{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "# Timelike orbits in Schwarzschild spacetime\n", "\n", "This Jupyter/SageMath worksheet is relative to the lectures\n", "[Geometry and physics of black holes](http://luth.obspm.fr/~luthier/gourgoulhon/bh16/).\n", "\n", "Click [here](https://raw.githubusercontent.com/egourgoulhon/BHLectures/master/sage/ges_orbits.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, with the command `sage -n jupyter`\n", "\n", "It uses the `integrated_geodesic` functionality introduced by Karim Van Aelst in **SageMath 8.1**, in the framework of the [SageManifolds](http://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "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": "code", "execution_count": 2, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "%display latex # LaTeX rendering turned on" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We define first the spacetime manifold $M$ and the standard Schwarzschild-Droste coordinates on it:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M')\n", "X. = M.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:\\phi')\n", "X" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "For graphical purposes, we introduce $\\mathbb{R}^3$ and some coordinate map $M\\to \\mathbb{R}^3$:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "M --> R^3\n", " (t, r, th, ph) |--> (x, y, z) = (r*cos(ph)*sin(th), r*sin(ph)*sin(th), r*cos(th))" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3 = Manifold(3, 'R^3', latex_name=r'\\mathbb{R}^3')\n", "X3. = R3.chart()\n", "to_R3 = M.diff_map(R3, {(X, X3): [r*sin(th)*cos(ph), \n", " r*sin(th)*sin(ph), r*cos(th)]})\n", "to_R3.display()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Next, we define the Schwarzschild metric:" ] }, { "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 = M.lorentzian_metric('g')\n", "m = var('m'); assume(m >= 0)\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": { "deletable": true, "editable": true }, "source": [ "We set the specific conserved energy and angular momentum:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "#var('E L', domain='real')\n", "E = 0.973\n", "L = 4.2*m" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u = -0.973000000000000/(2*m/r - 1) d/dt + sqrt((17.6400000000000*m^2/r^2 + 1)*(2*m/r - 1) + 0.946729000000000) d/dr + 4.20000000000000*m/(r^2*sin(th)^2) d/dph" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = M.vector_field(name='u')\n", "u[0] = E/(1-2*m/r)\n", "u[1] = sqrt(E^2 - (1-2*m/r)*(1+L^2/r^2))\n", "u[3] = L/(r^2*sin(th)^2)\n", "u.display()" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g(u,u): M --> R\n", " (t, r, th, ph) |--> -(-35.28*m^3 + 17.64*m^2*r - (-35.28*m^3 + 17.64*m^2*r - 2.0*m*r^2 + 1.0*r^3)*sin(th)^2)/((2*m*r^2 - r^3)*sin(th)^2)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u,u).display()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Pericenter and apocenter:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "0.05327100000000007*r^3 - 2.0*r^2 + 17.64*r - 35.28" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eq = (r^3*((1-2/r)*(1+L^2/m^2/r^2) - E^2)).simplify_full()\n", "eq" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Univariate Polynomial Ring in w over Real Field with 53 bits of precision\n" ] } ], "source": [ "R. = RR[]\n", "print(R)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "0.0532710000000001*w^3 - 2.00000000000000*w^2 + 17.6400000000000*w - 35.2800000000000" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eqp = R(eq.subs(r=w))\n", "eqp" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[2.85236553627370, 9.05774475732293, 25.6337691154627]" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "roots = sorted(eqp.real_roots())\n", "roots" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(9.05774475732293*m, 25.6337691154627*m)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r_per = roots[1]*m\n", "r_apo = roots[2]*m\n", "r_per, r_apo" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We pick an initial point and an initial tangent vector:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector u_0 at Point p_0 on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "r0 = r_per\n", "p0 = M.point((0, r0, pi/2, 1e-12), name='p_0')\n", "Tp0 = M.tangent_space(p0) \n", "u0 = u.at(p0)\n", "u0.set_name('u_0')\n", "print(u0)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u_0 = 1.248725471366881 d/dt + 0.0511928294380894/m d/dph" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u0.display()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Let us check that the scalar square of $u_0$ is $-1$, by means of the metric tensor at $p_0$:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Symmetric bilinear form g on the Tangent space at Point p_0 on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "g0 = g.at(p0)\n", "print(g0)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The scalar square is indeed equal to $-1$: " ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-0.9999999999999992" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g0(u0,u0).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "### Check\n", "\n", "Let us compute the conserved energy and angular momentum along the geodesic by \n", "taking scalar products of the 4-velocity $v_0$ with the Killing vectors $\\xi=\\frac{\\partial}{\\partial t}$ and $\\eta=\\frac{\\partial}{\\partial t}$:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(Vector field d/dt on the 4-dimensional differentiable manifold M,\n", " Vector field d/dph on the 4-dimensional differentiable manifold M)" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xi = X.frame()[0]\n", "eta = X.frame()[3]\n", "xi, eta" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector d/dt at Point p_0 on the 4-dimensional differentiable manifold M\n", "Tangent vector d/dph at Point p_0 on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "xi0 = xi.at(p0)\n", "eta0 = eta.at(p0)\n", "print(xi0)\n", "print(eta0)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The specific conserved energy $\\varepsilon$ is:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "0.9729999999999998" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "- g0(xi0, u0)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "while the specific conserved angular momentum $\\ell$ is" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "4.2*m" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g0(eta0, u0)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We declare the geodesic through $p_0$ having $v_0$ as inital vector, denoting by $s$ the affine parameter (proper time):" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Integrated geodesic in the 4-dimensional differentiable manifold M" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = var('s')\n", "geod = M.integrated_geodesic(g, (s, 0, 1500), u0); geod" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We ask for the numerical integration of the geodesic, providing some numerical value for the parameter $m$, and then plot it in terms of the Cartesian chart:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "sol = geod.solve(step=4, parameters_values={m: 1}) # numerical integration\n", "interp = geod.interpolate() # interpolation of the solution for the plot" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "graph = geod.plot_integrated(chart=X3, mapping=to_R3, ambient_coords=(x,y), plot_points=500, \n", " thickness=2, label_axes=False) " ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "image/png": 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Rq5Yk+DdpIuUqihTJ+/iJKEMMsmwtJQUYMgTxi1eiF1bgRcP36LmwNzB4sN4j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EAAAg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\na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fqRajt3rKyP0Ui5j0lJtIi5cUNri2yHPXuoGlMUSfLlyBHaseDYDLpxsjZsoNZLDx9q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\npoqEigQrBVkiOWAYQP36QEwMi0Ga0tI48eRIPgPs3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"text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph += circle((0,0), r_per/m, linestyle=':')\n", "graph += circle((0,0), r_apo/m, linestyle=':')\n", "graph += circle((0,0), 2, fill=True, edgecolor='grey', facecolor='grey')\n", "show(graph, aspect_ratio=1, axes_labels=[r'$x/m$', r'$y/m$'])" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "'ges_orbit_e0.973_l4.200.pdf'" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "file_name = \"ges_orbit_e{:.3f}_l{:.3f}.pdf\".format(float(E), float(L/m))\n", "graph.save(file_name, aspect_ratio=1, axes_labels=[r'$x/m$', r'$y/m$'])\n", "file_name" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Some details about the system solved to get the geodesic:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Geodesic in the 4-dimensional differentiable manifold M equipped with Lorentzian metric g on the 4-dimensional differentiable manifold M, and integrated over the Real interval (0, 1500) as a solution to the following geodesic equations, written with respect to Chart (M, (t, r, th, ph)):\n", "\n", "Initial point: Point p_0 on the 4-dimensional differentiable manifold M with coordinates [0, 9.05774475732293*m, 1/2*pi, 1.00000000000000e-12] with respect to Chart (M, (t, r, th, ph))\n", "Initial tangent vector: Tangent vector u_0 at Point p_0 on the 4-dimensional differentiable manifold M with components [1.248725471366881, 0, 0, 0.0511928294380894/m] with respect to Chart (M, (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" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "[[2*Dr*Dt*m/(2*m*r - r^2),\n", " -(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", " (Dph^2*r*cos(th)*sin(th) - 2*Dr*Dth)/r,\n", " -2*(Dph*Dth*r*cos(th) + Dph*Dr*sin(th))/(r*sin(th))],\n", " Tangent vector u_0 at Point p_0 on the 4-dimensional differentiable manifold M,\n", " Chart (M, (t, r, th, ph))]" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod.system(verbose=True)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We recognize in the above list the Christoffel symbols of the metric $g$:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Gam^t_t,r = -m/(2*m*r - r^2) \n", "Gam^r_t,t = -(2*m^2 - m*r)/r^3 \n", "Gam^r_r,r = m/(2*m*r - r^2) \n", "Gam^r_th,th = 2*m - r \n", "Gam^r_ph,ph = (2*m - r)*sin(th)^2 \n", "Gam^th_r,th = 1/r \n", "Gam^th_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_r,ph = 1/r \n", "Gam^ph_th,ph = cos(th)/sin(th) " ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.1", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 2 }