{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# 4-dimensional anti-de Sitter spacetime\n", "\n", "This Jupyter/SageMath worksheet presents the computation of the curvature tensor and the Ricci tensor of the 4-dimensional anti-de Sitter spacetime $\\mathrm{AdS}_4$.\n", "See the [SageManifolds home page](http://sagemanifolds.obspm.fr/) for more details about tensor calculus with the free computer algebra system SageMath. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We declare first the spacetime manifold $M$ and the chart $X$ of Poincaré coordinates:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, x, y, z)\\right)$ " ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M')\n", "X. = M.chart('t x y z:(0,+oo)')\n", "X" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad x :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( -\\infty, +\\infty \\right) ;\\quad z :\\ \\left( 0 , +\\infty \\right)$ " ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" } ], "source": [ "X.coord_range()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The AdS metric is then defined as" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -\\frac{L^{2}}{z^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{L^{2}}{z^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{L^{2}}{z^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y + \\frac{L^{2}}{z^{2}} \\mathrm{d} z\\otimes \\mathrm{d} z$ " ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g = M.lorentzian_metric('g')\n", "L = var('L', domain='real')\n", "g[0,0] = -L^2/z^2\n", "g[1,1], g[2,2], g[3,3] = L^2/z^2, L^2/z^2, L^2/z^2\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The Christoffel symbols are (the vanishing ones and those that can be deduced by symmetry on the last two indices are not displayed)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, z }^{ \\, t \\phantom{\\, t} \\phantom{\\, z} } & = & -\\frac{1}{z} \\\\ \\Gamma_{ \\phantom{\\, x} \\, x \\, z }^{ \\, x \\phantom{\\, x} \\phantom{\\, z} } & = & -\\frac{1}{z} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, z }^{ \\, y \\phantom{\\, y} \\phantom{\\, z} } & = & -\\frac{1}{z} \\\\ \\Gamma_{ \\phantom{\\, z} \\, t \\, t }^{ \\, z \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{1}{z} \\\\ \\Gamma_{ \\phantom{\\, z} \\, x \\, x }^{ \\, z \\phantom{\\, x} \\phantom{\\, x} } & = & \\frac{1}{z} \\\\ \\Gamma_{ \\phantom{\\, z} \\, y \\, y }^{ \\, z \\phantom{\\, y} \\phantom{\\, y} } & = & \\frac{1}{z} \\\\ \\Gamma_{ \\phantom{\\, z} \\, z \\, z }^{ \\, z \\phantom{\\, z} \\phantom{\\, z} } & = & -\\frac{1}{z} \\end{array}$ " ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The components of the Riemann curvature tensor are" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, x \\, t \\, x }^{ \\, t \\phantom{\\, x} \\phantom{\\, t} \\phantom{\\, x} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, y \\, t \\, y }^{ \\, t \\phantom{\\, y} \\phantom{\\, t} \\phantom{\\, y} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, z \\, t \\, z }^{ \\, t \\phantom{\\, z} \\phantom{\\, t} \\phantom{\\, z} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, x} \\, t \\, t \\, x }^{ \\, x \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, x} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, x} \\, y \\, x \\, y }^{ \\, x \\phantom{\\, y} \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, x} \\, z \\, x \\, z }^{ \\, x \\phantom{\\, z} \\phantom{\\, x} \\phantom{\\, z} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, y} \\, t \\, t \\, y }^{ \\, y \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, y} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, y} \\, x \\, x \\, y }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} \\phantom{\\, y} } & = & \\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, y} \\, z \\, y \\, z }^{ \\, y \\phantom{\\, z} \\phantom{\\, y} \\phantom{\\, z} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, z} \\, t \\, t \\, z }^{ \\, z \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, z} } & = & -\\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, z} \\, x \\, x \\, z }^{ \\, z \\phantom{\\, x} \\phantom{\\, x} \\phantom{\\, z} } & = & \\frac{1}{z^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, z} \\, y \\, y \\, z }^{ \\, z \\phantom{\\, y} \\phantom{\\, y} \\phantom{\\, z} } & = & \\frac{1}{z^{2}} \\end{array}$ " ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.riemann().display_comp(only_nonzero=True, only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "while those of the Ricci tensor are" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Ric}\\left(g\\right)_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{3}{z^{2}} \\\\ \\mathrm{Ric}\\left(g\\right)_{ \\, x \\, x }^{ \\phantom{\\, x}\\phantom{\\, x} } & = & -\\frac{3}{z^{2}} \\\\ \\mathrm{Ric}\\left(g\\right)_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & -\\frac{3}{z^{2}} \\\\ \\mathrm{Ric}\\left(g\\right)_{ \\, z \\, z }^{ \\phantom{\\, z}\\phantom{\\, z} } & = & -\\frac{3}{z^{2}} \\end{array}$ " ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.ricci().display_comp(only_nonzero=True, only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Another view of the same thing:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = \\frac{3}{z^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t -\\frac{3}{z^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x -\\frac{3}{z^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y -\\frac{3}{z^{2}} \\mathrm{d} z\\otimes \\mathrm{d} z$ " ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.ricci().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The Ricci scalar turns out to be constant (as on any maximally symmetric spacetime):" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, x, y, z\\right) & \\longmapsto & -\\frac{12}{L^{2}} \\end{array}$ " ] }, "execution_count": 9, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g.ricci_scalar().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Finally, we check that the **Einstein equation** with the negative cosmological constant $\\Lambda=-\\frac{3}{L^2}$ is satisfied:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Lambda = -3/L^2\n", "g.ricci() - 1/2*g.ricci_scalar()*g + Lambda*g == 0" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.0", "name": "sage-8.0" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 0 }