{
 "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 einsteinpy.symbolic import RicciTensor, RicciScalar\n",
    "from einsteinpy.symbolic.predefined import AntiDeSitter\n",
    "\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": [
       "$\\displaystyle \\left[\\begin{matrix}3 & 0 & 0 & 0\\\\0 & - 3 \\cos^{2}{\\left(t \\right)} & 0 & 0\\\\0 & 0 & - 3 \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)} & 0\\\\0 & 0 & 0 & - 3 \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡3      0                0                        0             ⎤\n",
       "⎢                                                               ⎥\n",
       "⎢         2                                                     ⎥\n",
       "⎢0  -3⋅cos (t)           0                        0             ⎥\n",
       "⎢                                                               ⎥\n",
       "⎢                     2        2                                ⎥\n",
       "⎢0      0       -3⋅cos (t)⋅sinh (χ)               0             ⎥\n",
       "⎢                                                               ⎥\n",
       "⎢                                          2       2        2   ⎥\n",
       "⎣0      0                0           -3⋅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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}