{
 "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": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & r^{2} & 0\\\\0 & 0 & r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "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": [
       "$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & - r & 0\\\\0 & 0 & - r \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & \\frac{1}{r} & 0\\\\\\frac{1}{r} & 0 & 0\\\\0 & 0 & - \\sin{\\left(\\theta \\right)} \\cos{\\left(\\theta \\right)}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & \\frac{1}{r}\\\\0 & 0 & \\frac{\\cos{\\left(\\theta \\right)}}{\\sin{\\left(\\theta \\right)}}\\\\\\frac{1}{r} & \\frac{\\cos{\\left(\\theta \\right)}}{\\sin{\\left(\\theta \\right)}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
      ],
      "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": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAArCAYAAACuAHIQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVQ4Ee2V0Q2CQAyGD+MA6gi4AToCjmAcgQ30EV51A3UDdQPdwMQNZATjBvj9yJFDiL6qoclv2+tv6d2V4mVZZlxJkiTA34MR9t2NdeWw2ENtwQ2MgQ9qYsnKMFWUP85Ryl6TTm3lzUJLdg/nS07Ds43ENat51CPqDekUXMCZ2AptSrKcT/IlG/z7Mrw4jquz4M3NtDfoHs4v9kY+n7UJXkrN5BkIsCfAx14Aze4r/sbdYMiCgga9RkXoSD6y1I+d/Hr1lUGijD7EYe49x8JGti1jgL0rgiLbjHpKaVe6jkAI8Qj62PZJRQ5j3Jq1qI9Q2kRU8JWszAcFmqQkk021CiqjUUoyURFVwqmRyeID5cdO2vYpfTUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{r}$"
      ],
      "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": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
      ],
      "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": [
       "$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\\\\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
      ],
      "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": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{a}{r} + 1 & 0 & 0 & 0\\\\0 & - \\frac{1}{c^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\0 & 0 & - \\frac{r^{2}}{c^{2}} & 0\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin^{2}{\\left(\\theta \\right)}}{c^{2}}\\end{matrix}\\right]$"
      ],
      "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": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & - \\frac{1}{c^{2}} & 0 & 0\\\\0 & 0 & - \\frac{r^{2}}{c^{2}} & 0\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin^{2}{\\left(\\theta \\right)}}{c^{2}}\\end{matrix}\\right]$"
      ],
      "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": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & -1 & 0 & 0\\\\0 & 0 & - r^{2} & 0\\\\0 & 0 & 0 & - r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "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": [
       "$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\\\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}- \\frac{a \\left(\\frac{a c^{2}}{2 r} - \\frac{c^{2}}{2}\\right)}{r^{2}} & 0 & 0 & 0\\\\0 & \\frac{a \\left(\\frac{a c^{2}}{2 r} - \\frac{c^{2}}{2}\\right)}{c^{2} r^{2} \\left(- \\frac{a}{r} + 1\\right)^{2}} & 0 & 0\\\\0 & 0 & \\frac{2 r \\left(\\frac{a c^{2}}{2 r} - \\frac{c^{2}}{2}\\right)}{c^{2}} & 0\\\\0 & 0 & 0 & \\frac{2 r \\left(\\frac{a c^{2}}{2 r} - \\frac{c^{2}}{2}\\right) \\sin^{2}{\\left(\\theta \\right)}}{c^{2}}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\sin{\\left(\\theta \\right)} \\cos{\\left(\\theta \\right)}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{\\cos{\\left(\\theta \\right)}}{\\sin{\\left(\\theta \\right)}}\\\\0 & \\frac{1}{r} & \\frac{\\cos{\\left(\\theta \\right)}}{\\sin{\\left(\\theta \\right)}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                        ⎡   ⎛   2    2⎞                      \n",
       "⎢                                        ⎢   ⎜a⋅c    c ⎟                      \n",
       "⎢                                        ⎢-a⋅⎜──── - ──⎟                      \n",
       "⎢                                        ⎢   ⎝2⋅r    2 ⎠                      \n",
       "⎢                                        ⎢───────────────         0           \n",
       "⎢                                        ⎢        2                           \n",
       "⎢                                        ⎢       r                            \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢                    ⎛   2    2⎞     \n",
       "⎢⎡                      a             ⎤  ⎢                    ⎜a⋅c    c ⎟     \n",
       "⎢⎢      0         ──────────────  0  0⎥  ⎢                  a⋅⎜──── - ──⎟     \n",
       "⎢⎢                   2 ⎛  a    ⎞      ⎥  ⎢                    ⎝2⋅r    2 ⎠     \n",
       "⎢⎢                2⋅r ⋅⎜- ─ + 1⎟      ⎥  ⎢       0         ────────────────   \n",
       "⎢⎢                     ⎝  r    ⎠      ⎥  ⎢                                2   \n",
       "⎢⎢                                    ⎥  ⎢                  2  2 ⎛  a    ⎞    \n",
       "⎢⎢      a                             ⎥  ⎢                 c ⋅r ⋅⎜- ─ + 1⎟    \n",
       "⎢⎢──────────────        0         0  0⎥  ⎢                       ⎝  r    ⎠    \n",
       "⎢⎢   2 ⎛  a    ⎞                      ⎥  ⎢                                    \n",
       "⎢⎢2⋅r ⋅⎜- ─ + 1⎟                      ⎥  ⎢                                    \n",
       "⎢⎢     ⎝  r    ⎠                      ⎥  ⎢                                    \n",
       "⎢⎢                                    ⎥  ⎢                                   2\n",
       "⎢⎢      0               0         0  0⎥  ⎢                                    \n",
       "⎢⎢                                    ⎥  ⎢       0                0          ─\n",
       "⎢⎣      0               0         0  0⎦  ⎢                                    \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢                                    \n",
       "⎢                                        ⎢       0                0           \n",
       "⎢                                        ⎢                                    \n",
       "⎣                                        ⎣                                    \n",
       "\n",
       "                                       ⎤                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                                      \n",
       "      0                    0           ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                                       ⎥                             ⎡0  0    \n",
       "                                       ⎥  ⎡0  0  0        0       ⎤  ⎢        \n",
       "      0                    0           ⎥  ⎢                       ⎥  ⎢        \n",
       "                                       ⎥  ⎢      1                ⎥  ⎢0  0    \n",
       "                                       ⎥  ⎢0  0  ─        0       ⎥  ⎢        \n",
       "                                       ⎥  ⎢      r                ⎥  ⎢        \n",
       "                                       ⎥  ⎢                       ⎥  ⎢        \n",
       "                                       ⎥  ⎢   1                   ⎥  ⎢0  0    \n",
       "   ⎛   2    2⎞                         ⎥  ⎢0  ─  0        0       ⎥  ⎢        \n",
       "   ⎜a⋅c    c ⎟                         ⎥  ⎢   r                   ⎥  ⎢        \n",
       "⋅r⋅⎜──── - ──⎟                         ⎥  ⎢                       ⎥  ⎢   1  co\n",
       "   ⎝2⋅r    2 ⎠                         ⎥  ⎣0  0  0  -sin(θ)⋅cos(θ)⎦  ⎢0  ─  ──\n",
       "──────────────             0           ⎥                             ⎣   r  si\n",
       "       2                               ⎥                                      \n",
       "      c                                ⎥                                      \n",
       "                                       ⎥                                      \n",
       "                    ⎛   2    2⎞        ⎥                                      \n",
       "                    ⎜a⋅c    c ⎟    2   ⎥                                      \n",
       "                2⋅r⋅⎜──── - ──⎟⋅sin (θ)⎥                                      \n",
       "                    ⎝2⋅r    2 ⎠        ⎥                                      \n",
       "      0         ───────────────────────⎥                                      \n",
       "                            2          ⎥                                      \n",
       "                           c           ⎦                                      \n",
       "\n",
       "             ⎤\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "0       0   ⎤⎥\n",
       "            ⎥⎥\n",
       "        1   ⎥⎥\n",
       "0       ─   ⎥⎥\n",
       "        r   ⎥⎥\n",
       "            ⎥⎥\n",
       "      cos(θ)⎥⎥\n",
       "0     ──────⎥⎥\n",
       "      sin(θ)⎥⎥\n",
       "            ⎥⎥\n",
       "s(θ)        ⎥⎥\n",
       "────    0   ⎥⎥\n",
       "n(θ)        ⎦⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\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": [
       "$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r \\left(- a + r\\right)} & 0 & 0\\\\\\frac{a}{2 r \\left(- a + r\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}\\frac{a c^{2} \\left(- a + r\\right)}{2 r^{3}} & 0 & 0 & 0\\\\0 & \\frac{a}{2 r \\left(a - r\\right)} & 0 & 0\\\\0 & 0 & a - r & 0\\\\0 & 0 & 0 & \\left(a - r\\right) \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\frac{\\sin{\\left(2 \\theta \\right)}}{2}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{1}{\\tan{\\left(\\theta \\right)}}\\\\0 & \\frac{1}{r} & \\frac{1}{\\tan{\\left(\\theta \\right)}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
      ],
      "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}