{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Visualizing frame dragging in Kerr space-time\n",
    "\n",
    "### Importing required modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from astropy import units as u\n",
    "import numpy as np\n",
    "from einsteinpy.metric import Kerr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining position/velocity of test particle\n",
    "  - Initial velocity is kept 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = 1.989e30 * u.kg\n",
    "pos_vec = [50e5 * u.km, np.pi / 2 * u.rad, np.pi * u.rad]\n",
    "vel_vec = [0 * u.km / u.s, 0 * u.rad / u.s, 0 * u.rad / u.s]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "end_lambda = ((1 * u.year).to(u.s)).value / 930\n",
    "# Choosing stepsize for ODE solver to be 0.02 minutes\n",
    "stepsize = ((0.02 * u.min).to(u.s)).value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "starting_time = 0 * u.s\n",
    "a = 0.3\n",
    "obj = Kerr.from_BL(pos_vec, vel_vec, starting_time, M, a)\n",
    "ans = obj.calculate_trajectory(\n",
    "    end_lambda=end_lambda, OdeMethodKwargs={\"stepsize\": stepsize}, return_cartesian=True\n",
    ")\n",
    "x, y = ans[1][:,1], ans[1][:,2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting the trajectory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.scatter(x,y, s=0.2)\n",
    "plt.scatter(0,0, c='black')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### It can be seen that as the particle approaches the massive body, it acquires axial velocity due to spin and frame-dragging effect of the body."
   ]
  }
 ],
 "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
}