{
 "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",
    "\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 Ricci Tensor(with both indices covariant)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 0 & 0 & 0\\\\0 & - 3 \\cos^{2}{\\left (t \\right )} & 0 & 0\\\\0 & 0 & \\left(\\sin^{2}{\\left (t \\right )} - 1\\right) \\sinh^{2}{\\left (\\chi \\right )} - 2 \\cos^{2}{\\left (t \\right )} \\sinh^{2}{\\left (\\chi \\right )} & 0\\\\0 & 0 & 0 & \\left(\\sin^{2}{\\left (t \\right )} - 1\\right) \\sin^{2}{\\left (\\theta \\right )} \\sinh^{2}{\\left (\\chi \\right )} - 2 \\sin^{2}{\\left (\\theta \\right )} \\cos^{2}{\\left (t \\right )} \\sinh^{2}{\\left (\\chi \\right )}\\end{matrix}\\right]$$"
      ],
      "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": [
       "$$-12$$"
      ],
      "text/plain": [
       "-12"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = RicciScalar.from_riccitensor(Ric)\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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}