{ "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://luth.obspm.fr/~luthier/gourgoulhon/bh16/).\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", "