{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Near-horizon geometry of the extremal Reissner-Nordström black hole" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook derives the near-horizon geometry of the extremal (i.e. maximally charged) Reissner-Nordström black hole. It is based on SageMath tools developed through the [SageManifolds project](https://sagemanifolds.obspm.fr/)." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 10.8.beta9, Release Date: 2025-11-11'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sage.version.banner" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Spacetime manifold and metric\n", "\n", "We declare $\\mathcal{M}$ as a 4-dimensional Lorentzian manifold and endow it with the **Boyer-Lindquist chart** (`BL`):\n", "\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', latex_name=r'\\mathcal{M}', structure='Lorentzian')\n", "BL. = M.chart(r\"t r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):periodic:\\phi\")\n", "BL" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 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{(periodic)}\$" ], "text/latex": [ "$\\displaystyle t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\text{(periodic)}$" ], "text/plain": [ "t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL.coord_range()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Initialization of the metric:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( -\\frac{m^{2}}{r^{2}} + \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{r^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\left( -\\frac{m^{2}}{r^{2}} + \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{r^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = (-m^2/r^2 + 2*m/r - 1) dt⊗dt + r^2/(m^2 - 2*m*r + r^2) dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = var('m', domain='real')\n", "assume(m>0)\n", "Q2 = m^2 # extremal Reissner-Nordström\n", "g = M.metric()\n", "g[0,0] = - (1 - 2*m/r + Q2/r^2)\n", "g[1,1] = -1/g[0,0]\n", "g[2,2], g[3,3] = r^2, r^2*sin(th)^2\n", "g.display()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = -\\frac{{\\left(m - r\\right)}^{2}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{r^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = -\\frac{{\\left(m - r\\right)}^{2}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{r^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = -(m - r)^2/r^2 dt⊗dt + r^2/(m - r)^2 dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.apply_map(factor)\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Curvature" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, t} \\, r \\, t \\, r }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{3 \\, m^{2} - 2 \\, m r}{m^{2} r^{2} - 2 \\, m r^{3} + r^{4}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, t} \\, {\\theta} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{m^{2} - m r}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, t} \\, {\\phi} \\, t \\, {\\phi} }^{ \\, t \\phantom{\\, {\\phi}} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(m^{2} - m r\\right)} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, r} \\, t \\, t \\, r }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{3 \\, m^{4} - 8 \\, m^{3} r + 7 \\, m^{2} r^{2} - 2 \\, m r^{3}}{r^{6}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{m^{2} - m r}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(m^{2} - m r\\right)} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, t \\, t \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{6}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{m r^{2} - r^{3}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(m^{2} - 2 \\, m r\\right)} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, t \\, t \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{6}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m}{m r^{2} - r^{3}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{m^{2} - 2 \\, m r}{r^{2}} \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, t} \\, r \\, t \\, r }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{3 \\, m^{2} - 2 \\, m r}{m^{2} r^{2} - 2 \\, m r^{3} + r^{4}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, t} \\, {\\theta} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{m^{2} - m r}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, t} \\, {\\phi} \\, t \\, {\\phi} }^{ \\, t \\phantom{\\, {\\phi}} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(m^{2} - m r\\right)} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, r} \\, t \\, t \\, r }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{3 \\, m^{4} - 8 \\, m^{3} r + 7 \\, m^{2} r^{2} - 2 \\, m r^{3}}{r^{6}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{m^{2} - m r}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(m^{2} - m r\\right)} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, t \\, t \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{6}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{m r^{2} - r^{3}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(m^{2} - 2 \\, m r\\right)} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, t \\, t \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{6}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m}{m r^{2} - r^{3}} \\\\ {\\mathrm{Riem}\\left(g\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{m^{2} - 2 \\, m r}{r^{2}} \\end{array}$" ], "text/plain": [ "Riem(g)^t_r,t,r = -(3*m^2 - 2*m*r)/(m^2*r^2 - 2*m*r^3 + r^4) \n", "Riem(g)^t_th,t,th = (m^2 - m*r)/r^2 \n", "Riem(g)^t_ph,t,ph = (m^2 - m*r)*sin(th)^2/r^2 \n", "Riem(g)^r_t,t,r = -(3*m^4 - 8*m^3*r + 7*m^2*r^2 - 2*m*r^3)/r^6 \n", "Riem(g)^r_th,r,th = (m^2 - m*r)/r^2 \n", "Riem(g)^r_ph,r,ph = (m^2 - m*r)*sin(th)^2/r^2 \n", "Riem(g)^th_t,t,th = (m^4 - 3*m^3*r + 3*m^2*r^2 - m*r^3)/r^6 \n", "Riem(g)^th_r,r,th = -m/(m*r^2 - r^3) \n", "Riem(g)^th_ph,th,ph = -(m^2 - 2*m*r)*sin(th)^2/r^2 \n", "Riem(g)^ph_t,t,ph = (m^4 - 3*m^3*r + 3*m^2*r^2 - m*r^3)/r^6 \n", "Riem(g)^ph_r,r,ph = -m/(m*r^2 - r^3) \n", "Riem(g)^ph_th,th,ph = (m^2 - 2*m*r)/r^2 " ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem = g.riemann()\n", "Riem.display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\left( \\frac{m^{4} - 2 \\, m^{3} r + m^{2} r^{2}}{r^{6}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{m^{2}}{m^{2} r^{2} - 2 \\, m r^{3} + r^{4}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\frac{m^{2}}{r^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{m^{2} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\left( \\frac{m^{4} - 2 \\, m^{3} r + m^{2} r^{2}}{r^{6}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{m^{2}}{m^{2} r^{2} - 2 \\, m r^{3} + r^{4}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\frac{m^{2}}{r^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{m^{2} \\sin\\left({\\theta}\\right)^{2}}{r^{2}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Ric(g) = (m^4 - 2*m^3*r + m^2*r^2)/r^6 dt⊗dt - m^2/(m^2*r^2 - 2*m*r^3 + r^4) dr⊗dr + m^2/r^2 dth⊗dth + m^2*sin(th)^2/r^2 dph⊗dph" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The Kretschmann scalar $K := R_{abcd} R^{abcd}$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} & \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{8 \\, {\\left(7 \\, m^{4} - 12 \\, m^{3} r + 6 \\, m^{2} r^{2}\\right)}}{r^{8}} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} & \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{8 \\, {\\left(7 \\, m^{4} - 12 \\, m^{3} r + 6 \\, m^{2} r^{2}\\right)}}{r^{8}} \\end{array}$" ], "text/plain": [ "M → ℝ\n", "(t, r, th, ph) ↦ 8*(7*m^4 - 12*m^3*r + 6*m^2*r^2)/r^8" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K = Riem.down(g)['_{abcd}']*Riem.up(g)['^{abcd}']\n", "K.display()" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{8 \\, {\\left(7 \\, m^{2} - 12 \\, m r + 6 \\, r^{2}\\right)} m^{2}}{r^{8}}\$" ], "text/latex": [ "$\\displaystyle \\frac{8 \\, {\\left(7 \\, m^{2} - 12 \\, m r + 6 \\, r^{2}\\right)} m^{2}}{r^{8}}$" ], "text/plain": [ "8*(7*m^2 - 12*m*r + 6*r^2)*m^2/r^8" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K.expr().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The Ricci scalar $g^{ab} R_{ab}$:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$" ], "text/plain": [ "r(g): M → ℝ\n", " (t, r, th, ph) ↦ 0" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci_scalar().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Eddington-Finkelstein coordinates" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Let us introduce the **tortoise coordinate** $r_*$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 2 \\, m \\log\\left(\\frac{{\\left| -m + r \\right|}}{m}\\right) - m + \\frac{m^{2}}{m - r} + r\$" ], "text/latex": [ "$\\displaystyle 2 \\, m \\log\\left(\\frac{{\\left| -m + r \\right|}}{m}\\right) - m + \\frac{m^{2}}{m - r} + r$" ], "text/plain": [ "2*m*log(abs(-m + r)/m) - m + m^2/(m - r) + r" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rstar(r) = r - m^2/(r - m) + 2*m*ln(abs(r - m)/m) - m \n", "rstar(r)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Note that the additive constant is chosen so that $r_* = 0$ for $r=2m$:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rstar(2*m)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pl = plot(rstar(r).subs({m: 1}), (r, 0, 5), \n", " axes_labels=[r'$r/m$', r'$r_*/m$']) \\\n", " + line([(1, -10), (1, 10)], color='black')\n", "show(pl, ymin=-6, ymax=6)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The chart of the **ingoing null Eddington-Finkelstein coordinates** $(v,r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathcal{M},(v, r, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathcal{M},(v, r, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (v, r, th, ph))" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF. = M.chart(r\"v r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):periodic:\\phi\")\n", "EF" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} v & = & 2 \\, m \\log\\left(\\frac{{\\left| -m + r \\right|}}{m}\\right) - m + \\frac{m^{2}}{m - r} + r + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} v & = & 2 \\, m \\log\\left(\\frac{{\\left| -m + r \\right|}}{m}\\right) - m + \\frac{m^{2}}{m - r} + r + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "v = 2*m*log(abs(-m + r)/m) - m + m^2/(m - r) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL_to_EF = BL.transition_map(EF, [t + rstar(r), r, th, ph])\n", "BL_to_EF.display()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} t & = & \\frac{2 \\, m^{2} \\log\\left(m\\right) - 2 \\, {\\left(m \\log\\left(m\\right) + m\\right)} r + r^{2} + {\\left(m - r\\right)} v - 2 \\, {\\left(m^{2} - m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{m - r} \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} t & = & \\frac{2 \\, m^{2} \\log\\left(m\\right) - 2 \\, {\\left(m \\log\\left(m\\right) + m\\right)} r + r^{2} + {\\left(m - r\\right)} v - 2 \\, {\\left(m^{2} - m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{m - r} \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "t = (2*m^2*log(m) - 2*(m*log(m) + m)*r + r^2 + (m - r)*v - 2*(m^2 - m*r)*log(abs(-m + r)))/(m - r)\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL_to_EF.inverse().display()" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rrrr}\n", "1 & \\frac{r^{2}}{m^{2} - 2 \\, m r + r^{2}} & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rrrr}\n", "1 & \\frac{r^{2}}{m^{2} - 2 \\, m r + r^{2}} & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$" ], "text/plain": [ "[ 1 r^2/(m^2 - 2*m*r + r^2) 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL_to_EF.jacobian()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2}} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r +\\mathrm{d} r\\otimes \\mathrm{d} v + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2}} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r +\\mathrm{d} r\\otimes \\mathrm{d} v + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = -(m^2 - 2*m*r + r^2)/r^2 dv⊗dv + dv⊗dr + dr⊗dv + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(EF)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Orthonormal tetrad associated to Eddington-Finkelstein coordinates:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "F(r) = (r - m)^2/r^2" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle E_{0} = \\left( \\frac{r}{{\\left| m - r \\right|}} \\right) \\frac{\\partial}{\\partial v }\$" ], "text/latex": [ "$\\displaystyle E_{0} = \\left( \\frac{r}{{\\left| m - r \\right|}} \\right) \\frac{\\partial}{\\partial v }$" ], "text/plain": [ "E_0 = r/abs(m - r) ∂/∂v" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\$\\displaystyle E_{1} = \\left( \\frac{r}{{\\left| m - r \\right|}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{{\\left| m - r \\right|}}{r} \\right) \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle E_{1} = \\left( \\frac{r}{{\\left| m - r \\right|}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{{\\left| m - r \\right|}}{r} \\right) \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "E_1 = r/abs(m - r) ∂/∂v + abs(m - r)/r ∂/∂r" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\$\\displaystyle E_{2} = \\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }\$" ], "text/latex": [ "$\\displaystyle E_{2} = \\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }$" ], "text/plain": [ "E_2 = 1/r ∂/∂th" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\$\\displaystyle E_{3} = \\frac{1}{r \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle E_{3} = \\frac{1}{r \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "E_3 = 1/(r*sin(th)) ∂/∂ph" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "EFf = EF.frame()\n", "E = M.vector_frame('E', [EFf[0]/sqrt(F(r)),\n", " EFf[0]/sqrt(F(r)) + sqrt(F(r))*EFf[1],\n", " EFf[2]/r,\n", " EFf[3]/(r*sin(th))])\n", "for v in E:\n", " show(v.display(EF))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Let us check that $E$ is an orthonormal frame:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = -E^{0}\\otimes E^{0}+E^{1}\\otimes E^{1}+E^{2}\\otimes E^{2}+E^{3}\\otimes E^{3}\$" ], "text/latex": [ "$\\displaystyle g = -E^{0}\\otimes E^{0}+E^{1}\\otimes E^{1}+E^{2}\\otimes E^{2}+E^{3}\\otimes E^{3}$" ], "text/plain": [ "g = -E^0⊗E^0 + E^1⊗E^1 + E^2⊗E^2 + E^3⊗E^3" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(E)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The dual tetrad:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle E^{0} = \\left( \\frac{{\\left| -m + r \\right|}}{r} \\right) \\mathrm{d} v + \\left( -\\frac{r}{{\\left| -m + r \\right|}} \\right) \\mathrm{d} r\$" ], "text/latex": [ "$\\displaystyle E^{0} = \\left( \\frac{{\\left| -m + r \\right|}}{r} \\right) \\mathrm{d} v + \\left( -\\frac{r}{{\\left| -m + r \\right|}} \\right) \\mathrm{d} r$" ], "text/plain": [ "E^0 = abs(-m + r)/r dv - r/abs(-m + r) dr" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\$\\displaystyle E^{1} = \\left( \\frac{r}{{\\left| -m + r \\right|}} \\right) \\mathrm{d} r\$" ], "text/latex": [ "$\\displaystyle E^{1} = \\left( \\frac{r}{{\\left| -m + r \\right|}} \\right) \\mathrm{d} r$" ], "text/plain": [ "E^1 = r/abs(-m + r) dr" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\$\\displaystyle E^{2} = r \\mathrm{d} {\\theta}\$" ], "text/latex": [ "$\\displaystyle E^{2} = r \\mathrm{d} {\\theta}$" ], "text/plain": [ "E^2 = r dth" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\$\\displaystyle E^{3} = r \\sin\\left({\\theta}\\right) \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle E^{3} = r \\sin\\left({\\theta}\\right) \\mathrm{d} {\\phi}$" ], "text/plain": [ "E^3 = r*sin(th) dph" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for f in E.dual_basis():\n", " show(f.display(EF))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "At this stage, two charts and three vector frames have been defined on $\\mathcal{M}$:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right), \\left(\\mathcal{M},(v, r, {\\theta}, {\\phi})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right), \\left(\\mathcal{M},(v, r, {\\theta}, {\\phi})\\right)\\right]$" ], "text/plain": [ "[Chart (M, (t, r, th, ph)), Chart (M, (v, r, th, ph))]" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{M}, \\left(E_{0},E_{1},E_{2},E_{3}\\right)\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{M}, \\left(E_{0},E_{1},E_{2},E_{3}\\right)\\right)\\right]$" ], "text/plain": [ "[Coordinate frame (M, (∂/∂t,∂/∂r,∂/∂th,∂/∂ph)),\n", " Coordinate frame (M, (∂/∂v,∂/∂r,∂/∂th,∂/∂ph)),\n", " Vector frame (M, (E_0,E_1,E_2,E_3))]" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "### Connection 1-forms with respect to the tetrad $E$:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "nabla = g.connection()\n", "print(nabla)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle (\\omega^i_j)=\\left(\\begin{array}{cccc} & \\omega^0_{\\ \\, 0} = 0 & \\omega^0_{\\ \\, 1} = \\left( -\\frac{m^{2} - m r}{r^{2} {\\left| -m + r \\right|}} \\right) E^{0} & \\omega^0_{\\ \\, 2} = 0 & \\omega^0_{\\ \\, 3} = 0 \\\\ & \\omega^1_{\\ \\, 0} = \\left( -\\frac{m^{2} - m r}{r^{2} {\\left| -m + r \\right|}} \\right) E^{0} & \\omega^1_{\\ \\, 1} = 0 & \\omega^1_{\\ \\, 2} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2} {\\left| -m + r \\right|}} \\right) E^{2} & \\omega^1_{\\ \\, 3} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2} {\\left| -m + r \\right|}} \\right) E^{3} \\\\ & \\omega^2_{\\ \\, 0} = 0 & \\omega^2_{\\ \\, 1} = \\left( \\frac{{\\left| -m + r \\right|}}{r^{2}} \\right) E^{2} & \\omega^2_{\\ \\, 2} = 0 & \\omega^2_{\\ \\, 3} = -\\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} E^{3} \\\\ & \\omega^3_{\\ \\, 0} = 0 & \\omega^3_{\\ \\, 1} = \\left( \\frac{{\\left| -m + r \\right|}}{r^{2}} \\right) E^{3} & \\omega^3_{\\ \\, 2} = \\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} E^{3} & \\omega^3_{\\ \\, 3} = 0 \\\\ \\end{array}\\right)\\)" ], "text/latex": [ "$\\displaystyle (\\omega^i_j)=\\left(\\begin{array}{cccc} & \\omega^0_{\\ \\, 0} = 0 & \\omega^0_{\\ \\, 1} = \\left( -\\frac{m^{2} - m r}{r^{2} {\\left| -m + r \\right|}} \\right) E^{0} & \\omega^0_{\\ \\, 2} = 0 & \\omega^0_{\\ \\, 3} = 0 \\\\ & \\omega^1_{\\ \\, 0} = \\left( -\\frac{m^{2} - m r}{r^{2} {\\left| -m + r \\right|}} \\right) E^{0} & \\omega^1_{\\ \\, 1} = 0 & \\omega^1_{\\ \\, 2} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2} {\\left| -m + r \\right|}} \\right) E^{2} & \\omega^1_{\\ \\, 3} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2} {\\left| -m + r \\right|}} \\right) E^{3} \\\\ & \\omega^2_{\\ \\, 0} = 0 & \\omega^2_{\\ \\, 1} = \\left( \\frac{{\\left| -m + r \\right|}}{r^{2}} \\right) E^{2} & \\omega^2_{\\ \\, 2} = 0 & \\omega^2_{\\ \\, 3} = -\\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} E^{3} \\\\ & \\omega^3_{\\ \\, 0} = 0 & \\omega^3_{\\ \\, 1} = \\left( \\frac{{\\left| -m + r \\right|}}{r^{2}} \\right) E^{3} & \\omega^3_{\\ \\, 2} = \\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} E^{3} & \\omega^3_{\\ \\, 3} = 0 \\\\ \\end{array}\\right)$" ], "text/plain": [ "(\\omega^i_j)=\\left(\\begin{array}{cccc} & \\omega^0_{\\ \\, 0} = 0 & \\omega^0_{\\ \\, 1} = \\left( -\\frac{m^{2} - m r}{r^{2} {\\left| -m + r \\right|}} \\right) E^{0} & \\omega^0_{\\ \\, 2} = 0 & \\omega^0_{\\ \\, 3} = 0 \\\\ & \\omega^1_{\\ \\, 0} = \\left( -\\frac{m^{2} - m r}{r^{2} {\\left| -m + r \\right|}} \\right) E^{0} & \\omega^1_{\\ \\, 1} = 0 & \\omega^1_{\\ \\, 2} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2} {\\left| -m + r \\right|}} \\right) E^{2} & \\omega^1_{\\ \\, 3} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r^{2} {\\left| -m + r \\right|}} \\right) E^{3} \\\\ & \\omega^2_{\\ \\, 0} = 0 & \\omega^2_{\\ \\, 1} = \\left( \\frac{{\\left| -m + r \\right|}}{r^{2}} \\right) E^{2} & \\omega^2_{\\ \\, 2} = 0 & \\omega^2_{\\ \\, 3} = -\\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} E^{3} \\\\ & \\omega^3_{\\ \\, 0} = 0 & \\omega^3_{\\ \\, 1} = \\left( \\frac{{\\left| -m + r \\right|}}{r^{2}} \\right) E^{3} & \\omega^3_{\\ \\, 2} = \\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)} E^{3} & \\omega^3_{\\ \\, 3} = 0 \\\\ \\end{array}\\right)" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = r'(\\omega^i_j)=\\left(\\begin{array}{cccc}'\n", "for i in M.irange():\n", " for j in M.irange():\n", " s += r'&' + latex(nabla.connection_form(i, j, E).display(E, EF)) \n", " s += r'\\\\'\n", "s += r'\\end{array}\\right)'\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "### Connection 1-forms with respect to the natural frame associated with Eddington-Finkelstein coordinates:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle (\\omega^i_j)=\\left(\\begin{array}{cccc} & \\omega^0_{\\ \\, 0} = \\left( -\\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} v & \\omega^0_{\\ \\, 1} = 0 & \\omega^0_{\\ \\, 2} = -r \\mathrm{d} {\\theta} & \\omega^0_{\\ \\, 3} = -r \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi} \\\\ & \\omega^1_{\\ \\, 0} = \\left( -\\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{5}} \\right) \\mathrm{d} v + \\left( \\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} r & \\omega^1_{\\ \\, 1} = \\left( \\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} v & \\omega^1_{\\ \\, 2} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r} \\right) \\mathrm{d} {\\theta} & \\omega^1_{\\ \\, 3} = -\\frac{{\\left(m^{2} - 2 \\, m r + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\mathrm{d} {\\phi} \\\\ & \\omega^2_{\\ \\, 0} = 0 & \\omega^2_{\\ \\, 1} = \\frac{1}{r} \\mathrm{d} {\\theta} & \\omega^2_{\\ \\, 2} = \\frac{1}{r} \\mathrm{d} r & \\omega^2_{\\ \\, 3} = -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\mathrm{d} {\\phi} \\\\ & \\omega^3_{\\ \\, 0} = 0 & \\omega^3_{\\ \\, 1} = \\frac{1}{r} \\mathrm{d} {\\phi} & \\omega^3_{\\ \\, 2} = \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\mathrm{d} {\\phi} & \\omega^3_{\\ \\, 3} = \\frac{1}{r} \\mathrm{d} r + \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\mathrm{d} {\\theta} \\\\ \\end{array}\\right)\\)" ], "text/latex": [ "$\\displaystyle (\\omega^i_j)=\\left(\\begin{array}{cccc} & \\omega^0_{\\ \\, 0} = \\left( -\\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} v & \\omega^0_{\\ \\, 1} = 0 & \\omega^0_{\\ \\, 2} = -r \\mathrm{d} {\\theta} & \\omega^0_{\\ \\, 3} = -r \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi} \\\\ & \\omega^1_{\\ \\, 0} = \\left( -\\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{5}} \\right) \\mathrm{d} v + \\left( \\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} r & \\omega^1_{\\ \\, 1} = \\left( \\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} v & \\omega^1_{\\ \\, 2} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r} \\right) \\mathrm{d} {\\theta} & \\omega^1_{\\ \\, 3} = -\\frac{{\\left(m^{2} - 2 \\, m r + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\mathrm{d} {\\phi} \\\\ & \\omega^2_{\\ \\, 0} = 0 & \\omega^2_{\\ \\, 1} = \\frac{1}{r} \\mathrm{d} {\\theta} & \\omega^2_{\\ \\, 2} = \\frac{1}{r} \\mathrm{d} r & \\omega^2_{\\ \\, 3} = -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\mathrm{d} {\\phi} \\\\ & \\omega^3_{\\ \\, 0} = 0 & \\omega^3_{\\ \\, 1} = \\frac{1}{r} \\mathrm{d} {\\phi} & \\omega^3_{\\ \\, 2} = \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\mathrm{d} {\\phi} & \\omega^3_{\\ \\, 3} = \\frac{1}{r} \\mathrm{d} r + \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\mathrm{d} {\\theta} \\\\ \\end{array}\\right)$" ], "text/plain": [ "(\\omega^i_j)=\\left(\\begin{array}{cccc} & \\omega^0_{\\ \\, 0} = \\left( -\\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} v & \\omega^0_{\\ \\, 1} = 0 & \\omega^0_{\\ \\, 2} = -r \\mathrm{d} {\\theta} & \\omega^0_{\\ \\, 3} = -r \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi} \\\\ & \\omega^1_{\\ \\, 0} = \\left( -\\frac{m^{4} - 3 \\, m^{3} r + 3 \\, m^{2} r^{2} - m r^{3}}{r^{5}} \\right) \\mathrm{d} v + \\left( \\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} r & \\omega^1_{\\ \\, 1} = \\left( \\frac{m^{2} - m r}{r^{3}} \\right) \\mathrm{d} v & \\omega^1_{\\ \\, 2} = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{r} \\right) \\mathrm{d} {\\theta} & \\omega^1_{\\ \\, 3} = -\\frac{{\\left(m^{2} - 2 \\, m r + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\mathrm{d} {\\phi} \\\\ & \\omega^2_{\\ \\, 0} = 0 & \\omega^2_{\\ \\, 1} = \\frac{1}{r} \\mathrm{d} {\\theta} & \\omega^2_{\\ \\, 2} = \\frac{1}{r} \\mathrm{d} r & \\omega^2_{\\ \\, 3} = -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\mathrm{d} {\\phi} \\\\ & \\omega^3_{\\ \\, 0} = 0 & \\omega^3_{\\ \\, 1} = \\frac{1}{r} \\mathrm{d} {\\phi} & \\omega^3_{\\ \\, 2} = \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\mathrm{d} {\\phi} & \\omega^3_{\\ \\, 3} = \\frac{1}{r} \\mathrm{d} r + \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\mathrm{d} {\\theta} \\\\ \\end{array}\\right)" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = r'(\\omega^i_j)=\\left(\\begin{array}{cccc}'\n", "for i in M.irange():\n", " for j in M.irange():\n", " s += r'&' + latex(nabla.connection_form(i, j, EF.frame()).display(EF)) \n", " s += r'\\\\'\n", "s += r'\\end{array}\\right)'\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Near-horizon coordinates" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The **near-horizon coordinates** $(T, R, \\theta, \\phi)$ are related to the Boyer-Lindquist coordinates by\n", "$$T = \\epsilon \\frac{t}{m}, \\quad R = \\frac{r-m}{\\epsilon m}$$\n", "where $\\epsilon$ is a constant parameter.\n", "The event horizon of the extremal Reissner-Nordström black hole is located at $r=m$, which corresponds to $R=0$." ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathcal{M},(T, R, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathcal{M},(T, R, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (T, R, th, ph))" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "NH. = M.chart(r\"T R th:(0,pi):\\theta ph:(0,2*pi):periodic:\\phi\")\n", "NH" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} T & = & \\frac{{\\epsilon} t}{m} \\\\ R & = & -\\frac{m - r}{{\\epsilon} m} \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} T & = & \\frac{{\\epsilon} t}{m} \\\\ R & = & -\\frac{m - r}{{\\epsilon} m} \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "T = eps*t/m\n", "R = -(m - r)/(eps*m)\n", "th = th\n", "ph = ph" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eps = var('eps', latex_name=r'\\epsilon')\n", "BL_to_NH = BL.transition_map(NH, [eps*t/m, (r-m)/(eps*m), th, ph])\n", "BL_to_NH.display()" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} t & = & \\frac{T m}{{\\epsilon}} \\\\ r & = & {\\left(R {\\epsilon} + 1\\right)} m \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} t & = & \\frac{T m}{{\\epsilon}} \\\\ r & = & {\\left(R {\\epsilon} + 1\\right)} m \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "t = T*m/eps\n", "r = (R*eps + 1)*m\n", "th = th\n", "ph = ph" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL_to_NH.inverse().display()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rrrr}\n", "\\frac{{\\epsilon}}{m} & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{{\\epsilon} m} & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rrrr}\n", "\\frac{{\\epsilon}}{m} & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{{\\epsilon} m} & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$" ], "text/plain": [ "[ eps/m 0 0 0]\n", "[ 0 1/(eps*m) 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL_to_NH.jacobian()" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{1}{m^{2}}\$" ], "text/latex": [ "$\\displaystyle \\frac{1}{m^{2}}$" ], "text/plain": [ "m^(-2)" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL_to_NH.jacobian_det()" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( -\\frac{R^{2} m^{2}}{R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1} \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{{\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\left( -\\frac{R^{2} m^{2}}{R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1} \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{{\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = -R^2*m^2/(R^2*eps^2 + 2*R*eps + 1) dT⊗dT + (R^2*eps^2 + 2*R*eps + 1)*m^2/R^2 dR⊗dR + (R^2*eps^2 + 2*R*eps + 1)*m^2 dth⊗dth + (R^2*eps^2 + 2*R*eps + 1)*m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(NH)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "From now on, we set the near-horizon coordinates to be the default ones on $\\mathcal{M}$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "M.set_default_chart(NH)\n", "M.set_default_frame(NH.frame())" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Hence `NH` becomes the default argument of `display()`:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( -\\frac{R^{2} m^{2}}{R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1} \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{{\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\left( -\\frac{R^{2} m^{2}}{R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1} \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{{\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + {\\left(R^{2} {\\epsilon}^{2} + 2 \\, R {\\epsilon} + 1\\right)} m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = -R^2*m^2/(R^2*eps^2 + 2*R*eps + 1) dT⊗dT + (R^2*eps^2 + 2*R*eps + 1)*m^2/R^2 dR⊗dR + (R^2*eps^2 + 2*R*eps + 1)*m^2 dth⊗dth + (R^2*eps^2 + 2*R*eps + 1)*m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## The near-horizon metric $h$ as the limit $\\epsilon\\to 0$ of $g$\n", "\n", "Let us define the **near-horizon metric** as the metric $h$ on $\\mathcal{M}$ that is the limit $\\epsilon\\to 0$ of the Reissner-Nordström metric $g$. The limit is taken by asking for a series expansion of $g$ with respect to $\\epsilon$ up to the 0-th order (i.e. keeping only $\\epsilon^0$ terms). This is achieved via the method `truncate`:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "g0 = g.truncate(eps, 0)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "We declare $h$ and initialize its `NH` components by writing simply `h[:] = g0[:]`, since `NH` is the default chart and `NH.frame()` the default vector frame:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = -R^{2} m^{2} \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = -R^{2} m^{2} \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -R^2*m^2 dT⊗dT + m^2/R^2 dR⊗dR + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h = M.lorentzian_metric('h')\n", "h[:] = g0[:] \n", "h.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Up to a global factor $m^2$, we recognize the **Bertotti-Robinson metric**, i.e. the product metric of \n", "$\\mathrm{AdS}_2\\times\\mathbb{S}^2$, with $(T,R)$ being **Poincaré coordinates** on $\\mathrm{AdS}_2$ and $(\\theta,\\phi)$ the standard coordinates on $\\mathbb{S}^2$ endowed with the standard (round) metric." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The components of $h$ with respect to Boyer-Lindquist coordinates are:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{m^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{m^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -(m^2 - 2*m*r + r^2)/m^2 dt⊗dt + m^2/(m^2 - 2*m*r + r^2) dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.display(BL)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = -\\frac{{\\left(m - r\\right)}^{2}}{m^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = -\\frac{{\\left(m - r\\right)}^{2}}{m^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -(m - r)^2/m^2 dt⊗dt + m^2/(m - r)^2 dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.apply_map(factor, frame=BL.frame(), chart=BL, keep_other_components=True)\n", "h.display(BL)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "while those with respect to Eddington-Finkelstein coordinates are:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{m^{2}} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v + \\frac{r^{2}}{m^{2}} \\mathrm{d} v\\otimes \\mathrm{d} r + \\frac{r^{2}}{m^{2}} \\mathrm{d} r\\otimes \\mathrm{d} v + \\left( \\frac{m^{3} + m^{2} r + m r^{2} + r^{3}}{m^{3} - m^{2} r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = \\left( -\\frac{m^{2} - 2 \\, m r + r^{2}}{m^{2}} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v + \\frac{r^{2}}{m^{2}} \\mathrm{d} v\\otimes \\mathrm{d} r + \\frac{r^{2}}{m^{2}} \\mathrm{d} r\\otimes \\mathrm{d} v + \\left( \\frac{m^{3} + m^{2} r + m r^{2} + r^{3}}{m^{3} - m^{2} r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -(m^2 - 2*m*r + r^2)/m^2 dv⊗dv + r^2/m^2 dv⊗dr + r^2/m^2 dr⊗dv + (m^3 + m^2*r + m*r^2 + r^3)/(m^3 - m^2*r) dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.display(EF)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = -\\frac{{\\left(m - r\\right)}^{2}}{m^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v + \\frac{r^{2}}{m^{2}} \\mathrm{d} v\\otimes \\mathrm{d} r + \\frac{r^{2}}{m^{2}} \\mathrm{d} r\\otimes \\mathrm{d} v + \\frac{{\\left(m^{2} + r^{2}\\right)} {\\left(m + r\\right)}}{{\\left(m - r\\right)} m^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = -\\frac{{\\left(m - r\\right)}^{2}}{m^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v + \\frac{r^{2}}{m^{2}} \\mathrm{d} v\\otimes \\mathrm{d} r + \\frac{r^{2}}{m^{2}} \\mathrm{d} r\\otimes \\mathrm{d} v + \\frac{{\\left(m^{2} + r^{2}\\right)} {\\left(m + r\\right)}}{{\\left(m - r\\right)} m^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -(m - r)^2/m^2 dv⊗dv + r^2/m^2 dv⊗dr + r^2/m^2 dr⊗dv + (m^2 + r^2)*(m + r)/((m - r)*m^2) dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.apply_map(factor, frame=EF.frame(), chart=EF, keep_other_components=True)\n", "h.display(EF)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Note that these components are singular at $r=m$, contrary to those of $g$. " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "### Curvature tensors of $h$" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Ric}\\left(h\\right) = R^{2} \\mathrm{d} T\\otimes \\mathrm{d} T -\\frac{1}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R +\\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Ric}\\left(h\\right) = R^{2} \\mathrm{d} T\\otimes \\mathrm{d} T -\\frac{1}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R +\\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Ric(h) = R^2 dT⊗dT - 1/R^2 dR⊗dR + dth⊗dth + sin(th)^2 dph⊗dph" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.ricci().display()" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\mathrm{r}\\left(h\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(T, R, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\mathrm{r}\\left(h\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(T, R, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$" ], "text/plain": [ "r(h): M → ℝ\n", " (t, r, th, ph) ↦ 0\n", " (v, r, th, ph) ↦ 0\n", " (T, R, th, ph) ↦ 0" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.ricci_scalar().display()" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, T} \\, R \\, T \\, R }^{ \\, T \\phantom{\\, R} \\phantom{\\, T} \\phantom{\\, R} } & = & -\\frac{1}{R^{2}} \\\\ {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, R} \\, T \\, T \\, R }^{ \\, R \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, R} } & = & -R^{2} \\\\ {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\sin\\left({\\theta}\\right)^{2} \\\\ {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -1 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, T} \\, R \\, T \\, R }^{ \\, T \\phantom{\\, R} \\phantom{\\, T} \\phantom{\\, R} } & = & -\\frac{1}{R^{2}} \\\\ {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, R} \\, T \\, T \\, R }^{ \\, R \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, R} } & = & -R^{2} \\\\ {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\sin\\left({\\theta}\\right)^{2} \\\\ {\\mathrm{Riem}\\left(h\\right)}_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -1 \\end{array}$" ], "text/plain": [ "Riem(h)^T_R,T,R = -1/R^2 \n", "Riem(h)^R_T,T,R = -R^2 \n", "Riem(h)^th_ph,th,ph = sin(th)^2 \n", "Riem(h)^ph_th,th,ph = -1 " ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.riemann().display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{C}\\left(h\\right) = 0\$" ], "text/latex": [ "$\\displaystyle \\mathrm{C}\\left(h\\right) = 0$" ], "text/plain": [ "C(h) = 0" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.weyl().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Killing vectors of the near-horizon geometry" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Two obvious Killing vectors of $h$ are $\\eta := \\frac{\\partial}{\\partial\\phi}$\n", "and $\\xi_1 := \\frac{\\partial}{\\partial T}$:" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\eta = \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle \\eta = \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "eta = ∂/∂ph" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eta = M.vector_field(0, 0, 0, 1, name='eta', latex_name=r'\\eta')\n", "eta.display()" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.lie_derivative(eta).display()" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{1} = \\frac{\\partial}{\\partial T }\$" ], "text/latex": [ "$\\displaystyle \\xi_{1} = \\frac{\\partial}{\\partial T }$" ], "text/plain": [ "xi1 = ∂/∂T" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi1 = M.vector_field(1, 0, 0, 0, name='xi1', latex_name=r'\\xi_{1}')\n", "xi1.display()" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.lie_derivative(xi1).display()" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{1} = \\frac{m}{{\\epsilon}} \\frac{\\partial}{\\partial t }\$" ], "text/latex": [ "$\\displaystyle \\xi_{1} = \\frac{m}{{\\epsilon}} \\frac{\\partial}{\\partial t }$" ], "text/plain": [ "xi1 = m/eps ∂/∂t" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi1.display(BL)" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{1} = \\frac{m}{{\\epsilon}} \\frac{\\partial}{\\partial v }\$" ], "text/latex": [ "$\\displaystyle \\xi_{1} = \\frac{m}{{\\epsilon}} \\frac{\\partial}{\\partial v }$" ], "text/plain": [ "xi1 = m/eps ∂/∂v" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi1.display(EF)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The above two Killing vectors correspond respectively to the **axisymmetry** and the **staticity** of the Reissner-Nordström metric. A third symmetry, which is not present in the original Reissner-Nordström metric, is the invariance under the **squeeze mapping** (also called **hyperbolic rotation**) $(T,R)\\mapsto (\\alpha T, R/\\alpha)$ for any $\\alpha>0$, as it is clear on the metric components of $h$ in NH coordinates. The corresponding Killing vector is " ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{2} = T \\frac{\\partial}{\\partial T } -R \\frac{\\partial}{\\partial R }\$" ], "text/latex": [ "$\\displaystyle \\xi_{2} = T \\frac{\\partial}{\\partial T } -R \\frac{\\partial}{\\partial R }$" ], "text/plain": [ "xi2 = T ∂/∂T - R ∂/∂R" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi2 = M.vector_field(T, -R, 0, 0, name='xi2', latex_name=r'\\xi_{2}')\n", "xi2.display()" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.lie_derivative(xi2).display()" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{2} = t \\frac{\\partial}{\\partial t } + \\left( m - r \\right) \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\xi_{2} = t \\frac{\\partial}{\\partial t } + \\left( m - r \\right) \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "xi2 = t ∂/∂t + (m - r) ∂/∂r" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi2.display(BL)" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{2} = \\left( 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left({\\left| -m + r \\right|}\\right) - 2 \\, r + v \\right) \\frac{\\partial}{\\partial v } + \\left( m - r \\right) \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\xi_{2} = \\left( 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left({\\left| -m + r \\right|}\\right) - 2 \\, r + v \\right) \\frac{\\partial}{\\partial v } + \\left( m - r \\right) \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "xi2 = (2*m*log(m) - 2*m*log(abs(-m + r)) - 2*r + v) ∂/∂v + (m - r) ∂/∂r" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi2.display(EF)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "We note that, contrary to those of $\\xi_1$, the components of $\\xi_2$ with respect to Boyer-Lindquist coordinates are independent from $\\epsilon$. Moreover the components of $\\xi_2$ with respect to Eddington-Finkelstein coordinates are singular at $r=m$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Finally, a fourth Killing vector is" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{3} = \\left( \\frac{1}{2} \\, T^{2} + \\frac{1}{2 \\, R^{2}} \\right) \\frac{\\partial}{\\partial T } -R T \\frac{\\partial}{\\partial R }\$" ], "text/latex": [ "$\\displaystyle \\xi_{3} = \\left( \\frac{1}{2} \\, T^{2} + \\frac{1}{2 \\, R^{2}} \\right) \\frac{\\partial}{\\partial T } -R T \\frac{\\partial}{\\partial R }$" ], "text/plain": [ "xi3 = (1/2*T^2 + 1/2/R^2) ∂/∂T - R*T ∂/∂R" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi3 = M.vector_field((T^2 + 1/R^2)/2, -R*T, 0, 0,\n", " name='xi3', latex_name=r'\\xi_{3}')\n", "xi3.display()" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.lie_derivative(xi3).display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "We shall see below that this Killing vector is actually related to the Killing vector $\\frac{\\partial}{\\partial \\tau}$ associated to the so-called *global NH coordinates* by \n", "$$ \\xi_3 = \\frac{\\partial}{\\partial \\tau} - \\frac{1}{2} \\frac{\\partial}{\\partial T}$$" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\xi_{3} = \\left( \\frac{{\\epsilon} m^{4} + {\\left({\\epsilon} m^{2} - 2 \\, {\\epsilon} m r + {\\epsilon} r^{2}\\right)} t^{2}}{2 \\, {\\left(m^{3} - 2 \\, m^{2} r + m r^{2}\\right)}} \\right) \\frac{\\partial}{\\partial t } + \\frac{{\\left({\\epsilon} m - {\\epsilon} r\\right)} t}{m} \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\xi_{3} = \\left( \\frac{{\\epsilon} m^{4} + {\\left({\\epsilon} m^{2} - 2 \\, {\\epsilon} m r + {\\epsilon} r^{2}\\right)} t^{2}}{2 \\, {\\left(m^{3} - 2 \\, m^{2} r + m r^{2}\\right)}} \\right) \\frac{\\partial}{\\partial t } + \\frac{{\\left({\\epsilon} m - {\\epsilon} r\\right)} t}{m} \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "xi3 = 1/2*(eps*m^4 + (eps*m^2 - 2*eps*m*r + eps*r^2)*t^2)/(m^3 - 2*m^2*r + m*r^2) ∂/∂t + (eps*m - eps*r)*t/m ∂/∂r" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi3.display(BL)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\xi_{3} = \\left( \\frac{4 \\, {\\epsilon} m^{3} \\log\\left(m\\right)^{2} + {\\epsilon} m^{3} - 3 \\, {\\epsilon} r^{3} + {\\left(8 \\, {\\epsilon} m \\log\\left(m\\right) + 5 \\, {\\epsilon} m\\right)} r^{2} + {\\left({\\epsilon} m - {\\epsilon} r\\right)} v^{2} + 4 \\, {\\left({\\epsilon} m^{3} - {\\epsilon} m^{2} r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)^{2} - {\\left(4 \\, {\\epsilon} m^{2} \\log\\left(m\\right)^{2} + 8 \\, {\\epsilon} m^{2} \\log\\left(m\\right) - {\\epsilon} m^{2}\\right)} r + 4 \\, {\\left({\\epsilon} m^{2} \\log\\left(m\\right) + {\\epsilon} r^{2} - {\\left({\\epsilon} m \\log\\left(m\\right) + {\\epsilon} m\\right)} r - {\\left({\\epsilon} m^{2} - {\\epsilon} m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)\\right)} v - 8 \\, {\\left({\\epsilon} m^{3} \\log\\left(m\\right) + {\\epsilon} m r^{2} - {\\left({\\epsilon} m^{2} \\log\\left(m\\right) + {\\epsilon} m^{2}\\right)} r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{2 \\, {\\left(m^{2} - m r\\right)}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{2 \\, {\\epsilon} m^{2} \\log\\left(m\\right) + {\\epsilon} r^{2} - 2 \\, {\\left({\\epsilon} m \\log\\left(m\\right) + {\\epsilon} m\\right)} r + {\\left({\\epsilon} m - {\\epsilon} r\\right)} v - 2 \\, {\\left({\\epsilon} m^{2} - {\\epsilon} m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{m} \\right) \\frac{\\partial}{\\partial r }\\)" ], "text/latex": [ "$\\displaystyle \\xi_{3} = \\left( \\frac{4 \\, {\\epsilon} m^{3} \\log\\left(m\\right)^{2} + {\\epsilon} m^{3} - 3 \\, {\\epsilon} r^{3} + {\\left(8 \\, {\\epsilon} m \\log\\left(m\\right) + 5 \\, {\\epsilon} m\\right)} r^{2} + {\\left({\\epsilon} m - {\\epsilon} r\\right)} v^{2} + 4 \\, {\\left({\\epsilon} m^{3} - {\\epsilon} m^{2} r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)^{2} - {\\left(4 \\, {\\epsilon} m^{2} \\log\\left(m\\right)^{2} + 8 \\, {\\epsilon} m^{2} \\log\\left(m\\right) - {\\epsilon} m^{2}\\right)} r + 4 \\, {\\left({\\epsilon} m^{2} \\log\\left(m\\right) + {\\epsilon} r^{2} - {\\left({\\epsilon} m \\log\\left(m\\right) + {\\epsilon} m\\right)} r - {\\left({\\epsilon} m^{2} - {\\epsilon} m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)\\right)} v - 8 \\, {\\left({\\epsilon} m^{3} \\log\\left(m\\right) + {\\epsilon} m r^{2} - {\\left({\\epsilon} m^{2} \\log\\left(m\\right) + {\\epsilon} m^{2}\\right)} r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{2 \\, {\\left(m^{2} - m r\\right)}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{2 \\, {\\epsilon} m^{2} \\log\\left(m\\right) + {\\epsilon} r^{2} - 2 \\, {\\left({\\epsilon} m \\log\\left(m\\right) + {\\epsilon} m\\right)} r + {\\left({\\epsilon} m - {\\epsilon} r\\right)} v - 2 \\, {\\left({\\epsilon} m^{2} - {\\epsilon} m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{m} \\right) \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "xi3 = 1/2*(4*eps*m^3*log(m)^2 + eps*m^3 - 3*eps*r^3 + (8*eps*m*log(m) + 5*eps*m)*r^2 + (eps*m - eps*r)*v^2 + 4*(eps*m^3 - eps*m^2*r)*log(abs(-m + r))^2 - (4*eps*m^2*log(m)^2 + 8*eps*m^2*log(m) - eps*m^2)*r + 4*(eps*m^2*log(m) + eps*r^2 - (eps*m*log(m) + eps*m)*r - (eps*m^2 - eps*m*r)*log(abs(-m + r)))*v - 8*(eps*m^3*log(m) + eps*m*r^2 - (eps*m^2*log(m) + eps*m^2)*r)*log(abs(-m + r)))/(m^2 - m*r) ∂/∂v + (2*eps*m^2*log(m) + eps*r^2 - 2*(eps*m*log(m) + eps*m)*r + (eps*m - eps*r)*v - 2*(eps*m^2 - eps*m*r)*log(abs(-m + r)))/m ∂/∂r" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi3.display(EF)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "### Symmetry group\n", "\n", "Since $h$ is the product metric of $\\mathrm{AdS}_2\\times\\mathbb{S}^2$,\n", "the group of isometries w.r.t. $h$ is the 6-dimensional group \n", "$$ G = \\mathrm{SL}(2,\\mathbb{R})\\times \\mathrm{SO}(3).$$\n", "Let us check explicitely that the Killing vectors $\\xi_1$, $\\xi_2$ and $\\xi_3$ generate $\\mathrm{SL}(2, \\mathbb{R})$. We have" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\xi_{1},\\xi_{2}\\right] = \\frac{\\partial}{\\partial T }\$" ], "text/latex": [ "$\\displaystyle \\left[\\xi_{1},\\xi_{2}\\right] = \\frac{\\partial}{\\partial T }$" ], "text/plain": [ "[xi1,xi2] = ∂/∂T" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi1.bracket(xi2).display()" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\xi_{1},\\xi_{3}\\right] = T \\frac{\\partial}{\\partial T } -R \\frac{\\partial}{\\partial R }\$" ], "text/latex": [ "$\\displaystyle \\left[\\xi_{1},\\xi_{3}\\right] = T \\frac{\\partial}{\\partial T } -R \\frac{\\partial}{\\partial R }$" ], "text/plain": [ "[xi1,xi3] = T ∂/∂T - R ∂/∂R" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi1.bracket(xi3).display()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\xi_{2},\\xi_{3}\\right] = \\left( \\frac{R^{2} T^{2} + 1}{2 \\, R^{2}} \\right) \\frac{\\partial}{\\partial T } -R T \\frac{\\partial}{\\partial R }\$" ], "text/latex": [ "$\\displaystyle \\left[\\xi_{2},\\xi_{3}\\right] = \\left( \\frac{R^{2} T^{2} + 1}{2 \\, R^{2}} \\right) \\frac{\\partial}{\\partial T } -R T \\frac{\\partial}{\\partial R }$" ], "text/plain": [ "[xi2,xi3] = 1/2*(R^2*T^2 + 1)/R^2 ∂/∂T - R*T ∂/∂R" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi2.bracket(xi3).display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Hence" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "all([xi1.bracket(xi2) == xi1,\n", " xi1.bracket(xi3) == xi2,\n", " xi2.bracket(xi3) == xi3])" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "To recognize a the Lie algebra of $\\mathrm{SL}(2,\\mathbb{R})$ , let us perform a slight change of basis:" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "vE = -sqrt(2)*xi3\n", "vF = sqrt(2)*xi1\n", "vH = 2*xi2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "This yields" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "all([vE.bracket(vF) == vH,\n", " vH.bracket(vE) == 2*vE,\n", " vH.bracket(vF) == -2*vF])" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "These are exactly the commutation relations of the standard basis of $\\mathfrak{sl}(2, \\mathbb{R})$. \n", "Indeed, we have, using the representation of $\\mathfrak{sl}(2, \\mathbb{R})$ by traceless $2\\times 2$ matrices:" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\left(\\begin{array}{rr}\n", "0 & 1 \\\\\n", "0 & 0\n", "\\end{array}\\right), \\left(\\begin{array}{rr}\n", "0 & 0 \\\\\n", "1 & 0\n", "\\end{array}\\right), \\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & -1\n", "\\end{array}\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\left(\\begin{array}{rr}\n", "0 & 1 \\\\\n", "0 & 0\n", "\\end{array}\\right), \\left(\\begin{array}{rr}\n", "0 & 0 \\\\\n", "1 & 0\n", "\\end{array}\\right), \\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & -1\n", "\\end{array}\\right)\\right)$" ], "text/plain": [ "(\n", "[0 1] [0 0] [ 1 0]\n", "[0 0], [1 0], [ 0 -1]\n", ")" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sl2 = lie_algebras.sl(QQ, 2, representation='matrix') # QQ instead of RR to deal with an exact field\n", "EE,FF,HH = sl2.gens()\n", "EE,FF,HH" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "all([EE.bracket(FF) == HH,\n", " HH.bracket(EE) == 2*EE,\n", " HH.bracket(FF) == -2*FF])" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Global near-horizon coordinates" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Let us introduce the (compactified) **global near-horizon coordinates** $(\\tau, \\chi, \\theta,\\phi)$, which are related to the NH coordinates $(T,R,\\theta,\\phi)$ exactly as the *conformal coordinates* $(\\tau,\\chi)$ are related to the *Poincaré coordinates* $(T,R)$ in $\\mathrm{AdS}_2$:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\mathcal{M},({\\tau}, {\\chi}, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\mathcal{M},({\\tau}, {\\chi}, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (ta, ch, th, ph))" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "GNH. = M.chart(r\"ta:\\tau ch:(-pi/2,pi/2):\\chi th:(0,pi):\\theta ph:(0,2*pi):periodic:\\phi\")\n", "GNH" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\tau} :\\ \\left( -\\infty, +\\infty \\right) ;\\quad {\\chi} :\\ \\left( -\\frac{1}{2} \\, \\pi , \\frac{1}{2} \\, \\pi \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\text{(periodic)}\$" ], "text/latex": [ "$\\displaystyle {\\tau} :\\ \\left( -\\infty, +\\infty \\right) ;\\quad {\\chi} :\\ \\left( -\\frac{1}{2} \\, \\pi , \\frac{1}{2} \\, \\pi \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\text{(periodic)}$" ], "text/plain": [ "ta: (-oo, +oo); ch: (-1/2*pi, 1/2*pi); th: (0, pi); ph: [0, 2*pi] (periodic)" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "GNH.coord_range()" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} T & = & \\frac{\\sin\\left({\\tau}\\right)}{\\cos\\left({\\tau}\\right) + \\sin\\left({\\chi}\\right)} \\\\ R & = & \\frac{\\cos\\left({\\tau}\\right) + \\sin\\left({\\chi}\\right)}{\\cos\\left({\\chi}\\right)} \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} T & = & \\frac{\\sin\\left({\\tau}\\right)}{\\cos\\left({\\tau}\\right) + \\sin\\left({\\chi}\\right)} \\\\ R & = & \\frac{\\cos\\left({\\tau}\\right) + \\sin\\left({\\chi}\\right)}{\\cos\\left({\\chi}\\right)} \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "T = sin(ta)/(cos(ta) + sin(ch))\n", "R = (cos(ta) + sin(ch))/cos(ch)\n", "th = th\n", "ph = ph" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "GNH_to_NH = GNH.transition_map(NH, [sin(ta)/(cos(ta) + sin(ch)),\n", " (cos(ta) + sin(ch))/cos(ch), th, ph])\n", "GNH_to_NH.display()" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " ta == pi*unit_step(-(cos(ta) + sin(ch))/cos(ch)) + arctan2(2*(cos(ta) + sin(ch))*sin(ta)/cos(ch)^2, 2*(cos(ta)^2 + cos(ta)*sin(ch))/cos(ch)^2) **failed**\n", " ch == arctan(sin(ch)/cos(ch)) **failed**\n", " th == th *passed*\n", " ph == ph *passed*\n", " T == (R^3*T^2*sin(pi*unit_step(-R)) - 2*R^3*T*cos(pi*unit_step(-R)) - R^3*sin(pi*unit_step(-R)) - R*sin(pi*unit_step(-R)))/(2*R^3*T*sin(pi*unit_step(-R)) - R^3*cos(pi*unit_step(-R)) + (R^3*cos(pi*unit_step(-R)) - R^2*abs(R))*T^2 - (R^2 - 1)*abs(R) - R*cos(pi*unit_step(-R))) **failed**\n", " R == -1/2*(2*R^3*T*sin(pi*unit_step(-R)) - R^3*cos(pi*unit_step(-R)) + (R^3*cos(pi*unit_step(-R)) - R^2*abs(R))*T^2 - (R^2 - 1)*abs(R) - R*cos(pi*unit_step(-R)))/(R*abs(R)) **failed**\n", " th == th *passed*\n", " ph == ph *passed*\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] } ], "source": [ "GNH_to_NH.set_inverse(atan2(2*R^2*T, R^2*(1 - T^2) +1) + pi*unit_step(-R),\n", " atan((R^2*(1 + T^2) - 1)/(2*R)), th, ph)" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} {\\tau} & = & \\pi \\mathrm{u}\\left(-R\\right) + \\arctan\\left(2 \\, R^{2} T, -{\\left(T^{2} - 1\\right)} R^{2} + 1\\right) \\\\ {\\chi} & = & \\arctan\\left(\\frac{{\\left(T^{2} + 1\\right)} R^{2} - 1}{2 \\, R}\\right) \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} {\\tau} & = & \\pi \\mathrm{u}\\left(-R\\right) + \\arctan\\left(2 \\, R^{2} T, -{\\left(T^{2} - 1\\right)} R^{2} + 1\\right) \\\\ {\\chi} & = & \\arctan\\left(\\frac{{\\left(T^{2} + 1\\right)} R^{2} - 1}{2 \\, R}\\right) \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "ta = pi*unit_step(-R) + arctan2(2*R^2*T, -(T^2 - 1)*R^2 + 1)\n", "ch = arctan(1/2*((T^2 + 1)*R^2 - 1)/R)\n", "th = th\n", "ph = ph" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "GNH_to_NH.inverse().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Let us draw the chart $(T,R)$ in terms of the chart $(\\tau,\\chi)$ with the curves $R=\\mathrm{const}$ in red and the\n", "curves $T = \\mathrm{const}$ in grey:" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 36 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_NH = NH.plot(chart=GNH, ambient_coords=(ch, ta), fixed_coords={th: pi/2, ph: 0},\n", " ranges={T: (-9, 9), R: (-9, 8)}, color={T: 'red', R: 'grey'},\n", " number_values=17, plot_points=200) \\\n", " + line([(-pi/2, 0), (pi/2, pi)], color='black', thickness=3) \\\n", " + text(r\"$\\mathscr{H}$\", (1.3, 2.5), color='black', fontsize=20)\n", "show(graph_NH, aspect_ratio=1, figsize=10) " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The expression of $h$ in terms of the global conformal coordinates is " ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = -\\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{m^{2} \\cos\\left({\\tau}\\right)^{2} \\sin\\left({\\chi}\\right)^{2} - m^{2} \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\tau}\\right)^{2} + 2 \\, m^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + m^{2}}{\\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right)^{2} + 2 \\, \\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\chi}\\right)^{2}} \\right) \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = -\\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{m^{2} \\cos\\left({\\tau}\\right)^{2} \\sin\\left({\\chi}\\right)^{2} - m^{2} \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\tau}\\right)^{2} + 2 \\, m^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + m^{2}}{\\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right)^{2} + 2 \\, \\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\chi}\\right)^{2}} \\right) \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -m^2/cos(ch)^2 dta⊗dta + (m^2*cos(ta)^2*sin(ch)^2 - m^2*cos(ch)^2*sin(ta)^2 + 2*m^2*cos(ta)*sin(ch) + m^2)/(cos(ch)^2*cos(ta)^2 + 2*cos(ch)^2*cos(ta)*sin(ch) + cos(ch)^2*sin(ch)^2) dch⊗dch + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.display(GNH)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The component $h_{\\chi\\chi}$ looks complicated, but that's only because it suffers from a lack of simplification. \n", "We may simplify it by enforcing the identities $\\sin^2\\chi = 1 - \\cos^2\\chi$ and $\\sin^2\\tau = 1 - \\cos^2\\tau$: " ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{m^{2} \\cos\\left({\\tau}\\right)^{2} \\sin\\left({\\chi}\\right)^{2} - m^{2} \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\tau}\\right)^{2} + 2 \\, m^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + m^{2}}{\\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right)^{2} + 2 \\, \\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\chi}\\right)^{2}}\$" ], "text/latex": [ "$\\displaystyle \\frac{m^{2} \\cos\\left({\\tau}\\right)^{2} \\sin\\left({\\chi}\\right)^{2} - m^{2} \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\tau}\\right)^{2} + 2 \\, m^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + m^{2}}{\\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right)^{2} + 2 \\, \\cos\\left({\\chi}\\right)^{2} \\cos\\left({\\tau}\\right) \\sin\\left({\\chi}\\right) + \\cos\\left({\\chi}\\right)^{2} \\sin\\left({\\chi}\\right)^{2}}$" ], "text/plain": [ "(m^2*cos(ta)^2*sin(ch)^2 - m^2*cos(ch)^2*sin(ta)^2 + 2*m^2*cos(ta)*sin(ch) + m^2)/(cos(ch)^2*cos(ta)^2 + 2*cos(ch)^2*cos(ta)*sin(ch) + cos(ch)^2*sin(ch)^2)" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h11 = h[GNH.frame(), 1, 1, GNH].expr()\n", "h11" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}}\$" ], "text/latex": [ "$\\displaystyle \\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}}$" ], "text/plain": [ "m^2/cos(ch)^2" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h11s = h11.subs({sin(ch)^2: 1 - cos(ch)^2, sin(ta)^2: 1 - cos(ta)^2}).simplify_full()\n", "h11s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "An alternative method is to rely on SymPy for trigonometric simplifications:" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}}\$" ], "text/latex": [ "$\\displaystyle \\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}}$" ], "text/plain": [ "m^2/cos(ch)^2" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h11s = h11._sympy_().trigsimp()._sage_()\n", "h11s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "We therefore set" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "h.add_comp(GNH.frame())[1, 1, GNH] = h11s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "and get a far nicer form:" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = -\\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}} \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = -\\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\frac{m^{2}}{\\cos\\left({\\chi}\\right)^{2}} \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -m^2/cos(ch)^2 dta⊗dta + m^2/cos(ch)^2 dch⊗dch + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.display(GNH)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The components of $h$ are singular only at the $\\mathrm{AdS}_2$ boundaries $\\chi = \\pm\\pi/2$, which are not part of the spacetime $\\mathcal{M}$. In particular, they are regular on the event horizon $R=0 \\iff \\tau = \\chi + \\frac{\\pi}{2}$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Couch-Torrence inversion $\\Phi$\n", "\n", "The Couch-Torrence inversion is the map" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(t, m - \\frac{m^{2}}{m - r}, {\\theta}, {\\phi}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(t, m - \\frac{m^{2}}{m - r}, {\\theta}, {\\phi}\\right) \\end{array}$" ], "text/plain": [ "Phi: M → M\n", " (t, r, th, ph) ↦ (t, m - m^2/(m - r), th, ph)" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi = M.diffeomorphism(M, {(BL, BL): (t, m^2/(r - m) + m, th, ph)}, \n", " name='Phi', latex_name=r'\\Phi')\n", "Phi.display(BL, BL)" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle r \\ {\\mapsto}\\ m - \\frac{m^{2}}{m - r}\$" ], "text/latex": [ "$\\displaystyle r \\ {\\mapsto}\\ m - \\frac{m^{2}}{m - r}$" ], "text/plain": [ "r |--> m - m^2/(m - r)" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi_r(r) = Phi.expr(BL, BL)[1]\n", "Phi_r" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The Couch-Torrence inversion takes a simple form in terms of the tortoise coordinate $r_*$:" ] }, { "cell_type": "code", "execution_count": 83, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(rstar(Phi_r(r)) + rstar(r)).simplify_log()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Hence we conclude that in terms of $r_*$, $\\Phi$ is simply $r_* \\mapsto - r_*$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The inverse of $\\Phi$ is " ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi^{-1}:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(t, -\\frac{m r}{m - r}, {\\theta}, {\\phi}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi^{-1}:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(t, -\\frac{m r}{m - r}, {\\theta}, {\\phi}\\right) \\end{array}$" ], "text/plain": [ "Phi^(-1): M → M\n", " (t, r, th, ph) ↦ (t, -m*r/(m - r), th, ph)" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.inverse().display(BL, BL)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "$\\Phi$ is an involution:" ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.inverse() == Phi" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The expression of $\\Phi$ in terms of Eddington-Finkelstein coordinates:" ] }, { "cell_type": "code", "execution_count": 86, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(\\frac{4 \\, m^{2} \\log\\left(m\\right) - 4 \\, {\\left(m \\log\\left(m\\right) + m\\right)} r + 2 \\, r^{2} + {\\left(m - r\\right)} v - 4 \\, {\\left(m^{2} - m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{m - r}, -\\frac{m r}{m - r}, {\\theta}, {\\phi}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(v, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(\\frac{4 \\, m^{2} \\log\\left(m\\right) - 4 \\, {\\left(m \\log\\left(m\\right) + m\\right)} r + 2 \\, r^{2} + {\\left(m - r\\right)} v - 4 \\, {\\left(m^{2} - m r\\right)} \\log\\left({\\left| -m + r \\right|}\\right)}{m - r}, -\\frac{m r}{m - r}, {\\theta}, {\\phi}\\right) \\end{array}$" ], "text/plain": [ "Phi: M → M\n", " (v, r, th, ph) ↦ ((4*m^2*log(m) - 4*(m*log(m) + m)*r + 2*r^2 + (m - r)*v - 4*(m^2 - m*r)*log(abs(-m + r)))/(m - r), -m*r/(m - r), th, ph)" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(EF, EF)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "The expression of $\\Phi$ in terms of near-horizon coordinates is particular simple and, up to a factor $\\epsilon^2$, amounts to an inversion in $R$:" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(T, R, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(T, \\frac{1}{R {\\epsilon}^{2}}, {\\theta}, {\\phi}\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathcal{M} \\\\ & \\left(T, R, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(T, \\frac{1}{R {\\epsilon}^{2}}, {\\theta}, {\\phi}\\right) \\end{array}$" ], "text/plain": [ "Phi: M → M\n", " (T, R, th, ph) ↦ (T, 1/(R*eps^2), th, ph)" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.display(NH, NH)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "### The Couch-Torrence inversion as a conformal isometry of $g$\n", "\n", "The pullback of $g$ by $\\Phi$ is" ] }, { "cell_type": "code", "execution_count": 88, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}^*g = -\\frac{m^{2}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{2} r^{2}}{m^{4} - 4 \\, m^{3} r + 6 \\, m^{2} r^{2} - 4 \\, m r^{3} + r^{4}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{m^{2} r^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{m^{2} r^{2} \\sin\\left({\\theta}\\right)^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle {\\Phi}^*g = -\\frac{m^{2}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{2} r^{2}}{m^{4} - 4 \\, m^{3} r + 6 \\, m^{2} r^{2} - 4 \\, m r^{3} + r^{4}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{m^{2} r^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{m^{2} r^{2} \\sin\\left({\\theta}\\right)^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Phi^*(g) = -m^2/r^2 dt⊗dt + m^2*r^2/(m^4 - 4*m^3*r + 6*m^2*r^2 - 4*m*r^3 + r^4) dr⊗dr + m^2*r^2/(m^2 - 2*m*r + r^2) dth⊗dth + m^2*r^2*sin(th)^2/(m^2 - 2*m*r + r^2) dph⊗dph" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pg = Phi.pullback(g)\n", "Pg.display(BL)" ] }, { "cell_type": "code", "execution_count": 89, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}^*g = -\\frac{m^{2}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2} r^{2}}{{\\left(m - r\\right)}^{4}} \\mathrm{d} r\\otimes \\mathrm{d} r + \\frac{m^{2} r^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{m^{2} r^{2} \\sin\\left({\\theta}\\right)^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle {\\Phi}^*g = -\\frac{m^{2}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2} r^{2}}{{\\left(m - r\\right)}^{4}} \\mathrm{d} r\\otimes \\mathrm{d} r + \\frac{m^{2} r^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{m^{2} r^{2} \\sin\\left({\\theta}\\right)^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Phi^*(g) = -m^2/r^2 dt⊗dt + m^2*r^2/(m - r)^4 dr⊗dr + m^2*r^2/(m - r)^2 dth⊗dth + m^2*r^2*sin(th)^2/(m - r)^2 dph⊗dph" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pg.apply_map(factor, frame=BL.frame(), chart=BL, keep_other_components=True)\n", "Pg.display(BL)" ] }, { "cell_type": "code", "execution_count": 90, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pg == m^2/(r - m)^2 * g" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "Hence we conclude that\n", "$$\\Phi^* g = \\frac{m^2}{(r - m)^2} g$$\n", "In other words, $\\Phi^*$ is a conformal isometry of $g$, with conformal factor $\\Omega = \\frac{m}{|r - m|}$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "### Action of $\\Phi$ on $h$:" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "Ph = Phi.pullback(h)" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}^*h = -\\frac{m^{2}}{R^{2} {\\epsilon}^{4}} \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle {\\Phi}^*h = -\\frac{m^{2}}{R^{2} {\\epsilon}^{4}} \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{m^{2}}{R^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Phi^*(h) = -m^2/(R^2*eps^4) dT⊗dT + m^2/R^2 dR⊗dR + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ph.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "As for $h$, the expression of $\\Phi^* h$ in Boyer-Lindquist coordinates is independent of $\\epsilon$:" ] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}^*h = \\left( -\\frac{m^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle {\\Phi}^*h = \\left( -\\frac{m^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{2}}{m^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Phi^*(h) = -m^2/(m^2 - 2*m*r + r^2) dt⊗dt + m^2/(m^2 - 2*m*r + r^2) dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ph.display(BL)" ] }, { "cell_type": "code", "execution_count": 94, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false }, "scrolled": true }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}^*h = -\\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle {\\Phi}^*h = -\\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "Phi^*(h) = -m^2/(m - r)^2 dt⊗dt + m^2/(m - r)^2 dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ph.apply_map(factor, frame=BL.frame(), chart=BL, keep_other_components=True)\n", "Ph.display(BL)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "For reference:" ] }, { "cell_type": "code", "execution_count": 95, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle h = -\\frac{{\\left(m - r\\right)}^{2}}{m^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle h = -\\frac{{\\left(m - r\\right)}^{2}}{m^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{m^{2}}{{\\left(m - r\\right)}^{2}} \\mathrm{d} r\\otimes \\mathrm{d} r + m^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + m^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "h = -(m - r)^2/m^2 dt⊗dt + m^2/(m - r)^2 dr⊗dr + m^2 dth⊗dth + m^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 95, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.display(BL)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "## Push-forwards of the SL(2,R) Killing vectors" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "source": [ "A generic point of $\\mathcal{M}$:" ] }, { "cell_type": "code", "execution_count": 96, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "p = M((T, R, th, ph), chart=NH, name='p')\n", "print(p)" ] }, { "cell_type": "code", "execution_count": 97, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(T, R, {\\theta}, {\\phi}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(T, R, {\\theta}, {\\phi}\\right)$" ], "text/plain": [ "(T, R, th, ph)" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "NH(p)" ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}_*\\xi_{1} = \\frac{\\partial}{\\partial T }\$" ], "text/latex": [ "$\\displaystyle {\\Phi}_*\\xi_{1} = \\frac{\\partial}{\\partial T }$" ], "text/plain": [ "Phi_*(xi1) = ∂/∂T" ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pxi1 = Phi.pushforward(xi1)\n", "Pxi1.at(Phi(p)).display()" ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}_*\\xi_{2} = T \\frac{\\partial}{\\partial T } + R \\frac{\\partial}{\\partial R }\$" ], "text/latex": [ "$\\displaystyle {\\Phi}_*\\xi_{2} = T \\frac{\\partial}{\\partial T } + R \\frac{\\partial}{\\partial R }$" ], "text/plain": [ "Phi_*(xi2) = T ∂/∂T + R ∂/∂R" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pxi2 = Phi.pushforward(xi2)\n", "Pxi2.at(Phi(p)).display()" ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\Phi}_*\\xi_{3} = \\left( \\frac{1}{2} \\, R^{2} {\\epsilon}^{4} + \\frac{1}{2} \\, T^{2} \\right) \\frac{\\partial}{\\partial T } + R T \\frac{\\partial}{\\partial R }\$" ], "text/latex": [ "$\\displaystyle {\\Phi}_*\\xi_{3} = \\left( \\frac{1}{2} \\, R^{2} {\\epsilon}^{4} + \\frac{1}{2} \\, T^{2} \\right) \\frac{\\partial}{\\partial T } + R T \\frac{\\partial}{\\partial R }$" ], "text/plain": [ "Phi_*(xi3) = (1/2*R^2*eps^4 + 1/2*T^2) ∂/∂T + R*T ∂/∂R" ] }, "execution_count": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pxi3 = Phi.pushforward(xi3)\n", "Pxi3.at(Phi(p)).display()" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 10.8.beta9", "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": "3.12.3" } }, "nbformat": 4, "nbformat_minor": 4 }