{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolically Understanding Christoffel Symbol and Riemann Curvature Tensor using EinsteinPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "from einsteinpy.symbolic import MetricTensor, ChristoffelSymbols, RiemannCurvatureTensor\n",
    "\n",
    "sympy.init_printing()  # enables the best printing available in an environment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining the metric tensor for 3d spherical coordinates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & r^{2} & 0\\\\0 & 0 & r^{2} \\sin^{2}{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0       0     ⎤\n",
       "⎢                 ⎥\n",
       "⎢    2            ⎥\n",
       "⎢0  r       0     ⎥\n",
       "⎢                 ⎥\n",
       "⎢        2    2   ⎥\n",
       "⎣0  0   r ⋅sin (θ)⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "syms = sympy.symbols('r theta phi')\n",
    "# define the metric for 3d spherical coordinates\n",
    "metric = [[0 for i in range(3)] for i in range(3)]\n",
    "metric[0][0] = 1\n",
    "metric[1][1] = syms[0]**2\n",
    "metric[2][2] = (syms[0]**2)*(sympy.sin(syms[1])**2)\n",
    "# creating metric object\n",
    "m_obj = MetricTensor(metric, syms)\n",
    "m_obj.tensor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the christoffel symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & - r & 0\\\\0 & 0 & - r \\sin^{2}{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & \\frac{1}{r} & 0\\\\\\frac{1}{r} & 0 & 0\\\\0 & 0 & - \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & \\frac{1}{r}\\\\0 & 0 & \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}}\\\\\\frac{1}{r} & \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                             ⎡             1   ⎤⎤\n",
       "⎢                     ⎡   1                ⎤  ⎢0    0       ─   ⎥⎥\n",
       "⎢⎡0  0       0     ⎤  ⎢0  ─        0       ⎥  ⎢             r   ⎥⎥\n",
       "⎢⎢                 ⎥  ⎢   r                ⎥  ⎢                 ⎥⎥\n",
       "⎢⎢0  -r      0     ⎥  ⎢                    ⎥  ⎢           cos(θ)⎥⎥\n",
       "⎢⎢                 ⎥  ⎢1                   ⎥  ⎢0    0     ──────⎥⎥\n",
       "⎢⎢             2   ⎥  ⎢─  0        0       ⎥  ⎢           sin(θ)⎥⎥\n",
       "⎢⎣0  0   -r⋅sin (θ)⎦  ⎢r                   ⎥  ⎢                 ⎥⎥\n",
       "⎢                     ⎢                    ⎥  ⎢1  cos(θ)        ⎥⎥\n",
       "⎢                     ⎣0  0  -sin(θ)⋅cos(θ)⎦  ⎢─  ──────    0   ⎥⎥\n",
       "⎣                                             ⎣r  sin(θ)        ⎦⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ch = ChristoffelSymbols.from_metric(m_obj)\n",
    "ch.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMt0Qq5nNdrvviSJmRMu65WEAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABWSURBVBgZY2BgEDJgYGBgds0DUQwMdVSkPOapFIANJYt4BwIPyNKKqYlxBXfUAQYGdo4orgkMDDZMCqwLGBg28wGFgOAexM9BYA6DFpjifgumeBVAFAC4Fhen6YFvfQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{r}$$"
      ],
      "text/plain": [
       "1\n",
       "─\n",
       "r"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ch.tensor()[1,1,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the Riemann Curvature tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWYAAADgCAMAAAAHZjxVAAAAP1BMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADFBd4eAAAAFHRSTlMAMquJZkQiEEAwdpnNVN2773zD6xmZA7EAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAyYSURBVHgB7V3tYts4DnQat9lrtx975/d/1rMkk+YAoEagE5e7O/nRihyDQwwhSrEn1unlsv58Oi0/0Fh7HvznxzY6H+W9mSchrtN4ubyerz+fVyWgwbXhr1iGPn+68Be+N/MkxHUaL5cvdxGgce9+8Oh8SOZmGg/y1fBJiJdpgLJN4/X167edzAl8avFctm1kFaw5IHgLT0K8I/Of30+ntx9dnQl8AjyVLUQ26pZDggM8CXFf5vOPJa3XnyU58z+BT4hnssVIQ3ttEhzhSYj7Mv9c7zy+X958oksPgQ2eyTY3spsdhk9C3Jf58m3J4PPlunVEPwQ+IZ7JFiM9N8ERnoS4K/Nbme7ZJ3rtIbDFE9kmR7azM+GTEHdl/nJ5XTJ4uXy1iaxtAp8MnsjWRDp2ght4EmIu86p2P9kYvsu84SPZHhvZzqzKPBVxV+a3rZo/d6qZwNdNY02zhCdkNpFWRjuyxU34JMRdmW/XsO/7l8AubMIT2ZpIK+OJ4ds1pUxsEuK+zD/XO+Zz94ZuHz5heCZbjPQyExzhSYj7Mp9/LRl++9PnufYQ+IR4JluM9PQER3gS4r7Mpx/LL9u/XnyeWw+BMTyTLUYG9IQZ4EmId2R+e/369VNX5ROBEU9lmxrZLwOET0K8I7NPYLwnle04jY+chFgymzeC/UqN9eD6SmbJPFZHEIVFBVBtNJ9k1L6HD5BY1axqfrikrp8BzPEhpKpZ1axqTiowybkbz1qXwFiXRO8k66u9+bfvza2tJCggAssO02i2U81gK2lCbocElh2mlawvM9pK2pj1mMDGtJLZInMjk4lNQtyXGW0lLhsCyw4Dvxf1ZS4+DdlhSomlFMHTqCuzsZUUqvI/gWWHOShz9TvIDnMrrZwiWZljV0rljGHZYbzMf1z+U/aC+726sZXcX7AdEdiaVpDUDra1b7/zJke2Y5nwSYj/ul4QQebaQFuJzYaZUgy+0LCfwkyIzchuVAyfhHiZBrx1UhtoK3HZEFh2GCis5aSqyi5a1gbaSpzMBJYd5qDMzJUCrhO3CicMT2yRJpKN7HGY2CTE/WpGP4vPBlwnHsbwVLapkT0zhE9CvCOzT2C8J5XtOI2PnIRYMjeXI79K4z24vpJZMo/XUo3EoqrdcFBvrqD3wQYSq5pVzQ8W1BKORRUPqGqOdUn0TiTz27n5g+EPWdpZiipenw9JGdf3y/WPWIEGGvG0BnqRNB7gQ5gnIV6mAflBI9ZjoHeSbOOZf0jKmLFkNmUWr0S+97jMxO9CYNlhmrXZqWbidyGw7DCNyustAOxNtUFcKQSWHebg+83E70Jg2WEOypwyf7QnyHaM4XhB8K9eesp5hJH+tQRHeBLi7t5M/C4Elh0G17crczViyA5zO6FyimRljv0ulTOGZYfxMheDxLpqZYc0thK7RRJYdhiUue/T2C4l5VtWrMzMlGJwJHWDwQITYjOyGwvDJyFeplEKGJI1fhaXjewwVhJUBNe3LzPxuxBYdpiD983oZ7FLlzSt4Nr6wZaeelKBnyV4LcEBnoS4X83oZ/HZguvEwxieyjY1smeG8EmId2T2CYz3pLIdp/GRkxBL5ma38qs03oPrK5kl83gt1UgsqtoNB/XaC70PNpBY1axqfrCglnAsqnjA31PNzbNm4mnleuvjVVjYP/TxMmvay2qDHeaP7aE7/2WiHMb/d/QpPu/NPBNxYIdpHulzWMudF9an2Oy8ZoX+oU/xWXPTJVCXQFb/B/B5L4H1uT3E70Jg2WGaKtjZNIjfhcCywzQqyw5zFaPcN+ccPq2K6zGG427Vr2bidyGw7DAH39ZHW4lbOwLfPrErz1rCtXWDrR2lqHIju7EwfBLibjUTvwuBZYfB9e3KXI0YssPcTpicIlmZY79L5Yxh2WG8zLLDrL8qJB0+9qJgwlFm2WHuN3TbxbNrAHrEh9Pdm2WHseWaUwSruS8z8bsQWHaYg/fNssO4cgajjUNRsKPVjH4WPyi4TjyM4UgavPraVX49wcjgtYQZ4EmI+5tGkN94VyrbcRofOQmxZG5OI79K4z24vpJZMo/XUo3EoqrdcFAvCtD7YAOJVc2q5gcLagnHoooHfFI1h+9pxDMa7Z0k23j6T5C5/55GPKXB3n+7zNqbtTcPnjpt2CSnkar5t1cz8bsQWHaY5qzaqWbidyGw7DCNyrLDXMUoN3ToZ2lV2o4JjjBeFPrVTPwuBJYd5uDb+mgrcWtLYNlhjslM/C4Elh3m4KZRjRiyw9zO5JwiWZljv0vljGHZYbzM8Cc+pWHcHXZvJrD9dpjlT1zYz405ObId1YRPQhz8ic/NrU/MIQQ23hFcWyvN1i73VbmR3VgYPgnxMo2S3zrj2sAvO3HZENh4RzLZ5kYmE5uEuC8z8bsQWHaYYzd09OtfiDdkzxziKhDPo9TIfiwIz1QzTtkPzPAd4n41M1cKuE6COQGeyhYi2cgeh/BJiHdk9gmM96SyHafxkZMQS2ZzC+BXaqwH11cyS+axOoIoLCqAaqPew9aedzhAYlWzqvndiyoe8EnVLDtM/TaAeB1GenHTkB1Gm8ZIFZkYLCoD3ppP2jSABhrxtAZ6J8k2nvmHpIwZLy2ggUY8rYFeJI0H+BDmSYj3ZCZ+FwLLDtNU047MxO9CYNlhGpVlh7mKUXYr9LO0Km3HBEcYd6t+NRO/C4Flhzn4tv72mVr5ehe3tgSWHeaYzMTvQmDZYQ5uGtWIITvM7UzOKZKVOfa7VM4Ylh3Gy1wcMOuqlcuusZXYvZnA1g6DpHawrX1jTo5sxzLhkxDLDnO/oUMjjV2/k3H4OBzDcX2XVingNbA2iCuFwLLDHLvTMH4Wt3ayw1hJUJGj1Zwyf1jKa3vHHBK8+tpVzyOIDF5LcIAx22Cw5xD3Nw3ZYdyigNHGoSgYru+OzH6c8R4kjcep1RzDY72TEEvmZrcaW8k4CtdXMkvmuE5SvVhUcegTditVs6o5Lr5U70TVbOwwelhSaiH9i4PHQzk7zHs/sug00zOLvCRLz3unHGQc7M16WFK8God7g8dDBTJ/gKFsli/QipXSnUasS6J3oksgrGbTIH4XAssO01TDzqZB/C4Elh2mUVl2mKsY5QRGP0ur0nZMcIRxt+pXM/G7EFh2mIOfnhC/C4FlhzkmM/G7EFh2mIObRjViyA5z26VzimRljv0ulTOGZYfxMssO8+EPS5Id5n5Dh34Wf0NHcIR9NZf7xnXg2iB+FwLLDnPsTkN2GFfO6Hch8NFqRj+LG5TBiCOpH2zpqecR+FmC1xIc4EmIl2nU/CBZdHf4bIk3BMNT2aZGJhObhHhHZp/AeE8q23EaHzkJsWQ2Z7NfqbEeXF/JLJnH6giisKgAqg24NtXeBw+QWNWsan6woJZwLKp4wCdVs7HD6JPteDVyvbi+zg7zIUs7S1HFSn1Iyiiz9mbtzXHxpXqxqOJQVXOsS6L3byAz8bsQWHaYphp29mbidyGw7DCNyrLDXMUoezP6WVqVtmOCI4y7Vb+aid+FwLLDHPz0hPhdCCw7zDGZid+FwLLDHNw0qhFDdpjbLp1TJCtz7HepnDEsOwyRubyPZL5lxV53CWy/HWZ564T93JiTI9tRTfgkxP07DXR32GzYV6UYHNfWDbZ2lPsqQmxGdmNh+CTEfZmJ34XAssPA+duXOWX+cDVl3DSZoiLEZmTHjOGTEPdlRj+Ly4bBiGeyxUhPzHDZYQLN7l1lb0YjzR2vR8QuA3BqfSGy0t0PCA4wEu9U8334x4+QNB6vyhzDY72TEEvm+1tHYwvZicL1lcySuVMomW4sqjjyCbuVqlnVHBdfqlfVbOV6wrlrKbf2E4i1aWjTiIsv1atNw8r1hHPXUm7tJxBr0/jtmwbxuxBYdpjm3NmpZuJ3IbDsMI3KssNcxSh7M/pZWpW2Y4IjjNfefjUTvwuBZYc5+OkJ8bsQWHaYYzITvwuBZYc5uGlUI4bsMLddOqdIVubY71I5Y1h2mIMyG1uJve4S2NphkNQOtrVvF/zkyHYsEz4J8TKNckOzzrg20FZis2GmFIMnsjWRjpjhOO9JiPsyE78LgWWHOXankXOd+Jqb0pXip7n0lBMYp+xfS3CE8TTqV3PKdeLnhOFIGry6yRYjg9eC38XjAE9CvCMzuDt8NgRGU0sq29TIZGKTEO/I7BMY70llO07jIychlsz3vdkv0gM9uL6SWTI/UEwlFIuq9OL/5U4Dex9sIbGq+YnV3Dy452U9vnx6cDHv4cEzbe5ge/TezJMQ12m8rc9DOX9ec4ZGq8Lo8Tb4mYe/N/MkxGUa/wchsrT7B+sUvgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡⎡0  0  0⎤  ⎡0  0  0⎤  ⎡0  0  0⎤⎤\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎥\n",
       "⎢                               ⎥\n",
       "⎢⎡0  0  0⎤  ⎡0  0  0⎤  ⎡0  0  0⎤⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎥\n",
       "⎢                               ⎥\n",
       "⎢⎡0  0  0⎤  ⎡0  0  0⎤  ⎡0  0  0⎤⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎣⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculating Riemann Tensor from Christoffel Symbols\n",
    "rm1 = RiemannCurvatureTensor.from_christoffels(ch)\n",
    "rm1.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡⎡0  0  0⎤  ⎡0  0  0⎤  ⎡0  0  0⎤⎤\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎥\n",
       "⎢                               ⎥\n",
       "⎢⎡0  0  0⎤  ⎡0  0  0⎤  ⎡0  0  0⎤⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎥\n",
       "⎢                               ⎥\n",
       "⎢⎡0  0  0⎤  ⎡0  0  0⎤  ⎡0  0  0⎤⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎣⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculating Riemann Tensor from Metric Tensor\n",
    "rm2 = RiemannCurvatureTensor.from_metric(m_obj)\n",
    "rm2.tensor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the christoffel symbols for Schwarzschild Spacetime Metric\n",
    "  - The expressions are unsimplified"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{a}{r} + 1 & 0 & 0 & 0\\\\0 & - \\frac{1}{c^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\0 & 0 & - \\frac{r^{2}}{c^{2}} & 0\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin^{2}{\\left (\\theta \\right )}}{c^{2}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  a                                      ⎤\n",
       "⎢- ─ + 1       0         0         0      ⎥\n",
       "⎢  r                                      ⎥\n",
       "⎢                                         ⎥\n",
       "⎢             -1                          ⎥\n",
       "⎢   0     ────────────   0         0      ⎥\n",
       "⎢          2 ⎛  a    ⎞                    ⎥\n",
       "⎢         c ⋅⎜- ─ + 1⎟                    ⎥\n",
       "⎢            ⎝  r    ⎠                    ⎥\n",
       "⎢                                         ⎥\n",
       "⎢                         2               ⎥\n",
       "⎢                       -r                ⎥\n",
       "⎢   0          0        ────       0      ⎥\n",
       "⎢                         2               ⎥\n",
       "⎢                        c                ⎥\n",
       "⎢                                         ⎥\n",
       "⎢                               2    2    ⎥\n",
       "⎢                             -r ⋅sin (θ) ⎥\n",
       "⎢   0          0         0    ────────────⎥\n",
       "⎢                                   2     ⎥\n",
       "⎣                                  c      ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "syms = sympy.symbols(\"t r theta phi\")\n",
    "G, M, c, a = sympy.symbols(\"G M c a\")\n",
    "# using metric values of schwarschild space-time\n",
    "# a is schwarzschild radius\n",
    "list2d = [[0 for i in range(4)] for i in range(4)]\n",
    "list2d[0][0] = 1 - (a / syms[1])\n",
    "list2d[1][1] = -1 / ((1 - (a / syms[1])) * (c ** 2))\n",
    "list2d[2][2] = -1 * (syms[1] ** 2) / (c ** 2)\n",
    "list2d[3][3] = -1 * (syms[1] ** 2) * (sympy.sin(syms[2]) ** 2) / (c ** 2)\n",
    "sch = MetricTensor(list2d, syms)\n",
    "sch.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & - \\frac{1}{c^{2}} & 0 & 0\\\\0 & 0 & - \\frac{r^{2}}{c^{2}} & 0\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin^{2}{\\left (\\theta \\right )}}{c^{2}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   0    0         0      ⎤\n",
       "⎢                          ⎥\n",
       "⎢   -1                     ⎥\n",
       "⎢0  ───   0         0      ⎥\n",
       "⎢     2                    ⎥\n",
       "⎢    c                     ⎥\n",
       "⎢                          ⎥\n",
       "⎢          2               ⎥\n",
       "⎢        -r                ⎥\n",
       "⎢0   0   ────       0      ⎥\n",
       "⎢          2               ⎥\n",
       "⎢         c                ⎥\n",
       "⎢                          ⎥\n",
       "⎢                2    2    ⎥\n",
       "⎢              -r ⋅sin (θ) ⎥\n",
       "⎢0   0    0    ────────────⎥\n",
       "⎢                    2     ⎥\n",
       "⎣                   c      ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# single substitution\n",
    "subs1 = sch.subs(a,0)\n",
    "subs1.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & -1 & 0 & 0\\\\0 & 0 & - r^{2} & 0\\\\0 & 0 & 0 & - r^{2} \\sin^{2}{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0    0        0     ⎤\n",
       "⎢                       ⎥\n",
       "⎢0  -1   0        0     ⎥\n",
       "⎢                       ⎥\n",
       "⎢         2             ⎥\n",
       "⎢0  0   -r        0     ⎥\n",
       "⎢                       ⎥\n",
       "⎢              2    2   ⎥\n",
       "⎣0  0    0   -r ⋅sin (θ)⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# multiple substitution\n",
    "subs2 = sch.subs([(a,0), (c,1)])\n",
    "subs2.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\\\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}\\frac{a c^{2} \\left(- \\frac{a}{r} + 1\\right)}{2 r^{2}} & 0 & 0 & 0\\\\0 & - \\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\0 & 0 & - r \\left(- \\frac{a}{r} + 1\\right) & 0\\\\0 & 0 & 0 & - r \\left(- \\frac{a}{r} + 1\\right) \\sin^{2}{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}}\\\\0 & \\frac{1}{r} & \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                        ⎡   2 ⎛  a    ⎞                      \n",
       "⎢                                        ⎢a⋅c ⋅⎜- ─ + 1⎟                      \n",
       "⎢                                        ⎢     ⎝  r    ⎠                      \n",
       "⎢⎡                      a             ⎤  ⎢──────────────        0             \n",
       "⎢⎢      0         ──────────────  0  0⎥  ⎢        2                           \n",
       "⎢⎢                   2 ⎛  a    ⎞      ⎥  ⎢     2⋅r                            \n",
       "⎢⎢                2⋅r ⋅⎜- ─ + 1⎟      ⎥  ⎢                                    \n",
       "⎢⎢                     ⎝  r    ⎠      ⎥  ⎢                     -a             \n",
       "⎢⎢                                    ⎥  ⎢      0         ──────────────      \n",
       "⎢⎢      a                             ⎥  ⎢                   2 ⎛  a    ⎞      \n",
       "⎢⎢──────────────        0         0  0⎥  ⎢                2⋅r ⋅⎜- ─ + 1⎟      \n",
       "⎢⎢   2 ⎛  a    ⎞                      ⎥  ⎢                     ⎝  r    ⎠      \n",
       "⎢⎢2⋅r ⋅⎜- ─ + 1⎟                      ⎥  ⎢                                    \n",
       "⎢⎢     ⎝  r    ⎠                      ⎥  ⎢                                   ⎛\n",
       "⎢⎢                                    ⎥  ⎢      0               0         -r⋅⎜\n",
       "⎢⎢      0               0         0  0⎥  ⎢                                   ⎝\n",
       "⎢⎢                                    ⎥  ⎢                                    \n",
       "⎢⎣      0               0         0  0⎦  ⎢                                    \n",
       "⎢                                        ⎢      0               0             \n",
       "⎣                                        ⎣                                    \n",
       "\n",
       "                              ⎤                                               \n",
       "                              ⎥                                               \n",
       "                              ⎥                                               \n",
       " 0                 0          ⎥                                               \n",
       "                              ⎥                             ⎡0  0    0       0\n",
       "                              ⎥  ⎡0  0  0        0       ⎤  ⎢                 \n",
       "                              ⎥  ⎢                       ⎥  ⎢                1\n",
       "                              ⎥  ⎢      1                ⎥  ⎢0  0    0       ─\n",
       " 0                 0          ⎥  ⎢0  0  ─        0       ⎥  ⎢                r\n",
       "                              ⎥  ⎢      r                ⎥  ⎢                 \n",
       "                              ⎥  ⎢                       ⎥  ⎢              cos\n",
       "                              ⎥  ⎢   1                   ⎥  ⎢0  0    0     ───\n",
       "                              ⎥  ⎢0  ─  0        0       ⎥  ⎢              sin\n",
       "  a    ⎞                      ⎥  ⎢   r                   ⎥  ⎢                 \n",
       "- ─ + 1⎟           0          ⎥  ⎢                       ⎥  ⎢   1  cos(θ)     \n",
       "  r    ⎠                      ⎥  ⎣0  0  0  -sin(θ)⋅cos(θ)⎦  ⎢0  ─  ──────    0\n",
       "                              ⎥                             ⎣   r  sin(θ)     \n",
       "             ⎛  a    ⎞    2   ⎥                                               \n",
       " 0        -r⋅⎜- ─ + 1⎟⋅sin (θ)⎥                                               \n",
       "             ⎝  r    ⎠        ⎦                                               \n",
       "\n",
       "    ⎤\n",
       "    ⎥\n",
       "    ⎥\n",
       "    ⎥\n",
       "   ⎤⎥\n",
       "   ⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎥⎥\n",
       "(θ)⎥⎥\n",
       "───⎥⎥\n",
       "(θ)⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎦⎥\n",
       "    ⎥\n",
       "    ⎥\n",
       "    ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch_ch = ChristoffelSymbols.from_metric(sch)\n",
    "sch_ch.tensor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the simplified expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r \\left(- a + r\\right)} & 0 & 0\\\\\\frac{a}{2 r \\left(- a + r\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}\\frac{a c^{2} \\left(- a + r\\right)}{2 r^{3}} & 0 & 0 & 0\\\\0 & \\frac{a}{2 r \\left(a - r\\right)} & 0 & 0\\\\0 & 0 & a - r & 0\\\\0 & 0 & 0 & \\left(a - r\\right) \\sin^{2}{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\frac{\\sin{\\left (2 \\theta \\right )}}{2}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{1}{\\tan{\\left (\\theta \\right )}}\\\\0 & \\frac{1}{r} & \\frac{1}{\\tan{\\left (\\theta \\right )}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                    ⎡   2                                    \n",
       "⎢                                    ⎢a⋅c ⋅(-a + r)                           \n",
       "⎢⎡                   a            ⎤  ⎢─────────────       0         0         \n",
       "⎢⎢     0        ────────────  0  0⎥  ⎢        3                               \n",
       "⎢⎢              2⋅r⋅(-a + r)      ⎥  ⎢     2⋅r                                \n",
       "⎢⎢                                ⎥  ⎢                                        \n",
       "⎢⎢     a                          ⎥  ⎢                    a                   \n",
       "⎢⎢────────────       0        0  0⎥  ⎢      0        ───────────    0         \n",
       "⎢⎢2⋅r⋅(-a + r)                    ⎥  ⎢               2⋅r⋅(a - r)              \n",
       "⎢⎢                                ⎥  ⎢                                        \n",
       "⎢⎢     0             0        0  0⎥  ⎢      0             0       a - r       \n",
       "⎢⎢                                ⎥  ⎢                                        \n",
       "⎢⎣     0             0        0  0⎦  ⎢                                        \n",
       "⎣                                    ⎣      0             0         0    (a - \n",
       "\n",
       "          ⎤                                               ⎤\n",
       "          ⎥  ⎡0  0  0      0     ⎤  ⎡0  0    0       0   ⎤⎥\n",
       "  0       ⎥  ⎢                   ⎥  ⎢                    ⎥⎥\n",
       "          ⎥  ⎢      1            ⎥  ⎢                1   ⎥⎥\n",
       "          ⎥  ⎢0  0  ─      0     ⎥  ⎢0  0    0       ─   ⎥⎥\n",
       "          ⎥  ⎢      r            ⎥  ⎢                r   ⎥⎥\n",
       "          ⎥  ⎢                   ⎥  ⎢                    ⎥⎥\n",
       "  0       ⎥  ⎢   1               ⎥  ⎢                1   ⎥⎥\n",
       "          ⎥  ⎢0  ─  0      0     ⎥  ⎢0  0    0     ──────⎥⎥\n",
       "          ⎥  ⎢   r               ⎥  ⎢              tan(θ)⎥⎥\n",
       "  0       ⎥  ⎢                   ⎥  ⎢                    ⎥⎥\n",
       "          ⎥  ⎢         -sin(2⋅θ) ⎥  ⎢   1    1           ⎥⎥\n",
       "      2   ⎥  ⎢0  0  0  ──────────⎥  ⎢0  ─  ──────    0   ⎥⎥\n",
       "r)⋅sin (θ)⎦  ⎣             2     ⎦  ⎣   r  tan(θ)        ⎦⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplified = sch_ch.simplify()\n",
    "simplified"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}