Worksheets/v1.1/SM_Kerr_Killing_tensor.ipynb

{
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   "cell_type": "markdown",
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    "editable": true
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   "source": [
    "# Walker-Penrose Killing tensor in Kerr spacetime\n",
    "\n",
    "This worksheet demonstrates a few capabilities of SageMath in computations regarding Kerr spacetime. More precisely, it focuses of the Killing tensor $K$ found by Walker & Penrose [[Commun. Math. Phys. **18**, 265 (1970)](https://doi.org/10.1007/BF01649445)].\n",
    "The employed differential geometry tools have been developed within the  [SageManifolds](http://sagemanifolds.obspm.fr) project (version 1.1, as included in SageMath 8.1).\n",
    "\n",
    "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.1/SM_Kerr_Killing_tensor.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath within the Jupyter notebook, via the command `sage -n jupyter`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "*NB:* a version of SageMath at least equal to 7.5 is required to run this worksheet:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 8.1, Release Date: 2017-12-07'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "First we set up the notebook to display mathematical objects using LaTeX rendering:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "%display latex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We also define a viewer for 3D plots (use `'threejs'` or `'jmol'` for interactive 3D graphics):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "viewer3D = 'threejs' # must be 'threejs', jmol', 'tachyon' or None (default)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To speed up the computations, we ask for running them in parallel on 8 cores:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Parallelism().set(nproc=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Spacetime manifold\n",
    "\n",
    "We declare the Kerr spacetime (or more precisely the Boyer-Lindquist domain of Kerr spacetime) as a 4-dimensional diffentiable manifold:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "M = Manifold(4, 'M', r'\\mathcal{M}')\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let us declare the **Boyer-Lindquist coordinates** via the method `chart()`, the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\\infty,+\\infty)$) and their LaTeX symbols:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "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/plain": [
       "Chart (M, (t, r, th, ph))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BL.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi') \n",
    "print(BL) ; BL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r\\right)</script></html>"
      ],
      "text/plain": [
       "(t, r)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BL[0], BL[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "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": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a\\right)</script></html>"
      ],
      "text/plain": [
       "(m, a)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('m, a', domain='real')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us introduce the spacetime metric:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "g = M.lorentzian_metric('g')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "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": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "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/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": 10,
     "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": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A matrix view of the components with respect to the manifold's default vector frame:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "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/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": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The list of the non-vanishing components:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "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/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": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.display_comp()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Levi-Civita Connection</h2>\n",
    "\n",
    "<p>The Levi-Civita connection $\\nabla$ associated with $g$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "nabla = g.connection() ; print(nabla)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us verify that the covariant derivative of $g$ with respect to $\\nabla$ vanishes identically:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} g = 0</script></html>"
      ],
      "text/plain": [
       "nabla_g(g) = 0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla(g).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Killing vectors</h2>\n",
    "<p>The default vector frame on the spacetime manifold is the coordinate basis associated with Boyer-Lindquist coordinates:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.default_frame() is BL.frame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "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/plain": [
       "Coordinate frame (M, (d/dt,d/dr,d/dth,d/dph))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BL.frame()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us consider the first vector field of this frame:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial t }</script></html>"
      ],
      "text/plain": [
       "Vector field d/dt on the 4-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = BL.frame()[0] ; xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector field d/dt on the 4-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(xi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The 1-form associated to it by metric duality is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} - 2 \\, m r + r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t + \\left( -\\frac{2 \\, a m r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "-(a^2*cos(th)^2 - 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) dt - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi_form = xi.down(g) ; xi_form.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Its covariant derivative is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (0,2) on the 4-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{a^{2} m \\cos\\left({\\theta}\\right)^{2} - m r^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, a^{2} m r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{a^{2} m \\cos\\left({\\theta}\\right)^{2} - m r^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} t + \\left( \\frac{{\\left(a^{3} m \\cos\\left({\\theta}\\right)^{2} - a m r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, a^{2} m r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, {\\left(a^{3} m r + a m r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{{\\left(a^{3} m \\cos\\left({\\theta}\\right)^{2} - a m r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( -\\frac{2 \\, {\\left(a^{3} m r + a m r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}</script></html>"
      ],
      "text/plain": [
       "(a^2*m*cos(th)^2 - m*r^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dt*dr + 2*a^2*m*r*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dt*dth - (a^2*m*cos(th)^2 - m*r^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dr*dt + (a^3*m*cos(th)^2 - a*m*r^2)*sin(th)^2/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dr*dph - 2*a^2*m*r*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dth*dt + 2*(a^3*m*r + a*m*r^3)*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dth*dph - (a^3*m*cos(th)^2 - a*m*r^2)*sin(th)^2/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dph*dr - 2*(a^3*m*r + a*m*r^3)*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) dph*dth"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nab_xi = nabla(xi_form) ; print(nab_xi) ; nab_xi.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us check that the Killing equation is satisfied:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nab_xi.symmetrize() == 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Similarly, let us check that $\\frac{\\partial}{\\partial\\phi}$ is a Killing vector:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "Vector field d/dph on the 4-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chi = BL.frame()[3] ; chi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla(chi.down(g)).symmetrize() == 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Principal null vectors\n",
    "\n",
    "We introduce the principal null vectors $k$ and $\\ell$ of Kerr spacetime:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\left( \\frac{a^{2} + r^{2}}{2 \\, {\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}} \\right) \\frac{\\partial}{\\partial t } + \\left( -\\frac{a^{2} - 2 \\, m r + r^{2}}{2 \\, {\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}} \\right) \\frac{\\partial}{\\partial r } + \\frac{a}{2 \\, {\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)}} \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "k = 1/2*(a^2 + r^2)/(a^2*cos(th)^2 + r^2) d/dt - 1/2*(a^2 - 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) d/dr + 1/2*a/(a^2*cos(th)^2 + r^2) d/dph"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = M.vector_field(name='k')\n",
    "k[:] = [(r^2+a^2)/(2*rho2), -Delta/(2*rho2), 0, a/(2*rho2)]\n",
    "k.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\ell = \\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/plain": [
       "el = (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": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "el = M.vector_field(name='el', latex_name=r'\\ell')\n",
    "el[:] = [(r^2+a^2)/Delta, 1, 0, a/Delta]\n",
    "el.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let us check that $k$ and $\\ell$ are null vectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(k,k).expr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(el,el).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Their scalar product is $-1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1</script></html>"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(k,el).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let us evaluate the \"acceleration\" of $k$, i.e. $\\nabla_k k$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{a^{4} m - m r^{4} - {\\left(a^{4} m - a^{4} r + a^{2} m r^{2} - a^{2} r^{3}\\right)} \\sin\\left({\\theta}\\right)^{2}}{2 \\, {\\left(a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}\\right)}} \\right) \\frac{\\partial}{\\partial t } + \\left( -\\frac{a^{4} m - 2 \\, a^{2} m^{2} r + 2 \\, m^{2} r^{3} - m r^{4} - {\\left(a^{4} m + 3 \\, a^{2} m r^{2} - a^{2} r^{3} - {\\left(a^{4} + 2 \\, a^{2} m^{2}\\right)} r\\right)} \\sin\\left({\\theta}\\right)^{2}}{2 \\, {\\left(a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}\\right)}} \\right) \\frac{\\partial}{\\partial r } + \\left( \\frac{a^{3} m - a m r^{2} - {\\left(a^{3} m - a^{3} r\\right)} \\sin\\left({\\theta}\\right)^{2}}{2 \\, {\\left(a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}\\right)}} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "1/2*(a^4*m - m*r^4 - (a^4*m - a^4*r + a^2*m*r^2 - a^2*r^3)*sin(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) d/dt - 1/2*(a^4*m - 2*a^2*m^2*r + 2*m^2*r^3 - m*r^4 - (a^4*m + 3*a^2*m*r^2 - a^2*r^3 - (a^4 + 2*a^2*m^2)*r)*sin(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) d/dr + 1/2*(a^3*m - a*m*r^2 - (a^3*m - a^3*r)*sin(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) d/dph"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_k = nabla(k).contract(k)\n",
    "acc_k.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We check that $k$ is a pregeodesic vector, i.e. that $\\nabla_k k = \\kappa_k k$ for some scalar field $\\kappa_k$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} m - m r^{2} - {\\left(a^{2} m - a^{2} r\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}}</script></html>"
      ],
      "text/plain": [
       "(a^2*m - m*r^2 - (a^2*m - a^2*r)*sin(th)^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} m - m r^{2} - {\\left(a^{2} m - a^{2} r\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}}</script></html>"
      ],
      "text/plain": [
       "(a^2*m - m*r^2 - (a^2*m - a^2*r)*sin(th)^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} m - m r^{2} - {\\left(a^{2} m - a^{2} r\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}}</script></html>"
      ],
      "text/plain": [
       "(a^2*m - m*r^2 - (a^2*m - a^2*r)*sin(th)^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in [0,1,3]:\n",
    "    show(acc_k[i] / k[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{a^{2} m - m r^{2} - {\\left(a^{2} m - a^{2} r\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\end{array}</script></html>"
      ],
      "text/plain": [
       "M --> R\n",
       "(t, r, th, ph) |--> (a^2*m - m*r^2 - (a^2*m - a^2*r)*sin(th)^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kappa_k = acc_k[[0]] / k[[0]]\n",
    "kappa_k.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_k == kappa_k * k"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Similarly let us evaluate the \"acceleration\" of $\\ell$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_l = nabla(el).contract(el)\n",
    "acc_l.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Hence $\\ell$ is a geodesic vector."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Walker-Penrose Killing tensor\n",
    "\n",
    "We need the 1-forms associated to $k$ and $\\ell$ by metric duality:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "uk = k.down(g)\n",
    "ul = el.down(g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The Walker-Penrose Killing tensor $K$ is then formed as\n",
    "$$ K = \\rho^2 (\\underline{\\ell}\\otimes \\underline{k} + (\\underline{k}\\otimes \\underline{\\ell}) + r^2 g $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field K of type (0,2) on the 4-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "K = rho2*(ul*uk+ uk*ul) + r^2*g\n",
    "K.set_name('K')\n",
    "print(K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} K_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{a^{2} r^{2} + {\\left(a^{4} - 2 \\, a^{2} m r\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ K_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, a^{3} m r \\sin\\left({\\theta}\\right)^{4} - {\\left(2 \\, a^{3} m r - a^{3} r^{2} - a r^{4} - {\\left(a^{5} + a^{3} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ K_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{a^{4} \\cos\\left({\\theta}\\right)^{4} + a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2}}{a^{2} - 2 \\, m r + r^{2}} \\\\ K_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4} \\\\ K_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a^{3} m r \\sin\\left({\\theta}\\right)^{4} - {\\left(2 \\, a^{3} m r - a^{3} r^{2} - a r^{4} - {\\left(a^{5} + a^{3} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ K_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{8} - 2 \\, a^{6} m r + a^{6} r^{2}\\right)} \\sin\\left({\\theta}\\right)^{8} - {\\left(2 \\, a^{8} - 4 \\, a^{6} m r + 3 \\, a^{6} r^{2} - 2 \\, a^{4} m r^{3} + a^{4} r^{4}\\right)} \\sin\\left({\\theta}\\right)^{6} + {\\left(a^{8} - 2 \\, a^{6} m r + a^{6} r^{2} - 2 \\, a^{4} m r^{3} - a^{4} r^{4} - a^{2} r^{6}\\right)} \\sin\\left({\\theta}\\right)^{4} + {\\left(a^{6} r^{2} + 3 \\, a^{4} r^{4} + 3 \\, a^{2} r^{6} + r^{8}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\end{array}</script></html>"
      ],
      "text/plain": [
       "K_t,t = (a^2*r^2 + (a^4 - 2*a^2*m*r)*cos(th)^2)/(a^2*cos(th)^2 + r^2) \n",
       "K_t,ph = -(2*a^3*m*r*sin(th)^4 - (2*a^3*m*r - a^3*r^2 - a*r^4 - (a^5 + a^3*r^2)*cos(th)^2)*sin(th)^2)/(a^2*cos(th)^2 + r^2) \n",
       "K_r,r = -(a^4*cos(th)^4 + a^2*r^2*cos(th)^2)/(a^2 - 2*m*r + r^2) \n",
       "K_th,th = a^2*r^2*cos(th)^2 + r^4 \n",
       "K_ph,t = -(2*a^3*m*r*sin(th)^4 - (2*a^3*m*r - a^3*r^2 - a*r^4 - (a^5 + a^3*r^2)*cos(th)^2)*sin(th)^2)/(a^2*cos(th)^2 + r^2) \n",
       "K_ph,ph = ((a^8 - 2*a^6*m*r + a^6*r^2)*sin(th)^8 - (2*a^8 - 4*a^6*m*r + 3*a^6*r^2 - 2*a^4*m*r^3 + a^4*r^4)*sin(th)^6 + (a^8 - 2*a^6*m*r + a^6*r^2 - 2*a^4*m*r^3 - a^4*r^4 - a^2*r^6)*sin(th)^4 + (a^6*r^2 + 3*a^4*r^4 + 3*a^2*r^6 + r^8)*sin(th)^2)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.display_comp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field nabla_g(K) of type (0,3) on the 4-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "DK = nabla(K)\n",
    "print(DK)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\nabla_{g} K_{ \\, t \\, r \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, r}\\phantom{\\, {\\phi}} } & = & a r \\sin\\left({\\theta}\\right)^{2} \\\\ \\nabla_{g} K_{ \\, t \\, {\\theta} \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\theta}}\\phantom{\\, {\\phi}} } & = & {\\left(a^{3} - 2 \\, a m r + a r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\nabla_{g} K_{ \\, t \\, {\\phi} \\, r }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}}\\phantom{\\, r} } & = & -a r \\sin\\left({\\theta}\\right)^{2} \\\\ \\nabla_{g} K_{ \\, t \\, {\\phi} \\, {\\theta} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}}\\phantom{\\, {\\theta}} } & = & -{\\left(a^{3} - 2 \\, a m r + a r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\nabla_{g} K_{ \\, r \\, t \\, {\\phi} }^{ \\phantom{\\, r}\\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -a r \\cos\\left({\\theta}\\right)^{2} + a r \\\\ \\nabla_{g} K_{ \\, r \\, r \\, {\\theta} }^{ \\phantom{\\, r}\\phantom{\\, r}\\phantom{\\, {\\theta}} } & = & \\frac{2 \\, {\\left(a^{4} \\cos\\left({\\theta}\\right)^{3} + a^{2} r^{2} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{2} - 2 \\, m r + r^{2}} \\\\ \\nabla_{g} K_{ \\, r \\, {\\theta} \\, r }^{ \\phantom{\\, r}\\phantom{\\, {\\theta}}\\phantom{\\, r} } & = & -\\frac{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{3} + a^{2} r^{2} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{2} - 2 \\, m r + r^{2}} \\\\ \\nabla_{g} K_{ \\, r \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, r}\\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & -a^{2} r \\cos\\left({\\theta}\\right)^{2} - r^{3} \\\\ \\nabla_{g} K_{ \\, r \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, r}\\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -a^{2} r - r^{3} + {\\left(a^{2} r + r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2} \\\\ \\nabla_{g} K_{ \\, {\\theta} \\, t \\, {\\phi} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & {\\left(a^{3} - 2 \\, a m r + a r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\nabla_{g} K_{ \\, {\\theta} \\, r \\, r }^{ \\phantom{\\, {\\theta}}\\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{3} + a^{2} r^{2} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{2} - 2 \\, m r + r^{2}} \\\\ \\nabla_{g} K_{ \\, {\\theta} \\, r \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, r}\\phantom{\\, {\\theta}} } & = & -a^{2} r \\cos\\left({\\theta}\\right)^{2} - r^{3} \\\\ \\nabla_{g} K_{ \\, {\\theta} \\, {\\theta} \\, r }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}}\\phantom{\\, r} } & = & 2 \\, a^{2} r \\cos\\left({\\theta}\\right)^{2} + 2 \\, r^{3} \\\\ \\nabla_{g} K_{ \\, {\\theta} \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -{\\left(a^{4} - 2 \\, a^{2} m r + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{3} \\\\ \\nabla_{g} K_{ \\, {\\phi} \\, t \\, r }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t}\\phantom{\\, r} } & = & -a r \\sin\\left({\\theta}\\right)^{2} \\\\ \\nabla_{g} K_{ \\, {\\phi} \\, t \\, {\\theta} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t}\\phantom{\\, {\\theta}} } & = & -{\\left(a^{3} - 2 \\, a m r + a r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\nabla_{g} K_{ \\, {\\phi} \\, r \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, r}\\phantom{\\, {\\phi}} } & = & -{\\left(a^{2} r + r^{3}\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\nabla_{g} K_{ \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\theta}}\\phantom{\\, {\\phi}} } & = & -{\\left(a^{4} - 2 \\, a^{2} m r + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{3} \\\\ \\nabla_{g} K_{ \\, {\\phi} \\, {\\phi} \\, r }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}}\\phantom{\\, r} } & = & 2 \\, {\\left(a^{2} r + r^{3}\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\nabla_{g} K_{ \\, {\\phi} \\, {\\phi} \\, {\\theta} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}}\\phantom{\\, {\\theta}} } & = & 2 \\, {\\left(a^{4} - 2 \\, a^{2} m r + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{3} \\end{array}</script></html>"
      ],
      "text/plain": [
       "nabla_g(K)_t,r,ph = a*r*sin(th)^2 \n",
       "nabla_g(K)_t,th,ph = (a^3 - 2*a*m*r + a*r^2)*cos(th)*sin(th) \n",
       "nabla_g(K)_t,ph,r = -a*r*sin(th)^2 \n",
       "nabla_g(K)_t,ph,th = -(a^3 - 2*a*m*r + a*r^2)*cos(th)*sin(th) \n",
       "nabla_g(K)_r,t,ph = -a*r*cos(th)^2 + a*r \n",
       "nabla_g(K)_r,r,th = 2*(a^4*cos(th)^3 + a^2*r^2*cos(th))*sin(th)/(a^2 - 2*m*r + r^2) \n",
       "nabla_g(K)_r,th,r = -(a^4*cos(th)^3 + a^2*r^2*cos(th))*sin(th)/(a^2 - 2*m*r + r^2) \n",
       "nabla_g(K)_r,th,th = -a^2*r*cos(th)^2 - r^3 \n",
       "nabla_g(K)_r,ph,ph = -a^2*r - r^3 + (a^2*r + r^3)*cos(th)^2 \n",
       "nabla_g(K)_th,t,ph = (a^3 - 2*a*m*r + a*r^2)*cos(th)*sin(th) \n",
       "nabla_g(K)_th,r,r = -(a^4*cos(th)^3 + a^2*r^2*cos(th))*sin(th)/(a^2 - 2*m*r + r^2) \n",
       "nabla_g(K)_th,r,th = -a^2*r*cos(th)^2 - r^3 \n",
       "nabla_g(K)_th,th,r = 2*a^2*r*cos(th)^2 + 2*r^3 \n",
       "nabla_g(K)_th,ph,ph = -(a^4 - 2*a^2*m*r + a^2*r^2)*cos(th)*sin(th)^3 \n",
       "nabla_g(K)_ph,t,r = -a*r*sin(th)^2 \n",
       "nabla_g(K)_ph,t,th = -(a^3 - 2*a*m*r + a*r^2)*cos(th)*sin(th) \n",
       "nabla_g(K)_ph,r,ph = -(a^2*r + r^3)*sin(th)^2 \n",
       "nabla_g(K)_ph,th,ph = -(a^4 - 2*a^2*m*r + a^2*r^2)*cos(th)*sin(th)^3 \n",
       "nabla_g(K)_ph,ph,r = 2*(a^2*r + r^3)*sin(th)^2 \n",
       "nabla_g(K)_ph,ph,th = 2*(a^4 - 2*a^2*m*r + a^2*r^2)*cos(th)*sin(th)^3 "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DK.display_comp()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let us check that $K$ is a Killing tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "DK.symmetrize().display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Equivalently, we may write, using index notation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DK['_(abc)'].display()"
   ]
  }
 ],
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