{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ricci Tensor and Scalar Curvature calculations using Symbolic module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy import cos, sin, sinh\n",
    "from einsteinpy.symbolic import MetricTensor, RicciTensor, RicciScalar\n",
    "from einsteinpy.symbolic.predefined import AntiDeSitter\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining the Anti-de Sitter spacetime Metric"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & \\cos^{2}{\\left(t \\right)} & 0 & 0\\\\0 & 0 & \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)} & 0\\\\0 & 0 & 0 & \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-1     0            0                     0            ⎤\n",
       "⎢                                                       ⎥\n",
       "⎢       2                                               ⎥\n",
       "⎢0   cos (t)         0                     0            ⎥\n",
       "⎢                                                       ⎥\n",
       "⎢                2        2                             ⎥\n",
       "⎢0      0     cos (t)⋅sinh (χ)             0            ⎥\n",
       "⎢                                                       ⎥\n",
       "⎢                                  2       2        2   ⎥\n",
       "⎣0      0            0          sin (θ)⋅cos (t)⋅sinh (χ)⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "metric = AntiDeSitter()\n",
    "metric.tensor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the Ricci Tensor(with both indices covariant)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 & 0 & 0 & 0\\\\0 & - 3 \\cos^{2}{\\left(t \\right)} & 0 & 0\\\\0 & 0 & \\left(\\sin^{2}{\\left(t \\right)} - 1\\right) \\sinh^{2}{\\left(\\chi \\right)} - 2 \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)} & 0\\\\0 & 0 & 0 & \\left(\\sin^{2}{\\left(t \\right)} - 1\\right) \\sin^{2}{\\left(\\theta \\right)} \\sinh^{2}{\\left(\\chi \\right)} - 2 \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡3      0                            0                                        \n",
       "⎢                                                                             \n",
       "⎢         2                                                                   \n",
       "⎢0  -3⋅cos (t)                       0                                        \n",
       "⎢                                                                             \n",
       "⎢               ⎛   2       ⎞     2           2        2                      \n",
       "⎢0      0       ⎝sin (t) - 1⎠⋅sinh (χ) - 2⋅cos (t)⋅sinh (χ)                   \n",
       "⎢                                                                             \n",
       "⎢                                                            ⎛   2       ⎞    \n",
       "⎣0      0                            0                       ⎝sin (t) - 1⎠⋅sin\n",
       "\n",
       "            0                             ⎤\n",
       "                                          ⎥\n",
       "                                          ⎥\n",
       "            0                             ⎥\n",
       "                                          ⎥\n",
       "                                          ⎥\n",
       "            0                             ⎥\n",
       "                                          ⎥\n",
       "2        2           2       2        2   ⎥\n",
       " (θ)⋅sinh (χ) - 2⋅sin (θ)⋅cos (t)⋅sinh (χ)⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric = RicciTensor.from_metric(metric)\n",
    "Ric.tensor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the Ricci Scalar(Scalar Curvature) from the Ricci Tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -12$"
      ],
      "text/plain": [
       "-12"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = RicciScalar.from_riccitensor(Ric)\n",
    "R.simplify()\n",
    "R.expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The curavture is -12 which is in-line with the theoretical results"
   ]
  }
 ],
 "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}