{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Visualizing Event Horizon and Ergosphere (Singularities) of Kerr Metric or Black Hole\n", "\n", "### Importing required modules" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import astropy.units as u\n", "import numpy as np\n", "import astropy.units as u\n", "import matplotlib.pyplot as plt\n", "\n", "from einsteinpy.coordinates import BoyerLindquistDifferential\n", "from einsteinpy.metric import Kerr" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Defining black holes and obtaining singularities" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "{'inner_ergosphere': ._in_ergo at 0x000002AAAB94D438>, 'inner_horizon': 247.98878315867296, 'outer_horizon': 5692.939432136259, 'outer_ergosphere': ._out_ergo at 0x000002AAAB94DCA8>}\n", "\n", "{'inner_ergosphere': ._in_ergo at 0x000002AAAA4D5A68>, 'inner_horizon': 1675.668821582463, 'outer_horizon': 4265.259393712469, 'outer_ergosphere': ._out_ergo at 0x000002AAAB952558>}\n" ] } ], "source": [ "# Metric or Black Hole parameters - Mass, M and Spin Parameter, a\n", "M = 4e30 * u.kg\n", "a1 = 0.4 * u.one\n", "a2 = 0.9 * u.one # Extremal Kerr Black Hole\n", "\n", "# Coordinate object to initialize metric with\n", "# Note that, for this example\n", "# the coordinate values below are irrelevant\n", "bl = BoyerLindquistDifferential(\n", " t=0. * u.s,\n", " r=1e3 * u.m,\n", " theta=np.pi / 2 * u.rad,\n", " phi=np.pi * u.rad,\n", " v_r=0. * u.m / u.s,\n", " v_th=0. * u.rad / u.s,\n", " v_p=0. * u.rad / u.s,\n", ")\n", "\n", "# Defining two Kerr Black Holes, one with a higher spin parameter\n", "kerr1 = Kerr(coords=bl, M=M, a=a1)\n", "kerr2 = Kerr(coords=bl, M=M, a=a2)\n", "\n", "# Getting the list of singularities\n", "sing_dict1 = kerr1.singularities()\n", "sing_dict2 = kerr2.singularities()\n", "\n", "# Let's check the contents of the dicts\n", "# 'ergosphere' entries should be functions\n", "print(sing_dict1, sing_dict2, sep=\"\\n\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Preparing singularities for plotting" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "scrolled": false }, "outputs": [], "source": [ "# Sampling Polar Angle for plotting in Polar Coordinates\n", "theta = np.linspace(0, 2 * np.pi, 100)\n", "\n", "# Ergospheres\n", "# These are functions\n", "Ei1, Eo1 = sing_dict1[\"inner_ergosphere\"], sing_dict1[\"outer_ergosphere\"]\n", "Ei2, Eo2 = sing_dict2[\"inner_ergosphere\"], sing_dict2[\"outer_ergosphere\"]\n", "\n", "# Creating lists of points on Ergospheres for different polar angles, for both black holes\n", "Ei1_list, Eo1_list = Ei1(theta), Eo1(theta)\n", "Ei2_list, Eo2_list = Ei2(theta), Eo2(theta)\n", "\n", "# For Black Hole 1 (a = 0.4)\n", "Xei1 = Ei1_list * np.sin(theta)\n", "Yei1 = Ei1_list * np.cos(theta)\n", "\n", "Xeo1 = Eo1_list * np.sin(theta)\n", "Yeo1 = Eo1_list * np.cos(theta)\n", "\n", "# For Black Hole 2 (a = 0.9)\n", "Xei2 = Ei2_list * np.sin(theta)\n", "Yei2 = Ei2_list * np.cos(theta)\n", "\n", "Xeo2 = Eo2_list * np.sin(theta)\n", "Yeo2 = Eo2_list * np.cos(theta)\n", "\n", "# Event Horizons\n", "Hi1, Ho1 = sing_dict1[\"inner_horizon\"], sing_dict1[\"outer_horizon\"]\n", "Hi2, Ho2 = sing_dict2[\"inner_horizon\"], sing_dict2[\"outer_horizon\"]\n", "\n", "# For Black Hole 1 (a = 0.4)\n", "Xhi1 = Hi1 * np.sin(theta)\n", "Yhi1 = Hi1 * np.cos(theta)\n", "\n", "Xho1 = Ho1 * np.sin(theta)\n", "Yho1 = Ho1 * np.cos(theta)\n", "\n", "# For Black Hole 2 (a = 0.9)\n", "Xhi2 = Hi2 * np.sin(theta)\n", "Yhi2 = Hi2 * np.cos(theta)\n", "\n", "Xho2 = Ho2 * np.sin(theta)\n", "Yho2 = Ho2 * np.cos(theta)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Plotting both black holes" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-6100.0, 6100.0)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15,7.5))\n", "\n", "ax1.fill(Xei1, Yei1, 'b', Xeo1, Yeo1, 'r', Xhi1, Yhi1, 'b', Xho1, Yho1, 'r', alpha=0.3)\n", "ax1.set_title(f\"$M = {M}, a = {a1}$\", fontsize=18)\n", "ax1.set_xlabel(\"X\", fontsize=18)\n", "ax1.set_ylabel(\"Y\", fontsize=18)\n", "ax1.set_xlim([-6100, 6100])\n", "ax1.set_ylim([-6100, 6100])\n", "\n", "ax2.fill(Xei2, Yei2, 'b', Xeo2, Yeo2, 'r', Xhi2, Yhi2, 'b', Xho2, Yho2, 'r', alpha=0.3)\n", "ax2.set_title(f\"$M = {M}, a = {a2}$\", fontsize=18)\n", "ax2.set_xlabel(\"X\", fontsize=18)\n", "ax2.set_ylabel(\"Y\", fontsize=18)\n", "ax2.set_xlim([-6100, 6100])\n", "ax2.set_ylim([-6100, 6100])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* The surfaces are clearly visible in the plots. Going radially inward, we have Outer Ergosphere, Outer Event Horizon, Inner Event Horizon and Inner Ergosphere. We can also observe the following:\n", " - As $a \\to 1$ (its maximum attainable value), the individual singularities become prominent.\n", " - As $a \\to 0$, some singularities appear to fade away, leaving us with a single surface, that is the Event Horizon of a Schwarzschild black hole." ] } ], "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": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 2 }