sage/Kerr_Carter_frame.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Carter frame in Kerr spacetime\n",
    "\n",
    "This Jupyter/SageMath notebook is relative to the lectures\n",
    "[Geometry and physics of black holes](https://relativite.obspm.fr/blackholes/).\n",
    "\n",
    "The corresponding tools have been developed within the  [SageManifolds](https://sagemanifolds.obspm.fr) project."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*NB:* a version of [SageMath](https://www.sagemath.org/) at least equal to 9.1 is required to run this notebook:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 9.2.beta14, Release Date: 2020-09-30'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we set up the notebook to display mathematical objects using LaTeX rendering:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%display latex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We ask for running tensor computations in parallel on 8 threads:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "Parallelism().set(nproc=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Spacetime manifold\n",
    "\n",
    "We declare the Kerr spacetime (or more precisely the part of it covered by Boyer-Lindquist coordinates) as a 4-dimensional Lorentzian manifold $\\mathcal{M}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4-dimensional Lorentzian manifold M\n"
     ]
    }
   ],
   "source": [
    "M = Manifold(4, 'M', latex_name=r'\\mathcal{M}', structure='Lorentzian')\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then introduce the standard <strong>Boyer-Lindquist coordinates</strong> as a chart `BL` (for *Boyer-Lindquist*) on $\\mathcal{M}$, via the method `chart()`, the argument of which is a string\n",
    "(delimited by `r\"...\"` because of the backslash symbols) expressing the coordinates names, their ranges (the default is $(-\\infty,+\\infty)$) and their LaTeX symbols:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Chart (M, (t, r, th, ph))\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\theta}, {\\phi})\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "Chart (M, (t, r, th, ph))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BL.<t,r,th,ph> = M.chart(r\"t r th:(0,pi):\\theta ph:(0,2*pi):\\phi\") \n",
    "print(BL); BL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "(t, r)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BL[0], BL[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Metric tensor</h2>\n",
    "\n",
    "<p>The 2 parameters $m$ and $a$ of the Kerr spacetime are declared as symbolic variables:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "(m, a)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('m, a', domain='real')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get the (yet undefined) spacetime metric:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = M.metric()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>The metric is set by its components in the coordinate frame associated with Boyer-Lindquist coordinates, which is the current manifold's default frame:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt*dt - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt*dph + (a^2*cos(th)^2 + r^2)/(a^2 - 2*m*r + r^2) dr*dr + (a^2*cos(th)^2 + r^2) dth*dth - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph*dt + (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph*dph"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho2 = r^2 + (a*cos(th))^2\n",
    "Delta = r^2 -2*m*r + a^2\n",
    "g[0,0] = -(1-2*m*r/rho2)\n",
    "g[0,3] = -2*a*m*r*sin(th)^2/rho2\n",
    "g[1,1], g[2,2] = rho2/Delta, rho2\n",
    "g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*sin(th)^2\n",
    "g.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>A matrix view of the components with respect to the manifold's default vector frame:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 & 0 & 0 & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\\n",
       "0 & \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} & 0 & 0 \\\\\n",
       "0 & 0 & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} & 0 \\\\\n",
       "-\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & 0 & {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "\\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 & 0 & 0 & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\\n",
       "0 & \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} & 0 & 0 \\\\\n",
       "0 & 0 & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} & 0 \\\\\n",
       "-\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} & 0 & 0 & {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}\n",
       "\\end{array}\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "[                                  2*m*r/(a^2*cos(th)^2 + r^2) - 1                                                                 0                                                                 0                          -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2)]\n",
       "[                                                                0                         (a^2*cos(th)^2 + r^2)/(a^2 - 2*m*r + r^2)                                                                 0                                                                 0]\n",
       "[                                                                0                                                                 0                                               a^2*cos(th)^2 + r^2                                                                 0]\n",
       "[                         -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2)                                                                 0                                                                 0 (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>The list of the non-vanishing components:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\\\ g_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\\\ g_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\end{array}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, m r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} - 1 \\\\ g_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\\\ g_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & {\\left(\\frac{2 \\, a^{2} m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} + a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} \\end{array}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "g_t,t = 2*m*r/(a^2*cos(th)^2 + r^2) - 1 \n",
       "g_t,ph = -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n",
       "g_r,r = (a^2*cos(th)^2 + r^2)/(a^2 - 2*m*r + r^2) \n",
       "g_th,th = a^2*cos(th)^2 + r^2 \n",
       "g_ph,t = -2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) \n",
       "g_ph,ph = (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.display_comp()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Carter frame\n",
    "The default vector frame on the spacetime manifold is the coordinate basis associated with Boyer-Lindquist coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.default_frame() is BL.frame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "Coordinate frame (M, (d/dt,d/dr,d/dth,d/dph))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BL.frame()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 4-velocity of the Carter obsever is defined by its components w.r.t. the Boyer-Lindquist frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_0 = \\left( \\frac{a^{2} + r^{2}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{a}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_0 = \\left( \\frac{a^{2} + r^{2}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{a}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps0 = (a^2 + r^2)/(sqrt(a^2*cos(th)^2 + r^2)*sqrt(a^2 - 2*m*r + r^2)) d/dt + a/(sqrt(a^2*cos(th)^2 + r^2)*sqrt(a^2 - 2*m*r + r^2)) d/dph"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho = sqrt(rho2)\n",
    "e0 = M.vector_field([(r^2 + a^2)/(rho*sqrt(Delta)), 0, 0, a/(rho*sqrt(Delta))],\n",
    "                    name='eps0', latex_name=r'\\varepsilon_0')\n",
    "e0.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We check that is a unit timelike vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(e0, e0).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and that it is future-directed, by computing its scalar product with the global null vector $k$, which generates the ingoing principal null geodesics:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\left( \\frac{a^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r } + \\left( \\frac{a}{a^{2} - 2 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\left( \\frac{a^{2} + r^{2}}{a^{2} - 2 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r } + \\left( \\frac{a}{a^{2} - 2 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "k = (a^2 + r^2)/(a^2 - 2*m*r + r^2) d/dt - d/dr + a/(a^2 - 2*m*r + r^2) d/dph"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = M.vector_field([(r^2 + a^2)/Delta, -1, 0, a/Delta],\n",
    "                   name='k')\n",
    "k.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}}{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}}{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-sqrt(a^2*cos(th)^2 + r^2)/sqrt(a^2 - 2*m*r + r^2)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(k, e0).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The spacelike vectors of the Carter frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_1 = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_1 = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps1 = sqrt(a^2 - 2*m*r + r^2)/sqrt(a^2*cos(th)^2 + r^2) d/dr"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1 = M.vector_field([0, sqrt(Delta)/rho, 0, 0],\n",
    "                    name='eps1', latex_name=r'\\varepsilon_1')\n",
    "e1.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_2 = \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_2 = \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\theta} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps2 = 1/sqrt(a^2*cos(th)^2 + r^2) d/dth"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2 = M.vector_field([0, 0, 1/rho, 0],\n",
    "                    name='eps2', latex_name=r'\\varepsilon_2')\n",
    "e2.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_3 = \\left( \\frac{a \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sin\\left({\\theta}\\right)} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_3 = \\left( \\frac{a \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sin\\left({\\theta}\\right)} \\right) \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps3 = a*sin(th)/sqrt(a^2*cos(th)^2 + r^2) d/dt + 1/(sqrt(a^2*cos(th)^2 + r^2)*sin(th)) d/dph"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3 = M.vector_field([a*sin(th)/rho, 0, 0, 1/(rho*sin(th))],\n",
    "                    name='eps3', latex_name=r'\\varepsilon_3')\n",
    "e3.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\varepsilon_{0},\\varepsilon_{1},\\varepsilon_{2},\\varepsilon_{3}\\right)\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\varepsilon_{0},\\varepsilon_{1},\\varepsilon_{2},\\varepsilon_{3}\\right)\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "Vector frame (M, (eps_0,eps_1,eps_2,eps_3))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CF = M.vector_frame('eps', [e0, e1, e2, e3], latex_symbol=r'\\varepsilon')\n",
    "CF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{0} = \\left( \\frac{a^{2} + r^{2}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{a}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{0} = \\left( \\frac{a^{2} + r^{2}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{a}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps_0 = (a^2 + r^2)/(sqrt(a^2*cos(th)^2 + r^2)*sqrt(a^2 - 2*m*r + r^2)) d/dt + a/(sqrt(a^2*cos(th)^2 + r^2)*sqrt(a^2 - 2*m*r + r^2)) d/dph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{1} = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{1} = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps_1 = sqrt(a^2 - 2*m*r + r^2)/sqrt(a^2*cos(th)^2 + r^2) d/dr"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{2} = \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{2} = \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial {\\theta} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps_2 = 1/sqrt(a^2*cos(th)^2 + r^2) d/dth"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{3} = \\left( \\frac{a \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sin\\left({\\theta}\\right)} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon_{3} = \\left( \\frac{a \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{1}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\sin\\left({\\theta}\\right)} \\right) \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps_3 = a*sin(th)/sqrt(a^2*cos(th)^2 + r^2) d/dt + 1/(sqrt(a^2*cos(th)^2 + r^2)*sin(th)) d/dph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for v in CF:\n",
    "    show(v.display())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the Carter frame is orthonormal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "-1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 1\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "-1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 1\n",
       "\\end{array}\\right)\n",
       "\\end{math}"
      ],
      "text/plain": [
       "[-1  0  0  0]\n",
       "[ 0  1  0  0]\n",
       "[ 0  0  1  0]\n",
       "[ 0  0  0  1]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix([[g(u, v).expr() for v in CF] for u in CF])  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Carter coframe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{0} = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} t + \\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\sin\\left({\\theta}\\right)^{2}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{0} = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} t + \\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\sin\\left({\\theta}\\right)^{2}}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} {\\phi}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps^0 = sqrt(a^2 - 2*m*r + r^2)/sqrt(a^2*cos(th)^2 + r^2) dt - sqrt(a^2 - 2*m*r + r^2)*a*sin(th)^2/sqrt(a^2*cos(th)^2 + r^2) dph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{1} = \\left( \\frac{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}}{\\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\mathrm{d} r</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{1} = \\left( \\frac{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}}{\\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\mathrm{d} r\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps^1 = sqrt(a^2*cos(th)^2 + r^2)/sqrt(a^2 - 2*m*r + r^2) dr"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{2} = \\left( \\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\theta}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{2} = \\left( \\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\theta}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps^2 = sqrt(a^2*cos(th)^2 + r^2) dth"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{3} = \\left( -\\frac{a \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} t + \\left( \\frac{{\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\varepsilon^{3} = \\left( -\\frac{a \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} t + \\left( \\frac{{\\left(a^{2} + r^{2}\\right)} \\sin\\left({\\theta}\\right)}{\\sqrt{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}} \\right) \\mathrm{d} {\\phi}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "eps^3 = -a*sin(th)/sqrt(a^2*cos(th)^2 + r^2) dt + (a^2 + r^2)*sin(th)/sqrt(a^2*cos(th)^2 + r^2) dph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for f in CF.coframe():\n",
    "    show(f.display())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4-acceleration of the Carter observer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "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": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector field a on the 4-dimensional Lorentzian manifold M\n"
     ]
    }
   ],
   "source": [
    "A = nabla(e0).contract(e0)  # use A to avoid any confusion with the Kerr parameter a\n",
    "A.set_name('a')\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}a = \\left( \\frac{a^{2} m \\sin\\left({\\theta}\\right)^{2} - a^{2} r \\sin\\left({\\theta}\\right)^{2} - a^{2} m + m r^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial r } -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}a = \\left( \\frac{a^{2} m \\sin\\left({\\theta}\\right)^{2} - a^{2} r \\sin\\left({\\theta}\\right)^{2} - a^{2} m + m r^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial r } -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial {\\theta} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "a = (a^2*m*sin(th)^2 - a^2*r*sin(th)^2 - a^2*m + m*r^2)/(a^2*cos(th)^2 + r^2)^2 d/dr - a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2)^2 d/dth"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.apply_map(factor)  # factor the components for a better display\n",
    "A.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a check, we note that the above formula agrees with that given by Eqs. (90)-(91) of \n",
    "[O. Semerak, Gen. Relat. Grav. **25**, 1041 (1993)](https://doi.org/10.1007/BF00763554)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Display in the Carter frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}a = \\left( \\frac{a^{2} m \\sin\\left({\\theta}\\right)^{2} - a^{2} r \\sin\\left({\\theta}\\right)^{2} - a^{2} m + m r^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\varepsilon_{1} -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{2}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}a = \\left( \\frac{a^{2} m \\sin\\left({\\theta}\\right)^{2} - a^{2} r \\sin\\left({\\theta}\\right)^{2} - a^{2} m + m r^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}} \\sqrt{a^{2} - 2 \\, m r + r^{2}}} \\right) \\varepsilon_{1} -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{2}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "a = (a^2*m*sin(th)^2 - a^2*r*sin(th)^2 - a^2*m + m*r^2)/((a^2*cos(th)^2 + r^2)^(3/2)*sqrt(a^2 - 2*m*r + r^2)) eps_1 - a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2)^(3/2) eps_2"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.apply_map(factor, frame=CF, keep_other_components=True)\n",
    "A.display(CF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4-rotation of the ZAMO frame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the rotation operator as\n",
    "$$ \\Omega_{\\rm rot}(v) = \\nabla_{\\varepsilon_0} v - \\Omega_{\\rm FW}(v)$$\n",
    "where\n",
    "$$\\Omega_{\\rm FW}(v) := (a\\cdot v) \\varepsilon_0 - (\\varepsilon_0\\cdot v) a$$\n",
    "is the Fermi-Walker operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rotation_operator(v):\n",
    "    return nabla(v).contract(e0) - g(A, v)*e0 + g(e0, v)*A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rotation_operator(e0) == 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us evaluate $\\Omega_{\\rm rot}(\\varepsilon_1)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} r \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial t } + \\frac{a r}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} r \\sin\\left({\\theta}\\right)^{2}}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial t } + \\frac{a r}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "a^2*r*sin(th)^2/(a^2*cos(th)^2 + r^2)^2 d/dt + a*r/(a^2*cos(th)^2 + r^2)^2 d/dph"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De1 = rotation_operator(e1)\n",
    "De1.apply_map(factor)\n",
    "De1.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{3}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{3}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "a*r*sin(th)/(a^2*cos(th)^2 + r^2)^(3/2) eps_3"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De1.apply_map(factor, frame=CF, keep_other_components=True)\n",
    "De1.display(CF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us now evaluate $\\Omega_{\\rm rot}(\\varepsilon_2)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2} \\sin\\left({\\theta}\\right)} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial t } + \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2} \\sin\\left({\\theta}\\right)} \\right) \\frac{\\partial}{\\partial {\\phi} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "sqrt(a^2 - 2*m*r + r^2)*a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2)^2 d/dt + sqrt(a^2 - 2*m*r + r^2)*a*cos(th)/((a^2*cos(th)^2 + r^2)^2*sin(th)) d/dph"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De2 = rotation_operator(e2)\n",
    "De2.apply_map(factor)\n",
    "De2.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{{\\left(a^{2} \\sin\\left({\\theta}\\right)^{2} - a^{2} - r^{2}\\right)} \\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{5}{2}}} \\right) \\varepsilon_{3}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{{\\left(a^{2} \\sin\\left({\\theta}\\right)^{2} - a^{2} - r^{2}\\right)} \\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{5}{2}}} \\right) \\varepsilon_{3}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-(a^2*sin(th)^2 - a^2 - r^2)*sqrt(a^2 - 2*m*r + r^2)*a*cos(th)/(a^2*cos(th)^2 + r^2)^(5/2) eps_3"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De2.apply_map(factor, frame=CF, keep_other_components=True)\n",
    "De2.display(CF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and finally $\\Omega_{\\rm rot}(e_3)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial r } + \\left( \\frac{{\\left(a^{2} \\sin\\left({\\theta}\\right)^{2} - a^{2} - r^{2}\\right)} \\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial r } + \\left( \\frac{{\\left(a^{2} \\sin\\left({\\theta}\\right)^{2} - a^{2} - r^{2}\\right)} \\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-sqrt(a^2 - 2*m*r + r^2)*a*r*sin(th)/(a^2*cos(th)^2 + r^2)^2 d/dr + (a^2*sin(th)^2 - a^2 - r^2)*sqrt(a^2 - 2*m*r + r^2)*a*cos(th)/(a^2*cos(th)^2 + r^2)^3 d/dth"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De3 = rotation_operator(e3)\n",
    "De3.apply_map(factor)\n",
    "De3.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us enforce further trigonometric simplification: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial r } + \\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial r } + \\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\right) \\frac{\\partial}{\\partial {\\theta} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-sqrt(a^2 - 2*m*r + r^2)*a*r*sin(th)/(a^2*cos(th)^2 + r^2)^2 d/dr - sqrt(a^2 - 2*m*r + r^2)*a*cos(th)/(a^2*cos(th)^2 + r^2)^2 d/dth"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De3.apply_map(lambda x: x.trig_reduce())\n",
    "De3.apply_map(lambda x: x.simplify_trig())\n",
    "De3.apply_map(factor)\n",
    "De3.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{1} + \\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\right) \\varepsilon_{2}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{1} + \\left( -\\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\right) \\varepsilon_{2}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-a*r*sin(th)/(a^2*cos(th)^2 + r^2)^(3/2) eps_1 - sqrt(a^2 - 2*m*r + r^2)*a*cos(th)/(a^2*cos(th)^2 + r^2)^(3/2) eps_2"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De3.apply_map(factor, frame=CF, keep_other_components=True)\n",
    "De3.display(CF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the orthonormal frame $(\\varepsilon_1, \\varepsilon_2, \\varepsilon_3)$ of the Carter observer's rest space, the matrix of the operator $\\Omega_{\\rm rot}$ must be of the type \n",
    "$$\n",
    "    \\left( \\begin{array}{ccc}\n",
    "    0 & - \\omega^3 & \\omega^3 \\\\\n",
    "    \\omega^3 & 0 & -\\omega^1 \\\\\n",
    "    - \\omega^2 & \\omega^1 & 0\n",
    "    \\end{array} \\right), \n",
    "$$\n",
    "where the $\\omega^i$'s are the components on the 4-rotation vector:\n",
    "$$\n",
    "    \\omega = \\omega^i \\varepsilon_i\n",
    "$$\n",
    "\n",
    "From the above expressions of the $\\Omega_{\\rm rot}(\\varepsilon_i)$'s, we get immediately $\\omega^3 = 0$ and we check that"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De1[CF, 3] == - De3[CF, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "De2[CF, 3] == - De3[CF, 2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore, we get the 4-rotation vector $\\omega$ as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\right) \\varepsilon_{1} -\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{2}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\left( \\frac{\\sqrt{a^{2} - 2 \\, m r + r^{2}} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\right) \\varepsilon_{1} -\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{\\frac{3}{2}}} \\varepsilon_{2}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "omega = sqrt(a^2 - 2*m*r + r^2)*a*cos(th)/(a^2*cos(th)^2 + r^2)^(3/2) eps_1 - a*r*sin(th)/(a^2*cos(th)^2 + r^2)^(3/2) eps_2"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega = - De3[CF, 2]*e1 + De3[CF, 1]*e2\n",
    "omega.set_name('omega', latex_name=r'\\omega')\n",
    "omega.apply_map(factor)\n",
    "omega.apply_map(factor, frame=CF, keep_other_components=True)\n",
    "omega.display(CF)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\frac{{\\left(a^{2} - 2 \\, m r + r^{2}\\right)} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial r } -\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\frac{{\\left(a^{2} - 2 \\, m r + r^{2}\\right)} a \\cos\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial r } -\\frac{a r \\sin\\left({\\theta}\\right)}{{\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}^{2}} \\frac{\\partial}{\\partial {\\theta} }\n",
       "\\end{math}"
      ],
      "text/plain": [
       "omega = (a^2 - 2*m*r + r^2)*a*cos(th)/(a^2*cos(th)^2 + r^2)^2 d/dr - a*r*sin(th)/(a^2*cos(th)^2 + r^2)^2 d/dth"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a check, we note that the above formula agrees with that given by Eq. (93) of \n",
    "[O. Semerak, Gen. Relat. Grav. **25**, 1041 (1993)](https://doi.org/10.1007/BF00763554)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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