{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Contravariant & Covariant indices in Tensors (Symbolic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from einsteinpy.symbolic import SchwarzschildMetric, MetricTensor, ChristoffelSymbols, RiemannCurvatureTensor\n",
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Analysing the schwarzschild metric along with performing various operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch = SchwarzschildMetric()\n",
    "sch.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{r}{- a + r} & 0 & 0 & 0\\\\0 & \\frac{c^{2} \\left(a - r\\right)}{r} & 0 & 0\\\\0 & 0 & - \\frac{c^{2}}{r^{2}} & 0\\\\0 & 0 & 0 & - \\frac{c^{2}}{r^{2} \\sin^{2}{\\left (\\theta \\right )}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  r                                 ⎤\n",
       "⎢──────      0        0        0     ⎥\n",
       "⎢-a + r                              ⎥\n",
       "⎢                                    ⎥\n",
       "⎢         2                          ⎥\n",
       "⎢        c ⋅(a - r)                  ⎥\n",
       "⎢  0     ──────────   0        0     ⎥\n",
       "⎢            r                       ⎥\n",
       "⎢                                    ⎥\n",
       "⎢                      2             ⎥\n",
       "⎢                    -c              ⎥\n",
       "⎢  0         0       ────      0     ⎥\n",
       "⎢                      2             ⎥\n",
       "⎢                     r              ⎥\n",
       "⎢                                    ⎥\n",
       "⎢                               2    ⎥\n",
       "⎢                             -c     ⎥\n",
       "⎢  0         0        0    ──────────⎥\n",
       "⎢                           2    2   ⎥\n",
       "⎣                          r ⋅sin (θ)⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch_inv = sch.inv()\n",
    "sch_inv.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNuzKriUQQ7912ZlSugimNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAT0lEQVQIHWNgVDIJYGAQY2D/wcCQycCwioFh/QSGfAGGbAaGfKAEA8N+EMH6B0RyK4BIJRDBqwAinzMwCjCwFjDwTGCQVlICarH///8zAwAXpw31OpgbMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch.order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'ll'"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch.config"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Obtaining Christoffel Symbols from Metric Tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\\\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}\\frac{a c^{2} \\left(- \\frac{a}{r} + 1\\right)}{2 r^{2}} & 0 & 0 & 0\\\\0 & - \\frac{a}{2 r^{2} \\left(- \\frac{a}{r} + 1\\right)} & 0 & 0\\\\0 & 0 & - r \\left(- \\frac{a}{r} + 1\\right) & 0\\\\0 & 0 & 0 & - r \\left(- \\frac{a}{r} + 1\\right) \\sin^{2}{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}}\\\\0 & \\frac{1}{r} & \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "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": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chr = ChristoffelSymbols.from_metric(sch_inv) # can be initialized from sch also\n",
    "chr.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'ull'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chr.config"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Changing the first index to covariant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r^{2}} & 0 & 0\\\\\\frac{a}{2 r^{2}} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}- \\frac{a}{2 r^{2}} & 0 & 0 & 0\\\\0 & \\frac{a}{2 c^{2} \\left(a - r\\right)^{2}} & 0 & 0\\\\0 & 0 & \\frac{r}{c^{2}} & 0\\\\0 & 0 & 0 & \\frac{r \\sin^{2}{\\left (\\theta \\right )}}{c^{2}}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & - \\frac{r}{c^{2}} & 0\\\\0 & - \\frac{r}{c^{2}} & 0 & 0\\\\0 & 0 & 0 & \\frac{r^{2} \\sin{\\left (2 \\theta \\right )}}{2 c^{2}}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & - \\frac{r \\sin^{2}{\\left (\\theta \\right )}}{c^{2}}\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin{\\left (2 \\theta \\right )}}{2 c^{2}}\\\\0 & - \\frac{r \\sin^{2}{\\left (\\theta \\right )}}{c^{2}} & - \\frac{r^{2} \\sin{\\left (2 \\theta \\right )}}{2 c^{2}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                    ⎡-a                                ⎤                     \n",
       "⎢                    ⎢────        0        0       0    ⎥                     \n",
       "⎢                    ⎢   2                              ⎥  ⎡0   0    0        \n",
       "⎢                    ⎢2⋅r                               ⎥  ⎢                  \n",
       "⎢⎡       a        ⎤  ⎢                                  ⎥  ⎢        -r        \n",
       "⎢⎢ 0    ────  0  0⎥  ⎢            a                     ⎥  ⎢0   0   ───       \n",
       "⎢⎢         2      ⎥  ⎢ 0    ─────────────  0       0    ⎥  ⎢          2       \n",
       "⎢⎢      2⋅r       ⎥  ⎢         2        2               ⎥  ⎢         c        \n",
       "⎢⎢                ⎥  ⎢      2⋅c ⋅(a - r)                ⎥  ⎢                  \n",
       "⎢⎢ a              ⎥  ⎢                                  ⎥  ⎢   -r             \n",
       "⎢⎢────   0    0  0⎥  ⎢                     r            ⎥  ⎢0  ───   0        \n",
       "⎢⎢   2            ⎥  ⎢ 0          0        ──      0    ⎥  ⎢     2            \n",
       "⎢⎢2⋅r             ⎥  ⎢                      2           ⎥  ⎢    c             \n",
       "⎢⎢                ⎥  ⎢                     c            ⎥  ⎢                  \n",
       "⎢⎢ 0     0    0  0⎥  ⎢                                  ⎥  ⎢              2   \n",
       "⎢⎢                ⎥  ⎢                              2   ⎥  ⎢             r ⋅si\n",
       "⎢⎣ 0     0    0  0⎦  ⎢                         r⋅sin (θ)⎥  ⎢0   0    0   ─────\n",
       "⎢                    ⎢ 0          0        0   ─────────⎥  ⎢                  \n",
       "⎢                    ⎢                              2   ⎥  ⎣                 2\n",
       "⎣                    ⎣                             c    ⎦                     \n",
       "\n",
       "                                                       ⎤\n",
       "         ⎡0       0             0              0      ⎤⎥\n",
       "0     ⎤  ⎢                                            ⎥⎥\n",
       "      ⎥  ⎢                                      2     ⎥⎥\n",
       "      ⎥  ⎢                                -r⋅sin (θ)  ⎥⎥\n",
       "0     ⎥  ⎢0       0             0         ─────────── ⎥⎥\n",
       "      ⎥  ⎢                                      2     ⎥⎥\n",
       "      ⎥  ⎢                                     c      ⎥⎥\n",
       "      ⎥  ⎢                                            ⎥⎥\n",
       "      ⎥  ⎢                                 2          ⎥⎥\n",
       "0     ⎥  ⎢                               -r ⋅sin(2⋅θ) ⎥⎥\n",
       "      ⎥  ⎢0       0             0        ─────────────⎥⎥\n",
       "      ⎥  ⎢                                       2    ⎥⎥\n",
       "      ⎥  ⎢                                    2⋅c     ⎥⎥\n",
       "      ⎥  ⎢                                            ⎥⎥\n",
       "n(2⋅θ)⎥  ⎢         2        2                         ⎥⎥\n",
       "──────⎥  ⎢   -r⋅sin (θ)   -r ⋅sin(2⋅θ)                ⎥⎥\n",
       "  2   ⎥  ⎢0  ───────────  ─────────────        0      ⎥⎥\n",
       "⋅c    ⎦  ⎢         2              2                   ⎥⎥\n",
       "         ⎣        c            2⋅c                    ⎦⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_chr = chr.change_config('lll') # changing the configuration to (covariant, covariant, covariant)\n",
    "new_chr.tensor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'lll'"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_chr.config"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Any arbitary index configuration would also work!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{a}{2 r \\left(- a + r\\right)} & 0 & 0\\\\\\frac{a c^{2} \\left(a - r\\right)}{2 r^{3}} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}\\frac{a}{2 r \\left(a - r\\right)} & 0 & 0 & 0\\\\0 & \\frac{a}{2 r \\left(a - r\\right)} & 0 & 0\\\\0 & 0 & - \\frac{1}{r} & 0\\\\0 & 0 & 0 & - \\frac{1}{r}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & - a + r & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\frac{1}{\\tan{\\left (\\theta \\right )}}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\left(- a + r\\right) \\sin^{2}{\\left (\\theta \\right )}\\\\0 & 0 & 0 & \\frac{\\sin{\\left (2 \\theta \\right )}}{2}\\\\0 & \\frac{1}{r} & \\frac{1}{\\tan{\\left (\\theta \\right )}} & 0\\end{matrix}\\right]\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                    ⎡     a                            ⎤     \n",
       "⎢⎡                   a            ⎤  ⎢───────────       0        0    0 ⎥     \n",
       "⎢⎢     0        ────────────  0  0⎥  ⎢2⋅r⋅(a - r)                       ⎥  ⎡0 \n",
       "⎢⎢              2⋅r⋅(-a + r)      ⎥  ⎢                                  ⎥  ⎢  \n",
       "⎢⎢                                ⎥  ⎢                  a               ⎥  ⎢0 \n",
       "⎢⎢   2                            ⎥  ⎢     0       ───────────   0    0 ⎥  ⎢  \n",
       "⎢⎢a⋅c ⋅(a - r)                    ⎥  ⎢             2⋅r⋅(a - r)          ⎥  ⎢  \n",
       "⎢⎢────────────       0        0  0⎥  ⎢                                  ⎥  ⎢0 \n",
       "⎢⎢       3                        ⎥  ⎢                          -1      ⎥  ⎢  \n",
       "⎢⎢    2⋅r                         ⎥  ⎢     0            0       ───   0 ⎥  ⎢  \n",
       "⎢⎢                                ⎥  ⎢                           r      ⎥  ⎢  \n",
       "⎢⎢     0             0        0  0⎥  ⎢                                  ⎥  ⎢0 \n",
       "⎢⎢                                ⎥  ⎢                               -1 ⎥  ⎣  \n",
       "⎢⎣     0             0        0  0⎦  ⎢     0            0        0   ───⎥     \n",
       "⎣                                    ⎣                                r ⎦     \n",
       "\n",
       "                                                     ⎤\n",
       "                     ⎡0  0    0            0        ⎤⎥\n",
       " 0    0       0   ⎤  ⎢                              ⎥⎥\n",
       "                  ⎥  ⎢                          2   ⎥⎥\n",
       " 0  -a + r    0   ⎥  ⎢0  0    0     (-a + r)⋅sin (θ)⎥⎥\n",
       "                  ⎥  ⎢                              ⎥⎥\n",
       " 1                ⎥  ⎢                  sin(2⋅θ)    ⎥⎥\n",
       " ─    0       0   ⎥  ⎢0  0    0         ────────    ⎥⎥\n",
       " r                ⎥  ⎢                     2        ⎥⎥\n",
       "                  ⎥  ⎢                              ⎥⎥\n",
       "             -1   ⎥  ⎢   1    1                     ⎥⎥\n",
       " 0    0     ──────⎥  ⎢0  ─  ──────         0        ⎥⎥\n",
       "            tan(θ)⎦  ⎣   r  tan(θ)                  ⎦⎥\n",
       "                                                     ⎥\n",
       "                                                     ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_chr2 = new_chr.change_config('lul')\n",
    "new_chr2.tensor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Obtaining Riemann Tensor from Christoffel Symbols and manipulating it's indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & \\frac{a}{r^{2} \\left(a - r\\right)} & 0 & 0\\\\0 & 0 & \\frac{a}{2 r} & 0\\\\0 & 0 & 0 & \\frac{a \\sin^{2}{\\left (\\theta \\right )}}{2 r}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0      0        0       0    ⎤\n",
       "⎢                             ⎥\n",
       "⎢       a                     ⎥\n",
       "⎢0  ──────────   0       0    ⎥\n",
       "⎢    2                        ⎥\n",
       "⎢   r ⋅(a - r)                ⎥\n",
       "⎢                             ⎥\n",
       "⎢                a            ⎥\n",
       "⎢0      0       ───      0    ⎥\n",
       "⎢               2⋅r           ⎥\n",
       "⎢                             ⎥\n",
       "⎢                         2   ⎥\n",
       "⎢                    a⋅sin (θ)⎥\n",
       "⎢0      0        0   ─────────⎥\n",
       "⎣                       2⋅r   ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rm = RiemannCurvatureTensor.from_christoffels(new_chr2)\n",
    "rm[0,0,:,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'ulll'"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rm.config"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & - \\frac{a c^{4}}{r^{3}} & 0 & 0\\\\0 & 0 & \\frac{a c^{4}}{2 r^{4} \\left(- a + r\\right)} & 0\\\\0 & 0 & 0 & \\frac{a c^{4}}{2 r^{4} \\left(- a + r\\right) \\sin^{2}{\\left (\\theta \\right )}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0    0           0                  0          ⎤\n",
       "⎢                                               ⎥\n",
       "⎢       4                                       ⎥\n",
       "⎢   -a⋅c                                        ⎥\n",
       "⎢0  ──────        0                  0          ⎥\n",
       "⎢      3                                        ⎥\n",
       "⎢     r                                         ⎥\n",
       "⎢                                               ⎥\n",
       "⎢                   4                           ⎥\n",
       "⎢                a⋅c                            ⎥\n",
       "⎢0    0     ─────────────            0          ⎥\n",
       "⎢              4                                ⎥\n",
       "⎢           2⋅r ⋅(-a + r)                       ⎥\n",
       "⎢                                               ⎥\n",
       "⎢                                      4        ⎥\n",
       "⎢                                   a⋅c         ⎥\n",
       "⎢0    0           0        ─────────────────────⎥\n",
       "⎢                             4             2   ⎥\n",
       "⎣                          2⋅r ⋅(-a + r)⋅sin (θ)⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rm2 = rm.change_config(\"uuuu\")\n",
    "rm2[0,0,:,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & \\frac{a c^{2}}{r^{3}} & 0 & 0\\\\0 & 0 & - \\frac{a c^{2}}{2 r^{3}} & 0\\\\0 & 0 & 0 & - \\frac{a c^{2}}{2 r^{3}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0   0      0       0   ⎤\n",
       "⎢                       ⎥\n",
       "⎢      2                ⎥\n",
       "⎢   a⋅c                 ⎥\n",
       "⎢0  ────    0       0   ⎥\n",
       "⎢     3                 ⎥\n",
       "⎢    r                  ⎥\n",
       "⎢                       ⎥\n",
       "⎢             2         ⎥\n",
       "⎢         -a⋅c          ⎥\n",
       "⎢0   0    ──────    0   ⎥\n",
       "⎢             3         ⎥\n",
       "⎢          2⋅r          ⎥\n",
       "⎢                       ⎥\n",
       "⎢                     2 ⎥\n",
       "⎢                 -a⋅c  ⎥\n",
       "⎢0   0      0     ──────⎥\n",
       "⎢                     3 ⎥\n",
       "⎣                  2⋅r  ⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rm3 = rm2.change_config(\"lulu\")\n",
    "rm3[0,0,:,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & \\frac{a}{r^{2} \\left(a - r\\right)} & 0 & 0\\\\0 & 0 & \\frac{a}{2 r} & 0\\\\0 & 0 & 0 & \\frac{a \\sin^{2}{\\left (\\theta \\right )}}{2 r}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0      0        0       0    ⎤\n",
       "⎢                             ⎥\n",
       "⎢       a                     ⎥\n",
       "⎢0  ──────────   0       0    ⎥\n",
       "⎢    2                        ⎥\n",
       "⎢   r ⋅(a - r)                ⎥\n",
       "⎢                             ⎥\n",
       "⎢                a            ⎥\n",
       "⎢0      0       ───      0    ⎥\n",
       "⎢               2⋅r           ⎥\n",
       "⎢                             ⎥\n",
       "⎢                         2   ⎥\n",
       "⎢                    a⋅sin (θ)⎥\n",
       "⎢0      0        0   ─────────⎥\n",
       "⎣                       2⋅r   ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rm4 = rm3.change_config(\"ulll\")\n",
    "rm4[0,0,:,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### It is seen that `rm` and `rm4` are same as they have the same configuration"
   ]
  }
 ],
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