{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "# Geodesics in Schwarzschild spacetime\n", "\n", "This worksheet aims at recovering some geodesics of Schwarzschild spacetime using numerical tools implemented in the SageMath class **IntegratedGeodesic**, within the [SageManifolds](http://sagemanifolds.obspm.fr) project (version 1.1, as included in SageMath 8.1).\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.1/SM_Schwarzschild_geod.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath within the Jupyter notebook, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "*NB:* a version of SageMath at least equal to 8.1 is required to run this worksheet:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.1, Release Date: 2017-12-07'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "First set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "## Spacetime manifold\n", "\n", "To set the Schwarzschild spacetime, declare a 4-dimensional differentiable manifold and a non-negative parameter $m$:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "Schw = Manifold(4, 'Schw', r'\\mathcal{Schw}')\n", "m = var('m') ; assume(m >= 0)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Define the Boyer-Lindquist coordinates on the union of two subregions $\\mathcal{R}_{\\mathrm{I}}$ and $\\mathcal{R}_{\\mathrm{II}}$:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "regI = Schw.open_subset('R_I', r'\\mathcal{R}_{\\mathrm{I}}')\n", "regII = Schw.open_subset('R_II', r'\\mathcal{R}_{\\mathrm{II}}')\n", "regI_II = regI.union(regII)\n", "BL. = regI_II.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:\\phi')\n", "BL_I = BL.restrict(regI, r>2*m)\n", "BL_II = BL.restrict(regII, r<2*m)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Finally set the Lorentzian metric of Schwarzschild spacetime:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = (2*m/r - 1) dt*dt - 1/(2*m/r - 1) dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = Schw.lorentzian_metric('g')\n", "g[0,0], g[1,1] = -(1-2*m/r), 1/(1-2*m/r)\n", "g[2,2], g[3,3] = r^2, (r*sin(th))^2\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true, "deletable": true, "editable": true }, "source": [ "## Geodesics in Boyer-Lindquist coordinates" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "### Defining the geodesic" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Declare the various variables needed to define a geodesic; start with the affine parameter and its extremal values:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "affine_param = var('s s_0 s_max')" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Then, declare the starting point of the geodesic:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "initial_pt_coords = var('t_0 r_0 th_0 ph_0') ; assume(r_0 > 2*m)\n", "p_0 = regI.point(initial_pt_coords, name='p_0')" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Declare the initial tangent vector:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "initial_tgt_vec_comps = var('Dt_0 Dr_0 Dth_0 Dph_0')\n", "v_0 = Schw.tangent_space(p_0)(initial_tgt_vec_comps)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The parametrised geodesic may now be initialised:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The curve was correctly set.\n", "Parameters appearing in the differential system defining the curve are [Dph_0, Dr_0, Dt_0, Dth_0, m, ph_0, r_0, s_0, s_max, t_0, th_0].\n" ] } ], "source": [ "geod = Schw.integrated_geodesic(g, affine_param, v_0, verbose=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Display the system of the geodesic equations:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Geodesic in the 4-dimensional differentiable manifold Schw equipped with Lorentzian metric g on the 4-dimensional differentiable manifold Schw, and integrated over the Real interval (s_0, s_max) as a solution to the following geodesic equations, written with respect to Chart (R_I_union_R_II, (t, r, th, ph)):\n", "\n", "Initial point: Point p_0 on the 4-dimensional differentiable manifold Schw with coordinates [t_0, r_0, th_0, ph_0] with respect to Chart (R_I_union_R_II, (t, r, th, ph))\n", "Initial tangent vector: Tangent vector at Point p_0 on the 4-dimensional differentiable manifold Schw with components [Dt_0, Dr_0, Dth_0, Dph_0] with respect to Chart (R_I_union_R_II, (t, r, th, ph))\n", "\n", "d(t)/ds = Dt\n", "d(r)/ds = Dr\n", "d(th)/ds = Dth\n", "d(ph)/ds = Dph\n", "d(Dt)/ds = 2*Dr*Dt*m/(2*m*r - r^2)\n", "d(Dr)/ds = -(4*Dth^2*m^2*r^3 - 4*Dth^2*m*r^4 + Dth^2*r^5 - 4*Dt^2*m^3 + 4*Dt^2*m^2*r + (Dr^2 - Dt^2)*m*r^2 + (4*Dph^2*m^2*r^3 - 4*Dph^2*m*r^4 + Dph^2*r^5)*sin(th)^2)/(2*m*r^3 - r^4)\n", "d(Dth)/ds = (Dph^2*r*cos(th)*sin(th) - 2*Dr*Dth)/r\n", "d(Dph)/ds = -2*(Dph*Dth*r*cos(th) + Dph*Dr*sin(th))/(r*sin(th))\n", "\n" ] } ], "source": [ "sys = geod.system(verbose=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "### Computing numerical solutions\n", "\n", "#### Timelike geodesics\n", "\n", "##### Bounded orbit" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Set a dictionnary providing numerical values for each of the parameters apprearing in the system defining the geodesic. \n", "\n", "The values suggested below make the initial tangent vector timelike with squared norm equal to -1. This way, the curve parameter `s` not only is an affine parameter of the (timelike) geodesic, but it actually is the proper time along it (up to numerical roundings)." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "params_values_bounded = {m:1, s_0:0, s_max:1500, t_0:0, r_0:8, th_0:pi/2, ph_0:1e-12, \n", " Dt_0:sqrt(80.81)/(4*sqrt(3)), Dr_0:0, Dth_0:0, Dph_0:4.1/64}" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Then integrate the geodesic for such values of the parameters:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Schw.default_chart().valid_coordinates(*p_0.coord())" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Performing numerical integration with method 'rk4_maxima'...\n", "Numerical integration completed.\n", "\n", "Checking all points are in the chart domain...\n", "All points are in the chart domain.\n", "\n", "The resulting list of points was associated with the key 'timelike_bounded_equatorial' (if this key already referred to a former numerical solution, such a solution was erased).\n" ] } ], "source": [ "sol_bounded = geod.solve(step=4, parameters_values=params_values_bounded, \n", " solution_key='timelike_bounded_equatorial', verbose=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The squared norm $g_{\\mu\\nu} \\dot{x}^{\\mu} \\dot{x}^{\\nu}$ of the vector tangent to any geodesic with respect to any affine parameter $s$ is constant throughout motion.\n", "\n", "In addition, up to a rotation of the system of polar coordinates, any geodesic in Schwarzschild spacetime is known to have constant coordinate $\\theta$ equal to $\\pi/2$ (which is why it was suggested above to set the initial condition on $\\theta$ to $\\pi/2$).\n", "In such a situation, energy $e$ and angular momentum $l$ per unit mass, defined as follows, are also conserved along a geodesic of Schwarzschild spacetime:\n", "\n", "$$e = \\left( 1 - \\frac{2m}{r} \\right) \\dot{t}$$\n", " \n", "$$l = r^{2} \\dot{\\phi}$$\n", "\n", "Therefore, using an interpolation of the previous numerical solution, one may check that these three quantities are indeed conserved along the numerical solution previoulsy computed (not to be disturbed by initial edge effects, the reference value used to check conservations of the various quantities is updated after 10 steps):" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Performing cubic spline interpolation by default...\n", "Interpolation completed and associated with the key 'timelike_bounded_equatorial' (if this key already referred to a former interpolation, such an interpolation was erased).\n" ] }, { "data": { "image/png": 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SwQIAAAAAS3kOFp599lnZ7Xa3V61atfKjbpKkKVOmyG63q3PnzvlW\nJgAAAICL886PQurUqaMFCxbIGOMo1DtfitWePXv06KOPKiYmJl/KAwAAAJBz+dINydvbW2FhYQoP\nD1d4eLhKlizpmpacnKx77rlH4eHhCg4OVuvWrbV+/fqLlpmZmalevXrpueeeU+XKlfOjmgAAAABy\nIV+ChW3btqlcuXKqWrWqevXqpX379rmmde3aVceOHdOPP/6o1atX69prr1Xr1q2VlJR0wTKfffZZ\nhYeHq2/fvvlRRQAAAAC5lOf+Qtdff70+/fRTVa9eXQcPHtTo0aMVExOjP/74Q2vWrNGqVat05MgR\nFStWTJL00ksv6euvv9aMGTN0zz33WJa5fPlyffLJJ1q3bl1eqwcAAADgEuU5WGjbtq3r/zp16qhJ\nkyaKiorStGnTdPr0aZ04ccKtW5IknTlzRjt27NC+fftcD0PbbDY98cQTGjx4sO6++2598MEHCg0N\nzWv1AAAAAFyi/HkS+TzBwcGqVq2atm/fruDgYJUtW1ZLlixxPfzsFBISopCQELe7ByVLltSOHTu0\nZ88edezY0fWZzMxMSVLx4sW1ZcuWCz7DEB8f7/GAdc+ePdWzZ8/8WkQAAADgPyHfg4XU1FTt2LFD\nvXv3Vo0aNXTo0CF5eXmpYsWKlvmrVKni9t7Pz08bNmxwSxs1apRSU1M1YcIEVahQ4YLznzJlioKC\ngvK2EAAAAADyHiw8+uij6tixo6KiovTXX3/pmWeekbe3t3r27KlSpUrphhtuUKdOnTR+/HhVq1ZN\nf/31l77//nt17txZ1157rUd5xYsX9/idhpCQENlsNtWsWTOv1QUAAACQQ3kOFvbv368777xTx44d\nU1hYmJo3b65ffvlFpUqVkiR9//33GjVqlPr166ejR48qMjJSMTExioiIyHPlAQAAABQcm8n6MEER\nlZKSouDgYCUnJ9MNCQAAAMgH+fI7CwAAAACuPgQLAAAAACwRLAAAAACwRLAAAAAAwBLBAgAAAABL\nBAsAAAAALBEsAAAAALBEsAAAAADAEsECAAAAAEsECwAAAAAsESwAAAAAsESwAAAAAMASwQIAAAAA\nSwQLAAAAACwRLAAAAACwRLAAAAAAwBLBAgAAAABLBAsAAAAALBEsAAAAALBEsAAAAADAEsECAAAA\nAEtXXbAQHx+vuLg4JSYmFnZVAAAAgCLNZowxhV2J/JCSkqLg4GAlJycrKCiosKsDAAAAFHlX3Z0F\nAAAAAPmDYAEAAACAJYIFAAAAAJYIFgAAAABYIlgAAAAAYIlgAQAAAIAlggUAAAAAlggWAAAAAFgi\nWAAAAABgiWABAAAAgCWCBQAAAACWCBYAAAAAWCJYAAAAAGCJYAEAAACAJYIFAAAAAJYIFgAAAABY\nIlgAAAAAYIlgAQAAAIAlggUAAAAAlggWAAAAAFgiWAAAAABgiWABAAAAgCWCBQAAAACWCBYAAAAA\nWLrqgoX4+HjFxcUpMTGxsKsCAAAAFGk2Y4wp7Erkh5SUFAUHBys5OVlBQUGFXR0AAACgyLvq7iwA\nAAAAyB8ECwAAAAAsESwAAAAAsESwAAAAAMASwQIAAAAASwQLAAAAACwRLAAAAACwRLAAAAAAwBLB\nAgAAAABLBAsAAAAALBEsAAAAALBEsAAAAADAEsECAAAAAEsECwAAAAAsESwAAAAAsESwAAAAAMAS\nwQIAAAAASwQLAAAAQBGUmCi9807BzoNgAQAAACiCvvtOmjatYOdBsAAAAAAUQWfPSiVKFOw8CBYA\nAACAIigtTSpevGDnQbAAAAAAFEHcWQAAAABgiWABAAAAgCW6IV2C+Ph4xcXFKTExsbCrAgAAABSY\ny3Fnwbtgi7/8pkyZoqCgoMKuBgAAAFCg6IYEAAAAwBLdkAAAAABY4s4CAAAAAEsECwAAAAAs0Q0J\nAAAAgCXuLAAAAACwRLAAAAAAwENGhpSZSbAAAAAAIIuzZx1/eWYBAAAAgBtnsMCdBQAAAABu0tIc\nfwkWAAAAALihGxIAAAAAS3RDAgAAAGCJbkgAAAAALNENCQAAAIAluiEBAAAAsEQ3JAAAAACW6IYE\nAAAAwBLdkAAAAABYohsSAAAAAEt0QwIAAABgiW5IAAAAACylpUl2u+TtXbDzIVgAAAAAipizZwu+\nC5JEsAAAAAAUOWfPFnwXJIlgAQAAAChy0tIIFi5JfHy84uLilJiYWNhVAQAAAArE5eqGVMCPRFx+\nU6ZMUVBQUGFXAwAAACgwdEMCAAAAYIluSAAAAAAscWcBAAAAgCWGTgUAAABgiW5IAAAAACzRDQkA\nAACAJbohAQAAALBENyQAAAAAluiGBAAAAMAS3ZAAAAAAWOLOAgAAAABLPLMAAAAAwBLdkAAAAABY\nohsSAAAAAEt0QwIAAABgiW5IAAAAACzRDQkAAACAJbohAQAAAPCQmSmdO0c3JBQRiYmJhV2F/yTa\nvXDQ7oWDdi8ctPvlR5sXjqLW7mlpjr/cWUCRUNR2sKsF7V44aPfCQbsXDtr98qPNC0dRafe1a6X7\n7nM8ryARLAAAAAD4/158UfrgA+nwYcd7uiFdYQoy6izoiLaoRMxWaPfCUVTbnTYvnPJp98Ipn3a/\n/GVfjvILEu1eOPKj7n//LX3zjeP/HTscf0uUKPh2ueqCBWMKrmx2sMJxudvdGM9XZqb1KyPD85We\n/u/ryy8TXf+fO+f5Skuzfp096/k6c8b9lZnp+Hv6tPXr1CnP18mT1q/UVM/XZ58l6sQJebxSUjxf\nycnWr+PHpUOHpL17pW3bpD/+kH7/XXr33UT9+ad08KCjbS5Verpj/kePSvv3S7t2SR9/nKi//nJc\ndTl2zFGPkycdbZiRUbDHiLwqyscBjjGFU35Blu08/jmPX2fOOPa3v/+WDhyQdu+Wtm517NerV0t/\n/unY13O6T2d3/E1Kcuy/Bw5I+/Y5yjxwQDpyRPrnH8dx6PRpR52y25+L8jotaEV1e7wc5RekS617\nSopjH8vMlD7/3LHdS9LOnY6/lyNY8C7Q0i+BMUYnTpzI9ee+/z5FkrRlS4pq1JCmTJFmzpSmTct9\nHb74Qho61HEQstn+faWlpcvPL8UtTfr374Vc7AvK6dPpCgxMydEXmUvJc/aso+75UVZW586ly8fH\nuu7ZfTY3eTMy0uXllZKjvLn/Ipgum82zXfJPuooVK6jy0+XrW7B1DwoquLrXqfNv2RUrStWqOV7V\nqztelStLfn6Odfr3345gYPt299fu3Y4DaNayy5e/cL29vCRvb8fLyytn+7DTyZPpCgkpmHYpyLKt\nys/PwOnUqXQFBxdc3fO7/POX/dSpgtzWC7b8Sy3beRHk/Ish56f92z65P4Z5eUlRUVKVKlJ0tFSz\nplSjhlSunBQa6gjajx2TDh5M1/jxKdqxw7Ev793reJ0+nbtlsdsd+3KxYv/uz3nZl3KyX1xoe8zr\nfnXqlOP7QF5cqA5nzqQrIODC5V/qMpw9my5//5Q8lXGhOqSl5e68l5s6OL/LZCfr976s/2c3zSk1\nNV2lS6fk6DPO/zMzHRfcJKllS0fw3KGDNHu2tHGjs95Senq6UlIubZsJDAyU7SInQZsxV9Z1tpSU\nFAUHBxd2NQAAAICrWnJysoKCgi6Y54oLFi71zsLvv6fo5psr6Kuv9ql16yANHCglJjoir3LlclfW\nddc5IriXX851NQDk0Zkzjr6Ye/c6uk5JUni4VLas4w5EsWKFWz8AuXP6tGOfPnjQ0cXIy8txhyE6\n2nF+tl91HaKB/PfPP9LixVKnTlLr1o7utZs3S8uXS3XqXHq5ObmzcMV1Q7LZbBeNcKw4A4KMjCAF\nBQW5vmT8/bfjFmhOOQ9qjz0mXUI1AORRUJAjOLjhhsKuCYD8EBQkRUQUdi2Aoi0oSEpIcPwfFSXN\nmeP4v1Spgv++etXE886GSkpy/E1OdvzdvTt35TgfIqlfP9+qBgAAAOSL8uX/fbaHoVNzwdfX8df5\nfIczaNi1K3flrFvnuCWal1s6AAAAQEGoUOHf//lRtlxwdrdy3lFw/r2UYKFatX+DDwAAAOBKUb78\nv/8TLFwCZ5DgvLOQ225I69bRBQkAAABXpvODBbohXYKUFMe4uikpjtEWnHcWzp61Go/dnTHS+vUE\nC2PHjlWTJk0UFBSkiIgI3XHHHdq6datbnrNnz+qBBx5Q6dKlFRgYqK5du+rIkSNuefbt26fbbrtN\n/v7+ioyM1GOPPabMi60EuIwdO1Z2u10PP/ywK412LxgHDhzQ3XffrdKlS8vPz0/169fX6tWr3fI8\n/fTTKlu2rPz8/HTLLbdo+/btbtOPHz+uu+66S8HBwQoNDdU999yjkydPXs7FKFIyMzP11FNPqUqV\nKvLz81N0dLTGjBnjkY92z5uffvpJcXFxKleunOx2u2bNmuWRJz/aeP369YqJiZGvr6+ioqL08n94\nOMELtXl6eroef/xx1atXTwEBASpXrpz69OmjgwcPupVBm+deTrZ1pwEDBshut2vChAlu6UWl3S/3\nnQWZq0RycrKRZO64I9mcOOH4SZmWLY2x241JSzOmVStjGjY05tAhR/7MTM8ydu1yfG727Mta9StO\nu3btzOTJk83GjRvN+vXrzW233WaioqLMqVOnXHkGDhxooqKizOLFi83q1avNDTfcYJo3b+6anpGR\nYerUqWPatGlj1q9fb3744QcTFhZmRo0aVRiLVOT8+uuvpnLlyqZBgwZm2LBhrnTaPf8dP37cVKpU\nyfTv39+sWrXK7N6928ybN8/s3LnTlWfcuHEmNDTUzJo1y2zYsMHcfvvtpkqVKubs2bOuPLfeeqtp\n2LCh+e2338zy5cvNNddcY+66667CWKQ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+/SYiIsIEBgYau93uMv1qeqdcNcaafnbYsGGmfPnypkCBAqZUqVKmbdu25t13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\nv3/K98Y/371798Q1vO8z18yf92oXXnjhlKKh097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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "interp_bounded = geod.interpolate(solution_key='timelike_bounded_equatorial', \n", " interpolation_key='timelike_bounded_equatorial', \n", " verbose=True)\n", "\n", "error_squar_norm_bounded = []\n", "error_e_bounded = []\n", "error_l_bounded = []\n", "\n", "i = 0\n", "for (S,T,R,TH,PH) in sol_bounded:\n", " P = geod(S, interpolation_key='timelike_bounded_equatorial')\n", " V = geod.tangent_vector_eval_at(S, interpolation_key='timelike_bounded_equatorial')\n", "\n", " squar_norm_bounded = numerical_approx((g.at(P)(V,V)).substitute({m:1}))\n", " e_bounded = numerical_approx((-g.at(P)[0,0]*V[0]).substitute({m:1}))\n", " l_bounded = numerical_approx((g.at(P)[3,3]*V[3]).substitute({m:1}))\n", " \n", " if i == 0:\n", " squar_norm_bounded_0 = squar_norm_bounded\n", " e_bounded_0 = e_bounded\n", " l_bounded_0 = l_bounded\n", " \n", " if i == 10:\n", " squar_norm_bounded_0 = squar_norm_bounded\n", " e_bounded_0 = e_bounded\n", " l_bounded_0 = l_bounded\n", "\n", " error_squar_norm_bounded += [(S,squar_norm_bounded - squar_norm_bounded_0)]\n", " error_e_bounded += [(S,e_bounded - e_bounded_0)]\n", " error_l_bounded += [(S,l_bounded - l_bounded_0)]\n", " \n", " i += 1\n", "\n", "plot_error_squar_norm_bounded = line(error_squar_norm_bounded)\n", "plot_error_e_bounded = line(error_e_bounded)\n", "plot_error_l_bounded = line(error_l_bounded)\n", "\n", "plot_error_squar_norm_bounded.show(title=\"Timelike geodesic - Bounded orbit: error on conservation of squared norm\",\n", " ymin=-1e-3, ymax=1e-3)\n", "plot_error_e_bounded.show(title=\"Timelike geodesic - Bounded orbit: error on conservation of e\", \n", " ymin=-1e-3, ymax=1e-3)\n", "plot_error_l_bounded.show(title=\"Timelike geodesic - Bounded orbit: error on conservation of l\",\n", " ymin=-2e-3, ymax=2e-3)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The curves are indeed constant with negligible numerical variations, the jolts in the middle being correlated with radial coordinate $r$ reaching a minimum.\n", "The jolts get smaller if one uses a shorter step of integration when calling method `solve`.\n", "\n", "One may finally plot the spatial part of the geodesic.\n", "To do so, start with defining the 3-dimensional Euclidean space with Cartesian coordinates:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "R3 = Manifold(3, 'R3', start_index=1)\n", "cart. = R3.chart()\n", "origin = R3.point((0,0,0), name='O')\n", "plot3D_origin = origin.plot(size=10)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Then define the mapping converting the spatial part of Boyer-Lindquist coordinates (i.e. the polar coordinates) into the Cartesian coordinates:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "BL_spatial_coords = Schw.diff_map(R3,{(BL, cart): [r*sin(th)*cos(ph), r*sin(th)*sin(ph),\n", " r*cos(th)]})" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": true, "editable": true }, "source": [ "This allows to plot the spatial part of the geodesic:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [], "source": [ "plot3D_projected_geod_bounded = geod.plot_integrated(interpolation_key='timelike_bounded_equatorial',\n", " mapping=BL_spatial_coords, \n", " plot_points=500, thickness=2,\n", " display_tangent=True, \n", " plot_points_tangent=30, scale=8, \n", " width_tangent=1, label_axes=False)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Plot the event horizon:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "plot3D_event_horizon = BL.plot(chart=cart, mapping=BL_spatial_coords, \n", " fixed_coords={t:0, r:2}, ranges={ph:(0,2*pi)}, \n", " number_values=15, color='yellow', label_axes=False)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Display all graphics:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "(plot3D_projected_geod_bounded + plot3D_event_horizon + plot3D_origin).show(viewer='threejs',\n", " online=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "As expected, one notices the advance of the periastron of such a bounded orbit.\n", "\n", "##### Circular case\n", "\n", "Schwarzschild spacetime admits circular orbits.\n", "The unique timelike geodesic (still parametrised with proper time) inducing a stable circular orbit with same angular momentum as that of the geodesic considered above is obtained with the following initial conditions:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "params_values_circular = {m:1, s_0:0, s_max:500, t_0:0, r_0:(16.81 + 4.1*sqrt(4.81))/2, \n", " th_0:pi/2, ph_0:0, Dt_0:1.1415, Dr_0:0, Dth_0:0, \n", " Dph_0:16.4/(16.81+4.1*sqrt(4.81))^2}" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Solve, interpolate and plot the orbit corresponding to such values of the parameters:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Performing numerical integration with method 'rk4_maxima'...\n", "Numerical integration completed.\n", "\n", "Checking all points are in the chart domain...\n", "All points are in the chart domain.\n", "\n", "The resulting list of points was associated with the key 'timelike_circular_equatorial' (if this key already referred to a former numerical solution, such a solution was erased).\n", "Performing cubic spline interpolation by default...\n", "Interpolation completed and associated with the key 'timelike_circular_equatorial' (if this key already referred to a former interpolation, such an interpolation was erased).\n" ] }, { "data": { "image/png": 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AAACwRWAAAAAAYIvAAAAAAMAWgQEAAACALQIDAAAAAFsEBgAAAAC2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\nad48KTXV7rb+7egwnAKFAfATj8dOl+zVS7rzTg6LQeDYtMn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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sol_circular = geod.solve(step=4, parameters_values=params_values_circular, \n", " solution_key='timelike_circular_equatorial', verbose=True)\n", "interp_circular = geod.interpolate(solution_key='timelike_circular_equatorial', \n", " interpolation_key='timelike_circular_equatorial', \n", " verbose=True)\n", "\n", "error_squar_norm_circular = []\n", "error_e_circular = []\n", "error_l_circular = []\n", "error_r_circular = []\n", "\n", "i = 0\n", "for (S,T,R,TH,PH) in sol_circular:\n", " P = geod(S, interpolation_key='timelike_circular_equatorial')\n", " V = geod.tangent_vector_eval_at(S, interpolation_key='timelike_circular_equatorial')\n", "\n", " squar_norm_circular = numerical_approx((g.at(P)(V,V)).substitute({m:1}))\n", " e_circular = numerical_approx((-g.at(P)[0,0]*V[0]).substitute({m:1}))\n", " l_circular = numerical_approx((g.at(P)[3,3]*V[3]).substitute({m:1}))\n", " \n", " if i == 0:\n", " squar_norm_circular_0 = squar_norm_circular\n", " e_circular_0 = e_circular\n", " l_circular_0 = l_circular\n", " R_0 = R\n", "\n", " error_squar_norm_circular += [(S,squar_norm_circular - squar_norm_circular_0)]\n", " error_e_circular += [(S,e_circular - e_circular_0)]\n", " error_l_circular += [(S,l_circular - l_circular_0)]\n", " error_r_circular += [(S,R - R_0)]\n", " \n", " i += 1\n", " \n", "plot_error_squar_norm_circular = line(error_squar_norm_circular)\n", "plot_error_e_circular = line(error_e_circular)\n", "plot_error_l_circular = line(error_l_circular)\n", "plot_error_r_circular = line(error_r_circular)\n", "\n", "plot_error_squar_norm_circular.show(title=\n", " \"Timelike geodesic - Circular orbit: error on conservation of squared norm\",\n", " ymin=-1e-3, ymax=1e-3)\n", "plot_error_e_circular.show(title=\"Timelike geodesic - Circular orbit: error on conservation of e\",\n", " ymin=-1e-3, ymax=1e-3)\n", "plot_error_l_circular.show(title=\"Timelike geodesic - Circular orbit: error on conservation of l\",\n", " ymin=-1e-3, ymax=1e-3)\n", "plot_error_r_circular.show(title=\"Timelike geodesic - Circular orbit: error on conservation of r\",\n", " ymin=-1e-3, ymax=1e-3)\n", "\n", "plot3D_projected_geod_circular = geod.plot_integrated(interpolation_key='timelike_circular_equatorial',\n", " mapping=BL_spatial_coords, \n", " plot_points=250, thickness=2,\n", " display_tangent=True, \n", " plot_points_tangent=10, scale=8,\n", " width_tangent=1, label_axes=False)\n", "\n", "(plot3D_projected_geod_circular + plot3D_event_horizon + plot3D_origin).show(viewer='threejs',\n", " online=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The second graph, which displays the evolution of radial coordinate $r$, does confirm that the orbit is circular, up to negligible oscillations.\n", "\n", "##### Unbounded trajectory\n", "\n", "In Schwarzschild spacetime, geodesics with sufficiently high radial velocity or angular momentum may induce unbounded trajectories.\n", "An example of such a geodesic (still parametrised with proper time) which has same angular momentum as that of the geodesics considered above is provided below:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Performing numerical integration with method 'rk4_maxima'...\n", "Numerical integration completed.\n", "\n", "Checking all points are in the chart domain...\n", "All points are in the chart domain.\n", "\n", "The resulting list of points was associated with the key 'timelike_unbounded_equatorial' (if this key already referred to a former numerical solution, such a solution was erased).\n", "Performing cubic spline interpolation by default...\n", "Interpolation completed and associated with the key 'timelike_unbounded_equatorial' (if this key already referred to a former interpolation, such an interpolation was erased).\n" ] }, { "data": { "image/png": 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rL7fLpOe+++5z+i9t1g530qbE+uabb1y2c+jQIX377bdq0aLFTf3I0DvvvOO0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\nUZgDAFAWx7yVZdmyZbnmmmsya9asbNmyJV/72td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z585V8+bN3dOqvvfeezp06JC++OKLTK3DmwcffFANGzbUU089pV27drmnMl2w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\nRI8+ExYWprvuuos+c5OKjo7WokWLtHP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Tpo3uu+8+7d+/3zGU6bJlyzR16lS3dQI5MWzYML3zzjuaOHGiNm7cqJYtWyo5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\nKt59993o6emJV199tfeYN954IyZOnBgdHR3R1NT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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "params_values_unbounded = {m:1, s_0:0, s_max:100, t_0:0, r_0:4, th_0:pi/2, ph_0:0, \n", " Dt_0:sqrt(16.405)/2, Dr_0:0.210, Dth_0:0, Dph_0:4.1/16}\n", "\n", "sol_unbounded = geod.solve(step=1, parameters_values=params_values_unbounded, \n", " solution_key='timelike_unbounded_equatorial', verbose=True)\n", "interp_unbounded = geod.interpolate(solution_key='timelike_unbounded_equatorial', \n", " interpolation_key='timelike_unbounded_equatorial', \n", " verbose=True)\n", "\n", "error_squar_norm_unbounded = []\n", "error_e_unbounded = []\n", "error_l_unbounded = []\n", "\n", "i = 0\n", "for (S,T,R,TH,PH) in sol_unbounded:\n", " P = geod(S, interpolation_key='timelike_unbounded_equatorial')\n", " V = geod.tangent_vector_eval_at(S, interpolation_key='timelike_unbounded_equatorial')\n", "\n", " squar_norm_unbounded = numerical_approx((g.at(P)(V,V)).substitute({m:1}))\n", " e_unbounded = numerical_approx((-g.at(P)[0,0]*V[0]).substitute({m:1}))\n", " l_unbounded = numerical_approx((g.at(P)[3,3]*V[3]).substitute({m:1}))\n", " \n", " if i == 0:\n", " squar_norm_unbounded_0 = squar_norm_unbounded\n", " e_unbounded_0 = e_unbounded\n", " l_unbounded_0 = l_unbounded\n", "\n", " error_squar_norm_unbounded += [(S,squar_norm_unbounded - squar_norm_unbounded_0)]\n", " error_e_unbounded += [(S,e_unbounded - e_unbounded_0)]\n", " error_l_unbounded += [(S,l_unbounded - l_unbounded_0)]\n", " \n", " if i == 10:\n", " squar_norm_unbounded_0 = squar_norm_unbounded\n", " e_unbounded_0 = e_unbounded\n", " l_unbounded_0 = l_unbounded\n", "\n", " error_squar_norm_unbounded += [(S,squar_norm_unbounded - squar_norm_unbounded_0)]\n", " error_e_unbounded += [(S,e_unbounded - e_unbounded_0)]\n", " error_l_unbounded += [(S,l_unbounded - l_unbounded_0)] \n", " \n", " i += 1\n", "\n", "plot_error_squar_norm_unbounded = line(error_squar_norm_unbounded)\n", "plot_error_e_unbounded = line(error_e_unbounded)\n", "plot_error_l_unbounded = line(error_l_unbounded)\n", "\n", "plot_error_squar_norm_unbounded.show(title= \n", " \"Timelike geodesic - Unbounded trajectory: error on \" +\n", " \"conservation of squared norm\", \n", " ymin=-1e-3, ymax=1e-3) \n", "plot_error_e_unbounded.show(title=\"Timelike geodesic - Unbounded trajectory: error on conservation of e\",\n", " ymin=-1e-3, ymax=1e-3)\n", "plot_error_l_unbounded.show(title=\"Timelike geodesic - Unbounded trajectory: error on conservation of l\",\n", " ymin=-1e-3, ymax=1e-3)\n", "\n", "plot3D_projected_geod_unbounded = geod.plot_integrated(interpolation_key='timelike_unbounded_equatorial',\n", " mapping=BL_spatial_coords, \n", " plot_points=200, thickness=2,\n", " display_tangent=True, \n", " plot_points_tangent=10, scale=8,\n", " width_tangent=1, label_axes=False)\n", "\n", "(plot3D_projected_geod_unbounded + plot3D_event_horizon + plot3D_origin).show(viewer='threejs',\n", " online=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "#### Null geodesics\n", "\n", "##### Unstable circular orbit\n", "\n", "The only null, bounded geodesics of Schwarzschild spacetime that do not cross the event horizon induce unstable circular orbits.\n", "In addition, the (constant) radius of such trajectories necessarily equals $3m$.\n", "\n", "Below is provided an example of such a (affinely parametrised) geodesic, for which conservation of squared norm, energy and angular momentum is checked:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Performing numerical integration with method 'rk4_maxima'...\n", "Numerical integration completed.\n", "\n", "Checking all points are in the chart domain...\n", "All points are in the chart domain.\n", "\n", "The resulting list of points was associated with the key 'null_circular_equatorial' (if this key already referred to a former numerical solution, such a solution was erased).\n", "Performing cubic spline interpolation by default...\n", "Interpolation completed and associated with the key 'null_circular_equatorial' (if this key already referred to a former interpolation, such an interpolation was erased).\n" ] }, { "data": { "image/png": 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