{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Demo black hole spacetime" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the **Schwarzschild spacetime** (which represents a **static black hole** in general relativity) as a **4-dimensional Lorentzian manifold** $M$ and we endow it with the chart $X$\n", "of standard Schwarzschild coordinates $(t,r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right)\\right]$ " ], "text/plain": [ "[Chart (M, (t, r, th, ph))]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', structure='Lorentzian')\n", "X. = M.chart(r\"t r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi\")\n", "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define next the **metric tensor** $g$ from its non-vanishing components. It depends on the parameter $m$, which is the black hole mass:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = (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": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = var('m')\n", "g = M.metric()\n", "g[0, 0] = -(1 - 2*m/r)\n", "g[1,1] = 1/(1-2*m/r)\n", "g[2,2] = r^2\n", "g[3,3] = r^2*sin(th)^2\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **volume form** (also called **Levi-Civita tensor**) associated with $g$ can then be evaluated:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-form eps_g on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\epsilon_{g} = r^{2} \\sin\\left({\\theta}\\right) \\mathrm{d} t\\wedge \\mathrm{d} r\\wedge \\mathrm{d} {\\theta}\\wedge \\mathrm{d} {\\phi}$ " ], "text/plain": [ "eps_g = r^2*sin(th) dt/\\dr/\\dth/\\dph" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(g.volume_form())\n", "g.volume_form().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Riemann curvature tensor** is computed as" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, r \\, t \\, r }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\theta} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\phi} \\, t \\, {\\phi} }^{ \\, t \\phantom{\\, {\\phi}} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, t \\, t \\, r }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, t \\, t \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, t \\, t \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}$ " ], "text/plain": [ "Riem(g)^t_r,t,r = -2*m/(2*m*r^2 - r^3) \n", "Riem(g)^t_th,t,th = -m/r \n", "Riem(g)^t_ph,t,ph = -m*sin(th)^2/r \n", "Riem(g)^r_t,t,r = -2*(2*m^2 - m*r)/r^4 \n", "Riem(g)^r_th,r,th = -m/r \n", "Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n", "Riem(g)^th_t,t,th = (2*m^2 - m*r)/r^4 \n", "Riem(g)^th_r,r,th = -m/(2*m*r^2 - r^3) \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_t,t,ph = (2*m^2 - m*r)/r^4 \n", "Riem(g)^ph_r,r,ph = -m/(2*m*r^2 - r^3) \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem = g.riemann()\n", "print(Riem)\n", "Riem.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The component $\\mathrm{Riem}(g)^t_{\\ \\, rtr} = \\mathrm{Riem}(g)^0_{\\ \\, 101}$ is returned by " ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}$ " ], "text/plain": [ "-2*m/(2*m*r^2 - r^3)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem[0,1,0,1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Kretschmann scalar** is defined as the following contraction of the Riemann tensor with itself:\n", "\n", "$$K = \\mathrm{Riem}(g)_{abcd} \\; \\mathrm{Riem}(g)^{abcd},$$\n", "\n", "where the Einstein summation convention on repeated indices is assumed. We evaluate it by using LaTeX notations for the indices:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\end{array}$ " ], "text/plain": [ "M --> R\n", "(t, r, th, ph) |--> 48*m^2/r^6" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K = Riem.down(g)['_{abcd}'] * Riem.up(g)['^{abcd}']\n", "print(K)\n", "K.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Ricci tensor** of $g$ is identically zero, reflecting the fact that the Schwarzschild metric is a solution of the **Einstein equation** in vacuum:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0$ " ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(g.ricci())\n", "g.ricci().display()" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.7.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": "2.7.15" } }, "nbformat": 4, "nbformat_minor": 2 }