{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolically Understanding Christoffel Symbol and Riemann Metric Tensor using EinsteinPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "from einsteinpy.utils import christoffel\n",
    "from pprint import pprint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining the metric tensor for 3d spherical coordinates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "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)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the christoffel symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(theta)**2]],\n",
       " [[0, 1/r, 0], [1/r, 0, 0], [0, 0, -sin(theta)*cos(theta)]],\n",
       " [[0, 0, 1/r], [0, 0, cos(theta)/sin(theta)], [1/r, cos(theta)/sin(theta), 0]]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ch = christoffel.christoffels(metric, syms)\n",
    "ch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/r"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ch[1][1][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the Riemann Curvature tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       "  [[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       "  [[0, 0, 0], [0, 0, 0], [0, 0, 0]]],\n",
       " [[[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       "  [[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       "  [[0, 0, 0], [0, 0, 0], [0, 0, 0]]],\n",
       " [[[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       "  [[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       "  [[0, 0, 0], [0, 0, 0], [0, 0, 0]]]]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rm = christoffel.riemann_curvature_tensor(metric, syms)\n",
    "rm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the christoffel symbols for Kerr Spacetime Metric\n",
    "  - The expressions are unsimplified"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0,\n",
       " R*a*r*(2*R*a*r**2*sin(theta)**2/(a**2*cos(theta)**2 + r**2)**2 - R*a*sin(theta)**2/(a**2*cos(theta)**2 + r**2))*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)*(-R**2*a**2*r**2*sin(theta)**4/(a**2*cos(theta)**2 + r**2)**2 + (R*r/(a**2*cos(theta)**2 + r**2) - 1)*(-a**2*(-R*r + a**2 + r**2)*sin(theta)**2 + (a**2 + r**2)**2)*sin(theta)**2/(a**2*cos(theta)**2 + r**2))) + (-2*R*r**2/(a**2*cos(theta)**2 + r**2)**2 + R/(a**2*cos(theta)**2 + r**2))*(-a**2*(-R*r + a**2 + r**2)*sin(theta)**2 + (a**2 + r**2)**2)*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)*(-R**2*a**2*r**2*sin(theta)**4/(a**2*cos(theta)**2 + r**2)**2 + (R*r/(a**2*cos(theta)**2 + r**2) - 1)*(-a**2*(-R*r + a**2 + r**2)*sin(theta)**2 + (a**2 + r**2)**2)*sin(theta)**2/(a**2*cos(theta)**2 + r**2))),\n",
       " R*a**2*r*(-a**2*(-R*r + a**2 + r**2)*sin(theta)**2 + (a**2 + r**2)**2)*sin(theta)**3*cos(theta)/((a**2*cos(theta)**2 + r**2)**3*(-R**2*a**2*r**2*sin(theta)**4/(a**2*cos(theta)**2 + r**2)**2 + (R*r/(a**2*cos(theta)**2 + r**2) - 1)*(-a**2*(-R*r + a**2 + r**2)*sin(theta)**2 + (a**2 + r**2)**2)*sin(theta)**2/(a**2*cos(theta)**2 + r**2))) + R*a*r*(-2*R*a**3*r*sin(theta)**3*cos(theta)/(a**2*cos(theta)**2 + r**2)**2 - 2*R*a*r*sin(theta)*cos(theta)/(a**2*cos(theta)**2 + r**2))*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)*(-R**2*a**2*r**2*sin(theta)**4/(a**2*cos(theta)**2 + r**2)**2 + (R*r/(a**2*cos(theta)**2 + r**2) - 1)*(-a**2*(-R*r + a**2 + r**2)*sin(theta)**2 + (a**2 + r**2)**2)*sin(theta)**2/(a**2*cos(theta)**2 + r**2))),\n",
       " 0]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kerr_ch = christoffel.kerr_christoffels()\n",
    "kerr_ch[0][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the simplified expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[0,\n",
       "   R*(-a**4*cos(theta)**2 - a**2*r**2*cos(theta)**2 + a**2*r**2 + r**4)/(2*(a**2*cos(theta)**2 + r**2)**2*(-R*r + a**2 + r**2)),\n",
       "   -2*R*a**2*r*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2)**2,\n",
       "   0],\n",
       "  [R*(-a**4*cos(theta)**2 - a**2*r**2*cos(theta)**2 + a**2*r**2 + r**4)/(2*(a**2*cos(theta)**2 + r**2)**2*(-R*r + a**2 + r**2)),\n",
       "   0,\n",
       "   0,\n",
       "   R*a*(a**6*cos(theta)**4 - a**4*r**2*cos(theta)**4 - 4*a**2*r**4*cos(theta)**2 - a**2*r**4 - 3*r**6)*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)**3*(-R*r + a**2 + r**2))],\n",
       "  [-2*R*a**2*r*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2)**2,\n",
       "   0,\n",
       "   0,\n",
       "   R*a**3*r*sin(theta)**3*cos(theta)/(a**2*cos(theta)**2 + r**2)**2],\n",
       "  [0,\n",
       "   R*a*(a**6*cos(theta)**4 - a**4*r**2*cos(theta)**4 - 4*a**2*r**4*cos(theta)**2 - a**2*r**4 - 3*r**6)*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)**3*(-R*r + a**2 + r**2)),\n",
       "   R*a**3*r*sin(theta)**3*cos(theta)/(a**2*cos(theta)**2 + r**2)**2,\n",
       "   0]],\n",
       " [[R*(-a**2*cos(theta)**2 + r**2)*(-R*r + a**2 + r**2)/(2*(a**2*cos(theta)**2 + r**2)**3),\n",
       "   0,\n",
       "   0,\n",
       "   R*a*(a**2*cos(theta)**2 - r**2)*(-R*r + a**2 + r**2)*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)**3)],\n",
       "  [0,\n",
       "   (r*(-R*r + a**2 + r**2) + (R - 2*r)*(a**2*cos(theta)**2 + r**2)/2)/((a**2*cos(theta)**2 + r**2)*(-R*r + a**2 + r**2)),\n",
       "   -a**2*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2),\n",
       "   0],\n",
       "  [0,\n",
       "   -a**2*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2),\n",
       "   -r*(-R*r + a**2 + r**2)/(a**2*cos(theta)**2 + r**2),\n",
       "   0],\n",
       "  [R*a*(a**2*cos(theta)**2 - r**2)*(-R*r + a**2 + r**2)*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)**3),\n",
       "   0,\n",
       "   0,\n",
       "   (2*r*(a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2) + (a**2*cos(theta)**2 + r**2)*(a**2*(R - 2*r)*sin(theta)**2 + 4*r*(a**2 + r**2)))*(R*r - a**2 - r**2)*sin(theta)**2/(2*(a**2*cos(theta)**2 + r**2)**3)]],\n",
       " [[-4*R*a**2*r*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2)**3,\n",
       "   0,\n",
       "   0,\n",
       "   4*R*a*r*(a**2 + r**2)*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2)**3],\n",
       "  [0,\n",
       "   a**2*sin(2*theta)/((-R*r + a**2 + r**2)*(a**2*cos(2*theta) + a**2 + 2*r**2)),\n",
       "   r/(a**2*cos(theta)**2 + r**2),\n",
       "   0],\n",
       "  [0,\n",
       "   r/(a**2*cos(theta)**2 + r**2),\n",
       "   -a**2*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2),\n",
       "   0],\n",
       "  [4*R*a*r*(a**2 + r**2)*sin(2*theta)/(a**2*cos(2*theta) + a**2 + 2*r**2)**3,\n",
       "   0,\n",
       "   0,\n",
       "   (a**2*(a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2)*sin(theta)**2 + (a**2*cos(theta)**2 + r**2)*(2*a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2))*sin(theta)*cos(theta)/(a**2*cos(theta)**2 + r**2)**3]],\n",
       " [[0,\n",
       "   R*a*(-a**2*cos(theta)**2 + r**2)/(2*(a**2*cos(theta)**2 + r**2)**2*(-R*r + a**2 + r**2)),\n",
       "   -R*a*r/((a**2*cos(theta)**2 + r**2)**2*tan(theta)),\n",
       "   0],\n",
       "  [R*a*(-a**2*cos(theta)**2 + r**2)/(2*(a**2*cos(theta)**2 + r**2)**2*(-R*r + a**2 + r**2)),\n",
       "   0,\n",
       "   0,\n",
       "   (R**2*a**2*r*(a**2*cos(theta)**2 - r**2)*sin(theta)**2 + (2*r*(a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2) + (a**2*cos(theta)**2 + r**2)*(a**2*(R - 2*r)*sin(theta)**2 + 4*r*(a**2 + r**2)))*(-R*r + a**2*cos(theta)**2 + r**2))/(2*(a**2*cos(theta)**2 + r**2)**3*(-R*r + a**2 + r**2))],\n",
       "  [-R*a*r/((a**2*cos(theta)**2 + r**2)**2*tan(theta)),\n",
       "   0,\n",
       "   0,\n",
       "   (R**2*a**2*r**2*(a**2 + r**2)*sin(theta)**2 + (a**2*(a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2)*sin(theta)**2 + (a**2*cos(theta)**2 + r**2)*(2*a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2))*(R*r - a**2*cos(theta)**2 - r**2))/((a**2*cos(theta)**2 + r**2)**3*(-R*r + a**2 + r**2)*tan(theta))],\n",
       "  [0,\n",
       "   (R**2*a**2*r*(a**2*cos(theta)**2 - r**2)*sin(theta)**2 + (2*r*(a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2) + (a**2*cos(theta)**2 + r**2)*(a**2*(R - 2*r)*sin(theta)**2 + 4*r*(a**2 + r**2)))*(-R*r + a**2*cos(theta)**2 + r**2))/(2*(a**2*cos(theta)**2 + r**2)**3*(-R*r + a**2 + r**2)),\n",
       "   (R**2*a**2*r**2*(a**2 + r**2)*sin(theta)**2 + (a**2*(a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2)*sin(theta)**2 + (a**2*cos(theta)**2 + r**2)*(2*a**2*(-R*r + a**2 + r**2)*sin(theta)**2 - (a**2 + r**2)**2))*(R*r - a**2*cos(theta)**2 - r**2))/((a**2*cos(theta)**2 + r**2)**3*(-R*r + a**2 + r**2)*tan(theta)),\n",
       "   0]]]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kerr_ch_simplified = christoffel.simplify_christoffels(kerr_ch)\n",
    "kerr_ch_simplified"
   ]
  }
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