{ "cells": [ { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "code", "execution_count": 5, "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": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (M, (t, x, th, ph))" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X. = M.chart(r't:\\tau x:(0,+oo):\\chi th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "X" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = -dt*dt + (-3*sqrt(1/2)*t*sqrt(m_0/xs^3) + 1)^(4/3) dx*dx + (-3*sqrt(1/2)*t*sqrt(m_0/xs^3) + 1)^(4/3)*x^2 dth*dth + (-3*sqrt(1/2)*t*sqrt(m_0/xs^3) + 1)^(4/3)*x^2*sin(th)^2 dph*dph" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.lorentzian_metric('g')\n", "m0 = var('m_0', domain='real'); assume(m0>0)\n", "xs = var('xs', latex_name=r'\\chi_{\\rm s}', domain='real'); assume(xs>0)\n", "a = (1 - 3*sqrt(m0/(2*xs^3)) * t)^(2/3)\n", "g[0,0] = -1\n", "g[1,1] = a^2\n", "g[2,2] = (a*x)^2\n", "g[3,3] = (a*x*sin(th))^2\n", "g.display()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g_t,t = -1 \n", "g_x,x = (-3*sqrt(1/2)*t*sqrt(m_0/xs^3) + 1)^(4/3) \n", "g_th,th = (-3*sqrt(1/2)*t*sqrt(m_0/xs^3) + 1)^(4/3)*x^2 \n", "g_ph,ph = (-3*sqrt(1/2)*t*sqrt(m_0/xs^3) + 1)^(4/3)*x^2*sin(th)^2 " ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display_comp()" ] }, { "cell_type": "code", "execution_count": 10, "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": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Ric(g)_t,t = -6*m_0/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3) \n", "Ric(g)_x,x = 3*2^(2/3)*m_0/((-3*sqrt(2)*sqrt(m_0)*t + 2*xs^(3/2))^(2/3)*xs^2) \n", "Ric(g)_th,th = 3*2^(2/3)*m_0*x^2/((-3*sqrt(2)*sqrt(m_0)*t + 2*xs^(3/2))^(2/3)*xs^2) \n", "Ric(g)_ph,ph = 3*2^(2/3)*m_0*x^2*sin(th)^2/((-3*sqrt(2)*sqrt(m_0)*t + 2*xs^(3/2))^(2/3)*xs^2) " ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display_comp()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Ric(g) = -6*m_0/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3) dt*dt + 3*2^(2/3)*m_0/((-3*sqrt(2)*sqrt(m_0)*t + 2*xs^(3/2))^(2/3)*xs^2) dx*dx + 3*2^(2/3)*m_0*x^2/((-3*sqrt(2)*sqrt(m_0)*t + 2*xs^(3/2))^(2/3)*xs^2) dth*dth + 3*2^(2/3)*m_0*x^2*sin(th)^2/((-3*sqrt(2)*sqrt(m_0)*t + 2*xs^(3/2))^(2/3)*xs^2) dph*dph" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-6*m_0/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric[0,0]" ] }, { "cell_type": "code", "execution_count": 14, "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": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "G_t,t = -12*m_0/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3) " ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "G.display_comp()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "u = d/dt" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = M.vector_field('u')\n", "u[0] = 1\n", "u.display()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g(u,u): M --> R\n", " (t, x, th, ph) |--> -1" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u,u).display()" ] }, { "cell_type": "code", "execution_count": 18, "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": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-dt" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u_form.display()" ] }, { "cell_type": "code", "execution_count": 20, "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", "T = rho(t,x)* (u_form * u_form)\n", "T.set_name('T')\n", "print(T)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "T = rho(t, x) dt*dt" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T.display()" ] }, { "cell_type": "code", "execution_count": 22, "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": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "E = -4*(12*sqrt(2)*pi*sqrt(m_0)*t*xs^(3/2)*rho(t, x) - 18*pi*m_0*t^2*rho(t, x) - 4*pi*xs^3*rho(t, x) + 3*m_0)/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3) dt*dt" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.display()" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "E_t,t = -4*(12*sqrt(2)*pi*sqrt(m_0)*t*xs^(3/2)*rho(t, x) - 18*pi*m_0*t^2*rho(t, x) - 4*pi*xs^3*rho(t, x) + 3*m_0)/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3) " ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.display_comp()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-4*(12*sqrt(2)*pi*sqrt(m_0)*t*xs^(3/2)*rho(t, x) - 18*pi*m_0*t^2*rho(t, x) - 4*pi*xs^3*rho(t, x) + 3*m_0)/(6*sqrt(2)*sqrt(m_0)*t*xs^(3/2) - 9*m_0*t^2 - 2*xs^3)" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E[0,0]" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "12*sqrt(2)*pi*sqrt(m_0)*t*xs^(3/2)*rho(t, x) - 18*pi*m_0*t^2*rho(t, x) - 4*pi*xs^3*rho(t, x) + 3*m_0 == 0" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eq = (-E[0,0]/4).expr().numerator() == 0\n", "eq" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[rho(t, x) == -3/2*m_0/(6*sqrt(2)*pi*sqrt(m_0)*t*xs^(3/2) - 9*pi*m_0*t^2 - 2*pi*xs^3)]" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(eq, rho(t,x))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.5.beta3", "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 }