{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Einstein Tensor calculations using Symbolic module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy import symbols, sin, cos, sinh\n",
    "from einsteinpy.symbolic import EinsteinTensor, MetricTensor\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": [],
   "source": [
    "syms = sympy.symbols(\"t chi theta phi\")\n",
    "t, ch, th, ph = syms\n",
    "m = sympy.diag(-1, cos(t) ** 2, cos(t) ** 2 * sinh(ch) ** 2, cos(t) ** 2 * sinh(ch) ** 2 * sin(th) ** 2).tolist()\n",
    "metric = MetricTensor(m, syms)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the Einstein Tensor (with both indices covariant)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3.0 & 0 & 0 & 0\\\\0 & 3.0 \\cos^{2}{\\left(t \\right)} & 0 & 0\\\\0 & 0 & 3.0 \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)} & 0\\\\0 & 0 & 0 & 3.0 \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-3.0       0                0                         0              ⎤\n",
       "⎢                                                                     ⎥\n",
       "⎢             2                                                       ⎥\n",
       "⎢ 0    3.0⋅cos (t)           0                         0              ⎥\n",
       "⎢                                                                     ⎥\n",
       "⎢                          2        2                                 ⎥\n",
       "⎢ 0         0       3.0⋅cos (t)⋅sinh (χ)               0              ⎥\n",
       "⎢                                                                     ⎥\n",
       "⎢                                                2       2        2   ⎥\n",
       "⎣ 0         0                0            3.0⋅sin (θ)⋅cos (t)⋅sinh (χ)⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "einst = EinsteinTensor.from_metric(metric)\n",
    "einst.tensor()"
   ]
  }
 ],
 "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
}