", " parameters={a: ap}, number_values=11,\n", " color=coordcol, thickness=2, max_range=rmax, \n", " label_axes=False)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ "xyz_n2 = [s.subs({a: ap, ph: pi}) for s in X_to_KS2(t, r, th, ph)[1:]]\n", "xyz_n2[1] += surf_shift" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [], "source": [ "graph2_pi = parametric_plot3d(xyz_n2, (r, 0, rmax), (th, 0, pi), color='pink',\n", " opacity=opacity)\n", "graph2_pi += X_plot2_pi" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ph_slice2 = graph2 + graph2_pi\n", "ph_slice2" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ph_slice + ph_slice2" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.1.beta8", "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 }