{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Tolman-Oppenheimer-Volkoff equations\n", "\n", "This Jupyter/SageMath worksheet is relative to the lectures\n", "[General relativity computations with SageManifolds](https://indico.cern.ch/event/505595/) given at the NewCompStar School 2016 (Coimbra, Portugal).\n", " \n", "These computations are based on [SageManifolds](http://sagemanifolds.obspm.fr) (v0.9)\n", "\n", "Click [here](https://raw.githubusercontent.com/egourgoulhon/NewCompStarSchool/master/WorkSheets/TOV.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`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This worksheet is divided in two parts:\n", "\n", "1. Deriving the TOV system from the Einstein equation\n", "2. Solving the TOV system to get stellar models\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Deriving the TOV system from the Einstein equation\n", "\n", "### Spacetime\n", "\n", "We declare the spacetime manifold $M$:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional differentiable manifold M\n" ] } ], "source": [ "M = Manifold(4, 'M')\n", "print(M)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "M?" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "M??" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We declare the chart of spherical coordinates $(t,r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "M.chart?" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X. = M.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "X" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Metric tensor\n", "\n", "The static and spherically symmetric metric ansatz, with the unknown functions $\\nu(r)$ and $m(r)$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{r}{r - 2 \\, m\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = -e^(2*nu(r)) dt*dt + r/(r - 2*m(r)) dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.lorentzian_metric('g')\n", "nu = function('nu')\n", "m = function('m')\n", "g[0,0] = -exp(2*nu(r))\n", "g[1,1] = 1/(1-2*m(r)/r)\n", "g[2,2] = r^2\n", "g[3,3] = (r*sin(th))^2\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One can display the metric components as a list:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t } \\phantom{\\, t } } & = & -e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r } \\phantom{\\, r } } & = & \\frac{r}{r - 2 \\, m\\left(r\\right)} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta} } \\phantom{\\, {\\theta} } } & = & r^{2} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & r^{2} \\sin\\left({\\theta}\\right)^{2} \\end{array}$ " ], "text/plain": [ "g_t,t = -e^(2*nu(r)) \n", "g_r,r = r/(r - 2*m(r)) \n", "g_th,th = r^2 \n", "g_ph,ph = r^2*sin(th)^2 " ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By default, only the nonzero components are shown; to get all the components, set the option `only_nonzero` to `False`:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t } \\phantom{\\, t } } & = & -e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\\\ g_{ \\, t \\, r }^{ \\phantom{\\, t } \\phantom{\\, r } } & = & 0 \\\\ g_{ \\, t \\, {\\theta} }^{ \\phantom{\\, t } \\phantom{\\, {\\theta} } } & = & 0 \\\\ g_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t } \\phantom{\\, {\\phi} } } & = & 0 \\\\ g_{ \\, r \\, t }^{ \\phantom{\\, r } \\phantom{\\, t } } & = & 0 \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r } \\phantom{\\, r } } & = & \\frac{r}{r - 2 \\, m\\left(r\\right)} \\\\ g_{ \\, r \\, {\\theta} }^{ \\phantom{\\, r } \\phantom{\\, {\\theta} } } & = & 0 \\\\ g_{ \\, r \\, {\\phi} }^{ \\phantom{\\, r } \\phantom{\\, {\\phi} } } & = & 0 \\\\ g_{ \\, {\\theta} \\, t }^{ \\phantom{\\, {\\theta} } \\phantom{\\, t } } & = & 0 \\\\ g_{ \\, {\\theta} \\, r }^{ \\phantom{\\, {\\theta} } \\phantom{\\, r } } & = & 0 \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta} } \\phantom{\\, {\\theta} } } & = & r^{2} \\\\ g_{ \\, {\\theta} \\, {\\phi} }^{ \\phantom{\\, {\\theta} } \\phantom{\\, {\\phi} } } & = & 0 \\\\ g_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi} } \\phantom{\\, t } } & = & 0 \\\\ g_{ \\, {\\phi} \\, r }^{ \\phantom{\\, {\\phi} } \\phantom{\\, r } } & = & 0 \\\\ g_{ \\, {\\phi} \\, {\\theta} }^{ \\phantom{\\, {\\phi} } \\phantom{\\, {\\theta} } } & = & 0 \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & r^{2} \\sin\\left({\\theta}\\right)^{2} \\end{array}$ " ], "text/plain": [ "g_t,t = -e^(2*nu(r)) \n", "g_t,r = 0 \n", "g_t,th = 0 \n", "g_t,ph = 0 \n", "g_r,t = 0 \n", "g_r,r = r/(r - 2*m(r)) \n", "g_r,th = 0 \n", "g_r,ph = 0 \n", "g_th,t = 0 \n", "g_th,r = 0 \n", "g_th,th = r^2 \n", "g_th,ph = 0 \n", "g_ph,t = 0 \n", "g_ph,r = 0 \n", "g_ph,th = 0 \n", "g_ph,ph = r^2*sin(th)^2 " ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display_comp(only_nonzero=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also display the metric components as a matrix, via the `[]` operator:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} & 0 & 0 & 0 \\\\\n", "0 & \\frac{r}{r - 2 \\, m\\left(r\\right)} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[ -e^(2*nu(r)) 0 0 0]\n", "[ 0 r/(r - 2*m(r)) 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `[]` operator can also be used to access to individual elements:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-e^{\\left(2 \\, \\nu\\left(r\\right)\\right)}$ " ], "text/plain": [ "-e^(2*nu(r))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[0,0]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Einstein equation\n", "\n", "Let us start by evaluating the Ricci tensor of $g$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "Ric = g.ricci()\n", "print(Ric)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = \\left( \\frac{{\\left(r^{2} e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} - 2 \\, r e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} - {\\left(r e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\frac{\\partial\\,m}{\\partial r} - 2 \\, r e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} + 3 \\, e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + {\\left(r^{2} e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} - 2 \\, r e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2}}{r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{{\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} - 2 \\, r \\frac{\\partial\\,m}{\\partial r} - {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} + 2 \\, m\\left(r\\right)}{r^{3} - 2 \\, r^{2} m\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{r \\frac{\\partial\\,m}{\\partial r} - {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + m\\left(r\\right)}{r} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{{\\left(r \\frac{\\partial\\,m}{\\partial r} - {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + m\\left(r\\right)\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "Ric(g) = ((r^2*e^(2*nu(r)) - 2*r*e^(2*nu(r))*m(r))*(d(nu)/dr)^2 - (r*e^(2*nu(r))*d(m)/dr - 2*r*e^(2*nu(r)) + 3*e^(2*nu(r))*m(r))*d(nu)/dr + (r^2*e^(2*nu(r)) - 2*r*e^(2*nu(r))*m(r))*d^2(nu)/dr^2)/r^2 dt*dt - ((r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 - 2*r*d(m)/dr - (r^2*d(m)/dr - r*m(r))*d(nu)/dr + (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 + 2*m(r))/(r^3 - 2*r^2*m(r)) dr*dr + (r*d(m)/dr - (r^2 - 2*r*m(r))*d(nu)/dr + m(r))/r dth*dth + (r*d(m)/dr - (r^2 - 2*r*m(r))*d(nu)/dr + m(r))*sin(th)^2/r dph*dph" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Ricci scalar is naturally obtained by the method `ricci_scalar()`:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": true }, "outputs": [], "source": [ "g.ricci_scalar?" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{2 \\, {\\left({\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} - {\\left(r \\frac{\\partial\\,m}{\\partial r} - 2 \\, r + 3 \\, m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} - 2 \\, \\frac{\\partial\\,m}{\\partial r}\\right)}}{r^{2}} \\end{array}$ " ], "text/plain": [ "r(g): M --> R\n", " (t, r, th, ph) |--> -2*((r^2 - 2*r*m(r))*(d(nu)/dr)^2 - (r*d(m)/dr - 2*r + 3*m(r))*d(nu)/dr + (r^2 - 2*r*m(r))*d^2(nu)/dr^2 - 2*d(m)/dr)/r^2" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci_scalar().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a check we can also compute it by taking the trace of the Ricci tensor with respect to $g$:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci_scalar() == g.inverse()['^{ab}']*Ric['_{ab}']" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Einstein tensor\n", "$$ G_{ab} := R_{ab} - \\frac{1}{2} R \\, g_{ab},$$\n", "or in index-free notation:\n", "$$ G := \\mathrm{Ric}(g) - \\frac{1}{2} r(g) \\, g $$" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms G on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "G = Ric - 1/2*g.ricci_scalar() * g\n", "G.set_name('G')\n", "print(G)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\frac{2 \\, e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\frac{\\partial\\,m}{\\partial r}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, {\\left({\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} - m\\left(r\\right)\\right)}}{r^{3} - 2 \\, r^{2} m\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{{\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} - r \\frac{\\partial\\,m}{\\partial r} - {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} + m\\left(r\\right)}{r} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{{\\left({\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} - r \\frac{\\partial\\,m}{\\partial r} - {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} + m\\left(r\\right)\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "G = 2*e^(2*nu(r))*d(m)/dr/r^2 dt*dt + 2*((r^2 - 2*r*m(r))*d(nu)/dr - m(r))/(r^3 - 2*r^2*m(r)) dr*dr + ((r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 - r*d(m)/dr - (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr + (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 + m(r))/r dth*dth + ((r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 - r*d(m)/dr - (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr + (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 + m(r))*sin(th)^2/r dph*dph" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "G.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The energy-momentum tensor\n", "\n", "We consider a perfect fluid matter model. \n", "Let us first defined the fluid 4-velocity $u$: " ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = e^{\\left(-\\nu\\left(r\\right)\\right)} \\frac{\\partial}{\\partial t }$ " ], "text/plain": [ "u = e^(-nu(r)) d/dt" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = M.vector_field('u')\n", "u[0] = exp(-nu(r))\n", "u.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[e^{\\left(-\\nu\\left(r\\right)\\right)}, 0, 0, 0\\right]$ " ], "text/plain": [ "[e^(-nu(r)), 0, 0, 0]" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u[:]" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Free module X(M) of vector fields on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "print(u.parent())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $u$ is a normalized timelike vector, i.e. that $g_{ab} u^a u^b = -1$, or, in index-free notation, $g(u,u)=-1$:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g\\left(u,u\\right)$ " ], "text/plain": [ "Scalar field g(u,u) on the 4-dimensional differentiable manifold M" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u,u)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(u,u) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "print(g(u,u))" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(u,u\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}$ " ], "text/plain": [ "g(u,u): M --> R\n", " (t, r, th, ph) |--> -1" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u,u).display()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}C^{\\infty}\\left(M\\right)$ " ], "text/plain": [ "Algebra of differentiable scalar fields on the 4-dimensional differentiable manifold M" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u,u).parent()" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Algebra of differentiable scalar fields on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "print(g(u,u).parent())" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u,u) == -1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To form the energy-momentum tensor, we need the 1-form $\\underline{u}$ that is metric-dual to the vector $u$, i.e. \n", "$u_a = g_{ab} u^b$:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "u_form = u.down(g)\n", "print(u_form)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-e^{\\nu\\left(r\\right)} \\mathrm{d} t$ " ], "text/plain": [ "-e^nu(r) dt" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u_form.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The energy-momentum tensor is then\n", "$$ T_{ab} = (\\rho + p) u_a u_b + p \\, g_{ab},$$\n", "or in index-free notation:\n", "$$ T = (\\rho + p) \\underline{u}\\otimes\\underline{u} + p \\, g$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since the tensor product $\\otimes$ is taken with the `*` operator, we write:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms T on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "rho = function('rho')\n", "p = function('p')\n", "T = (rho(r)+p(r))* (u_form * u_form) + p(r) * g\n", "T.set_name('T')\n", "print(T)" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}T = e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\rho\\left(r\\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{r p\\left(r\\right)}{r - 2 \\, m\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} p\\left(r\\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} p\\left(r\\right) \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "T = e^(2*nu(r))*rho(r) dt*dt + r*p(r)/(r - 2*m(r)) dr*dr + r^2*p(r) dth*dth + r^2*p(r)*sin(th)^2 dph*dph" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T.display()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}T\\left(u,u\\right)$ " ], "text/plain": [ "Scalar field T(u,u) on the 4-dimensional differentiable manifold M" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T(u,u)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field T(u,u) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "print(T(u,u))" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} T\\left(u,u\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\rho\\left(r\\right) \\end{array}$ " ], "text/plain": [ "T(u,u): M --> R\n", " (t, r, th, ph) |--> rho(r)" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T(u,u).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Einstein equation\n", "\n", "The Einstein equation is \n", "$$ G = 8\\pi T$$\n", "We rewrite it as $E = 0$ with $E := G - 8\\pi T$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms E on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "E = G - 8*pi*T\n", "E.set_name('E')\n", "print(E)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}E = -\\frac{2 \\, {\\left(4 \\, \\pi r^{2} e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\rho\\left(r\\right) - e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\frac{\\partial\\,m}{\\partial r}\\right)}}{r^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, {\\left(4 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + m\\left(r\\right)\\right)}}{r^{3} - 2 \\, r^{2} m\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( -\\frac{8 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} + r \\frac{\\partial\\,m}{\\partial r} + {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} - m\\left(r\\right)}{r} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -\\frac{{\\left(8 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} + r \\frac{\\partial\\,m}{\\partial r} + {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} - m\\left(r\\right)\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "E = -2*(4*pi*r^2*e^(2*nu(r))*rho(r) - e^(2*nu(r))*d(m)/dr)/r^2 dt*dt - 2*(4*pi*r^3*p(r) - (r^2 - 2*r*m(r))*d(nu)/dr + m(r))/(r^3 - 2*r^2*m(r)) dr*dr - (8*pi*r^3*p(r) - (r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 + r*d(m)/dr + (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr - (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 - m(r))/r dth*dth - (8*pi*r^3*p(r) - (r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 + r*d(m)/dr + (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr - (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 - m(r))*sin(th)^2/r dph*dph" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.display()" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} E_{ \\, t \\, t }^{ \\phantom{\\, t } \\phantom{\\, t } } & = & -\\frac{2 \\, {\\left(4 \\, \\pi r^{2} e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\rho\\left(r\\right) - e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\frac{\\partial\\,m}{\\partial r}\\right)}}{r^{2}} \\\\ E_{ \\, r \\, r }^{ \\phantom{\\, r } \\phantom{\\, r } } & = & -\\frac{2 \\, {\\left(4 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + m\\left(r\\right)\\right)}}{r^{3} - 2 \\, r^{2} m\\left(r\\right)} \\\\ E_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta} } \\phantom{\\, {\\theta} } } & = & -\\frac{8 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} + r \\frac{\\partial\\,m}{\\partial r} + {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} - m\\left(r\\right)}{r} \\\\ E_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & -\\frac{{\\left(8 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} + r \\frac{\\partial\\,m}{\\partial r} + {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} - m\\left(r\\right)\\right)} \\sin\\left({\\theta}\\right)^{2}}{r} \\end{array}$ " ], "text/plain": [ "E_t,t = -2*(4*pi*r^2*e^(2*nu(r))*rho(r) - e^(2*nu(r))*d(m)/dr)/r^2 \n", "E_r,r = -2*(4*pi*r^3*p(r) - (r^2 - 2*r*m(r))*d(nu)/dr + m(r))/(r^3 - 2*r^2*m(r)) \n", "E_th,th = -(8*pi*r^3*p(r) - (r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 + r*d(m)/dr + (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr - (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 - m(r))/r \n", "E_ph,ph = -(8*pi*r^3*p(r) - (r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 + r*d(m)/dr + (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr - (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 - m(r))*sin(th)^2/r " ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.display_comp()" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(4 \\, \\pi r^{2} e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\rho\\left(r\\right) - e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} \\frac{\\partial\\,m}{\\partial r}\\right)}}{r^{2}}$ " ], "text/plain": [ "-2*(4*pi*r^2*e^(2*nu(r))*rho(r) - e^(2*nu(r))*d(m)/dr)/r^2" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E[0,0]" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[D[0]\\left(m\\right)\\left(r\\right) = 4 \\, \\pi r^{2} \\rho\\left(r\\right)\\right]$ " ], "text/plain": [ "[D[0](m)(r) == 4*pi*r^2*rho(r)]" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE0 = solve(E[0,0].expr()==0, diff(m(r),r))\n", "EE0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The notation `D[0](m)` for the first derivative of $m(r)$ is not very handy; this will be improved in SageMath 7.4; meanwhile, we may use the `ExpressionNice` utility to restore some textbook notation:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial\\,m}{\\partial r} = 4 \\, \\pi r^{2} \\rho\\left(r\\right)$ " ], "text/plain": [ "d(m)/dr == 4*pi*r^2*rho(r)" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sage.manifolds.utilities import ExpressionNice\n", "EE0 = ExpressionNice(EE0[0])\n", "EE0" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(4 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + m\\left(r\\right)\\right)}}{r^{3} - 2 \\, r^{2} m\\left(r\\right)}$ " ], "text/plain": [ "-2*(4*pi*r^3*p(r) - (r^2 - 2*r*m(r))*d(nu)/dr + m(r))/(r^3 - 2*r^2*m(r))" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E[1,1]" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[D[0]\\left(\\nu\\right)\\left(r\\right) = \\frac{4 \\, \\pi r^{3} p\\left(r\\right) + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}\\right]$ " ], "text/plain": [ "[D[0](nu)(r) == (4*pi*r^3*p(r) + m(r))/(r^2 - 2*r*m(r))]" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE1 = solve(E[1,1].expr()==0, diff(nu(r),r))\n", "EE1" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial\\,\\nu}{\\partial r} = \\frac{4 \\, \\pi r^{3} p\\left(r\\right) + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}$ " ], "text/plain": [ "d(nu)/dr == (4*pi*r^3*p(r) + m(r))/(r^2 - 2*r*m(r))" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE1 = ExpressionNice(EE1[0])\n", "EE1" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E[3,3] == E[2,2]*sin(th)^2" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{8 \\, \\pi r^{3} p\\left(r\\right) - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\left(\\frac{\\partial\\,\\nu}{\\partial r}\\right)^{2} + r \\frac{\\partial\\,m}{\\partial r} + {\\left(r^{2} \\frac{\\partial\\,m}{\\partial r} - r^{2} + r m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} - {\\left(r^{3} - 2 \\, r^{2} m\\left(r\\right)\\right)} \\frac{\\partial^2\\,\\nu}{\\partial r^2} - m\\left(r\\right)}{r}$ " ], "text/plain": [ "-(8*pi*r^3*p(r) - (r^3 - 2*r^2*m(r))*(d(nu)/dr)^2 + r*d(m)/dr + (r^2*d(m)/dr - r^2 + r*m(r))*d(nu)/dr - (r^3 - 2*r^2*m(r))*d^2(nu)/dr^2 - m(r))/r" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E[2,2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The energy-momentum conservation equation\n", "The energy-momentum tensor must obey\n", "$$ \\nabla_b T^b_{\\ \\, a} = 0$$\n", "We first form the tensor $T^b_{\\ \\, a}$ by raising the first index of $T_{ab}$:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (1,1) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "Tu = T.up(g, 0)\n", "print(Tu)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We get the Levi-Civita connection $\\nabla$ associated with the metric $g$:" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "nabla = g.connection()\n", "print(nabla)" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t } \\, t \\, r }^{ \\, t \\phantom{\\, t } \\phantom{\\, r } } & = & \\frac{\\partial\\,\\nu}{\\partial r} \\\\ \\Gamma_{ \\phantom{\\, t } \\, r \\, t }^{ \\, t \\phantom{\\, r } \\phantom{\\, t } } & = & \\frac{\\partial\\,\\nu}{\\partial r} \\\\ \\Gamma_{ \\phantom{\\, r } \\, t \\, t }^{ \\, r \\phantom{\\, t } \\phantom{\\, t } } & = & \\frac{{\\left(r e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} - 2 \\, e^{\\left(2 \\, \\nu\\left(r\\right)\\right)} m\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r}}{r} \\\\ \\Gamma_{ \\phantom{\\, r } \\, r \\, r }^{ \\, r \\phantom{\\, r } \\phantom{\\, r } } & = & \\frac{r \\frac{\\partial\\,m}{\\partial r} - m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)} \\\\ \\Gamma_{ \\phantom{\\, r } \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta} } \\phantom{\\, {\\theta} } } & = & -r + 2 \\, m\\left(r\\right) \\\\ \\Gamma_{ \\phantom{\\, r } \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & -{\\left(r - 2 \\, m\\left(r\\right)\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta} } \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r } \\phantom{\\, {\\theta} } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta} } \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta} } \\phantom{\\, r } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta} } \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi} } \\phantom{\\, {\\phi} } } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r } \\phantom{\\, {\\phi} } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta} } \\phantom{\\, {\\phi} } } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi} } \\phantom{\\, r } } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi} } \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi} } \\phantom{\\, {\\theta} } } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$ " ], "text/plain": [ "Gam^t_t,r = d(nu)/dr \n", "Gam^t_r,t = d(nu)/dr \n", "Gam^r_t,t = (r*e^(2*nu(r)) - 2*e^(2*nu(r))*m(r))*d(nu)/dr/r \n", "Gam^r_r,r = (r*d(m)/dr - m(r))/(r^2 - 2*r*m(r)) \n", "Gam^r_th,th = -r + 2*m(r) \n", "Gam^r_ph,ph = -(r - 2*m(r))*sin(th)^2 \n", "Gam^th_r,th = 1/r \n", "Gam^th_th,r = 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) \n", "Gam^ph_ph,r = 1/r \n", "Gam^ph_ph,th = cos(th)/sin(th) " ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We apply `nabla` to `Tu` to get the tensor $(\\nabla T)^b_{\\ \\, ac} = \\nabla_c T^b_{\\ \\, a}$ (MTW index convention):" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (1,2) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "dTu = nabla(Tu)\n", "print(dTu)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The divergence $\\nabla_b T^b_{\\ \\, a}$ is then computed as the trace of the tensor $(\\nabla T)^b_{\\ \\, ac}$ on the first index ($b$, position `0`) and last index ($c$, position `2`):" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "divT = dTu.trace(0,2)\n", "print(divT)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also take the trace by using the index notation:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divT == dTu['^b_{ab}']" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "print(divT)" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( {\\left(p\\left(r\\right) + \\rho\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + \\frac{\\partial\\,p}{\\partial r} \\right) \\mathrm{d} r$ " ], "text/plain": [ "((p(r) + rho(r))*d(nu)/dr + d(p)/dr) dr" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divT.display()" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0, {\\left(p\\left(r\\right) + \\rho\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + \\frac{\\partial\\,p}{\\partial r}, 0, 0\\right]$ " ], "text/plain": [ "[0, (p(r) + rho(r))*d(nu)/dr + d(p)/dr, 0, 0]" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divT[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The only non trivially vanishing components is thus" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(p\\left(r\\right) + \\rho\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r} + \\frac{\\partial\\,p}{\\partial r}$ " ], "text/plain": [ "(p(r) + rho(r))*d(nu)/dr + d(p)/dr" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "divT[1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence the energy-momentum conservation equation $\\nabla_b T^b_{\\ \\, a}=0$ reduces to" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[D[0]\\left(p\\right)\\left(r\\right) = -{\\left(p\\left(r\\right) + \\rho\\left(r\\right)\\right)} D[0]\\left(\\nu\\right)\\left(r\\right)\\right]$ " ], "text/plain": [ "[D[0](p)(r) == -(p(r) + rho(r))*D[0](nu)(r)]" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE2 = solve(divT[1].expr()==0, diff(p(r),r))\n", "EE2" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial\\,p}{\\partial r} = -{\\left(p\\left(r\\right) + \\rho\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r}$ " ], "text/plain": [ "d(p)/dr == -(p(r) + rho(r))*d(nu)/dr" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE2 = ExpressionNice(EE2[0])\n", "EE2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The TOV system\n", "\n", "Let us collect all the independent equations obtained so far:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{\\partial\\,m}{\\partial r} = 4 \\, \\pi r^{2} \\rho\\left(r\\right), \\frac{\\partial\\,\\nu}{\\partial r} = \\frac{4 \\, \\pi r^{3} p\\left(r\\right) + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}, \\frac{\\partial\\,p}{\\partial r} = -{\\left(p\\left(r\\right) + \\rho\\left(r\\right)\\right)} \\frac{\\partial\\,\\nu}{\\partial r}\\right)$ " ], "text/plain": [ "(d(m)/dr == 4*pi*r^2*rho(r),\n", " d(nu)/dr == (4*pi*r^3*p(r) + m(r))/(r^2 - 2*r*m(r)),\n", " d(p)/dr == -(p(r) + rho(r))*d(nu)/dr)" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE0, EE1, EE2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Solving the TOV system\n", "\n", "In order to solve the TOV system, we need to specify a fluid equation of state. For simplicity, we select a polytropic one:\n", "$$ p = k \\rho^2$$" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}k \\rho\\left(r\\right)^{2}$ " ], "text/plain": [ "k*rho(r)^2" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('k', domain='real')\n", "p_eos(r) = k*rho(r)^2\n", "p_eos(r)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We substitute this expression for $p$ in the TOV equations:" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}D[0]\\left(\\nu\\right)\\left(r\\right) = \\frac{4 \\, \\pi k r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}$ " ], "text/plain": [ "D[0](nu)(r) == (4*pi*k*r^3*rho(r)^2 + m(r))/(r^2 - 2*r*m(r))" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE1_rho = EE1.substitute_function(p, p_eos)\n", "EE1_rho" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, k \\rho\\left(r\\right) D[0]\\left(\\rho\\right)\\left(r\\right) = -{\\left(k \\rho\\left(r\\right)^{2} + \\rho\\left(r\\right)\\right)} D[0]\\left(\\nu\\right)\\left(r\\right)$ " ], "text/plain": [ "2*k*rho(r)*D[0](rho)(r) == -(k*rho(r)^2 + rho(r))*D[0](nu)(r)" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE2_rho = EE2.substitute_function(p, p_eos)\n", "EE2_rho" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}D[0]\\left(\\rho\\right)\\left(r\\right) = -\\frac{{\\left(k \\rho\\left(r\\right) + 1\\right)} D[0]\\left(\\nu\\right)\\left(r\\right)}{2 \\, k}$ " ], "text/plain": [ "D[0](rho)(r) == -1/2*(k*rho(r) + 1)*D[0](nu)(r)/k" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE2_rho = (EE2_rho / (2*k*rho(r))).simplify_full()\n", "EE2_rho" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We subsitute the expression of equation `EE1_rho` for $\\partial \\nu/\\partial r$, in order to get rid of any derivative in the right-hand sides:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}D[0]\\left(\\rho\\right)\\left(r\\right) = -\\frac{4 \\, \\pi k^{2} r^{3} \\rho\\left(r\\right)^{3} + 4 \\, \\pi k r^{3} \\rho\\left(r\\right)^{2} + k m\\left(r\\right) \\rho\\left(r\\right) + m\\left(r\\right)}{2 \\, {\\left(k r^{2} - 2 \\, k r m\\left(r\\right)\\right)}}$ " ], "text/plain": [ "D[0](rho)(r) == -1/2*(4*pi*k^2*r^3*rho(r)^3 + 4*pi*k*r^3*rho(r)^2 + k*m(r)*rho(r) + m(r))/(k*r^2 - 2*k*r*m(r))" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE2_rho = EE2_rho.subs({diff(nu(r),r): EE1_rho.rhs()}).simplify_full()\n", "EE2_rho" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system to solve for $(m(r), \\nu(r), \\rho(r))$ is thus" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{\\partial\\,m}{\\partial r} = 4 \\, \\pi r^{2} \\rho\\left(r\\right), \\frac{\\partial\\,\\nu}{\\partial r} = \\frac{4 \\, \\pi k r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}, \\frac{\\partial\\,\\rho}{\\partial r} = -\\frac{4 \\, \\pi k^{2} r^{3} \\rho\\left(r\\right)^{3} + 4 \\, \\pi k r^{3} \\rho\\left(r\\right)^{2} + k m\\left(r\\right) \\rho\\left(r\\right) + m\\left(r\\right)}{2 \\, {\\left(k r^{2} - 2 \\, k r m\\left(r\\right)\\right)}}\\right)$ " ], "text/plain": [ "(d(m)/dr == 4*pi*r^2*rho(r),\n", " d(nu)/dr == (4*pi*k*r^3*rho(r)^2 + m(r))/(r^2 - 2*r*m(r)),\n", " d(rho)/dr == -1/2*(4*pi*k^2*r^3*rho(r)^3 + 4*pi*k*r^3*rho(r)^2 + k*m(r)*rho(r) + m(r))/(k*r^2 - 2*k*r*m(r)))" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EE1_rho = ExpressionNice(EE1_rho)\n", "EE2_rho = ExpressionNice(EE2_rho)\n", "EE0, EE1_rho, EE2_rho" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Numerical resolution\n", "\n", "Let us use a standard 4th-order Runge-Kutta method:" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": true }, "outputs": [], "source": [ "desolve_system_rk4?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We gather all equations in a list for the ease of manipulation:" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": true }, "outputs": [], "source": [ "eqs = [EE0, EE1_rho, EE2_rho]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get a numerical solution, we have of course to specify some numerical value for the EOS constant $k$; let us choose $k=1/4$:" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[4 \\, \\pi r^{2} \\rho\\left(r\\right), \\frac{\\pi r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}, -\\frac{\\pi r^{3} \\rho\\left(r\\right)^{3} + 4 \\, \\pi r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right) \\rho\\left(r\\right) + 4 \\, m\\left(r\\right)}{2 \\, {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)}}\\right]$ " ], "text/plain": [ "[4*pi*r^2*rho(r),\n", " (pi*r^3*rho(r)^2 + m(r))/(r^2 - 2*r*m(r)),\n", " -1/2*(pi*r^3*rho(r)^3 + 4*pi*r^3*rho(r)^2 + m(r)*rho(r) + 4*m(r))/(r^2 - 2*r*m(r))]" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k0 = 1/4\n", "rhs = [eq.rhs().subs(k=k0) for eq in eqs]\n", "rhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The integration for $m(r)$ and $\\rho(r)$ has to stop as soon as $\\rho(r)$ become negative. An easy way to ensure this is to use the Heaviside function:" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h(x) = (1+sign(x))/2\n", "plot(h(x), (x,-4,4), thickness=2, aspect_ratio=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and to multiply the r.h.s. of the $dm/dr$ and $d\\rho/dr$ equations by $h(\\rho)$:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, \\pi r^{2} {\\left(\\mathrm{sgn}\\left(\\rho\\left(r\\right)\\right) + 1\\right)} \\rho\\left(r\\right), \\frac{\\pi r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}, -\\frac{{\\left(\\pi r^{3} \\rho\\left(r\\right)^{3} + 4 \\, \\pi r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right) \\rho\\left(r\\right) + 4 \\, m\\left(r\\right)\\right)} {\\left(\\mathrm{sgn}\\left(\\rho\\left(r\\right)\\right) + 1\\right)}}{4 \\, {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)}}\\right]$ " ], "text/plain": [ "[2*pi*r^2*(sgn(rho(r)) + 1)*rho(r),\n", " (pi*r^3*rho(r)^2 + m(r))/(r^2 - 2*r*m(r)),\n", " -1/4*(pi*r^3*rho(r)^3 + 4*pi*r^3*rho(r)^2 + m(r)*rho(r) + 4*m(r))*(sgn(rho(r)) + 1)/(r^2 - 2*r*m(r))]" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rhs[0] = rhs[0] * h(rho(r))\n", "rhs[2] = rhs[2] * h(rho(r))\n", "rhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us add an extra equation, for the purpose of getting the star's radius as an output of the integration, via the equation $dR/dr = 1$, again multiplied by the heaviside function of $\\rho$:" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, \\pi r^{2} {\\left(\\mathrm{sgn}\\left(\\rho\\left(r\\right)\\right) + 1\\right)} \\rho\\left(r\\right), \\frac{\\pi r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right)}{r^{2} - 2 \\, r m\\left(r\\right)}, -\\frac{{\\left(\\pi r^{3} \\rho\\left(r\\right)^{3} + 4 \\, \\pi r^{3} \\rho\\left(r\\right)^{2} + m\\left(r\\right) \\rho\\left(r\\right) + 4 \\, m\\left(r\\right)\\right)} {\\left(\\mathrm{sgn}\\left(\\rho\\left(r\\right)\\right) + 1\\right)}}{4 \\, {\\left(r^{2} - 2 \\, r m\\left(r\\right)\\right)}}, \\frac{1}{2} \\, \\mathrm{sgn}\\left(\\rho\\left(r\\right)\\right) + \\frac{1}{2}\\right]$ " ], "text/plain": [ "[2*pi*r^2*(sgn(rho(r)) + 1)*rho(r),\n", " (pi*r^3*rho(r)^2 + m(r))/(r^2 - 2*r*m(r)),\n", " -1/4*(pi*r^3*rho(r)^3 + 4*pi*r^3*rho(r)^2 + m(r)*rho(r) + 4*m(r))*(sgn(rho(r)) + 1)/(r^2 - 2*r*m(r)),\n", " 1/2*sgn(rho(r)) + 1/2]" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rhs.append(1 * h(rho(r)))\n", "rhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the purpose of the numerical integration via `desolve_system_rk4`, we have to replace the symbolic functions $m(r)$, $\\nu(r)$ and $\\rho(r)$ by some symbolic variables, $m_1$, $\\nu_1$ and\n", "$\\rho_1$, say:" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, \\pi r^{2} \\rho_{1} {\\left(\\mathrm{sgn}\\left(\\rho_{1}\\right) + 1\\right)}, -\\frac{\\pi r^{3} \\rho_{1}^{2} + m_{1}}{2 \\, m_{1} r - r^{2}}, \\frac{{\\left(\\pi r^{3} \\rho_{1}^{3} + 4 \\, \\pi r^{3} \\rho_{1}^{2} + m_{1} \\rho_{1} + 4 \\, m_{1}\\right)} {\\left(\\mathrm{sgn}\\left(\\rho_{1}\\right) + 1\\right)}}{4 \\, {\\left(2 \\, m_{1} r - r^{2}\\right)}}, \\frac{1}{2} \\, \\mathrm{sgn}\\left(\\rho_{1}\\right) + \\frac{1}{2}\\right]$ " ], "text/plain": [ "[2*pi*r^2*rho_1*(sgn(rho_1) + 1),\n", " -(pi*r^3*rho_1^2 + m_1)/(2*m_1*r - r^2),\n", " 1/4*(pi*r^3*rho_1^3 + 4*pi*r^3*rho_1^2 + m_1*rho_1 + 4*m_1)*(sgn(rho_1) + 1)/(2*m_1*r - r^2),\n", " 1/2*sgn(rho_1) + 1/2]" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('m_1 nu_1 rho_1 r_1')\n", "rhs = [y.subs({m(r): m_1, nu(r): nu_1, rho(r): rho_1}) for y in rhs]\n", "rhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The integration parameters:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": true }, "outputs": [], "source": [ "rho_c = 1\n", "r_min = 1e-8\n", "r_max = 1\n", "np = 200\n", "delta_r = (r_max - r_min) / (np-1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The numerical resolution, with the initial conditions \n", "$$(r_{\\rm min}, m(r_{\\rm min}), \\nu(r_{\\rm min}), \\rho(r_{\\rm min}), R(r_{\\rm min})) = (r_{\\rm min},0,0,\\rho_{\\rm c},r_{\\rm min}) $$\n", "set in the parameter `ics`:" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": true }, "outputs": [], "source": [ "sol = desolve_system_rk4(rhs, vars=(m_1, nu_1, rho_1, r_1), ivar=r, \n", " ics=[r_min, 0, 0, rho_c, r_min], \n", " end_points=(r_min, r_max), step=delta_r)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution is returned as a list, the first 10 elements of which being:" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[1.00000000000000 \\times 10^{-8}, 0, 0, 1, 1.00000000000000 \\times 10^{-8}\\right], \\left[0.00502513557789, 5.31396735484 \\times 10^{-07}, 0.000105774675722, 0.999735571504, 0.00502513557789\\right], \\left[0.0100502611558, 4.24965314706 \\times 10^{-06}, 0.000384231107483, 0.999039516928, 0.0100502611558\\right], \\left[0.0150753867337, 1.43327909521 \\times 10^{-05}, 0.000846989113836, 0.997882978864, 0.0150753867337\\right], \\left[0.0201005123116, 3.39411395241 \\times 10^{-05}, 0.00149437530532, 0.996265461165, 0.0201005123116\\right], \\left[0.0251256378894, 6.62085388025 \\times 10^{-05}, 0.00232621990806, 0.99418783561, 0.0251256378894\\right], \\left[0.0301507634673, 0.000114233579607, 0.00334221530611, 0.991651444462, 0.0301507634673\\right], \\left[0.0351758890452, 0.000181070903418, 0.00454196301924, 0.988657981618, 0.0351758890452\\right], \\left[0.0402010146231, 0.000269722576805, 0.0059249834935, 0.985209466401, 0.0402010146231\\right], \\left[0.045226140201, 0.000383129555491, 0.00749071832026, 0.98130823596, 0.045226140201\\right]\\right]$ " ], "text/plain": [ "[[1.00000000000000e-8, 0, 0, 1, 1.00000000000000e-8],\n", " [0.00502513557788945,\n", " 5.31396735484216e-07,\n", " 0.000105774675722407,\n", " 0.9997355715039982,\n", " 0.005025135577889449],\n", " [0.0100502611557789,\n", " 4.24965314706328e-06,\n", " 0.000384231107483355,\n", " 0.9990395169277086,\n", " 0.0100502611557789],\n", " [0.01507538673366835,\n", " 1.43327909521466e-05,\n", " 0.000846989113836226,\n", " 0.9978829788643846,\n", " 0.01507538673366835],\n", " [0.0201005123115578,\n", " 3.39411395241295e-05,\n", " 0.001494375305317491,\n", " 0.9962654611649957,\n", " 0.0201005123115578],\n", " [0.02512563788944725,\n", " 6.62085388024929e-05,\n", " 0.002326219908059372,\n", " 0.99418783560987,\n", " 0.02512563788944725],\n", " [0.0301507634673367,\n", " 0.000114233579606946,\n", " 0.003342215306107709,\n", " 0.9916514444622955,\n", " 0.0301507634673367],\n", " [0.03517588904522615,\n", " 0.000181070903418294,\n", " 0.00454196301924163,\n", " 0.9886579816183987,\n", " 0.03517588904522614],\n", " [0.0402010146231156,\n", " 0.000269722576804864,\n", " 0.005924983493499082,\n", " 0.9852094664010502,\n", " 0.04020101462311559],\n", " [0.04522614020100505,\n", " 0.00038312955549096,\n", " 0.007490718320263326,\n", " 0.9813082359597236,\n", " 0.04522614020100504]]" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sol[:10]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Each element is of the form $[r, m(r), \\nu(r), \\rho(r), R(r)]$. So to get the list of $(r, \\rho(r))$ values, we write" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(1.00000000000000 \\times 10^{-8}, 1\\right), \\left(0.00502513557789, 0.999735571504\\right), \\left(0.0100502611558, 0.999039516928\\right), \\left(0.0150753867337, 0.997882978864\\right), \\left(0.0201005123116, 0.996265461165\\right), \\left(0.0251256378894, 0.99418783561\\right), \\left(0.0301507634673, 0.991651444462\\right), \\left(0.0351758890452, 0.988657981618\\right), \\left(0.0402010146231, 0.985209466401\\right), \\left(0.045226140201, 0.98130823596\\right)\\right]$ " ], "text/plain": [ "[(1.00000000000000e-8, 1),\n", " (0.00502513557788945, 0.9997355715039982),\n", " (0.0100502611557789, 0.9990395169277086),\n", " (0.01507538673366835, 0.9978829788643846),\n", " (0.0201005123115578, 0.9962654611649957),\n", " (0.02512563788944725, 0.99418783560987),\n", " (0.0301507634673367, 0.9916514444622955),\n", " (0.03517588904522615, 0.9886579816183987),\n", " (0.0402010146231156, 0.9852094664010502),\n", " (0.04522614020100505, 0.9813082359597236)]" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rho_sol = [(s[0], s[3]) for s in sol]\n", "rho_sol[:10]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may then use this list to have some plot of $\\rho(r)$, thanks to the function `line`, which transforms a list into a graphical object:" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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0BX7RFZhwtS8MDXBeenq6rrlGOuwwaepU22kQZOnp6bYjwCH0BX7RFZhwtS9s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\nxAtDAwAjo0ZJ//639MgjtpMAAIB4YWi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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = line(rho_sol, axes_labels=[r'$r$', r'$\\rho$'], gridlines=True)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution for $m(r)$:" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(1.00000000000000 \\times 10^{-8}, 0\\right), \\left(0.00502513557789, 5.31396735484 \\times 10^{-07}\\right), \\left(0.0100502611558, 4.24965314706 \\times 10^{-06}\\right), \\left(0.0150753867337, 1.43327909521 \\times 10^{-05}\\right), \\left(0.0201005123116, 3.39411395241 \\times 10^{-05}\\right), \\left(0.0251256378894, 6.62085388025 \\times 10^{-05}\\right), \\left(0.0301507634673, 0.000114233579607\\right), \\left(0.0351758890452, 0.000181070903418\\right), \\left(0.0402010146231, 0.000269722576805\\right), \\left(0.045226140201, 0.000383129555491\\right)\\right]$ " ], "text/plain": [ "[(1.00000000000000e-8, 0),\n", " (0.00502513557788945, 5.31396735484216e-07),\n", " (0.0100502611557789, 4.24965314706328e-06),\n", " (0.01507538673366835, 1.43327909521466e-05),\n", " (0.0201005123115578, 3.39411395241295e-05),\n", " (0.02512563788944725, 6.62085388024929e-05),\n", " (0.0301507634673367, 0.000114233579606946),\n", " (0.03517588904522615, 0.000181070903418294),\n", " (0.0402010146231156, 0.000269722576804864),\n", " (0.04522614020100505, 0.00038312955549096)]" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m_sol = [(s[0], s[1]) for s in sol]\n", "m_sol[:10]" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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80qVLNXbsWF1xxRVauXKlRowYoREjRmj16tW7jrnhhhu0cOFCzZw5U7W1tfrlL3+pCRMm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\nCIaOwA56Aju82hOGBg+yLOkPf5B++lNpxAhpxw5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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = line(m_sol, axes_labels=[r'$r$', r'$m$'], gridlines=True)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution for $\\nu(r)$ (has to be rescaled by adding a constant to ensure $\\nu(+\\infty) = 1$):" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(1.00000000000000 \\times 10^{-8}, 0\\right), \\left(0.00502513557789, 0.000105774675722\\right), \\left(0.0100502611558, 0.000384231107483\\right), \\left(0.0150753867337, 0.000846989113836\\right), \\left(0.0201005123116, 0.00149437530532\\right), \\left(0.0251256378894, 0.00232621990806\\right), \\left(0.0301507634673, 0.00334221530611\\right), \\left(0.0351758890452, 0.00454196301924\\right), \\left(0.0402010146231, 0.0059249834935\\right), \\left(0.045226140201, 0.00749071832026\\right)\\right]$ " ], "text/plain": [ "[(1.00000000000000e-8, 0),\n", " (0.00502513557788945, 0.000105774675722407),\n", " (0.0100502611557789, 0.000384231107483355),\n", " (0.01507538673366835, 0.000846989113836226),\n", " (0.0201005123115578, 0.001494375305317491),\n", " (0.02512563788944725, 0.002326219908059372),\n", " (0.0301507634673367, 0.003342215306107709),\n", " (0.03517588904522615, 0.00454196301924163),\n", " (0.0402010146231156, 0.005924983493499082),\n", " (0.04522614020100505, 0.007490718320263326)]" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nu_sol = [(s[0], s[2]) for s in sol]\n", "nu_sol[:10]" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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mdKTSUmn2bPNzAACqgpUGAIgykYgZEs46S2rc2FxKlYEBAHA4GBoAIIqEw1J6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\noGPz9NDAmoaOp6FBSktz7suwZ4/Uu7ftRAAAAOC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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = line(nu_sol, axes_labels=[r'$r$', r'$\\tilde{\\nu}$'], gridlines=True)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution for $R(r)$:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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alBPuvRfKyuD++y0MkiRJteWkQVnviy+ge3c46ih48EFIJEInkiRJyixOGpT1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\nNJIkScoVtSoNTZo0IT8/n7K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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r_sol = [(s[0], s[4]) for s in sol]\n", "line(r_sol)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The total gravitational mass of the star is obtained via the last element (index: `-1`) of the list of $(r,m(r))$ values:" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.095595655983$ " ], "text/plain": [ "0.09559565598299921" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M_grav = m_sol[-1][1]\n", "M_grav" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, the stellar radius is obtained through the last element of the list of $(r,R(r))$ values:" ] }, { "cell_type": "code", "execution_count": 83, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.431323288769$ " ], "text/plain": [ "0.4313232887688435" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R = r_sol[-1][1]\n", "R" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The star's compactness:" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.221633420852$ " ], "text/plain": [ "0.22163342085205887" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M_grav/R" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Sequence of stellar models\n", "\n", "Let us perform a loop on the central density.\n", "First we set up a list of values for $\\rho_c$:" ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false }, "outputs": [ { "data": { 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0.470000000000000,\n", " 0.546666666666667,\n", " 0.623333333333333,\n", " 0.700000000000000,\n", " 0.776666666666667,\n", " 0.853333333333333,\n", " 0.930000000000000,\n", " 1.00666666666667,\n", " 1.08333333333333,\n", " 1.16000000000000,\n", " 1.23666666666667,\n", " 1.31333333333333,\n", " 1.39000000000000,\n", " 1.46666666666667,\n", " 1.54333333333333,\n", " 1.62000000000000,\n", " 1.69666666666667,\n", " 1.77333333333333,\n", " 1.85000000000000,\n", " 1.92666666666667,\n", " 2.00333333333333,\n", " 2.08000000000000,\n", " 2.15666666666667,\n", " 2.23333333333333,\n", " 2.31000000000000,\n", " 2.38666666666667,\n", " 2.46333333333333,\n", " 2.54000000000000,\n", " 2.61666666666667,\n", " 2.69333333333333,\n", " 2.77000000000000,\n", " 2.84666666666667,\n", " 2.92333333333333,\n", " 3.00000000000000]" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rho_c_min = 0.01\n", "rho_c_max = 3\n", "n_conf = 40\n", "rho_c_list = [rho_c_min + i * (rho_c_max-rho_c_min)/(n_conf-1) for i in range(n_conf)]\n", "rho_c_list" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The loop:" ] }, { "cell_type": "code", "execution_count": 86, "metadata": { "collapsed": true }, "outputs": [], "source": [ "M_list = list()\n", "R_list = list()\n", "for rho_c in rho_c_list:\n", " sol = desolve_system_rk4(rhs, vars=(m_1, nu_1, rho_1, r_1), ivar=r, \n", " ics=[r_min, 0, 0, rho_c, r_min], \n", " end_points=(r_min, r_max), step=delta_r)\n", " M_list.append( sol[-1][1] )\n", " R_list.append( sol[-1][4] )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The mass along the sequence:" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.00307931536045, 0.0234854698535, 0.0393069279916, 0.0516981286546, 0.0614830416515, 0.0692798437746, 0.0755155592693, 0.0805202557445, 0.0845711377823, 0.0878316691012, 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8qamppktAhMjQDeRoPzJ0Aznaz9UMwxpp6Natm1q3bq0tW7bUO751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\nhw7V2LFjtXfvXtPRGo9nmfvuu897+umn634OBoNe+/btvfn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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = line(zip(rho_c_list, M_list), axes_labels=[r'$\\rho_c$', r'$M$'], gridlines=True)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, we draw $M$ as a function of $R$:" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": 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MGgQnnbSMQw5J8dpr8b2Pr4q5D9+H78P3Ub33kfXSGerZs2e6pKRk4+/Xr1+f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\n7y98ux4AAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "graph = line(zip(R_list, M_list), axes_labels=[r'$R$', r'$M$'], gridlines=True)\n", "graph" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we save the plot in a pdf file to use it in our next publication ;-)" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": true }, "outputs": [], "source": [ "graph.save('plot_M_R.pdf')" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.3", "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": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }