{ "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\n", "from einsteinpy.coordinates import BoyerLindquistDifferential" ] }, { "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", "a = 0.3 * u.m\n", "BL_obj = BoyerLindquistDifferential(50e5 * u.km, np.pi / 2 * u.rad, np.pi * u.rad,\n", " 0 * u.km / u.s, 0 * u.rad / u.s, 0 * u.rad / u.s,\n", " a)" ] }, { "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": [ { "name": "stderr", "output_type": "stream", "text": [ "/home/shreyas/Softwares/anaconda3/lib/python3.7/site-packages/scipy/integrate/_ivp/common.py:32: UserWarning: The following arguments have no effect for a chosen solver: `first_step`.\n", " .format(\", \".join(\"`{}`\".format(x) for x in extraneous)))\n" ] } ], "source": [ "obj = Kerr.from_coords(BL_obj, M)\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", 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" ] }, "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": "3.7.2" } }, "nbformat": 4, "nbformat_minor": 2 }