{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Anti-de Sitter spacetime\n", "\n", "This worksheet demonstrates a few capabilities of [SageManifolds](http://sagemanifolds.obspm.fr) (version 0.9.1) in computations regarding anti-de Sitter spacetime.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v0.9.1/SM_AdS.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath within the Jupyter notebook, via the command `sage -n jupyter`" ] }, { "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": [ "We also define a viewer for 3D plots (use `'jmol'` for interactive 3D graphics):" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "viewer3D = 'tachyon' # must be 'jmol', 'tachyon' or None (default)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime manifold\n", "\n", "We declare the anti-de Sitter spacetime as a 4-dimensional differentiable manifold:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/plain": [ "4-dimensional differentiable manifold M" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', r'\\mathcal{M}')\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": 4, "metadata": { "collapsed": false }, "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/plain": [ "Chart (M_0, (ta, rh, th, ph))" ] }, "execution_count": 4, "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": "markdown", "metadata": {}, "source": [ "

$\\mathbb{R}^5$ as an ambient space

\n", "

The AdS metric can be defined as that induced by the immersion of $\\mathcal{M}$ in $\\mathbb{R}^5$ equipped with a flat pseudo-Riemannian metric of signature $(-,-,+,+,+)$. We therefore introduce $\\mathbb{R}^5$ as a 5-dimensional manifold covered by canonical coordinates:

" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R5, (U, V, X, Y, Z))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^5,(U, V, X, Y, Z)\\right)$ " ], "text/plain": [ "Chart (R5, (U, V, X, Y, Z))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R5 = Manifold(5, 'R5', r'\\mathbb{R}^5')\n", "X5. = R5.chart()\n", "print(X5) ; X5" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The AdS immersion into $\\mathbb{R}^5$ is defined as a differential map $\\Phi$ from $\\mathcal{M}$ to $\\mathbb{R}^5$, by providing its expression in terms of $\\mathcal{M}$'s default chart (which is X_hyp = $(\\mathcal{M}_0,(\\tau,\\rho,\\theta,\\phi))$ ) and $\\mathbb{R}^5$'s default chart (which is X5 = $(\\mathbb{R}^5,(U,V,X,Y,Z))$ ):

" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Differentiable map Phi from the 4-dimensional differentiable manifold M to the 5-dimensional differentiable manifold R5\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi:& \\mathcal{M} & \\longrightarrow & \\mathbb{R}^5 \\\\ \\mbox{on}\\ \\mathcal{M}_0 : & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(U, V, X, Y, Z\\right) = \\left(\\frac{\\cosh\\left({\\rho}\\right) \\sin\\left(b {\\tau}\\right)}{b}, \\frac{\\cos\\left(b {\\tau}\\right) \\cosh\\left({\\rho}\\right)}{b}, \\frac{\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)}{b}, \\frac{\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)}{b}, \\frac{\\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)}{b}\\right) \\end{array}$ " ], "text/plain": [ "Phi: M --> R5\n", "on M_0: (ta, rh, th, ph) |--> (U, V, X, Y, Z) = (cosh(rh)*sin(b*ta)/b, cos(b*ta)*cosh(rh)/b, cos(ph)*sin(th)*sinh(rh)/b, sin(ph)*sin(th)*sinh(rh)/b, cos(th)*sinh(rh)/b)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('b', domain='real')\n", "assume(b>0)\n", "Phi = M.diff_map(R5, [sin(b*ta)/b * cosh(rh),\n", " cos(b*ta)/b * cosh(rh),\n", " sinh(rh)/b *sin(th)*cos(ph),\n", " sinh(rh)/b *sin(th)*sin(ph),\n", " sinh(rh)/b *cos(th)],\n", " name='Phi', latex_name=r'\\Phi')\n", "print(Phi) ; Phi.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The constant $b$ is a scale parameter. Considering AdS metric as a solution of vacuum Einstein equation with negative cosmological constant $\\Lambda$, one has $b = \\sqrt{-\\Lambda/3}$.

\n", "

Let us evaluate the image of a point via the map $\\Phi$:

" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "p = M.point((ta, rh, th, ph), name='p') ; print(p)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right)$ " ], "text/plain": [ "(ta, rh, th, ph)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p.coord()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point Phi(p) on the 5-dimensional differentiable manifold R5\n" ] } ], "source": [ "q = Phi(p) ; print(q)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{\\cosh\\left({\\rho}\\right) \\sin\\left(b {\\tau}\\right)}{b}, \\frac{\\cos\\left(b {\\tau}\\right) \\cosh\\left({\\rho}\\right)}{b}, \\frac{\\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)}{b}, \\frac{\\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)}{b}, \\frac{\\cos\\left({\\theta}\\right) \\sinh\\left({\\rho}\\right)}{b}\\right)$ " ], "text/plain": [ "(cosh(rh)*sin(b*ta)/b,\n", " cos(b*ta)*cosh(rh)/b,\n", " cos(ph)*sin(th)*sinh(rh)/b,\n", " sin(ph)*sin(th)*sinh(rh)/b,\n", " cos(th)*sinh(rh)/b)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q.coord()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The image of $\\mathcal{M}$ by the immersion $\\Phi$ is a hyperboloid of one sheet, of equation $-U^2-V^2+X^2+Y^2+Z^2=-b^{-2}$. Indeed:

" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{b^{2}}$ " ], "text/plain": [ "-1/b^2" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(Uq,Vq,Xq,Yq,Zq) = q.coord()\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 $\\phi=0$ (red) and $\\theta=\\pi/2$ and $\\phi=\\pi$ (green) (the brown lines are the lines $\\tau={\\rm const}$):

" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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ZJ5E/Kw8mFjR1wVp0sUJ1i0LR/uXenJAwAHBnmPZLVR35/I7PppaVfRAY6AuI\ngfGAGmgAghjmrvT0m1a66s/ASFt/hbx043izMiHhV4WCD/gCtY6OLiw76NKlmszM/2ZmTrWMz9D3\nc9yvhAQCpbNnByqVMGv9Xbl+fc7mzb5OTv2vXPEG1DKZ9t139UUY857ramhoMOPRu8k6yjKwhg9B\nN/Eb51d5phIAQgEvQAtcAwA4A3aACNA/8W0f4PX0uc8BOgBAC9ACSbhk0ZOLmFGMfz8DXL3kjhVF\nyESGeTXn+iek3wjxACJufP2/GRjozOcvOXPmfzf9/vvH997rBngDWqAfwwxgGO9Nm/renxf692/R\naMKuXKnV6fSDUhWwMj3dq7PqvAkdYxgPpbIeGCKTmSXfl02ZMkwsrj1+3IHHa9Bqr5SXX2lq+hX4\n9KOPRs6fb/r+3MQq4oiWZQyMmccw8Qx5mFT6VmI04AacBU4COsAXcAGuQeIvwRWgHmgGGoFGQA20\nADqAB/AAIWAP2AMugAeUlcroxOgJCydI50kPnjDDPoU24JuNGwE80boT5LGEhNNAWHr6TXfTcFyT\nTtf25QeEbL/33iGANzBOLn+0pGRUTo5Bkn3h5MmX1GqhQPDKrl0bS0rCgEnAc0BJTMxmQsoTEvp+\niL4YwbL5gAdQnpho+l3gXx44sP+ZM00FBQSo7t+//O9/v/zii5eB7YDzyy+buDPWy2pG7pY/8YhZ\nwCgLlLAHRAABeIAOaIKfj9/MQTMLcgsgxHvL3hs9enSHDy+vLvf3uT4wJ1MItIB96+k5/a9gHaAD\nWgA+ZNEyxdOWMl/NKmyMjs7JymqbBLmMEE9AfsuLf0lQ0JmmprSqKpSWbg8K8gT4QEhmpl9UlAE7\nc/ncuU/vu8/Hzu75Dz8kDzzQdvvXhDgDHtdfOBAzzKgcc56JSSdkEnANGGKqlJD5+wfb27sLBISQ\n02Kx7r77IBYvXbpUJBJBJqtJShIALfHxnomJpulPZ6xi5G414Q7g8OHDY8eONXcvOkYmETgDPIB/\nfZKEZISEffvm+QZpaWmFhYUzZszoLOJvlXY8bfZrsyEEhIC2tZ6jARoBHRIeT3g//n0e/QTWqrPZ\nMlGEoHWG+7mEhIsKhTPDhNwYnWe///7tOXOKr12b2Ng4HNAAs9LTYYQiyaZp08QnT4bedddDHS3P\n+Ukq5bOsMwDgCuDGMGPNF/F7CRkL1ANBRg6K71etOrh58wCRyInPP3jt2l+ETHj++ZsmGR8iZCTQ\nAHiYO7Usf6wJ6wp3y3xCZVtkSSlJEAM8oAXQgPvuNk9pS0uLft5kREREbPcmFOcU5uguDkRqAAAg\nAElEQVTEuntfvRccIAL08aUFtEATJIMl7y9+/97+9/b1h7F+nYX7fH//yxUV+nDfTMh9HV099QF3\nd4FaHdLQ4AAQYFMnG7730Rfz5zf++KMjn/9Cl+sbKxISzigUzoAIuAhMNN/7lCVkEFABjDJaHxb6\n+wfa27sLhfVa7Rdq9ZS4uCeffLLDU6blhHgD9ikpMN/KkoaGhqKiIkuenK1nTeFugR+FyCMEjq3F\nkxZw/+3Zk5mfn79jx45ly5bZ2dl18yE66BQ7FIs+XQRx63G567M3/Lz8Ns7fGHW/IQsItuFyefn8\ngAAAWRx3NSGhTKHIBV645ZUfKRY/3tgoACQMM9Rog+VFAQHDxOKqlpa3Tp++7Z1/lUrVLDsAaALK\ngSfM8W79hWG8lUovoBYINXgHLl9eMW5ciL19g073bVOTJ8MEDhnSxZKRbwmZBFw196W0LS2IOmau\nvYZ7YevWrebuwnVaThv/RTymANOBJ4HHEZ/a+ws/njlzRiaTvfLKK5mZmd1/VOaPmVGrovA4EAVE\nA7HALGA6/Gb7bUjb0OvOWLv09PQOb58JrGEYjuPS9JcSvVV5+ScODjnASYYxag8/Hjz434MG9egh\n3wKHgELgFyDHyN3rUK1MVgpcBPYYMDGqqhL8/D4aPHhLePhdTk73jxyZl5d3+0cVF5cB9UCp+Xai\nN8sVJnqBhnuPHfzzoN90P0wFooAZYBYZ4M1WVVX1wQcfTJw4ceLEiT/99FP3H5izZ5fXBKHzw3xE\nATFADBAFPA7ZFlnfe2Uz9m/Y0HZ1jvTWlG+vOT09ztFxeVDQ7ieeMGpP3gsN/XTIkG0zZ/b0gfsZ\nJhs4DuwBjpgj31tksgtAFaCWGeCltSwwMGnQoKf79Rvn7Dw+JOT3X37p/mPPSCRXgBLzDUwtJIhu\ny5rC/euvv96zZ495+3Aw/yDuAaa1JvtCQ77NqqqqLl++PHPmzOeff/7QoUPdfNSpQ4eeDfR9dIw7\nIoEoIBaIASKBx5F2LM2A3bNev7//vj7c6+RyGfC1XH7THT4EnnV3Xx0czNXWGq8bn0dGfjpkyL9C\nQ3vfREnJH0Ax8AvAdfIZxXiKJZJLQGnfUjV52rT1ISFvBAQ87O4+2MFhXkxMz/tRfA5oAMot8DJS\nlsSaZllMnTr1888/N2MHFioWTpBNQL/rq0xlj8tyNhiyMuvj4+Ph4ZGVlfXFF1+cOnVq+fLlf3a+\nDr5N2LhxX5ac3/p22rIjgaPOidEEcIA94IzY1bHkcRK9PLr8QrkB+2l1KioqAECn+5FlKwFpu3mN\nTQcPfkPIIYCIRABys7ON1IePXnrpytGjAB7py5qgwMBQufwy0A84ERNzQio1WP+6IZRlCwFX4FSv\nNlYr+fbb5UFBp48cKWpo2HnxovPw4ScbGz/uxSYHoaHZgFq/R4LJHT582PQH7R1rCncA165dM+PR\nD1YchAfAh5+Ln2qHSvGcEWeax8bGLlu2rLCwcPny5V3vdKjnM3ny6pISeb/J/zoWGnTBHvWADhAC\nLsg6lTVh2YSNWRuN11vLkZ2dXVBQcNONP27fDoDdsSOTZT39/LwmTNDfXldevueee1TAVT7fUyjk\nEyIw2rV1ag8c8BAKJ86d63/XXX1px2PTpvEcd1a/3I1lj5s23+/juKuAN4DW5WDd9J/IyPT4+D1X\nrnx7+XLz0KF/NTbu/OWXXncjluPOAy4AZLJeN9I7Fr6H+w3M/dGhZ8xY7ZIslGAGEANMR+bPPTjz\n2Xf5+fmvv/76xIkTX3nlldve+fLevetDQpIGDXpQ6obpwGzgaWA2EAX5/91cjrA9VVVVt964hmH0\nJ1Rn3liTOcAwSiCRYcp27Xo9IGBZYKCRejV/wID/DBmiCAszZKPp6YeAMoBLTy+7pdBkRGlpVUBV\nt6Mj6YknFgwc+Lin5xBHxxcefNBQvSiQSGqBCyZPMGs5m8pxnDVNhdRrbGx0dHQ08UGZfzDKciUc\nAB24DPM8Y1lZWVlZWcHBwWvXrhUKhV3feUlQkLdQWGmv+szv/FV3zfUN4jRAM+ST5ZvmGWABvRWJ\nImQDsAhA+yt1ZGRkx8RkAys5bteqVflffjl13ryxb75pjA6sDw0dYGf3zIcf8tqtRzWA0tIjQUGe\nQB0w3DjrrTr0DSESQAf0u116LAoIONvUVNTY6MDjHTl1Cr6+BuxGFSFuQLNE4mrC3YktfwfDNlZW\nljEXZYkS9gBntmQHEBUVlZmZ+e677xYVFS1fvvzNN9+srq7u7M4LDx5Uh4fza8nq08FzCvujEdAA\nAkAMxT6Ff4x/VnaWKTtvXpOBq7fcuD0mpgpYWVoKwEUgEPF4Rbm5xjj65hkzfIXCy2q1gZMdQGDg\nXRxXBbgDf8XEGLjxzj3OcUWAE3C+8yrW/IEDhzg6fnf5cplKlbJs2ZG6OsMmO4A/AA5wNWyjtxMe\nHm7aA/ae9YW76ZcPkL8TiAGClFUpJj50h0aOHLl69erw8PBXX301Ojr6s88+u/U+ngMGvPb11ytZ\ntjwkhFTj4WPuuASoAAI4oIJXEa2IXvjhwvKqO+JE63igDADwTGuleJ9UOggoBxAQAGDHhx/qOO6p\nzveCv66sDBkZO6XS0oSEnVIpMjJWE4KMDGRkdPaIf0+f3nT8uIrjFhUXG+aHucV4jqsCHIESQvJM\nVYK/Py2tAXBCB8X3E998w7i6Zl+96ioQhDo45NbW3mWcz0P2EsklQNWTi6z2kSXvgHIr6yvLmHh5\nGLOQUZYqIQIawO22uOdq5cqVOTk5YWFho0ePfvHFFzu8T8l33y14+mkBIccDGs4MbIYLIAB0QCOi\nRkVlLs80cZ9Nr5yQfwNngWfk8qmbNgH4kZDLQFTri/+fAQGuAsGy9jv9tvpKKi1jWVdACHgAaF0R\n7AioATuAtG4D0QgIgBrgKiAvLdX/2vhXWJiHQDDiuefGGyfg2nxLiD/gAfSTy4WG2LfytioYxlWp\nrAMGnDmD4GAAJ7/55oW4uEtqtZ+9/T8eeWT6J5/Azc2ofThMyAigCvA3SY5ZUU0G1jhyN/HHImWp\nEo4ALDHZAaxcuXLr1q1RUVH79u175ZVXvvnmm1vvE/TYY99evtzPzm7mZZ/7/nTtX24H9fV95LNO\nZpHpxOYn0uhaX+j6ZN9CiD0QVVradgcxn69ut9MvgJaMjHcJSSPEjmWHAgMBZ6AF8GMYLfCgXO7F\nMI+Vlo6Vyz0ZRn+dFTfADQgBxgDZgYF7CFner5+XWj1gzBhjJzuAv3NcBcADKhWKP00yfvdj2Qr9\nlJXERAAb33xzSlSUhuOe8fU9sH379IwMYyc7gPMSSTPgDKAbezn0nTVNlbHGkbspkYcI+gEE/3xU\n/q9nLf0k5K5du3766SehUPj+++93eIc3AgM9BQJHPv8n15qdwZdg13pZKBWYfkxOos1e528uIfop\ntPqzqTmE8BlmfOvuMd+tWZO/ebPd8OH//PprACgr+zIw0AtwADigCfCIirons7ufb3YR8oRc/oNC\nccDDY7hQGFJVVcTnv5SSYpqzndmEBANq/Vb1JjniMULSgR1CoZ+Tk8TFxV0g+KfRClAdOkTIaEAo\nkcD4p1VpWcboTFOZIQ8QeAH2cL8imF88IGL4uKd27DD2QQ0iPT29sLCwsrLS3t5+xYoVPj4+bd/a\nPH/+xf37HXi8a3zN537nS3yaod+yTAtUoWxLme1d7KksM3NRa8xlcdyfUqmKZZvk8omttYt3pk6t\nyMv74MgR+Ph8QYgv4ADUA1MzMhAd3cujXrjw+vjx47Vat3Pn3AAdcA2oAaJbyzXGwxISCNQCQ4yz\nq2V7E4KDKysrh9nb+4jFo5ycXv7oI/uHHzbqEW+1l5BxgD0gNn6U0XA3OtM8xWQKgQfuueD6bKWv\nA4+n4TgNxxU3Nb1z5Ag8PY19dIOYNGkSgOwbV13WnDu3eeZMTXW1I493VtS8Mbwcjq37FdeD22V9\nr4eb6HQ6Hu+GeuOLhFwFsjIyAByaNWsPsKzdy/6VgQOLa2oebWoKBoSADrhPLvfsW9n6yxkztAUF\nKp3ulb/+QkbG3pgYZ6AZEAPlfn4zy417HvsXQgYD5wEBw4ww2vaW8pdeys/KGiEWT+a4B8+dK7nl\nmoUm8xch1cC9xcUIDTXeUaxmM8hW1ldzB7Bz505jHyL1aCpcAR7GvvLs6GeeOdfSouY4Ox4vQiz+\n8t57FYMGLQkKOn34cHVlpbF70h3lF8pzjudk7cva/ftuAPL/yJl/MOQR8jP5mdWy9gPt/Yb6Pfrk\noxu2boh+I3p/4e9RX2UIRw1r0GqDVfYvnOqPRkAL2AGuINGkFKW3PaIluynZAfjp/8cwf2za9Csw\nuv0EjytXztTUxDQ1jQY8gcly+XSO62OyX9637+rRo4SQlz/5BABmzZrMcRPS030YRggMqqhQEvKp\n0ZbCArif41oYJhAgLItSA/9rqtXqe0aN+runp/3evXP79w8XiwfPmVMCGL2+3rnfgbuBkrAw83XB\nElnlyN0EyGSCfoAO3LbW5+fSpYyXX676809nPp9PCMdxDVrtRbV6xaFD8PIyVz9LNCWvrX2NLWQr\naivAA8iNV+ZD64bvLcBZ4CLgCAiAYIBD+CXHmde8XQWCc/YtO/tdLPFtvn6ZpybgGsr+r6ztsn/W\n7eDBf95zDw94j+PeI+RQ+3VMwHwXl3F1dSFAAMOEJST0vg7TzuLAwGAHh36jRz9xyzVaizZuvLBo\nkQdAgFJgqjEr44WE9AOqgXADvcevXrnyyvz5x7/7zlMonNO/v0qnezY5WfDQQwB+IeQuQCSRCE24\nnqi9c4R4A0JjppnVjdxpuHeMxBLYA3Xgtnfw/LwdGuotFIp4PALogBq1ev6HH4oeecRk3cval7Xw\n3wsr6ipg3zodj9ea4wD03VK3Xvav7fZ6oKh1tnsQwAMG4MEKt8lXPJv5uk1+5Vf7a66X4JvgJ/bb\nGLcx6gGrv/RH/q5d6yIjAQwEKoFRDLO0tVKhysxcGB//cH195BdfGCTWARx59132iy94wPyTJ7u4\nG0uIK6ABXBkmwGiVkyJCfICrDBPa50PMmzOnMDcXpaUvDhjAcZzHyJFT28/uT0+/FBsrBhxap0Wa\n2DFCwgFBfDyMdnlVs6yN7wsa7h0gUwncAQ7c1q6enHcZRnfxopdQ6MDj6YBGrbb/6NGPzJ3raOSU\nTz+eHvtqLNxuyHQ/Fz8/Z7+M9zJIHeFEHGki7AmWiWD8vf03pm/0H+Dv5+VXxiubNXRWlCJq+4fb\n0QCoAX/ABS4O/Dmn+w/Qir6zv/yTy1W4A86AGH4OfpmvZkqHmnRrKoP7+K23DrzzTtuXGzIyAqKj\nAXCZmS8//bRzv35uAsGSjma490ZNzabx410Fghf++APOzre5c0bGkZgYe6AGuKemxigTBzMyzsfE\nCIA/gYd7+05fmJDAHjjQ7/z56V5ePEJq1Or4I0du/em+J2QicAXwM0eklMhkXklJWsDVOEe3umSH\n9Ya7UT8ikVkEjkATuPTbPDk1587lfvNN84EDp48c8RAKeUC9Vluj0YSPGze981WLvca8xiiPKCEG\nRAAALdLeTosZElPWUhZg17M5GOUXy7O+y2J/ZrO+zoIDHH14z9T6hnAO5wWqzeILV8Ua2AHuAA+Z\nL2dGTbLi8fvqp546lvK/pcVtNZmdhKQKBMP9/DQct7akxCDHWujvP9zJyW/MmEe7uWayrOynwEA7\nwBPgM8wgIwzhGxISahQKNRAsl6OH5xKWL1/++4EDDuXlf3dwEPP5dRrNnA8/7GI+TCEhIQBfJhP2\nZVvj3qokxAsQGee0qnUtX9KzyhOqMOZSJuYtBiJAe/tkB+A+YMAjL700NSNDfurUoKeeqtVq+YT4\ni0RXjx79YPDg1SEh369ebaiOkQeJslgJd0AEyXBJwt8SDq47GDMkBkBPkx2Av7f/wmcXZn6RWVVU\nlRCTMIQ/avOVC4nXKoRNvH/UDYQWaAAqgYuIVkQv/PdCQ/0Upqfj8drKBKMYRv+XrYT8F+A5OtoR\nErtggUEOdHXfviGOjiqdrrvJDiAg4AGOk8jlHEBY9jcjnGUVb9pUwTAuQIVC0cU2CTdZvnz5S3Pm\nlPz224xLl2aIxWI+33vUqAV//dX1TMcImUwNlButMNK1YwCAXOOcVrWu5Ut6dOR+MzKTwBloAJfZ\ny2fmvdBQD6FQQIiAEA3HNep00uefHzNnDry9e9dgCUqCpwTDCeADzTi762wQgnrXVGeqq6t9fHyk\nE6Uniws9r6ndBghyB9dBC3DANUCHhFkJGxda5ULWxPvvf/TXX/U74uuH7f8h5AzgxDAXyssHikQT\n586d8NZbfT/QpkGDXAWCF3qVAlc2bjy9aJE70ASMMMJZ1m8IuReoBQJu935PT0//Q6ks+vnnkZcv\nhzk4iHi8Bq325S7PH7RXQogvYC+TweSD928YRqpUigAnI2Sadc1w17PikXtTU5NRmhYBfMTHxfe6\ngddPn5574sRzBw9eVqtVOp0Ln3/0//7vP/fdtz4k5Ke1a9H5Vo4dYuKZ4GnBcAEAqMDt4gye7AD0\nC51yfs7ZmbI7/OGHSzX2AiXBNaAfEAZ4YNM3m/z/5p+TbwWrWHU3biTgJ5H8deMdBgOngaU5OQ48\nHgcYJNmXBwW58PkPzJ3bu4d7LFx4N8edBzigOCbmSkJC37vU3uMcVwO4AT92/uFAf1mY377+uiUr\n68nGxuFiMQGeSU7ufrID8JXJGoBicwzeH2fZFsABRrmChwmmXxuctY7cAaSkpMTFxRm2TVmqLGlf\nEoDizcWhMFDl7tq1zyUSPiEE0HCcFqjz8Ljk6vr29993/TgttEHRQRXqCv0mAR3O2zGS4uLi1xYt\n+vXX/ZcdGuENSAAAF+Gn8ZNGSDcs3GBFsyTPJiSsVyiu6ufJZGR8HRj4I6AoLf3myy/ZL74YM378\njI62FihMSMjevt0BuFpREcQwrkAIw6iAIQkJty4x/XLGjKtHjzrz+XNOnOhjb09IpUKW5Yywf8Cv\nUulAlvUAeHK5S7viu0ajKSws3L59+6DAwD/ee28Ix7kKBBzH3T93btCiRb040GFCIoALEkmwyadF\nfkvIA8B5INSgsWaNZ1Nh1eG+dOnStWvXGrZNMpPADZIwCfum4V+XiYMG2fN4Yj6fA1Q6Xa1Gc0Wj\n6eJUHnmIwOP6osnunAAwhtGhIRWXyy+LNPAHREAYcBWoR+ZbmVEPW+hZ1ltXqEYRAiCrtPTCpk3r\nFYqk0lIEBDzt6zvE0bFhyJC237JfSKXuQA3L8gAR9IvY/jcjiQAtAAEaAA1QB4yTy0dv2gRAMWiQ\nM58/5/BhiMUG+AEyMkpiYrQAxzBhBj3Fmk2I/p+xbULL0aNHt2/fPnPmzOQpU0Ls7V0FAkc+v6ql\n5bU+7A9ziGGGK5XngRBzZMtfhGgMvVbWGs+mwnrLMsbiAHCIn9X7mkwXZKdOzTt5MuKZZ66o1QTw\ntrMb7Oj4fljYawEByrffvunO5AECb0AIqKBOVxujP92Rf/rMwf05cZIHUA6cBfbpL+yG6I+io9dE\nl1db4nbwt65QBeAK1GzalKdQnMf1PdxdBII6rVaf7OlS6eeEeLOsI8sOAPoDfEDIMA/K5ZM5TsAw\nD5eWPpSRoV/J6wx4A4OAawrFfkLW+Po6VFVduHLFMMkOYNascwyjBexZ9geDnmKdxHEXAS2wXypd\nsWLF8uXLIyIiRjs7/7l69TCxeKBIRADp88/3JdkB3M2yFYAfUN16+tpkimSyFmAIYNgd3q3xbCpg\nbddQbU+lUhm2wfi0eDwLvGCK5+TCnj0rg4L+M2TI1vDwreHh/x406P3Q0GPvvKP/ruxzGaKAWGA6\nyi6WmaA/t/Xs6ln2Y3kYDfQHvAA/YAYwGRu2bzB3126ntHQm8DOgAOKBtQyjv/lZX98JTk75Cxd+\nBOwFfgC2ARndfkf8yjD/BV4dODDL1TUb+BXYbNB3U6NcfgqoBLj0dAM2+83y5c8C9wBJjz7Kcdxn\nkZEbQkM3Dx26eejQkvffN9RR0oF64Lw54mUv0ALUSSQGbNOMl27uCysuy/z+++8ODg539e1a8u2R\naQReQCO4NBM9Jz9/+qnblSvZn33mKhAICNEBDVpthmv1z/2uwg2yGTLFdDPMF+6CdI2UPcaiBKgA\ntEAAMBBMAJO5JNNytpPkOI60G/BymZkvzZqlA/oDJ1snzJTv3fvatGnjW1qCOM4FeLjnc8ABVP/w\nw+cvv+wjFHqfPOkB/YpmVAExBnpPFUilLiwrADzlcjtDXH/j2LFj27dvL/jii0UVFUOBz0NDnfl8\nBz7/vEr1hqH3Q/+LkADAPjUVsbGGbblrTampiItrBDwN9K/Q1NTk4OBgkKZMzIrLMv7+/gb8uJRT\nkAN7QAjJcImh2rytiS++OOqNN2SnTj1XVFSpUjVptY48noOYB1egGc4Zf/315Zcm60x35CzLAR8Y\nA0wERgIXACXYXeyEf0zYuN1SJkqSG0sZhGGuArXtkh1A0U8/RapUozlOrF+62avo/HjevH52dnVa\n7RMcd29p6Si5XAsMBX4j5LghLpcxLCengWG4Hk5R75BWq12xYgWAVatWbc/LE4pEuQMG9LOz4xHy\nTHKywZMdwGCZjAecnT3b4C13zWH27Cr95TsMVJmxrv1k2rPicLe3t8/PzzdUa5JhkrZln2bxxunT\n806e/JGrUbdwoy87xZ3vF5Z/dv/bb28IC3v/iScKDh0ycX/KL5QvfP/62qWE//xvZh6XxuES4AoM\nBh4DhgFiVPxSsejVRRu3bSyvNH8VPvPG2S+rYmPd274oKwNwYdeuvampA4D75fLJvR3f/f722/1F\nIpVOJz91CgACAoSbNj1cWnoOEAMalv2WEHT7Eh+dGZqTUwg0ARUxMejV/Mhjx461xfqIESNy33ln\nA8McDQ520mimFBW9uGYNDH7Zbj2ZrBQ4CyAtzSjtd64G4AHXkpIM0poVXRH7JlYc7j4+Pm6G244j\nvSAdQkALdrl5trUDwCQwW0ZWHQipqavT3ouwWq1WzOP1s7PzOnXqx7g4xbRp5/fsMd7R9RfLjl4a\nLX1FSh4kAc8HbNq3iTxGyGSi+FpB/kbIE4Q8SMiDBGeBa4AG4ANBwBAgCPDAon8umvDohPLy8mvX\nrhmvn7fV/uIkANxycv4X7gEBxxMS4p991pXPP+ztLehtraNkz57DmzfzgAlz5tzwjYCAxzhOxzA1\nQCiQP2sWK5VW923Gy8McdxYQAeUKRY/2720f63w+XzFt2tqQkNzNm73t7PqNGtVw6ZIDgI1G+7wV\nHHwZuM9YrXflGKAD6pRKg7RmrWdTrXoqJIA333zznXZ7QvUF+TvBQEAL7nOzPSHkSQIXoPmGiY//\nCgtzFQjseTxwXINO16TVSl94Qbp4sQGPm7U/a+EnCyuuVkAICABB69Q/tP75vy4CACoANTAIcALs\ngGagGnAFmoBC4BJmPjTTx8MnKipKf7UQU6qurs7JyZk2bVrbLaek0iyWzQM2ZmT4R0fHE1Lv4zPY\nyenNPtQi1oaEDBCJXvjoI3T+A56USptY1h4oBh7JyBD1bdfJg4SEAle6t3+vPtMjIiJmtc6Ul/n7\nRzg6OvD5DVrtpLlzw197DcAFQtwBkdES4LJcLkxMFACOpg2ZCzKZMClJC/gY4rhHjhwx4Ik9U7Lu\ncE9OTh4+fPjEiRP73hSJJXBF1JCozIS+fpTuHXmaPPG7RPCQOS8zasLNU8i/mjWrPC9PzOOJeDwN\nx6l0umadbsxzz92/ZElfDpq1Nyt6XTScACFg17ozMAdmGCN/Ws4eZZkIhj3OMiOYwpOFEcMi2HyW\nPcKyBSwqAADhgPj6pgioAZxxfUf4avhV+E2MmCgSie66664FBtq8pTv0+yi0vyWqtQT/AsPsYNkr\nfP6QgQOd+PwleXlwde3FIfRBqeG4BX/91fU9SxISTioU+ouaDuvLRfsAAMcJ6Q+AYTw7/yiQkZFR\nWFg4c+bMESNG6G9J8Pf3tbMbIBJxgPvIkVPbbTH/HSEPAWXAIKOFQDEhAYCdyfcBzidkOCDo889l\njJWSJmPd4Z6dnX3gwIE1a9b0vSkSS+ACxpPJWW+GFfZZ+7OiP4uGAyRhEnZJp3Whgi+/zPvjj4YD\nB+x4PAI063T1Wq37yJFzerU2WvqSlK1k9dstgABqMCFM5luZ/l7dmvdSpiqDEIGzA+EGCIAm4DLg\nAIgBAlwFI2Ym+k5UqVSVlZXe3t7Jycm96GQvHD9+fPjw4W1f6sO9PxAMHAQ4sXikt/eJxsbUqqpb\nH5ublTUuqqvFWVf379/2yiv2PN7c7i1GPZeTUx4d7VlRoQUKgcg+vN2qEhKqFQp/oJhhxt2S7/pY\nbz9a/2nduv2ffhpob+/I40156SXP0aNv/ZxxlhAvwLlXk4W64wTD+CuVNSbfB1hJyFhAkJKCvp3R\npeFuTlOnTv3vf//bx0ZSj6bGbYqDAAnjEjbOM8OsD6lMyl5lobvNDvJ6p3Nz8zdvrty3z5HHExBC\ngCadrkajkb7wwqTuDeQTPkhQfKuA4/Xlr8wQJmps1MLpvdz6kUQReAE8oBk4DzgDTgAPaAI04NK5\nw4cPL168WKVSDRs27KYLdptAFCHPAeeBvQAAb09PsaPj7OTkMVOn6u+wWyp1BkpZlgcIAC2gBYRA\nA1ACvCiXh7QG3761a/O//NJVIPAeNSqyJ9NXDkulTizbDNQD9/ThHdeQkNCoUHA3Fhy0Wu3q1asB\nrFq1qu3GP99999fPP3cVCJp0upc+/rjT8lFCQr1CcaUbG4r10tmz9SEhasDdtDmTxzDeSmVdn69C\nZdXhLjB3B/oqwCDXkifXr2fkN9DPAK310MbtG9mLLBwhm96tDY9Cx40LHTdO//eC99777fPPhYT4\niURlKSkbN2+u0Wi8Ro92HznymU52G5bOk7LnWTgDBH7ufuUf9XV+C5fFkVgCNwUR2zAAACAASURB\nVKAFEAM1cL7Gr3PWwgvQQPquNOONjL179wKYNGlSdHT0pEmThg0bFtXlANmwfgHaLsbRYmdXWFe3\nWKXaKZXWsawYcACagf6AFuADmtZrWwHoD5QpFMcVCjXgOmFCdmVlkL39wy++GNzDTVfG5uTkSKVi\nlg0E9hLS6yk64k2btCyrYdnjhAznOH1tHTfGOvv22z9/9pmvnZ2rQPD0oUO3WTe7adNFhcIXQGkp\nAgN716uuBAefB/yBCwzja8KtZvwlEo1SOaDP7Vjv2VTYwMg9OTn57rvvHj9+fF8aST2eGpccBw24\nT83wbEgXStmrLADui14evXH//sR58zwEAnsej0+ISqerUqu9R40KnDz50Xnz2u6Wcywnen10hbpC\nP2D3E/kd3HDQ39swi4/IUwSOgANwDaiGI3iN3jo4AhxQA1SBy+YA7N69OyUlRV8ZnzhxosHL8QUF\nBTk5OXPb7c7YVnMfDZzy8KjUaIbV1Q3huECAB7QAoQxTCgxeuDDspt835eXIyfnvrFkOgD2wp3//\n+6qra3Q6ycKFwe+/35vOlZUdDgzsB4gYxrsPU2g+JeQP4LSf37z335/VbnOxA2vX7v/00zAHByGP\np+W457sZTKWldUFBGqMNruvkcl5iYoOBTm923zVChH0+l2u9Z1NhA+EOICsrq4/DQDKTwAdoArfZ\nDM8GiSJwhZ/Irzy5r4PoY++9V3rkyPn8fBGPB6BRq61Wq4MmT67Tau9a8Pym3Zuy/sy6fimSDMP/\npCSWwAsgQCOqPyvf+MWmd/ZvvL5ZcS1wBvAA993140ZHR1dXVwPYsmWLYT5+tVq5cuXKlSvbvpxD\nSC0A4B+E/MvObrpK5QXYAyEM4yyV+nZvLuBcb+/R7u7uVVWDamtbgCuAM8M80IuAzsz8ftasUQCf\nYfr1Kt91Ol3kPff4sexC4A8gtvX9+w8/v3Cx2IXPb9Tp7p8zZ+hrr3W/zaOEDAbsjXS17vT03NjY\nOuAB00bNH4SMAYR9KLtbdbLDNsJ90qRJ2dnZfWmBzCDoh/gZ8YkPmXof6vTj6bHvx4IY+PdK+oIF\nVT/84C4UAtBxXLNOt9j5TE2QBiJEjY3a8JSx9uwlcwnEgA6oun61k+sVGwD1QA0gROlXpQEIAJCd\nnZ2dnZ2Tk+Pt7e3t7b3JQOf02p9QPSCVfsOy+qk9w0Si4RqNu1arBv5WVgb/7j4DV/fv/3jePC+h\ncG5y8u9Ll9ayrAoIAo4BT/fi7ZOZmTNrVjBQCYztycOPHz+elZUVERExZfLkmilT3Fm2Aeh/4QLs\n7BLHj3cVCPiEPP3JJ7iv55PLExIaFAoV4GGcNPidkPHANYnEy4SVmQOETAL4fQh3Y+w7a0pWvIip\njQFO0NkDBMqDhln10COxy2MhhGyWgS8vEJOcHJed7R4TU+XqWq/VOvL5NWEaOCCk1j7icMP+5etq\nzp0z7BH1GH8GjQAf8IF0vRRAaVopagEd4AS4AXYIfCowYXsCgEmTJq1cuXLr1q1hYWF5eXmTJk1a\nuNAA1/MrLCxs+/sfQAVAAD6Pd97NrXzgwHOE/I3jup/sADa9+KKnUOg2ciQeeuienJwpHHePXF4H\nDAP2ErK9pxs3RkdLOa4U8AVOdW+XguPHj69YsSIrK2vVqlWzZs1ycXMLzMmpAVyATxjmM6nUQyjU\ncNzThYW9SXYAmzZdBpyA3i2CvS17iUQLtBhoVVE3BUokjUC51Z4O7TtbGLnL5fIXX3xx2LBhvW6B\nxBB4IuX1lNmBpt4KgzxD4IC0JWkxATFGOkT0yujG3T82e+mKHRqfqfQd1OTYpNNdUKl8x4x5ZMmS\nYENcPEz6rJQZyyg+UQDAYMAbcADUQA3QADQA7oAnQIBmoBkQAbX/G8LrZf8/e+cdH1WZvv3vc6Yk\nk0oagZBJCAlVukDOWBDRVRTbKmQorq4iKhbIhF3b4op1LT8yAesKKrpSJqjr2hVRd3V3TpAuEDrJ\nJNRAep3MzHn/mAQCqZP27mb3+if5JOc8pz7XuZ/rbj/8sGvXrnXr1gGLFy+Oiopq3zNtWDjss7S0\nd61WwBkaOjYsbEd5+YcFBT6N9kBs7KigoFpVva9RQyKHyVSmKIBelgdaLL6GsduFGAyOxMTRzZfY\n9VrrnOsy9WLXSy8deeGFY1FRTlUtcbl+17E6vVgsVRkZJdCnKwjh8OHKAQPo9mymQiFEB3wJ/9Gh\nMvQMcn/xxRcrKysbyqy+QswR+LHKsmrWwG4l99T3U5euX4o/6p+76inknciLuy+O4LrE1x8tluxN\nm/SnTmmEcKtqtcdzwun0HzEiZvTo3/jeyztza6Z5vrnOa3qk/q86SIA+oAcnFIMKpyACDBAMlVAC\nvcBF6tRU6/Tz1ZgDBw4sXry4tLQ0KSlp+vTpJh+LcDUk9+lCxEMu9O7Xr6i2Nrey8p9lZQB5eT+t\nW+dQlFlLlrRgxc+PjR0cEKAR4t5mWs1lCpEAwd4Y9tRU4YuytNVkClMUP9gJv2o0DT0ez5NPPjl9\n+vSGMftevPbrX4vdu/VCaIWYtmfP8U5qPHRYCD+IycnpirCZA0IkgKYbi0RWrl5dPHt2rw58Uf6n\nuf//R1VV1WOPPdYRxVbcLdBx4NXOa63XxuPeKOgNTtR3u+opGG815qv5aFDfqztE8dGj2V9+aX/r\nLf/Tp4MkSQjhUdUKt/uY0/nAm29GX3VVW4bN3J5p32PPWJNRFzbogf0kxiQezD2o7lLx+qj7gBbK\n4ShIcArCqevoVFOf0eqG06ifNnH5O3fu/OCDD7y/+/TlPo/cgeqgILdeX+hy9a2sHO1yuSERQsAN\nlRACJRAEp2G4LB+DW+x24N1bbqn45RcVWs1E/UyIIeAEp8Uy2pdqLd8KMRhUCE9NDap/gb20TlPW\n+qlvvnlr3rxInU4vSU6P56rp0wueeWYY+OfmNm7+5yu+FWISlLSYAdtu2GV5bFbWLhjbjYTjECKm\nvXmq/+lmOz2D3IHZs2evWrWq3buLBwQC9eXuvhXiVkEgax5bMyO+SzSZtKVpVrsVf+RE2f54EzN2\n6U03Rep0p7dvD5QkL1+UuN3lLlfMmDF3NZ/1Kq4UBNQ5Kqgh9+Nz1JUzMGWYlIMKEhQjG2X7M3a8\nFeGPKOigEsohFDxQghwn25c0zSmZmZleGb2NFN+Y3PNCQgZXVwe73Re73TrwBwlqQAu1oKlP9wgA\nAW4ohpIpU8p37nRpNGkrV7ZQQOYMtplMekXRQjbc6NOcsljWZmRcBn1VNTMzc9euXTRF68DmF17Y\n+PbbgRqNW1UDhg83r1sH7DaZ4hSlBPp1fCLbbIUzZlSAsSs4Ye3aspkzayCyGwlngxCXQVVycrDv\njtz/dG8qPYbcFy1aNH/+/HZ7VsW9Aon5l89fOr37omXW7l4784WZ+HH4zcP96d/p49t32FOWpeS7\n83GirmrpKZccO3Zq27a/PvigXpLCtFqgxuM54XQmXXiheflyGpTezNyeaX7ITChokC+QbYtsTdK6\nFyc5GX1nNN4cmirUFXXnYPnckvFpBjqohjIwgBYqkSNl+9JmbUZVVb2UN3369Ja1+LPknpc3PS6u\nTK9PgEucTu9luODGJUto6LlV1SOKEhgbu23hwhPr1ulhY0REVGBgUl6eW1VDZXly2yzZjSkpYevW\n6UEry/18MX5/EuIzWA/X/fGPTdL6e9OmObZsMfr5AUEjRpzT0dvhyImPj4LAzpjIe4VIAk3XcMI+\nIYxg6EbC+VSIyRA4fz5LfZ7X/yP3fxesWrVq1KhRjdXJNkI8IFBRX+3WW5G6LnXp+qV4zrJe5yLl\nmZR1e9bhx1rLWvPwtsYve77/Pn3u3FCt1hspX+V2l7vdV959txg37J3DGzK+zEALErmrm7bWG0Pc\nKQgCN+ShflJ3peImUSfKV0EhhIMLXHAceZycuTjT6Ne0Dn6G4ttixU8Xwi3EUNCqai9wwVj4CF5t\n8Z2/b+TI+IqK3hB2+HA/VXVCHsxoc+jkTiECIaHN1VrefPNN2yuvVPzyyyR4vtFenz311I6VK6P1\ner0kRY8efdXy5Y0zTgssFn1GRi1EdlicsQlxPZRA3y6ghU1CxENNcnJsdwVEbpflQVlZp9tV2eY/\ntwHTGfQQcqcD3o/Ve1bPfmU2btTXu/VWiKmC/lCBurJLjituEl7SdCxxtCMNdceLL25YvjxIo/GT\npFqPZ6++8qWBefjT97T+6Fc1bR8nc3+m+XEzfQHkKNn+WJ1JK8yCcNBCTX3FMW+6Ux4IcLHkjiVp\n05uNjDyj1TQ25M9Y7tOFOBoUpPF4oisrB8ID8LYsb1eUdc2/83+cOjVoz55QrVY/fPilM2bsSUvT\n5edHQzVIsjyhbfb4l0KMgmy4osXJ5dXWS0pKbpg6NfGPfwxQlOMwogFBPxwfP9Bg8JOkWlW9dM6c\ngb/7XXNDbRNiSGdkIeVbLLqMjJCusa83CjEK3MnJAd1F7iULFlQvWwZE+3g5VVVVwP/I/d8F7Sb3\nBR8uWLZhGbXdXXtA/EYQTHJcsvJI57/rtp22Gf83A0HmPZnT5Q6l79Z+913G3XdH6nR3jt9DDeb8\n3uNPBBe6XM9u397GkrmmJ03KUYUgqIY8KIQi0EMShEM1+EMu9K3X8U+DDmqYPThlybz06H79mhv5\n1Vdfzc/PLygomD179uX1HYXS0tLS09P/YDL9uGOHU6s1uFzGmhp/t9v7chQDECrLvesdp2dwd0zM\nwICAQI2m96hR0+qlj53p6fsXLowDfyiBi9pSuTczc5/ZHAQxzWzs9RUPGzZs2LBhZ5abm4UYAkdg\nkKqWb9jwzrx5QRqNVoiYMWOuaM2fdMRiCczIKIb+HZ7OW4WIgeiuyFZdu7Z85syazutu2joOHixJ\nSvIDfx+P2AO8qfQkcm+3RiYvkrNKs6jo7jYd4jZBMMmRycqTnU/u4hZBeKfF4QRO0eh6id6n9XMP\nxwRqNIGS5AGXqubX1Dz21lsnnU7jlCmtnM8MQXR9ZeAT4C2Xq0O+XlYKFfRQBscgkroE10pwggbK\nMEuTlr76fssU7w0GByZNmlRQUPDqq6/OTkwsO3YsvLZ2lMvVG4JBA4AEan2ovRtcUAMzHY55sjzA\nYPCXpOvuvTfh3Fyqkry8bKu1xGqNbTO/u9LS9lutAoY02tgrKDUpK+ULEQbWhIQ+er1WiMlz58a1\nOavrkBCxoO+wMlNtsUgZGce7pk5ksRBaCOpGzikQIsT3hiT/I/d/L7T7eSz4YMGyfyyjsqu07+Yg\n5gqkropwFzMEISy5cUna1I7mfIqpgnDkEbL9oTo796uZM3f//LNBownUaFRVdalqkcvVf+zYyFGj\nGtccNt1hUk4paCEMQkFAGa/MfuX+SXUlw8RtgjCQoBx1uWp6yaQcVHBBBWggAIoI36wdLk94/uH/\nM41qKeb90UcfBTZs2DBOr3dkZRlcLn8Ihv7ggUA4DCNAB4Nk+biieIff6efn6N3blJd3XIiYq666\nqZlehkfs9o0pKRfk5wOHYEobJo5DCB30red3L6E/8cQTopmk1iXx8ROqqiRJyu7V667Nm1sp6Hgu\nii0WbUZGPgzp4IzOzXX07x/eNRR8SIj47o12PyxELOgOHCDRhyjnHuBNpSeRe7sdIAs+WbBswzJc\n3epQlR+Vs8qzKOsSwd2Sacn4JgMVx1KHMahDNWRMD5sUh4JfM+f5ww9P3HFHPz8/nRACajwep6rG\njhkz9o474qdMAcQFAkAHUajrVXGXIAQ8cAr1/bMDilsFEQAUY/u9zfyUmUjQQjFUQzhUgQOcWGZZ\n0u9vJZB8SFJS4cGDgASXQyBMhxL4EDwwDX4N+vrXfkZ4eHxwcD+9ftDBg0GqWgrXtmiY70lLc1ut\n/qDKclJrEny+ybRFUZZD72nTjBdc0AKtz4+NTQoICNVogqurL87PV5OT+/z0U8uDN8ZxIU7CyA4r\nKjuFGNIZbYwa41shLgfNggVkZHT64E3in0KMh4Lk5H6+CP09g9x7Qm0ZLwwGQztD3SsadQrteqTe\nlwpdddyMDzOQuGrIVR1kdkDJVQhAHiQ3/e9Jk548fPjuPXtG3H57fk2NS1WDNZqyX375dP58OTQ0\neGB9t4Ba1PUqkHpdKtWggXBMT5+1wdX3VcoACMb8rJm9cAJUCAN/qIIASIRwrD9a/aZIk8cNavaM\njx4dnJMzFiIhCLbBLtgGpTARLgGzzbYLyMsDXvv1rweFhMT6+dWqau+LL3ZDNPxsNn/dfLmYIenp\nBdOnF4GfouS3ljq7ac6cRVAEAR98sHjx4iaZPaV37+cSE0cFBYVrtWVu97jZsw0ul+Gf/2x55CZx\nWpYHQd6MjuZMhMgycNzXmjltwLgFC8o6fdAW4SW4Pj7u1QOYnZ5E7sDQoUPbsVdycjIe8CEApBMw\nM20man3P0k5FLrkEgsSSu5Z0cChxrSAAnJwJcWkO4x55ZNGhQ/fv22d+5ZXL7rpLL0nROp33X3HF\nfsn5IXJo6IpFi6w3WfU5glrQoRxT0j5uIBkdg0rQQSSMgANQDC4IBQkKoQqioBZnmPr90P1+10oX\nR4bmff31aeeRy269OK82L682L2KAGDS2X3a0+x/juAD6BGKAk7ASlkA09IE1M2Z8odf/cfDgP0RG\nOnfuDNPpJt5118IDB8b++ONlqnqhzeaBaNgkxPvNsNukzMxyWS6DHEXZ2Mw2qqouXrx4W37+Do9n\nGTwFZY2/BCUlzw4YcHlYmNHPTwhx62uvPbBvX9wzz+wCPWzynVsvsNmqINLX3RohLjW1FPw7PE5j\nVIIByn2POm83SsADmvnz275LR9Ih/63Qc2QZOrCYEgtEN8syYpogBk6hru7kg9q222a8PANP+1t/\neGF515KxIQMduW+1NaT9DHpfog8O1uQfqQkv1sWX+AMD/P0PVVcDpyNqD0ysIgS0UIT6jpr2cpr1\nDSsREAVREADVkIv6uSpuFYSBqC9g0BtUKIXT0BtKGL49cOewiqByjfGA38Ggqpgav5yAagAtBIMe\ng5MqF2OyiICtQvgLEW4w/Co8fIDHM/DIkW0azbiHHpr83HPnnf8vJpNQlGI4BtObmSA5aWnlVmsU\nuGU55lx9xqutDxs2LCUlBahISyuwWjVgbBDG/lj//r202j46Xa2qznnzTSZOPLu/w3EsPj7M9xgP\n4BshJkOBLPftWAmBPCEiwdAFdWaOCNGrk1Ku2oJtsjwoK6sCotp8xJ6hyfA/cgcW/HXBsh+6W3MX\ndwiCOPxy5+emmh41KcXK2sfXmmM6JLyKmwRRUHO2KE1bT2CBSSlUkM4G6qxYtOjC4OAPXn99a1lZ\nocuV27/6+HhnnfheCpvBBUAceq1wxqvooJzZo2e/f//74nZBMEhwul6oEZANRvAHJzggGsroU6i/\nPL9XlFtfSG2m8eT4wyH+flKVxx0t9C63esBYJTmEpoRyt1vAEJ1uSFnZ1VALp+C2xrMgM/MXs9kP\njsKk5jOYHEIcgTK4yuPZtXu3N2insbaeazKFKkopHHzkEU1w8PaVK8O0Wg+UuVxNFq45ajK5FcXo\ne+jLRpMpSVEqO9yQ+gchLoFDMKiz+WGHEEkQcOgQCQmdO3KT+EqWL8rKyodhbb6Q//R6YWfwP1mG\nrN1Z0O2yuxY8XTKwckBB0EFmd+AgHAS57+X6fAInFXTYnjrbP/quZ54Z8/DDz+bkfHH6tFJSktJn\nUlCOpq7sewgkASSdNqTtMj6+q3//XH8qIZRVu1cl3ZbEVjgCAnqDBgrBDQbYAxXgBwOgGvQcH+D8\nOqbwa0NhgL/GOUj956SSLGNpkL9mvDbkEr/QI4k1u35VcXqsyxinu+G223LKyr6C38HHMAxea6yB\npKQYbbZaMMLf4+K8Gn1jHJZlf1gL1w4fvm7dOm8TqMbaerzdngthcGrlyj3vvReq1Za6XLft3t1c\nSbIYuz0QKn23mifY7SUQDTgcvu7bEMNTU8u7RDWsKzZBd9V2H5GcrIHI5OS279I+Gvk3RI8i99mz\nZ1dXV/u616o/rIKuotpm4Qc6Nu7Y2Lmjpn+e7m175MbdkXHMfzKjRR4s+yrIiKnCq6ukxDcbcLL0\n66/LvnWRX1+7qw9BWk1Ele6fJSXfFxVN3N6LSnCDPwfFQdxwCCoACAMNnKJO1fGDYtBAFGjAn8IR\nrr0XVa6MPeb1wZbHu3+4oPhd/+NiYlKpcKPlSGLN30eVGSYFbn7EMhfGT5tWcdVVd0M1/EOIonOD\nynulpFxgsx2FOPglLo6GFV3qMfFf/5obEpIN0bt3J379dQt35oeYmK+NxgiNRidE5KhR97VWaXKP\nNzDf9+4ZZbIMtLH4QXOITE11QlRHhmgGv3jLzXVXtEw4AAVt/pasWrXqPz0x9Qx6FLnTbje3ptvv\nhAc85Ofnd+6o+Y58/EHlo5+arenYFihbFbT1akmbYXnXQhSobXYknIBaKKX8UrdSUqKUlMx84IF3\nHcfMrkmUggoRMIY+Jfqkvxvqwt7DwE1QjgYVdHVNP3BBb6iEUhA4B6u44SDocUare5MrM059I39L\n7A5wQTjPbbD651pHhfIEPPnqq/0uu+z9kJCFcK/VuuE8t2dKyqWqWgLBsP3c+EKvy/TJJ5/8ubj4\nbVleBCObCbZbfvPNLyUlhYaEVAQGjjxxYvKePdeuWdPqvblIVWuhwHcS7C3LTjjeQfaMj6/tGp9q\nELjArwtGbhJ/y8oS1MXZ/rehp5F7O5BIImq33wkBLkwDfGtD0SrseXZU5GHy9Es6VHIAA3iwP+yb\nUy7jqwz0bYo7EjfWt9Kuhd5gQNwugLuefhpYu+Z7dkMVSBDOzNfu37+/Un1TpQx0EEN5H3din0TK\noFddlhO10Kue671fhRg4CdXg4Wgcyq8YtY17CuVAB+ggjMtv4L1/fpCUlPTpDz9sKSkZKsuH4TFF\nyc/M5Fx9drSqZkM0ZNXrLZmZmd5OGl4RZmhm5mnoD5z7bShcv/75xERNdnaUThc9evRN06aVulxR\ntFUzOS3LYfCjj2EzfaxWbyuUDiLHq5/YbK1t6BuGLljgBuRmgms7GzPmz6+tX/i1BdnZ2V14Nt2L\n/5E7AG4Q2LM7v0dBs9CCFtOQTiZ3QgHWPri2I2NY1loIQB7o2/TL3J/pre7bcnlhIHN7JqGgRc1U\nyQUXCOiF6dm6u2GaY6IU8vAGTVr/Zs0jL682ry5cUg8xFBUdpByqIRA0cKw+bjIECkCFQDASuwND\nAdRAMJ+nkB9H+W9tNynghGoWXUfmwTqx5T27/SOL5QKYZjaPkaTXz+25cY2qboVnYYIQixcv3r17\n9+LFi88WLDMaJzgcRXBYUc4UE34oLs72wAN99XoJfvvGG9esWRP69NMJFks17GqbmF4AlTCmjQ+g\nAQ63V9JpiIvXrpVgU4ej5hvDAwXdFg2ZnKzz5VN38803d+HJdC96VLQMUF1d7e/v82pSzBP4k5mS\nOd3UMYO3bcghJ2FBAk5Ov346vE4V7ByIWYJIcpblxNP+CDaRIohGTpTtqT587cQ0QW+obj0EU0wX\nhGG5yZJ+bTre8pB9wQ8qcLziiLs8jgTIR/1GNT1vUo4q6KEItkE19IWBEACVjPoOdwA7TXij2Q25\nVA2EYChDv1c4L1C9Y3KSy3P5fhSEQTVvJ1vMd1lHX8D+CyAaKrE9aktJPOsh+FaIIlgSEjJxzpyA\nkBBvRKOqqjOmTDn8zTc6eCgk5MaSksbXpaalFVutTngvMlLXq1eoVutR1Svvvjv+XCn/uBACTsry\niDZEKx4TIhz8fA2bsdlqZsz4EGZ1ZHbbbKUzZlR2QVfV40IEdVuFmYMHi5OSKtrWzKTHxMl40dMs\nd39//61bt/q0S/pf0wEkuofZgXc+e8erSHQus0MnOQ8CAGypPq7HA0GQ+3YbomtCQMXL7IBqUymt\ni4GJuzuOcvBgW2kD7I/YKQUP9IIhyCZZ/VaVo2VqIYjtMt9vdVius1ADERBJ7FaohmCcQ9TQfQIH\nuKEvoyR+sxlOgz93/ss6+Uqm70K1qeEHwYA5w5z21Vn+HW2x9IG/lpYmWK360tLdu3ffeuutKSkp\nt8yZ863F8hbIpaWbm8pNFY8++n5ExNa+fcMjIoI1moLa2jl79sQ3qvy1HUJA07Zs+ApZBj7zNWzG\nbC6DG33bpxFkWVdfhrlz4e722DQfavT0IPQ0cgc+/PBDn7ZP+3UaAgSHONRFp3Qehg4diuiaQLNA\nUOmI2e7AgR5UfIqTMT1iwgDO1vcypZrwwzLrHMVAfVulCgQEQALkkhJTZ0qrK1VqQEAIpqtNgH2x\nnQLwQG+iZsalX5tOEahUxVEZTexWcIKOkkR15gGoAQMZl6CtwfNnB6dBT9YFWK+HE3mn33D0OwIS\n1i+smXl1+kxkevqlDkcRXAoRVuvixYsDAwNXrFiRkpISkp4eJsta6NuYmouKFo0fr4uIKA0MvPHg\nwTlPPfXQgQNN3oGrVbUIBtIm2STJbi+FK1rdrhEOgh8c8rG3+DmIjz8Buvbv3yy83Q27B9nLllW0\n+So+ar615H8ieiC533LLLT7vUwMeltq6SQdUdikICO6CoaWOzhvzE2b0UOnjbqGgI3d562a7sl6h\niPTLGxX/KgQn+EH0+eai5QqLN5Ld+pM1jzzgvdMyRwDoQ8A1YupHUAwShYNBJXxHnf6+ZhIvboQy\n0PLOVPpeGzf5UygDlar+mF5PwWjc+lc4Bm7MS8yZx+qDHY3GU7KshRuhLD29b9++ofWV66Pt9iMQ\nDNvq/Zw5X31lMRpfk+VEg0HA5Ftv9Xe59rZYuuukLNfCL20LaHF6HZs+xq17y9YPSE31aa/zcMIb\nIdwx7b4xjoO+c0dsHmcS4NqCHhPh7kUPJPd2QmLZ18u651DKdgV311gvtfXNntsL2SSjRR7tmzdV\n2a5A62Z72stpAE3o1ahr1LrwdkOj8igVUF4X2B53TxwQvF555Z9QAjqqr2eSTAAAIABJREFUEtg+\nhr/my96YyPxkxuiNEUfqSrY/dDkbbFAIWk6M5nQ0U9dhcEAtSqGSeTyzXJar10ExCMzPn2XkiXa7\nU5Zd8JeFC7ecG8A+UlUPeE3vzMzH+/dfc//9FwQGulQ1ctSoe/bujbzzzmKIg+bynoDRdrurLnmr\ndfRLTa2EPT4qM+Gy7IaTHXOHerwE0dmRLRqvLHP4cOcO2yRik5OlujiD1tEeu/DfGD2Q3IcOHepz\nKpMHNMy/xofqQh1B6vRUNJ1P7pb3LV65vCPIWJ4BvlvuIbQla8r6hhVwfNy0EWq5ykI1CAhF3HFW\nlbV+afV27UBAIOJ2EQMjT/BspeyNlskfyfIduyduDcUJBjaMzHtxC/2O1DXjuOJ6Xh9uMThAz/ap\n7BvI5C+R42V0mJ8xT0VxwlvHZE5DISLt7HFH2u1bIRCONxJh3gwLezs6+r20tBg/v15abf+xY+fv\n23f92rUARuNBWd4LjhZdoC5ZliAvqXWG3wM6aLZTSTMYb7c74Bcf9zoPQ1NTuyK3r6Y7eSc52Vt3\nrlW0Lxbj3xk9kNz9/f19fUjJo5KBTvduNgd5iIyr8+XM1FtTkTrsqwoHgf0RH+JkMndn+rTMtm9u\nenDrOisnoBp0EEjaZ/WuSAlCsFxpoRR00ItNUfSt5bH37Y5lDtzg5osBpZHl+j65empBz5xreWkj\nFEA4xDJvo3XyF3gLHuy/jNgbZfsjdorAn2yZpUMp+l5R31MJBh1i3tk7eL2q7gA95Dbwi37/7LOR\nISGa0NCqwMCJ+/bN27LlytWrG17IZZmZ0d63qamkVi/CMzM1YDh4sNXbNcRqVbxZPz4qMwIu8mmH\nRnCCG/Z3QTRkt4XoKbNn6yGkDeUHfPXV/fujB5I7sGjRIp+2z/o5C1i2pptkGQDJd+u4LfBAecdG\naIenNxh0rcfF55HnjVpIMTVTmcAPwqEEXBCI9XMrkLkrk3CoIT0lnaK6sgT334ItCsCIUZ8jiIAI\ntulLXv3GadgDbojg9xcxZQNylIwG+rF9FKO+BicE8OcoBfj8HagBHY9fTp4B8vLU11RvHH1Dfl/i\ncJTDZquVDz4AHoqPV95+O8bPr9bjmZSb2x+a6JdkNNbIsga2taC8G407oe157hI+VxTIre8s2G54\nk12jOzZIY9R2Y7SMDvQQ0QZy70npS170THL3Fav+tApP12RbN4UMWwai851KM56c0QmhkAE+k7s5\n1QzYLK2EThoxthJYFwA6HCsddQmFAZieNym7FSQsMy2Aulati4wMYdE1AHfHxEQd1HAKIjkU6tya\nwLTv6wrBH+nN0Fcs7w6zeKX5/HEYahj1VV1NcXGnuFZVR30GpaBl6VSmx8UBjpcd6CCEzJyzzlV/\no/G0EJtvvfVPiYlGP78ASZq3ZcuC/fuP19R4wNFUBml/u32PtzZLC8q7zeYHu9qQgJqcmloBO32s\nKOD9TNd2IGDGA1VdYIScBAXoljymMp+LaPQc9Exy99UxIsfJqB22c9qMDHNGV9TXSL09FU+HjSKn\n77sEtOnWpf01jRFYUpsPvTCAhBGj5VqL9/4oJxTrOisS1vfrjFbLFRaqAAjnppFRiQZDcn5I+H5Q\nwZ+3x6NGMnVLLAUQgvVH68ArTPLndQa7cgmxuQw4VOe2/cMb9w/MwZAHHkjk8+vJy8w0YrT8yoLA\nnFFndF8cGuo4ceJofPwPsbEjT54scrkW7N/vtdYvU9Uyb92SpuSXYFnuDfktttJ2tq23RqDV6gFf\nK+SaUlPdHRP/JHB2gc2jhwvppgoErqwsLRS2Vjhs69atPcybSk8l9zFjxviayuQtL9Ntoe7oO9+i\nkPvLSB2ei+04MR14Wg+VsWfZEdh/bl7Nd9VpSuk3pHMSasEPQqEQ23N1y4J0czqHoAr8+NulpxJ/\nf1s/p3PyPzAcBydH48gax+Qk2fakzVsYUnowLqKQOq0mjn3DOPilSimE8Ny+1/L7M/lbOAoqVUNY\n9KUVSL86nTIIxO8O6d5+/QYFBPhLUqnb3a+sTC4v/+Ohc94QrSxLcKwp+SXJYvFA7xaSlVJSDsCJ\nlu9aPbzdSnyS3d/PyCiArb70Dj0fZnN5F9g83vBEukDKbww3eCCoDVuOGdOOQg//1uiZ5O4rBjCA\nGtAwgAHdcLgccqil45Et5+F3b/4O0XSgoQ9w+1b9OPOXTDRtUnKUXAWBKaVplSCPPHRnv0y2xTac\nICAMJIKqNG/fcsuj8fEL4+JGbwqiBtxQiuemoVcPHw4M8sa2V7I/gTf2fZCSmGLYC27ozefXM3U9\nnACJ/VdyxWMmdYVKOfih3EBVCAP3QgU4WVdZx4OzTl4+Y0XvBzfE1vZSgzUa/+DgJVu3Djt5MhB2\nn6ui9Lbbi0GCkkaZqKSk7GmNGQtgEC35Xc+g0HeSnbt2bYTvYTbnQYLWfb4+wqv8negWy92rU+pb\n09x3797dDSfTzeix5N4e37dgwecLuuBczkd/+qN2vlPJqDPWJXl2BDrfnAEpI1LQtXnxr2k2gj7l\n6RQkqA9hTUlMkWNknKABA4/ceXvNzp0J/v4D/P0Dqhi7XkMB9MP8vDk1RLkJpg+QbctsVIGH/QMB\nKpc7qAAV+rH9Av78dygHHd8dVwDDobpj2a9k1C5i/4V+r3BHibgp/r/t2zf4X3vj/f39Jemb0YUP\n/bgh5b77iIgYabOVN+UCDZRlFUKaKhRTAEBx86p3MahwpMWMJy/6ybIKVW3Y8ixMpipffLZNwg2D\nOzZCY3iVv+g21D3uOAScAloj957nTaUHk7vPCprXhd9tmXPVnU/uyZclIzr8SN2+m4iibca+d043\nF92vBQl5hExJyb39+v0qPDxn7Za6zMIg8kNroiaMuuONN+7ft++fZWUDDrvDHXVK+oEk3L3Ypigp\n41IsMywYYCBp69MwGkdkgQtU9l+O30mu2Aq1EIqYK/bebjPkgoYqI1tGihF7DFdsDbtlXdS1v0Ro\ng0WQRlOsdWWOO5mfUGN66cq6M0xJ2QkxsPVc4z3EYpGgrCn140qH4xjomhdGbtm5sxDawisDLBaP\nr69nXFxFhz07Ad4nlutzQ66W0W3RMoEQBsya1fJmPSw31YseS+6+Yv5N87vNoQrg7vxY3xkLZnTO\n8/Q1cUW0KYOJEBCY+jYyYwsLf3nxxeElgUgEr9+7Ijl5fHDw2ODg+EP+4Ye0XuO9KNa1ZGie7vLL\nz+xk+qne7g3ksUsYCED6VemcBAnrZ1Zg9E7kLaCFIF67glc1slwlY4AQjCkp47/WR3ysvXRtaFxZ\naERISIzkF67T+UuSbeLJ/9u5842DR555bSVu8ovz7TvqrPJJDkcpxHNuDExKijcftgl1xWgs9KYM\nNxZt6lELl7S4QR1k+RAU+CigdzAsljNGSKe2yQ70vmJd30M1a8GC2La5kGbPnt3VJ9P96LHk7qt7\n5FThKdws+7gbQ907+1uifqrWJ4x3AO04K6ltsowKAnuB/b5+/W7t00cODZVDQyeFhVmTk7evXHnJ\n346MPxp81bFwt6pWeDzDxo9XDh48/Y9ayqAW/FHyzvKaAQy1xK6HH6GWIwko9Zmeslvm7yBhspqO\n+PHSRvp9BcVsjGfbLuXLFz/su0p/0bshE+JDwgNCJpeH9Xf6D6g2/H1AcZjHc9CvdOmt+eUh7ryQ\nEiCldwrloGf34XpB1mjMhQLYem726RiLJQyONqWZVHvvaHP68gUX5IMEP7Uawx4XF+z7wrKqXdFP\nDeHfBVZ2t8UmJi9dGuBLp44ehh5L7sDqc/MGW8bU8VOh+2SZ5FHJqKw90qGuGk1A7YwFga/8LoGH\nIo4CG559ltLS2u+/X3nLLVnPP/9UQsLihISlAwe+mJR0b04MUDXxieSQkMT6NpXVHs+Rmpqxd9yx\ni4r94VVPjcm5Z+/e+fv23f7BB0RGApZJlrrZGYiYVcczXrdrXp6qPyX4kaBtmn9OEJODdYD9XXvQ\nCY3+S7Hzi5/3D9Kt0gSFngq4fG2vq9aG/Skq6Pb4kXJliFHjn1RtqA1WQ7RanSTlJFTlXVhzsafg\nppIySiEA6w/1kZdTLejILzzbDTEPekPf8+5AenoN9Gnq3gTKcjUcbF4rj5Jl4JI2FOdqR6Xcjvvs\nj7dtVeYruiKBrzFqFyxwtuH8fSKK/yD0tGYdDbFo0SKfWqqKNIGKau2OG5JDTsKjCRSgrujMw4k0\ngQ71hfaPKRYKJJwvHdPVk9U/nntu4n33oaqnN22KGD9+5Zw5v33wwT/NmVPt8QDhWu0Xw05v6l92\n148x/cr0Bo1GLwQgCeFRVY0QJS5XkEbjVFUJVo84ebfhykvGmKJGj9Y30FiAtA/SrJuslKG+ev7J\ni2sEQ8Efqhj0g6HPYb1WklBVvSS5VFWAS1U1Qrj9VL1T0oD3WH6SBPhLkgqSEBqQhNhjrByRHwhU\nBLrXTS244T3dSf9aZToEkfwLH3xG2oPyOoNCLeoSB8IIiPGCo9h22lLC6iLWs4VIAp3FQoNuTd8L\ncSloz/0jQGZmidl8ChKbmWhVaWnHrNYg6N3aTNwqRB/o68uE/acQoyBo7Vp88sQ2wEEh+kJAp7JE\nrhD+EN31zLNGiGvAlZwc2aKc5StR/KegYyUEexjc4MchDnVDQKRySIFOXvGW1R719h397OAHYw8E\nx1x99cdPPVXj8ZhmzPjGat3x888333KLgMseeGDbG2+4VXXf5s0eVc3etKnY5YrS6QyStNeviisJ\n82ifH3FRdK2+1uNRwSBJm955J1xb96oI+Mt998X6+Xn7zpa4XFPyInq5dQkV/nqtVOZyXXnvvZmv\nvTZ43Ljr771XM348QXVBxkEzNBUJnkpjnvm+JiL/LNMs1ixr0+6/k2AEP9Czb3TVkJOBAS7Jm61V\nV0pHCED1qKfDasPK9dUejwQGSUKILy45PfWnCEBV1SORzs3jym7rZ5r/8dfCIgii168v1L+jhO+l\ncDxZQ8j5iXsPsq4/RJAwO+7wahWwvWQz/9Zsftyc8kodue+GAVBitzesNVgLKnjs9vPXwikp+WZz\nYvNPrRD6tM2SFb7XmnN5F3LtZXbvCJ1rue9NTQ3z3a3TPsSDBC0zOz2uGOQZ9GTL3VeIOwXRrL53\n9cz4mZ0++MZPP51w/fUAp04RGXnTgxP/ZvgRJxsin1YU5bH33ivatKlWVQ9s3uxwOofGxY265Zaf\nX3ttxIQJOiE+euONpAsv/PiNNx5bsWLFyy+f2r5dK4S/JBW5XG5VDdJoAjQanRCFwvXElYcTXYZJ\nO3tNOB3iNWndqqoTAvCXJEAIoaqqG6rc7iCNRq1PUvf+Ignx8tj8WbujQys1CCHAo6o1qnqspiZU\nq/WXpAdWrMDjObh5c+LChfzwA5MmARdfG/qv0aXqc62/SOJRYXvElhLadNKmSBNIqP/XyHK/QNAf\nEiEIqojao7t+e2TD99b7W07/6h/6F//m++hqj6eX01kYrt80qCx3ZPXIv+u+2uT+tcmTdTFokHvL\n9lS7mC2Ihxqu/IL7TnCzGQLhFEXv8k2ezfyyGYkl0Za01HQVVTJK3Ir6p/oj5uWVxsUBIQ3nTl5e\nYVycG6IaTagvhbgC9DYbzWSrHhMiCIJbm4ntMKI/E2Jyx+zufUKEdG6nvZycowkJ1TCg65lnkxBJ\n0KvFA3kryPawepBe/I/cz2LaimkfHvxw9KagrevLsp577m9//nPcmDEaIQC3qu7atMkgSeZ586o8\nHtvrrwOSECPGjz+1fbtTVVVVDdRonB6PC/yF8ECwRlPmdtd4PGE6nQaEEE6PRyeEl161Qtx7w77R\nx4Lu/jlGW69geMd0q6oEHtAJ4VFV71+8//Uayw0fmPdfHtBArarmhFd/NbBwjtLXy8uBGk2F213l\n8dR4PLFG44VTp36zfLll69aP58y56f77X7zzzocOHNj2wgujH3446/nnkx95hLKyfnMjw3S6b+76\nMOayq9t438QMQSK5z+a2nKSa9nGaVbFSgPpW06+cWCjwo/FHIu3lNOsbVoZCAmihmDdH/+HU75dt\nV1XrunV9p0x5bvnD6RvST/dy4cTxtsOI8VshFl1C1kgIY/w+flzH3/wx31FXmEx9Wf06O3PKO2Z0\nhB9geybTriRLhlr+upxVhXxwO/TGcJTK91VVVaVYiZuxPWlLCa9j551CDAXNuSLMMSFCIcDhwGhs\neP7ZJlOCoiDL/k02Tc3L2xQXFwn9W5uJWUJcAEHNfyQa40chRrTGbi3gtMVSnJFhgJjOY4mNsjwg\nKysyLY0lS7x/OfbVV32nTPH+/u3TT1/5+OPA1089NT45OXz8+JfvuOOeBx8scrnWr1jRb/TovK1b\nU+bNq3a7v1i+vNLt3rlp08x589avWOH0eIQQBkmq8nj8JanG4wnVaq88cMDY2uWvXr16VmuBkv+h\n6OHk/sILLzz88MNt3Nj0B9Pe0k33bo7pX2jw3haN13qtl08kUXe7vIQr8Fq3dThD0AJqVdVL2Q2j\nBL1DeXcsd7vXXnTyph2RQRUajRDa+pGPO5233n//3s2bt23ceMu99x7esuWau++WJkz4ZM4cl6re\nvGIFmzczbhyqSkhIzYYNflfUt2ArLSUkJGXxpHW1f5cjZHtaHY+oJ06IaB/q+om7BRHYLLaU3m1l\nEMt6S8YPGY5nHUaMLW8pfieoRV3a9CtnspqU04r6zPn/zavNixsdRwBcVJ+Tc5I92fInijLUYrku\nPR0wPWlSihQCkY3yv+75l1i3rtJsDrwNjOBiwef8ficjr6FwNMBknbzhSXvM1eLYheBi1ifcd4hL\n7oFeGI4w9R0+uAFGgJPDl63rP3VaaHxo6S2laFBfqju3n4QwwSEY2GD6KEKMa0p2/0SIK8FflqVm\nOmIfE0IPEa3NxN1CxEOgL+S+U4h+ENY+zb24eOef/hT44osaiCssLNm0acvGjQGSVO5252/devtb\nb3m2bHnnlVd0QuglaffPP1d4PJE6nUdVhRBuVS1zubyrvZr65aNOCI0Q0ZWV1x49+o+wMEdkpCSE\nVogaj0crhHfheEbUkoTwVnvy1E9Dr39Fqje2vFaOVD/jAK8lRP0y1K2qBpfropwcY21LalZPFdzp\n8Zp7WVlZ2zc2DTEd/2lbjarmU33pWNPQceNix4z54s03r7377o/feOOm5cvZvDlny5afs7Kmr1tX\n8+23fhMmEBLi2rBBO25c+caNQb/61Zmhqr/91v/KK5s8SsmxY6F9+wK/v09svrZszQNrZsScX2Rj\nEFwPwLD6v9xgq6+52MAPeZbZgZAQoN/FY/ju7/L4s4F3PjE7YHvKZl5q9ilgxjTMlPFdRsqbKfa7\nW6sC33Iejgcg81RmSmRT5FWJHCgrxxViIAj7diUYttvt1wFgf8Iu7hAEoexTLrr7Ivtye67ZPC2b\nD6LBj6WX89JOfhhluTveruQp35Uop8k7+rUq0gQBrJ5EoUME50llvdxVwewdGpi0y3MgsYpAJmTM\nuvWlV6665KoPxAf4kXksM6VvCnCJw1EYF3de8qcWTsF3Vuusc8n9Bput3GzerCjjm7nu6rYVPzkN\nRnyot1X1ySdakCD3l196R0YarrhigdF49dy53y5ffrHJlKUolR5PlE6nQq2qxuj1FR6PVgg/IQAP\nGCTJr6rqUq32aEjIepNJCCEaOIlWmkwCvIpfLQwKCHCrqqin2oZ1o72c67V4AGdIyEY4ERysrd9e\nL0SZ2x2g0VS63U6Px1+StEK4VHXwuHHDx43b8fPP186b9+lrr11/zz3Pz52bMm+eW1WrPR69EHs3\nbxZw/ZmpUVJS+/PPuvp592Rg4HVCGN99t+W7NGzYsJY3+M9FD7fcfequkncqL+7ZOPxQn++Oe2Kc\nb8wPzl8zb82M2E6roGT5zJLxY4bcX7bP86HbRkNkOjLNr5vlvrJ9fltHcOCIfyRejm/9oCJVEIj6\nbNO3NzMv0/yGmWrUJU3J7mCxWKx/txILEpwg6y+85GRd/QssbhAMBh2xlbF5GXnAdyHisUvIGgMu\nFvzAxI28+nv5OxTcGHKo/FD1M0vOQSrljF0fNOCk4YPbC/AjfKv2ru91f7mp6pgBA9Kitz3OJ54o\nvbjU+qPVcrnlTPfX40JUQUKD6WMTYirkwPDz5lRmZr7Z7GpeeFkvhAmCWpuJO4RIgGBV/fLpp695\n/PG/TJtWq6o7Nm70hicFajQ6IRAiSqer8XiqPJ4ASRpeUjLs5MmvEhNrdTqvGMgZ6e9cm9ebmOyq\nV/m83hpgeHHxKYMhT6/XC1HqdgdJ0nX33PPLzz/rJck4ZszeTZtuXLHi87lz48aMGfHwwzteeGH4\n+PHS5MnedWQT13D06N8GDjRVVrYaGtRxfCaEqQ3roR6MHk7uPsGebb/o9YsIZPWDq2fGdL5P9Tyk\nf56+8KeFlKO+3GmPwJZtm7FiBjV4XvGIdsXiOHDE/yGeKtR0H85KLBToWv8oijmCfjiealbAEY8I\nNE2wv+l2k7JJAeRbZKVEIRhqeOJvOPfzXAONW9wl6ANObPfaUgakbBPimdF8eAUY4DSrPuWtYr6b\nBVFQxcgPDK5asdtcSTDGjX7xWzU/pVQSDgXsf51JkzkyASByPUP2h4xfPcf6o5Vq1Iy6c8sWIg4C\nGxy9PC3NabVWQux5c2rLlpMXXqiF8Gbm2lYhEhpIw2UbNgRPmLDt9dcVu73yl1+KXK4QrVYF05Ej\no6urv4qPr6jPEpDqVcGGT9prIGug3O0O0WrDS0rExIn+krRz48Zyt/s3DzzgVNXEe+5BCFwuhLC/\n+qosy2LcOIQ4E9rkxRsazc0eT+8ORFKej7Vr98+cGdItcZAbhBgD4atWtVB7oOe11muIHi7L4IvD\nxDTU5M3wnLV41sw3u5zcF0xdsPDHhR3NIDwX5qHmGWIGQeQX5hvDW1HAm0QccXUF031C2+qgyaNk\n5aRi/Yc1fWJ601sUQu8m/mx/1+413u1P2cVcQQDo+fNYvs9jelzcGeNdjpOVMgU91vXWlHtS+sny\n85uVD4dCfwhjznjp4q/94zfX5F7hJoD8hKrYbIP/Mak6yJNnrPnkIdtv7zFvnwGRrA/mwcM8Mhii\ncBURBHv+Yff2hzqDIlBhmN1+htxPQDAcgdhzT/74J5+oTVViPvDtt9+/9tr45OQgUGFJUpJekvRC\neCP0BQQIYdDpInQ6r3yxNz5eFBWd0Oura2svGD9ewJS33iIggIqK2qws3eTJAOXlZwna4ciNj3fC\nwDVrgMnNPBTTo4828x9GejxaOhRJeR4OKkognOqC7k7n49Aho9fd1eLcf+aZZ3qq4E7PzlD1wqdi\nnqse6b6WTBo0OCGEvFPNNutpDypAsPCjhe0fweV7amMNaDC91ErTH/t8O26sf2s+1b4ENGSeaKkE\nru2RumrAx+N4+9zwHPsf7d6Kx0quYnrRFGW3l9cy9x9wCmqo7uvx9NMP3EX4foDCcZgmjqpa66YE\n+jBvl3XbJgenoJL3npdNh7l5F1RRWsLm0tKa1xRvmZU86h5WKSSCp0HZgMT0dBdNLEn6LF7sgiqw\n9OnzcHz8C4mJLyUlvTxo0Lf336/Lzt62cmV2fPzR4OBwnS5YowFqVXXS3LmX3nVX6MiRt2dn356d\nfdvGjbc988yww4cHnzz5wL59vztw4Jo1a6asWUNAAEBgYB2zQ0PTu8RqFTCgA5V14zu7VIBHUXRQ\n06ljNolPZ80yQNSqVV1/qH9f9Hxy9wmzYmd5W1fb97VTs/YNThAtkp3viI2IRSKvpAMfDDdAZlHr\nRcbPIHVqKhKmCW3o6FbT0pdDtak4MS9pwlS03GsBTLebUhJT5DiZWtDy0tXEBPBZg6pblussVIOE\nkq8AT4WGOt199CcEWgjmUFhZr2p3wLG6Xqx/Nijk5VEGKspBhVAjZaBHcSgTXY7SKpDwDEPSYaiG\nIhDYj9e9GLLFUgGnzk2QKaVBJ8GSkuJvv7UYjS8mJp4OCnJqtRf6+w8JCIjU6bxuzCqPp++YMRPM\n5srS0tCqqjuysn6bnX3P3r1z9+zpn5Y2YOHCG9fWV6cICuLmmyOdTp/WVNsUJdD3vKezsNkqac8q\nrgUcy8oSMHZBlxfWLsvKakut4x7sTeW/gdx9Tj+rAS32Pd1C7pWgw364M49lijShouzpQP+dAgDz\nwz4sxk1DTDixvteGr5SzNQGnrOmIGnmcDHiV98wFmZTibTu1sRfvNjCf67o4qeBP7K/94oODIyUp\nNg+c4EfuBDcgK1AIEvRl3cC430bJuCCIWwcI2zM2XKAHjXH95w6cUEVAUqzBhaEGKjC/UHdbPHC4\n0fzxpoM91K/fUwkJy8aPt91///DAwEi9flO/ftlhYZEm0x3Z2XP27Pltdvb8ffss+/dfvXr1sGHD\nxhYVOV2upj2Q9ThgsfgqoWoVxQ3+tlZ62zaHY4qigb6d2FIjJyfK+4uPzWDbgQTvW9aaHnvzzTd3\n9Zn8f0TPJ3efu2d5QHNODcKuQ+rNqQiU4s48VmyvWKRz1GFfkXp7KgLZ5MOsThmQgtqmfqCyUcZN\n2nfNVriVB8lImF45fxGQYqqLjzTdbjJipKauD59yHYXBTD+TcFBcfEWfi6kCN0einc7rhv7f9u1j\nt6oGB9RCCMolAOF7oRD8eOkqIjIVb9mGLweT0j/Fa6yKBwRRRqpAED0oNt1mi8gBw9kPT7jFEtsg\niWH/l1/ueu45NJoquKiqKsbPzyBJNR5P3zFjLjGbL96/f1RBwZQm21OYTAKKWrtvmzMytBCXmtra\nhmcR6D29uJYyy1rAjxkZwaDvPHJ3KYoOjnTWcM3jB1nu14ZymFu3bu3B3lT+G8gdqKnxQeVLTkhG\nsG7nuq47nzMQJwWuTq79m3xRcnsabjSAdaqVapTdPn5yykFP2tet1CW3P2xHxfpFsza+fb4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cOSEaj7/WmrZDnXQkOHa2b2QSDmT9u5MHsQQjFmtr0Y22/hpos29h4mg/YbGsJGjbpr/fqmsu4m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\n49MfycvL48w9NZFIhJegGgHD3QT8fr8m9ZkylCnvV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BbEcCcisiCGOxGRBTHciYgsiOFO\nRGRBDHciIgtiuBMRWRDDnYjIgv4/yhK5409wzEQAAAAASUVORK5CYII=\n", "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph1 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:0}, \n", " ranges={ta:(0,2*pi), rh:(0,2)}, nb_values=9, color={ta:'red', rh:'brown'},\n", " thickness=2, parameters={b:1}, label_axes=False)\n", "graph2 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:pi},\n", " ranges={ta:(0,2*pi), rh:(0,2)}, nb_values=9, color={ta:'green', rh:'brown'},\n", " thickness=2, parameters={b:1}, label_axes=False)\n", "from sage.manifolds.utilities import set_axes_labels\n", "show(set_axes_labels(graph1+graph2,'V','X','U'), aspect_ratio=1, viewer=viewer3D)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Spacetime metric

\n", "

First, we introduce on $\\mathbb{R}^5$ the flat pseudo-Riemannian metric $h$ of signature $(-,-,+,+,+)$:

" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "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/plain": [ "h = -dU*dU - dV*dV + dX*dX + dY*dY + dZ*dZ" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h = R5.metric('h', signature=1)\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": [ "

As mentionned above, the AdS metric $g$ on $\\mathcal{M}$ is that induced by $h$, i.e.$g$ is the pullback of $h$ by the map $\\Phi$:

" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "g = M.lorentzian_metric('g')\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": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -\\cosh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\frac{1}{b^{2}} \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho} + \\frac{\\sinh\\left({\\rho}\\right)^{2}}{b^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}}{b^{2}} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = -cosh(rh)^2 dta*dta + b^(-2) drh*drh + sinh(rh)^2/b^2 dth*dth + sin(th)^2*sinh(rh)^2/b^2 dph*dph" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-\\cosh\\left({\\rho}\\right)^{2} & 0 & 0 & 0 \\\\\n", "0 & \\frac{1}{b^{2}} & 0 & 0 \\\\\n", "0 & 0 & \\frac{\\sinh\\left({\\rho}\\right)^{2}}{b^{2}} & 0 \\\\\n", "0 & 0 & 0 & \\frac{\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}}{b^{2}}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[ -cosh(rh)^2 0 0 0]\n", "[ 0 b^(-2) 0 0]\n", "[ 0 0 sinh(rh)^2/b^2 0]\n", "[ 0 0 0 sin(th)^2*sinh(rh)^2/b^2]" ] }, "execution_count": 16, "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": 17, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = -\\frac{\\partial}{\\partial {\\tau} }\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\rho}+\\frac{\\partial}{\\partial {\\tau} }\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\tau} -\\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\tau} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\theta} + \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\tau} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\tau} -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\tau} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\phi} + \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\tau} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\tau} -b^{2} \\cosh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\rho} }\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\rho} + b^{2} \\cosh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\rho} }\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\tau} -\\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\rho} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\theta} + \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\rho} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\rho} -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\rho} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\phi} + \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\rho} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\rho} -b^{2} \\cosh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\theta} + b^{2} \\cosh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\tau} +\\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\theta} -\\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\rho} -\\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta} -b^{2} \\cosh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\phi} + b^{2} \\cosh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\tau} +\\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\phi} -\\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\rho} + \\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} -\\sinh\\left({\\rho}\\right)^{2} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}$ " ], "text/plain": [ "Riem(g) = -d/dta*drh*dta*drh + d/dta*drh*drh*dta - sinh(rh)^2 d/dta*dth*dta*dth + sinh(rh)^2 d/dta*dth*dth*dta - sin(th)^2*sinh(rh)^2 d/dta*dph*dta*dph + sin(th)^2*sinh(rh)^2 d/dta*dph*dph*dta - b^2*cosh(rh)^2 d/drh*dta*dta*drh + b^2*cosh(rh)^2 d/drh*dta*drh*dta - sinh(rh)^2 d/drh*dth*drh*dth + sinh(rh)^2 d/drh*dth*dth*drh - sin(th)^2*sinh(rh)^2 d/drh*dph*drh*dph + sin(th)^2*sinh(rh)^2 d/drh*dph*dph*drh - b^2*cosh(rh)^2 d/dth*dta*dta*dth + b^2*cosh(rh)^2 d/dth*dta*dth*dta + d/dth*drh*drh*dth - d/dth*drh*dth*drh - sin(th)^2*sinh(rh)^2 d/dth*dph*dth*dph + sin(th)^2*sinh(rh)^2 d/dth*dph*dph*dth - b^2*cosh(rh)^2 d/dph*dta*dta*dph + b^2*cosh(rh)^2 d/dph*dta*dph*dta + d/dph*drh*drh*dph - d/dph*drh*dph*drh + sinh(rh)^2 d/dph*dth*dth*dph - sinh(rh)^2 d/dph*dth*dph*dth" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem = g.riemann()\n", "print(Riem)\n", "Riem.display()" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "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}} } & = & -b^{2} \\cosh\\left({\\rho}\\right)^{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}} } & = & -b^{2} \\cosh\\left({\\rho}\\right)^{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}} } & = & -b^{2} \\cosh\\left({\\rho}\\right)^{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 = -b^2*cosh(rh)^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 = -b^2*cosh(rh)^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 = -b^2*cosh(rh)^2 \n", "Riem(g)^ph_rh,rh,ph = 1 \n", "Riem(g)^ph_th,th,ph = sinh(rh)^2 " ] }, "execution_count": 18, "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": 19, "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" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 3 \\, b^{2} \\cosh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} -3 \\mathrm{d} {\\rho}\\otimes \\mathrm{d} {\\rho} -3 \\, \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} -3 \\, \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "Ric(g) = 3*b^2*cosh(rh)^2 dta*dta - 3 drh*drh - 3*sinh(rh)^2 dth*dth - 3*sin(th)^2*sinh(rh)^2 dph*dph" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric = g.ricci()\n", "print(Ric)\n", "Ric.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "3 \\, b^{2} \\cosh\\left({\\rho}\\right)^{2} & 0 & 0 & 0 \\\\\n", "0 & -3 & 0 & 0 \\\\\n", "0 & 0 & -3 \\, \\sinh\\left({\\rho}\\right)^{2} & 0 \\\\\n", "0 & 0 & 0 & -3 \\, \\sin\\left({\\theta}\\right)^{2} \\sinh\\left({\\rho}\\right)^{2}\n", "\\end{array}\\right)$ " ], "text/plain": [ "[ 3*b^2*cosh(rh)^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": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci scalar:

" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field r(g) on the 4-dimensional differentiable manifold M\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ \\mbox{on}\\ \\mathcal{M}_0 : & \\left({\\tau}, {\\rho}, {\\theta}, {\\phi}\\right) & \\longmapsto & -12 \\, b^{2} \\end{array}$ " ], "text/plain": [ "r(g): M --> R\n", "on M_0: (ta, rh, th, ph) |--> -12*b^2" ] }, "execution_count": 21, "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": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 22, "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 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:

" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Lambda = -3*b^2\n", "Ric - 1/2*R*g + Lambda*g == 0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spherical coordinates\n", "Let us introduce spherical coordinates $(\\tau,r,\\theta,\\phi)$ on the AdS spacetime via the coordinate change\n", "$$ r = \\frac{\\sinh(\\rho)}{b} $$" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (M_0, (ta, r, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}_0,({\\tau}, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (M_0, (ta, r, th, ph))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_spher. = M0.chart(r'ta:\\tau r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "print(X_spher) ; X_spher" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\tau} & = & {\\tau} \\\\ r & = & \\frac{\\sinh\\left({\\rho}\\right)}{b} \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$ " ], "text/plain": [ "ta = ta\n", "r = sinh(rh)/b\n", "th = th\n", "ph = ph" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_spher = X_hyp.transition_map(X_spher, [ta, sinh(rh)/b, th, ph])\n", "hyp_to_spher.display()" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\tau} & = & {\\tau} \\\\ {\\rho} & = & {\\rm arcsinh}\\left(b r\\right) \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$ " ], "text/plain": [ "ta = ta\n", "rh = arcsinh(b*r)\n", "th = th\n", "ph = ph" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_spher.set_inverse(ta, asinh(b*r), th, ph)\n", "spher_to_hyp = hyp_to_spher.inverse()\n", "spher_to_hyp.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The expression of the metric tensor in the new coordinates is

" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -b^{2} r^{2} - 1 \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{1}{b^{2} r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "text/plain": [ "g = (-b^2*r^2 - 1) dta*dta + 1/(b^2*r^2 + 1) dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_spher.frame(), X_spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, the expression of the Riemann tensor is" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, r \\, {\\tau} \\, r }^{ \\, {\\tau} \\phantom{\\, r} \\phantom{\\, {\\tau}} \\phantom{\\, r} } & = & -\\frac{b^{2}}{b^{2} r^{2} + 1} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\theta} \\, {\\tau} \\, {\\theta} }^{ \\, {\\tau} \\phantom{\\, {\\theta}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\theta}} } & = & -b^{2} r^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\tau}} \\, {\\phi} \\, {\\tau} \\, {\\phi} }^{ \\, {\\tau} \\phantom{\\, {\\phi}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\phi}} } & = & -b^{2} r^{2} \\sin\\left({\\theta}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\tau} \\, {\\tau} \\, r }^{ \\, r \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, r} } & = & -b^{4} r^{2} - b^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -b^{2} r^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -b^{2} r^{2} \\sin\\left({\\theta}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\tau} \\, {\\tau} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\theta}} } & = & -b^{4} r^{2} - b^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{b^{2}}{b^{2} r^{2} + 1} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -b^{2} r^{2} \\sin\\left({\\theta}\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\tau} \\, {\\tau} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} \\phantom{\\, {\\phi}} } & = & -b^{4} r^{2} - b^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{b^{2}}{b^{2} r^{2} + 1} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & b^{2} r^{2} \\end{array}$ " ], "text/plain": [ "Riem(g)^ta_r,ta,r = -b^2/(b^2*r^2 + 1) \n", "Riem(g)^ta_th,ta,th = -b^2*r^2 \n", "Riem(g)^ta_ph,ta,ph = -b^2*r^2*sin(th)^2 \n", "Riem(g)^r_ta,ta,r = -b^4*r^2 - b^2 \n", "Riem(g)^r_th,r,th = -b^2*r^2 \n", "Riem(g)^r_ph,r,ph = -b^2*r^2*sin(th)^2 \n", "Riem(g)^th_ta,ta,th = -b^4*r^2 - b^2 \n", "Riem(g)^th_r,r,th = b^2/(b^2*r^2 + 1) \n", "Riem(g)^th_ph,th,ph = -b^2*r^2*sin(th)^2 \n", "Riem(g)^ph_ta,ta,ph = -b^4*r^2 - b^2 \n", "Riem(g)^ph_r,r,ph = b^2/(b^2*r^2 + 1) \n", "Riem(g)^ph_th,th,ph = b^2*r^2 " ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(X_spher.frame(), X_spher, only_nonredundant=True)" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.4.beta4", "language": "", "name": "sagemath" }, "language": "python", "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 }