{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "

Chaotic photon orbits and shadows of a non-Kerr
object described by the Hartle-Thorne spacetime

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "These notebooks accompany the paper\n", "\"Chaotic photon orbits and shadows of a non-Kerr object described by the Hartle-Thorne spacetime\" ([arXiv:2111.09367 [gr-qc]](https://arxiv.org/abs/2111.09367)) and give a demonstration of the calculations present in the paper. \n", "The code is in some extent based on programs available as examples in the [SageManifolds project's](https://sagemanifolds.obspm.fr/examples.html) website. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

$\\delta q=1$ , $\\chi=0.327352$

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With the Hartle \\& Thorne metric being non-integrable, the analytical calculation of the shadow curve is not possible so we use numerical methods to integrate the geodesic system of equations. Even if it sounds counter intuitive at first, it is a lot easier to follow backwards light rays that start from the observer and asymptote to the black hole's event horizon. This method is called backwards ray-tracing." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|SageMath|\\phantom{\\verb!x!}\\verb|version|\\phantom{\\verb!x!}\\verb|9.2,|\\phantom{\\verb!x!}\\verb|Release|\\phantom{\\verb!x!}\\verb|Date:|\\phantom{\\verb!x!}\\verb|2020-10-24|$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|SageMath|\\phantom{\\verb!x!}\\verb|version|\\phantom{\\verb!x!}\\verb|9.2,|\\phantom{\\verb!x!}\\verb|Release|\\phantom{\\verb!x!}\\verb|Date:|\\phantom{\\verb!x!}\\verb|2020-10-24|\n", "\\end{math}" ], "text/plain": [ "'SageMath version 9.2, Release Date: 2020-10-24'" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%display latex\n", "version()\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import pandas as pd\n", "import multiprocessing as mp\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We increase the lower limit of the radial coordinate $r$ in order to cloak the pathologies of the metric, in this case the nullification of $g_{rr}$, behind the compact object's surface." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 2.08177675496000 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad \\mathit{ph} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 2.08177675496000 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad \\mathit{ph} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}\n", "\\end{math}" ], "text/plain": [ "t: (-oo, +oo); r: (2.08177675496000, +oo); th: (0, pi); ph: [0, 2pi] (periodic)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', latex_name=r'\\mathcal{M}', structure='Lorentzian')\n", "\n", "X.=M.chart(r\"t r:(2.08177675496,+inf) th:(0,pi):\\theta ph:(0,2pi):periodic\\phi\") \n", "\n", "\n", "X.coord_range()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "var('m,b,a,Ω,q', domain='real')\n", "g = M.metric()\n", "m=1" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "g[0,0] =((2m)/r - 1) + (1/(16m^2r^5))(a^2)(((64m^6)r)(sin(th)^2) - (32m^6)r -(1 - 3(cos(th)^2))(((16m^5)(2m - r))(m + r) - ((5r^3)q)(((2m)(m - r))(2m^2 - 3r^2 + (6m)r) + ((3r^2)(r - 2m)^2)log(1 - (2m)/r)))) \n", "\n", "g[0,3] = -2a(m^2)((sin(th))^2)/r\n", "\n", "##\n", "g[1,1]= 1/(1 - (2m)/r) - (1/(32m^2r^3(r - 2m)^2))(a^2)((2m)(80m^6 + (8m^4)r^2 - (24m^5)r + ((10m^3)r^3)q + ((20m^2)r^4)q - ((45m)r^5)q + (15r^6)q) + (((15r^5)(r - 2m)^2)q)log(1 - (2m)/r) + (3cos(2th))((2m)(80m^6 + (8m^4)r^2 - (56m^5)r + ((10m^3)r^3)q + ((20m^2)r^4)q - ((45m)r^5)q + (15r^6)q) + (((15r^5)(r - 2m)^2)q)log(1 - (2m)/r))) \n", "\n", "\n", "\n", "g[2,2] =r^2 + (1/8)((a^2r^2)(3(cos(th)^2) - 1))(-(((8m^3)(2m + r))/r^4) - (5q)(-((2m)/r) + (3r)/m + ((3r^2)/(2m^2) - 3)log(1 - (2m)/r) + 3))\n", "\n", "\n", "\n", "g[3,3] = (r^2)(sin(th)^2) + (1/8)(((a^2r^2)(3(cos(th)^2) - 1))(-(((8m^3)(2m+ r))/r^4)-(5q)(-((2m)/r) + (3r)/m + ((3r^2)/(2m^2) - 3)log(1 - (2m)/r) + 3)))(sin(th)^2)\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def g00(m,r,th,a):return g[0,0](m,r,th,a)\n", "def g03(m,r,th,a):return g[0,3](m,r,th,a)\n", "def g11(m,r,th,a):return g[1,1](m,r,th,a)\n", "def g22(m,r,th,a):return g[2,2](m,r,th,a)\n", "def g33(m,r,th,a):return g[3,3](m,r,th,a)\n", "def D(m,r,th,a): return (g03(m,r,th,a))^2-g00(m,r,th,a)g33(m,r,th,a)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\mathcal{M} & \\longrightarrow & \\mathbb{E}^{3} \\\\ & \\left(t, r, {\\theta}, \\mathit{ph}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left(\\mathit{ph}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left(\\mathit{ph}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\end{array}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\mathcal{M} & \\longrightarrow & \\mathbb{E}^{3} \\\\ & \\left(t, r, {\\theta}, \\mathit{ph}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left(\\mathit{ph}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left(\\mathit{ph}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\end{array}\n", "\\end{math}" ], "text/plain": [ "M --> E^3\n", " (t, r, th, ph) |--> (x, y, z) = (rcos(ph)sin(th), rsin(ph)sin(th), rcos(th))" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E. = EuclideanSpace()\n", "phi = M.diff_map(E, [rsin(th)cos(ph), rsin(th)sin(ph), rcos(th)])\n", "phi.display()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We further restrict the coordinate chart, using the Killing Horizon equation $\\mathcal{D}=0$." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "temp(a,q)=D(m,r,th,a)\n", "killh(r,th)=temp(0.327352,1)\n", "X.add_restrictions(killh(r,th)>0.001)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "def initial_vector(r0, b,al, ph0=0, E=1, inward=False):\n", " t0,th0=0,pi/2\n", " L = -bE\n", " vth0 = al/(100r0)\n", " vt0=(Eg33(m,r0,th0,a)+Lg03(m,r0,th0,a))/(D(m,r0,th0,a))\n", " vr0=sqrt((((1/D(m,r0,th0,a))((L^2)g00(m,r0,th0,a)+2ELg03(m,r0,th0,a)+(E^2)g33(m,r0,th0,a)))\n", " -((vth0)^2)g22(m,r0,th0,a))/g11(m,r0,th0,a))\n", " if inward:\n", " \n", " vr0 = - vr0\n", " \n", " vph0 = -(1/D(m,r0,th0,a))(Eg03(m,r0,th0,a)+L(g00(m,r0,th0,a)))\n", " p0 = M((t0, r0, th0, ph0), name='p_0')\n", " \n", " return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')\n" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "r0 = 100 # source distance\n", "th0=pi/2\n", "s = var('s') # affine parameter " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We set an $N \\times M$ grid for the impact parameters $\\{b,\\alpha\\}$, in the region of interest, where the pocket is admitted. " ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "b_max = -4.301\n", "b_min = -4.30434\n", "nb = 300\n", "\n", "db = (b_max - b_min) / (nb - 1)\n", "b_sel = [float(b_min + idb) for i in range(nb)]\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We do not need to compute for the negative values of the impact parameter $\\alpha$ due to the spacetime symmetry." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "al_max = 0.25\n", "al_min = 0\n", "nal = 300\n", "dal = (al_max - al_min) / (nal - 1)\n", "al_sel = [float(al_min + idal) for i in range(nal)]\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We integrate the geodesic equations for every possible pair of the impact parameters $b$ and $\\alpha$. We specifically set conditions to decide the fate of a light ray at the end of the integration. The $\\{b,\\alpha\\}$ that correspond to light rays that are deflected by the compact object to infinity, are saved as light.csv, while those that fall into the surface of the object as shadow.csv. For the latter, we use additional conditions to decide which specific exit to the object they follow. We color code the pairs as: (i) purple for light rays that plunge into the object from the upper throat; (ii) cyan for light rays that plunge into the object from the lower throat; (iii) green for those that escape to infinity." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import multiprocessing as mp\n", "\n", "def integration(b):\n", " #we use 'pos' as a counter to avoid the files not saving correctly due to a long title.\n", " pos = b_sel.index(b) \n", " \n", "\n", " shadow=[]\n", " light=[] \n", " \n", "\n", " \n", " \n", " \n", "\n", "\n", " for al in al_sel:\n", " color=0\n", " \n", " \n", " print(b,al)\n", " ph0=0\n", " v0 = initial_vector(r0, b,al, ph0=ph0, inward=True)\n", " geod = M.integrated_geodesic(g, (s, 0, 300), v0, across_charts=True)\n", " \n", " sol = geod.solve_across_charts(step=0.05, method='odeint', parameters_values={a:0.327352,q:1},\n", " verbose='False')\n", " \n", " \n", " \n", " \n", " \n", " if (sol[0][1][(len(sol[0][1]))-1][2])>100:\n", " light.append([b,al])\n", " \n", " else:\n", " if sol[0][1][len(sol[0][1])-1][3]>pi/2 and sol[0][1][len(sol[0][1])-1][0] < 5000:\n", " \n", " color='purple'\n", " \n", " if sol[0][1][len(sol[0][1])-1][3] 5000:\n", " color='red'\n", " \n", " \n", " \n", " shadow.append([color,b,al])\n", " \n", " my_df=pd.DataFrame(shadow)\n", " my_df.to_csv(r'G:\\desktop\\shadows\\shadow(%f).csv' %pos,index=False,header=False) \n", " \n", " \n", " my_df=pd.DataFrame(light)\n", " my_df.to_csv(r'G:\\desktop\\lights\\light(%f).csv' %pos,index=False,header=False)\n", " \n", " \n", " \n", " \n", " \n", " \n", "pool = mp.Pool(10)\n", "sols = pool.map(integration,[b for b in b2]) \n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "shad_tot=[]\n", "for i in range(len(b_sel)):\n", " try:\n", "\n", " shad=pd.read_csv(r'G:\\desktop\\shadows\\shadow(%f).csv'%i,delimiter=',',header=None,index_col=None)\n", " shad=shad.values.tolist()\n", " shad_tot.append(shad)\n", " except: None\n", "\n", " \n", "lig_tot=[]\n", "for i in range(len(b_sel)):\n", " \n", " try:\n", " lig=pd.read_csv(r'G:\\desktop\\lights\\light(%f).csv'%i,delimiter=',',header=None,index_col=None)\n", " lig=lig.values.tolist()\n", " lig_tot.append(lig)\n", " except: None\n", " \n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "\n", "lig_points=[]\n", "for i in range(len(lig_tot)):\n", " for j in range(len(lig_tot[i])):\n", " lig_points.append([lig_tot[i][j][0],lig_tot[i][j][1]])\n", " \n", "shad_points=[]\n", "for i in range(len(shad_tot)):\n", " for j in range(len(shad_tot[i])):\n", " shad_points.append([shad_tot[i][j][1],shad_tot[i][j][2]])\n", "\n", "#we create the symmetric as to the horizontal axis points, for the negative values of the impact parameter 'a' \n", "shadow_symm=[]\n", "light_symm=[]\n", "\n", "for i in range(len(shad_points)):\n", " shadow_symm.append([shad_points[i][0],shad_points[i][1](-1)])\n", " \n", "for i in range(len(lig_points)):\n", " light_symm.append([lig_points[i][0],lig_points[i][1](-1)])\n", "\n", "colors=[]\n", "for i in range(len(shad_tot)):\n", " for j in range(len(shad_tot[i])):\n", " \n", " colors.append(shad_tot[i][j][0])\n" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph1=scatter_plot(shad_points,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=100,zorder=1,marker=\"s\",edgecolor=colors,facecolor=colors,frame=True)\n", "graph2=scatter_plot(shadow_symm,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=80,zorder=1,marker=\"s\",edgecolor=colors,facecolor=colors,frame=True) \n", "graph3=scatter_plot(lig_points,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=100,zorder=1,marker=\"s\",edgecolor='green',facecolor='green',frame=True)\n", "graph4=scatter_plot(light_symm,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=80,zorder=1,marker=\"s\",edgecolor='green',facecolor='green',frame=True) \n", "\n", "\n", "graph=graph1+graph2+graph3+graph4\n", "show(graph,xmin=-4.30434,xmax=-4.301,ymin=0,ymax=0.25,fontsize=100,figsize=50,ticks=[0.0005,0.05],axes_labels=[r'$b$', r'$\\alpha$'],frame=True) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
$\\delta q=1$ , $\\chi=0.4$
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this regime, we are past the critical spin parameter $\\chi^$ where\n", "the equatorial light-ring has bifurcated to two non equatorial ones, symmetrical as to\n", "the $\\theta = \\frac{\\pi}{2}$ plane. Note that in order to avoid the pathologies of the metric, specifically the $g_{rr}$ component's nullification, the lower limit of the radius in the coordinate chart, that represents the surface of the object, is now set to $ r_s = 2.11586644885$." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "verbose 0 (2209: graphics.py, get_minmax_data) xmin was NaN (setting to 0)\n", "verbose 0 (2209: graphics.py, get_minmax_data) xmax was NaN (setting to 0)\n", "verbose 0 (2209: graphics.py, get_minmax_data) ymin was NaN (setting to 0)\n", "verbose 0 (2209: graphics.py, get_minmax_data) ymax was NaN (setting to 0)\n" ] }, { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "shad_tot=[]\n", "shad_tot=pd.read_csv(r'G:\\desktop\\shadows04\\shadsall.csv',delimiter=',')\n", "shad_tot=shad_tot.values.tolist()\n", "\n", "lig_tot=[]\n", "lig_tot=pd.read_csv(r'G:\\desktop\\lights04\\lightall.csv',delimiter=',')\n", "lig_tot=lig_tot.values.tolist()\n", "\n", "\n", "shadow_symm=[]\n", "light_symm=[]\n", "for i in range(len(shad_tot)):\n", " shadow_symm.append([shad_tot[i][0],shad_tot[i][1](-1)])\n", "for i in range(len(lig_tot)):\n", " light_symm.append([lig_tot[i][0],lig_tot[i][1](-1)])\n", "\n", "mrkrsh=200\n", "mrkrl=200\n", "\n", "\n", "graph1=scatter_plot(lig_tot,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=mrkrl,zorder=1,marker=\"s\",edgecolor='black',facecolor='green',frame=True)\n", "graph2=scatter_plot(shad_tot,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=mrkrsh,zorder=1,marker=\"s\",edgecolor='black',facecolor='black',frame=True)\n", "graph3=scatter_plot(shadow_symm,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=mrkrsh,zorder=1,marker=\"s\",edgecolor='black',facecolor='black',frame=True)\n", "graph4=scatter_plot(light_symm,aspect_ratio='automatic',figsize=[50,50],alpha=1,markersize=mrkrl,zorder=1,marker=\"s\",edgecolor='black',facecolor='green',frame=True)\n", "\n", "graphsh=graph1+graph2+graph3+graph4\n", "graphsh.fontsize(120)\n", "graphsh.axes_width(5)\n", "show(graphsh,xmin=-6,xmax=8,ymin=-6,ymax=6,figsize=120,axes_labels=[r'$b$', r'$\\alpha$'],frame=True, legend_loc=(1.02, 0.68))\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.7" } }, "nbformat": 4, "nbformat_minor": 4 }