{ "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 0x7f733c8b0f70>, 'inner_horizon': 247.98878315867296, 'outer_horizon': 5692.939432136259, 'outer_ergosphere': ._out_ergo at 0x7f730f324af0>}\n", "\n", "{'inner_ergosphere': ._in_ergo at 0x7f733c062310>, 'inner_horizon': 1675.668821582463, 'outer_horizon': 4265.259393712469, 'outer_ergosphere': ._out_ergo at 0x7f733c0620d0>}\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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 2 }