{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Anti-de Sitter spacetime\n", "\n", "This notebook demonstrates some capabilities of SageMath in computations regarding the 4-dimensional anti-de Sitter spacetime. The corresponding tools have been developed within the [SageManifolds](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 8.8 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.4, Release Date: 2021-08-22'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We initialize a time counter for benchmarking:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import time\n", "comput_time0 = time.perf_counter()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime manifold\n", "\n", "We declare the anti-de Sitter spacetime as a 4-dimensional Lorentzian manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$$" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', r'\\mathcal{M}', structure='Lorentzian')\n", "print(M); M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We consider hyperbolic coordinates $(\\tau,\\rho,\\theta,\\phi)$ on $\\mathcal{M}$. Allowing for the standard coordinate singularities at $\\rho=0$, $\\theta=0$ or $\\theta=\\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\\mathbb{R}^4$). If we restrict ourselves to regular coordinates (i.e. to considering only mathematically well defined charts), the hyperbolic coordinates cover only an open part of $\\mathcal{M}$, which we call $\\mathcal{M}_0$, on which $\\rho$ spans the open interval $(0,+\\infty)$, $\\theta$ the open interval $(0,\\pi)$ and $\\phi$ the open interval $(0,2\\pi)$. Therefore, we declare:

" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (M_0, (ta, rh, th, ph))\n" ] }, { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}_0,({\\tau}, {\\rho}, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}_0,({\\tau}, {\\rho}, {\\theta}, {\\phi})\\right)$$" ], "text/plain": [ "Chart (M_0, (ta, rh, th, ph))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M0 = M.open_subset('M_0', r'\\mathcal{M}_0' )\n", "X_hyp. = M0.chart(r'ta:\\tau rh:(0,+oo):\\rho th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "print(X_hyp); X_hyp" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\tau} :\\ \\left( -\\infty, +\\infty \\right) ;\\quad {\\rho} :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\tau} :\\ \\left( -\\infty, +\\infty \\right) ;\\quad {\\rho} :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "ta: (-oo, +oo); rh: (0, +oo); th: (0, pi); ph: (0, 2*pi)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_hyp.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $\\mathbb{R}^{2,3}$ as an ambient space\n", "The AdS metric can be defined as that induced by the immersion of $\\mathcal{M}$ in $\\mathbb{R}^{2,3}$, the latter being nothing but $\\mathbb{R}^5$ equipped with a flat pseudo-Riemannian metric of signature $(-,-,+,+,+)$. Let us construct $\\mathbb{R}^{2,3}$ as a 5-dimensional manifold covered by canonical coordinates:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R23, (U, V, X, Y, Z))\n" ] }, { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^{2,3},(U, V, X, Y, Z)\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^{2,3},(U, V, X, Y, Z)\\right)$$" ], "text/plain": [ "Chart (R23, (U, V, X, Y, Z))" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R23 = Manifold(5, 'R23', r'\\mathbb{R}^{2,3}', structure='pseudo-Riemannian', signature=1, \n", " metric_name='h')\n", "X23. = R23.chart()\n", "print(X23); X23" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the pseudo-Riemannian metric of $\\mathbb{R}^{2,3}$:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}h = -\\mathrm{d} U\\otimes \\mathrm{d} U-\\mathrm{d} V\\otimes \\mathrm{d} V+\\mathrm{d} X\\otimes \\mathrm{d} X+\\mathrm{d} Y\\otimes \\mathrm{d} Y+\\mathrm{d} Z\\otimes \\mathrm{d} Z\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}h = -\\mathrm{d} U\\otimes \\mathrm{d} U-\\mathrm{d} V\\otimes \\mathrm{d} V+\\mathrm{d} X\\otimes \\mathrm{d} X+\\mathrm{d} Y\\otimes \\mathrm{d} Y+\\mathrm{d} Z\\otimes \\mathrm{d} Z$$" ], "text/plain": [ "h = -dU⊗dU - dV⊗dV + dX⊗dX + dY⊗dY + dZ⊗dZ" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h = R23.metric()\n", "h[0,0], h[1,1], h[2,2], h[3,3], h[4,4] = -1, -1, 1, 1, 1\n", "h.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The AdS immersion into $\\mathbb{R}^{2,3}$ is defined as a differential map $\\Phi$ from $\\mathcal{M}$ to $\\mathbb{R}^{2,3}$, by providing its expression in terms of $\\mathcal{M}$'s default chart (which is X_hyp = $(\\mathcal{M}_0,(\\tau,\\rho,\\theta,\\phi))$ ) and $\\mathbb{R}^{2,3}$'s default chart (which is X23 = $(\\mathbb{R}^{2,3},(U,V,X,Y,Z))$ ):" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Differentiable map Phi from the 4-dimensional Lorentzian manifold M to the 5-dimensional pseudo-Riemannian manifold R23\n" ] }, { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathbb{R}^{2,3} \\\\ \\mbox{on}\\ \\mathcal{M}_0 : & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(U, V, X, Y, Z\\right) = \\left({\\ell} \\cos\\left(\\frac{{\\tau}}{{\\ell}}\\right) \\cosh\\left({\\rho}\\right), {\\ell} \\cosh\\left({\\rho}\\right) \\sin\\left(\\frac{{\\tau}}{{\\ell}}\\right), {\\ell} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)\\right) \\end{array}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathbb{R}^{2,3} \\\\ \\mbox{on}\\ \\mathcal{M}_0 : & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(U, V, X, Y, Z\\right) = \\left({\\ell} \\cos\\left(\\frac{{\\tau}}{{\\ell}}\\right) \\cosh\\left({\\rho}\\right), {\\ell} \\cosh\\left({\\rho}\\right) \\sin\\left(\\frac{{\\tau}}{{\\ell}}\\right), {\\ell} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)\\right) \\end{array}$$" ], "text/plain": [ "Phi: M → R23\n", "on M_0: (ta, rh, th, ph) ↦ (U, V, X, Y, Z) = (l*cos(ta/l)*cosh(rh), l*cosh(rh)*sin(ta/l), l*cos(ph)*sin(th)*sinh(rh), l*sin(ph)*sin(th)*sinh(rh), l*cos(th)*sinh(rh))" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('l', latex_name=r'\\ell', domain='real')\n", "assume(l>0)\n", "Phi = M.diff_map(R23, [l*cosh(rh)*cos(ta/l),\n", " l*cosh(rh)*sin(ta/l),\n", " l*sinh(rh)*sin(th)*cos(ph),\n", " l*sinh(rh)*sin(th)*sin(ph),\n", " l*sinh(rh)*cos(th)],\n", " name='Phi', latex_name=r'\\Phi')\n", "print(Phi); Phi.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The constant $\\ell$ is the AdS length parameter. Considering AdS metric as a solution of vacuum Einstein equation with negative cosmological constant $\\Lambda$, one has $\\ell = \\sqrt{-3/\\Lambda}$.\n", "\n", "Let us evaluate the image of a point via the map $\\Phi$:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "p = M((ta, rh, th, ph), name='p'); print(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinates of $p$ in the chart X_hyp:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right)$$" ], "text/plain": [ "(ta, rh, th, ph)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_hyp(p)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(p) on the 5-dimensional pseudo-Riemannian manifold R23\n" ] } ], "source": [ "q = Phi(p); print(q)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\ell} \\cos\\left(\\frac{{\\tau}}{{\\ell}}\\right) \\cosh\\left({\\rho}\\right), {\\ell} \\cosh\\left({\\rho}\\right) \\sin\\left(\\frac{{\\tau}}{{\\ell}}\\right), {\\ell} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\ell} \\cos\\left(\\frac{{\\tau}}{{\\ell}}\\right) \\cosh\\left({\\rho}\\right), {\\ell} \\cosh\\left({\\rho}\\right) \\sin\\left(\\frac{{\\tau}}{{\\ell}}\\right), {\\ell} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)\\right)$$" ], "text/plain": [ "(l*cos(ta/l)*cosh(rh),\n", " l*cosh(rh)*sin(ta/l),\n", " l*cos(ph)*sin(th)*sinh(rh),\n", " l*sin(ph)*sin(th)*sinh(rh),\n", " l*cos(th)*sinh(rh))" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X23(q)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The image of $\\mathcal{M}$ by the immersion $\\Phi$ is a hyperboloid of one sheet, of equation $$-U^2-V^2+X^2+Y^2+Z^2=-\\ell^2.$$\n", "Indeed:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\ell}^{2}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\ell}^{2}$$" ], "text/plain": [ "-l^2" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(Uq,Vq,Xq,Yq,Zq) = X23(q)\n", "s = - Uq^2 - Vq^2 + Xq^2 + Yq^2 + Zq^2\n", "s.simplify_full()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may use the immersion $\\Phi$ to draw the coordinate grid $(\\tau,\\rho)$ in terms of the coordinates $(U,V,X)$ for $\\theta=\\pi/2$ and $\\phi=0$ ($X\\geq 0$ part) or $\\phi=\\pi$ \n", "($X\\leq 0$ part). The red (rep. grey) curves are those for which $\\rho={\\rm const}$ \n", "(resp. $\\tau={\\rm const}$):" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_hyp = X_hyp.plot(X23, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:0}, \n", " ranges={ta:(0,2*pi), rh:(0,2)}, number_values=9, \n", " color={ta:'red', rh:'grey'}, thickness=2, parameters={l:1}, \n", " label_axes=False) # phi = 0 => X > 0 part\n", "graph_hyp += X_hyp.plot(X23, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:pi},\n", " ranges={ta:(0,2*pi), rh:(0,2)}, number_values=9, \n", " color={ta:'red', rh:'grey'}, thickness=2, parameters={l:1}, \n", " label_axes=False) # phi = pi => X < 0 part\n", "show(graph_hyp, aspect_ratio=1, axes_labels=['V','X','U'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To have a nicer picture, we add the plot of the hyperboloid obtained by parametric_plot with $(\\tau,\\rho)$ as parameters and the expressions of $(U,V,X)$ in terms of $(\\tau,\\rho)$ deduced from the coordinate representation of $\\Phi$:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\ell} \\cos\\left(\\frac{{\\tau}}{{\\ell}}\\right) \\cosh\\left({\\rho}\\right), {\\ell} \\cosh\\left({\\rho}\\right) \\sin\\left(\\frac{{\\tau}}{{\\ell}}\\right), {\\ell} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\ell} \\cos\\left(\\frac{{\\tau}}{{\\ell}}\\right) \\cosh\\left({\\rho}\\right), {\\ell} \\cosh\\left({\\rho}\\right) \\sin\\left(\\frac{{\\tau}}{{\\ell}}\\right), {\\ell} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right), {\\ell} \\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)\\right)$$" ], "text/plain": [ "Coordinate functions (l*cos(ta/l)*cosh(rh), l*cosh(rh)*sin(ta/l), l*cos(ph)*sin(th)*sinh(rh), l*sin(ph)*sin(th)*sinh(rh), l*cos(th)*sinh(rh)) on the Chart (M_0, (ta, rh, th, ph))" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi.coord_functions() # the default pair of charts (X_hyp, X23) is assumed" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\cos\\left({\\tau}\\right) \\cosh\\left({\\rho}\\right), \\cosh\\left({\\rho}\\right) \\sin\\left({\\tau}\\right), \\sinh\\left({\\rho}\\right)\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\cos\\left({\\tau}\\right) \\cosh\\left({\\rho}\\right), \\cosh\\left({\\rho}\\right) \\sin\\left({\\tau}\\right), \\sinh\\left({\\rho}\\right)\\right)$$" ], "text/plain": [ "(cos(ta)*cosh(rh), cosh(rh)*sin(ta), sinh(rh))" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ug = Phi.coord_functions()[0](ta,rh,pi/2,0).subs({l:1}) # l=1 substituted to have numerical values\n", "Vg = Phi.coord_functions()[1](ta,rh,pi/2,0).subs({l:1})\n", "Xg = Phi.coord_functions()[2](ta,rh,pi/2,0).subs({l:1})\n", "Ug, Vg, Xg" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "hyperboloid = parametric_plot3d([Vg, Xg, Ug], (ta,0,2*pi), (rh,-2,2), color=(1.,1.,0.9))\n", "graph_hyp += hyperboloid\n", "show(graph_hyp, aspect_ratio=1, axes_labels=['V','X','U'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime metric" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As mentionned above, the AdS metric $g$ on $\\mathcal{M}$ is that induced by the flat metric $h$ on $\\mathbb{R}^{2,3}$, i.e.$g$ is the pullback of $h$ by the differentiable map $\\Phi$:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "g = M.metric()\n", "g.set( Phi.pullback(h) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The expression of $g$ in terms of $\\mathcal{M}$'s default frame is found to be

" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -\\cosh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + {\\ell}^{2} \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho} + {\\ell}^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + {\\ell}^{2} \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -\\cosh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + {\\ell}^{2} \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho} + {\\ell}^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + {\\ell}^{2} \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$$" ], "text/plain": [ "g = -cosh(rh)^2 dta⊗dta + l^2 drh⊗drh + l^2*sinh(rh)^2 dth⊗dth + l^2*sin(th)^2*sinh(rh)^2 dph⊗dph" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-\\cosh\\left({\\rho}\\right)^{2} & 0 & 0 & 0 \\\\\n", "0 & {\\ell}^{2} & 0 & 0 \\\\\n", "0 & 0 & {\\ell}^{2} \\sinh\\left({\\rho}\\right)^{2} & 0 \\\\\n", "0 & 0 & 0 & {\\ell}^{2} \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}\n", "\\end{array}\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-\\cosh\\left({\\rho}\\right)^{2} & 0 & 0 & 0 \\\\\n", "0 & {\\ell}^{2} & 0 & 0 \\\\\n", "0 & 0 & {\\ell}^{2} \\sinh\\left({\\rho}\\right)^{2} & 0 \\\\\n", "0 & 0 & 0 & {\\ell}^{2} \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ -cosh(rh)^2 0 0 0]\n", "[ 0 l^2 0 0]\n", "[ 0 0 l^2*sinh(rh)^2 0]\n", "[ 0 0 0 l^2*sin(th)^2*sinh(rh)^2]" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Curvature

\n", "

The Riemann tensor of $g$ is

" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Riem = g.riemann()\n", "print(Riem)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\rho} \\, {\\tau} \\, {\\rho} }^{ \\, {\\tau} \\phantom{\\, {\\rho}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\rho}} } & = & -1 \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\theta} \\, {\\tau} \\, {\\theta} }^{ \\, {\\tau} \\phantom{\\, {\\theta}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\theta}} } & = & -\\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\phi} \\, {\\tau} \\, {\\phi} }^{ \\, {\\tau} \\phantom{\\, {\\phi}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\phi}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\rho}} \\, {\\tau} \\, {\\tau} \\, {\\rho} }^{ \\, {\\rho} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\rho}} } & = & -\\frac{\\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\rho}} \\, {\\theta} \\, {\\rho} \\, {\\theta} }^{ \\, {\\rho} \\phantom{\\, {\\theta}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\theta}} } & = & -\\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\rho}} \\, {\\phi} \\, {\\rho} \\, {\\phi} }^{ \\, {\\rho} \\phantom{\\, {\\phi}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\phi}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\tau} \\, {\\tau} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\theta}} } & = & -\\frac{\\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\rho} \\, {\\rho} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\rho}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\theta}} } & = & 1 \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\tau} \\, {\\tau} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\phi}} } & = & -\\frac{\\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\rho} \\, {\\rho} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\rho}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\phi}} } & = & 1 \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\sinh\\left({\\rho}\\right)^{2} \\end{array}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\rho} \\, {\\tau} \\, {\\rho} }^{ \\, {\\tau} \\phantom{\\, {\\rho}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\rho}} } & = & -1 \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\theta} \\, {\\tau} \\, {\\theta} }^{ \\, {\\tau} \\phantom{\\, {\\theta}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\theta}} } & = & -\\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\phi} \\, {\\tau} \\, {\\phi} }^{ \\, {\\tau} \\phantom{\\, {\\phi}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\phi}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\rho}} \\, {\\tau} \\, {\\tau} \\, {\\rho} }^{ \\, {\\rho} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\rho}} } & = & -\\frac{\\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\rho}} \\, {\\theta} \\, {\\rho} \\, {\\theta} }^{ \\, {\\rho} \\phantom{\\, {\\theta}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\theta}} } & = & -\\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\rho}} \\, {\\phi} \\, {\\rho} \\, {\\phi} }^{ \\, {\\rho} \\phantom{\\, {\\phi}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\phi}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\tau} \\, {\\tau} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\theta}} } & = & -\\frac{\\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\rho} \\, {\\rho} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\rho}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\theta}} } & = & 1 \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\tau} \\, {\\tau} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\phi}} } & = & -\\frac{\\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\rho} \\, {\\rho} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\rho}} \\phantom{\\, {\\rho}} \\phantom{\\, {\\phi}} } & = & 1 \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\sinh\\left({\\rho}\\right)^{2} \\end{array}$$" ], "text/plain": [ "Riem(g)^ta_rh,ta,rh = -1 \n", "Riem(g)^ta_th,ta,th = -sinh(rh)^2 \n", "Riem(g)^ta_ph,ta,ph = -sin(th)^2*sinh(rh)^2 \n", "Riem(g)^rh_ta,ta,rh = -cosh(rh)^2/l^2 \n", "Riem(g)^rh_th,rh,th = -sinh(rh)^2 \n", "Riem(g)^rh_ph,rh,ph = -sin(th)^2*sinh(rh)^2 \n", "Riem(g)^th_ta,ta,th = -cosh(rh)^2/l^2 \n", "Riem(g)^th_rh,rh,th = 1 \n", "Riem(g)^th_ph,th,ph = -sin(th)^2*sinh(rh)^2 \n", "Riem(g)^ph_ta,ta,ph = -cosh(rh)^2/l^2 \n", "Riem(g)^ph_rh,rh,ph = 1 \n", "Riem(g)^ph_th,th,ph = sinh(rh)^2 " ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci tensor:

" ] }, { "cell_type": "code", "execution_count": 24, "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) = \\frac{3 \\, \\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} -3 \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho} -3 \\, \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -3 \\, \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = \\frac{3 \\, \\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} -3 \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho} -3 \\, \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -3 \\, \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$$" ], "text/plain": [ "Ric(g) = 3*cosh(rh)^2/l^2 dta⊗dta - 3 drh⊗drh - 3*sinh(rh)^2 dth⊗dth - 3*sin(th)^2*sinh(rh)^2 dph⊗dph" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric = g.ricci()\n", "print(Ric)\n", "Ric.display()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{3 \\, \\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} & 0 & 0 & 0 \\\\\n", "0 & -3 & 0 & 0 \\\\\n", "0 & 0 & -3 \\, \\sinh\\left({\\rho}\\right)^{2} & 0 \\\\\n", "0 & 0 & 0 & -3 \\, \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}\n", "\\end{array}\\right)\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{3 \\, \\cosh\\left({\\rho}\\right)^{2}}{{\\ell}^{2}} & 0 & 0 & 0 \\\\\n", "0 & -3 & 0 & 0 \\\\\n", "0 & 0 & -3 \\, \\sinh\\left({\\rho}\\right)^{2} & 0 \\\\\n", "0 & 0 & 0 & -3 \\, \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 3*cosh(rh)^2/l^2 0 0 0]\n", "[ 0 -3 0 0]\n", "[ 0 0 -3*sinh(rh)^2 0]\n", "[ 0 0 0 -3*sin(th)^2*sinh(rh)^2]" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci scalar:

" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field r(g) on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ \\mbox{on}\\ \\mathcal{M}_0 : & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{12}{{\\ell}^{2}} \\end{array}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ \\mbox{on}\\ \\mathcal{M}_0 : & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{12}{{\\ell}^{2}} \\end{array}$$" ], "text/plain": [ "r(g): M → ℝ\n", "on M_0: (ta, rh, th, ph) ↦ -12/l^2" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R = g.ricci_scalar()\n", "print(R)\n", "R.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We recover the fact that AdS spacetime has a constant curvature. It is indeed a **maximally symmetric space**. In particular, the Riemann tensor is expressible as\n", "$$R^i_{\\ \\, jlk} = \\frac{R}{n(n-1)} \\left( \\delta^i_{\\ \\, k} g_{jl} - \\delta^i_{\\ \\, l} g_{jk} \\right),$$\n", "where $n$ is the dimension of $\\mathcal{M}$: $n=4$ in the present case. Let us check this formula here, under the form $R^i_{\\ \\, jlk} = -\\frac{R}{6} g_{j[k} \\delta^i_{\\ \\, l]}$:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "delta = M.tangent_identity_field() \n", "Riem == - (R/6)*(g*delta).antisymmetrize(2,3) # 2,3 = last positions of the \n", " # type-(1,3) tensor g*delta" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may also check that AdS metric is a solution of the vacuum **Einstein equation** with (negative) cosmological constant $\\Lambda = - 3/\\ell^2$:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Lambda = -3/l^2\n", "Ric - 1/2*R*g + Lambda*g == 0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Radial null geodesics\n", "\n", "Null geodesics that are radial with respect to coordinates $(\\tau,\\rho,\\theta,\\phi)$ obey\n", "$$\\tau = \\pm 2 \\ell \\left( \\mathrm{atan} \\left(\\mathrm{e}^\\rho\\right) - \\frac{\\pi}{4} \\right) + \\tau_0,$$\n", "where $\\tau_0$ is a constant (the value of $\\tau$ at $\\rho=0$). Note that, due to the homogeneity of AdS spacetime, any null geodesic is a \"radial\" geodesic with respect to some coordinate system $(\\tau',\\rho',\\theta',\\phi')$, as in Minkowski spacetime, any null geodesic is a straight line and one can always find a Minkowskian coordinate system $(t',x',y',z')$ with respect to which the null geodesic is radial.\n", "\n", "Let us consider two finite families of radial null geodesics having $\\theta=\\pi/2$ and $\\phi=0$ or $\\pi$: \n", "- null_geod1 has $\\phi=\\pi$ when $\\tau< 0$ and $\\phi=0$ when $\\tau>0$\n", "- null_geod2 has $\\phi=0$ when $\\tau<0$ and $\\phi=\\pi$ when $\\tau>0$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": [ "lamb = var('lamb', latex_name=r'\\lambda')\n", "null_geod1 = [M.curve({X_hyp: [2*sgn(lamb)*l*(atan(exp(abs(lamb))) - pi/4) + 2*pi*(i-4)/8, \n", " abs(lamb), pi/2, pi*unit_step(-lamb)]}, \n", " (lamb, -oo, +oo)) for i in range(9)]\n", "null_geod2 = [M.curve({X_hyp: [2*sgn(lamb)*l*(atan(exp(abs(lamb))) - pi/4) + 2*pi*(i-4)/8, \n", " abs(lamb), pi/2, pi*unit_step(lamb)]}, \n", " (lamb, -oo, +oo)) for i in range(9)]\n", "null_geods = null_geod1 + null_geod2" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Curve in the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(null_geods[0])" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\Bold{R} & \\longrightarrow & \\mathcal{M} \\\\ & {\\lambda} & \\longmapsto & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) = \\left(-\\pi - \\frac{1}{2} \\, {\\left(\\pi - 4 \\, \\arctan\\left(e^{\\left({\\left| {\\lambda} \\right|}\\right)}\\right)\\right)} {\\ell} \\mathrm{sgn}\\left({\\lambda}\\right), {\\left| {\\lambda} \\right|}, \\frac{1}{2} \\, \\pi, \\pi \\mathrm{u}\\left(-{\\lambda}\\right)\\right) \\end{array}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\Bold{R} & \\longrightarrow & \\mathcal{M} \\\\ & {\\lambda} & \\longmapsto & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) = \\left(-\\pi - \\frac{1}{2} \\, {\\left(\\pi - 4 \\, \\arctan\\left(e^{\\left({\\left| {\\lambda} \\right|}\\right)}\\right)\\right)} {\\ell} \\mathrm{sgn}\\left({\\lambda}\\right), {\\left| {\\lambda} \\right|}, \\frac{1}{2} \\, \\pi, \\pi \\mathrm{u}\\left(-{\\lambda}\\right)\\right) \\end{array}$$" ], "text/plain": [ "ℝ → M\n", " lamb ↦ (ta, rh, th, ph) = (-pi - 1/2*(pi - 4*arctan(e^abs(lamb)))*l*sgn(lamb), abs(lamb), 1/2*pi, pi*unit_step(-lamb))" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "null_geods[0].display()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\Bold{R} & \\longrightarrow & \\mathcal{M} \\\\ & {\\lambda} & \\longmapsto & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) = \\left(-\\pi - \\frac{1}{2} \\, {\\left(\\pi - 4 \\, \\arctan\\left(e^{\\left({\\left| {\\lambda} \\right|}\\right)}\\right)\\right)} {\\ell} \\mathrm{sgn}\\left({\\lambda}\\right), {\\left| {\\lambda} \\right|}, \\frac{1}{2} \\, \\pi, \\pi \\mathrm{u}\\left({\\lambda}\\right)\\right) \\end{array}\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & \\Bold{R} & \\longrightarrow & \\mathcal{M} \\\\ & {\\lambda} & \\longmapsto & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) = \\left(-\\pi - \\frac{1}{2} \\, {\\left(\\pi - 4 \\, \\arctan\\left(e^{\\left({\\left| {\\lambda} \\right|}\\right)}\\right)\\right)} {\\ell} \\mathrm{sgn}\\left({\\lambda}\\right), {\\left| {\\lambda} \\right|}, \\frac{1}{2} \\, \\pi, \\pi \\mathrm{u}\\left({\\lambda}\\right)\\right) \\end{array}$$" ], "text/plain": [ "ℝ → M\n", " lamb ↦ (ta, rh, th, ph) = (-pi - 1/2*(pi - 4*arctan(e^abs(lamb)))*l*sgn(lamb), abs(lamb), 1/2*pi, pi*unit_step(lamb))" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "null_geods[9].display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To graphically display these geodesics, we introduce a Cartesian-like coordinate system\n", "$(\\tau,x_\\rho,y_\\rho,z_\\rho)$ linked to $(\\tau,\\rho,\\theta,\\phi)$ by the standard formulas:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\tau} & = & {\\tau} \\\\ {x_\\rho} & = & {\\rho} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\\\ {y_\\rho} & = & {\\rho} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\\\ {z_\\rho} & = & {\\rho} \\cos\\left({\\theta}\\right) \\end{array}\\right.\$" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\tau} & = & {\\tau} \\\\ {x_\\rho} & = & {\\rho} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\\\ {y_\\rho} & = & {\\rho} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\\\ {z_\\rho} & = & {\\rho} \\cos\\left({\\theta}\\right) \\end{array}\\right.$$" ], "text/plain": [ "ta = ta\n", "x_rho = rh*cos(ph)*sin(th)\n", "y_rho = rh*sin(ph)*sin(th)\n", "z_rho = rh*cos(th)" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_hyp_graph. = M0.chart(r'ta:\\tau x_rho:x_\\rho y_rho:y_\\rho z_rho:z_\\rho')\n", "hyp_to_hyp_graph = X_hyp.transition_map(X_hyp_graph, [ta, rh*sin(th)*cos(ph), \n", " rh*sin(th)*sin(ph), rh*cos(th)])\n", "hyp_to_hyp_graph.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us plot the null geodesics in terms of the coordinates $(\\tau,x_\\rho)$:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 36 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_2d = Graphics()\n", "for geod in null_geods:\n", " geod.expr(geod.domain().canonical_chart(), X_hyp_graph)\n", " graph_2d += geod.plot(X_hyp_graph, ambient_coords=(x_rho,ta), prange=(-4,4),\n", " parameters={l:1}, color='green', thickness=1.5)\n", "graph_2d += X_hyp_graph.plot(X_hyp_graph, ambient_coords=(x_rho,ta), \n", " fixed_coords={th:0, ph:pi}, \n", " ranges={ta:(-pi,pi), x_rho:(-4,4)}, \n", " number_values={ta: 9, x_rho: 9},\n", " color={ta:'red', x_rho:'grey'}, parameters={l:1})\n", "show(graph_2d, aspect_ratio=1, ymin=-pi, ymax=pi)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also get a 3D view of the radial null geodesics via the isometric immersion $\\Phi$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "