{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Predefined Metrics in Symbolic Module\n", "\n", "### Importing some of the predefined tensors. All the metrics are comprehensively listed in EinsteinPy documentation." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from einsteinpy.symbolic.predefined import Schwarzschild, DeSitter, AntiDeSitter, Minkowski, find\n", "from einsteinpy.symbolic import RicciTensor, RicciScalar\n", "import sympy\n", "from sympy import simplify\n", "\n", "sympy.init_printing() # for pretty printing" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Printing the metrics for visualization\n", "All the functions return instances of :py:class:`~einsteinpy.symbolic.metric.MetricTensor`" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 - \\frac{r_{s}}{r} & 0 & 0 & 0\\\\0 & - \\frac{1}{c^{2} \\left(1 - \\frac{r_{s}}{r}\\right)} & 0 & 0\\\\0 & 0 & - \\frac{r^{2}}{c^{2}} & 0\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin^{2}{\\left(\\theta \\right)}}{c^{2}}\\end{matrix}\\right]$" ], "text/plain": [ "⎡ rₛ ⎤\n", "⎢1 - ── 0 0 0 ⎥\n", "⎢ r ⎥\n", "⎢ ⎥\n", "⎢ -1 ⎥\n", "⎢ 0 ─────────── 0 0 ⎥\n", "⎢ 2 ⎛ rₛ⎞ ⎥\n", "⎢ c ⋅⎜1 - ──⎟ ⎥\n", "⎢ ⎝ r ⎠ ⎥\n", "⎢ ⎥\n", "⎢ 2 ⎥\n", "⎢ -r ⎥\n", "⎢ 0 0 ──── 0 ⎥\n", "⎢ 2 ⎥\n", "⎢ c ⎥\n", "⎢ ⎥\n", "⎢ 2 2 ⎥\n", "⎢ -r ⋅sin (θ) ⎥\n", "⎢ 0 0 0 ────────────⎥\n", "⎢ 2 ⎥\n", "⎣ c ⎦" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sch = Schwarzschild()\n", "sch.tensor()" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & 1.0 & 0 & 0\\\\0 & 0 & 1.0 & 0\\\\0 & 0 & 0 & 1.0\\end{matrix}\\right]$" ], "text/plain": [ "⎡-1 0 0 0 ⎤\n", "⎢ ⎥\n", "⎢0 1.0 0 0 ⎥\n", "⎢ ⎥\n", "⎢0 0 1.0 0 ⎥\n", "⎢ ⎥\n", "⎣0 0 0 1.0⎦" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Minkowski(c=1).tensor()" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & e^{\\frac{2 x}{\\alpha}} & 0 & 0\\\\0 & 0 & e^{\\frac{2 x}{\\alpha}} & 0\\\\0 & 0 & 0 & e^{\\frac{2 x}{\\alpha}}\\end{matrix}\\right]$" ], "text/plain": [ "⎡-1 0 0 0 ⎤\n", "⎢ ⎥\n", "⎢ 2⋅x ⎥\n", "⎢ ─── ⎥\n", "⎢ α ⎥\n", "⎢0 ℯ 0 0 ⎥\n", "⎢ ⎥\n", "⎢ 2⋅x ⎥\n", "⎢ ─── ⎥\n", "⎢ α ⎥\n", "⎢0 0 ℯ 0 ⎥\n", "⎢ ⎥\n", "⎢ 2⋅x⎥\n", "⎢ ───⎥\n", "⎢ α ⎥\n", "⎣0 0 0 ℯ ⎦" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "DeSitter().tensor()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & \\cos^{2}{\\left(t \\right)} & 0 & 0\\\\0 & 0 & \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)} & 0\\\\0 & 0 & 0 & \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}\\end{matrix}\\right]$" ], "text/plain": [ "⎡-1 0 0 0 ⎤\n", "⎢ ⎥\n", "⎢ 2 ⎥\n", "⎢0 cos (t) 0 0 ⎥\n", "⎢ ⎥\n", "⎢ 2 2 ⎥\n", "⎢0 0 cos (t)⋅sinh (χ) 0 ⎥\n", "⎢ ⎥\n", "⎢ 2 2 2 ⎥\n", "⎣0 0 0 sin (θ)⋅cos (t)⋅sinh (χ)⎦" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "AntiDeSitter().tensor()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calculating the scalar (Ricci) curavtures\n", "They should be constant for De-Sitter and Anti-De-Sitter spacetimes." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "scalar_curvature_de_sitter = RicciScalar.from_metric(DeSitter())\n", "scalar_curvature_anti_de_sitter = RicciScalar.from_metric(AntiDeSitter())" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\frac{2 e^{- \\frac{2 x}{\\alpha}}}{\\alpha^{2}}$" ], "text/plain": [ " -2⋅x \n", " ───── \n", " α \n", "-2⋅ℯ \n", "──────────\n", " 2 \n", " α " ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "scalar_curvature_de_sitter.expr" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAq0AAAArCAYAAABSKrFHAAAABHNCSVQICAgIfAhkiAAAEf5JREFUeJztnX2wHlV9xz+XJF4So0EUIYwF0tQXIgmJUKy8xIvWgLxodWpBMBokdQpxajpSaUyo0VHeVBpmeOmMY3PVKlqiVVSGqImRQYNhnJBEE4XEUoUkkKABlCAkpH98d3vPc+7uPrvPs+/5fWZ27n12zz67v9855/uct/3twNKlSzEMwzAMwzCMOnNIidf6M2ANsBnYCLyrxGu3DfNlNtrmr7bZUxbmt/S02Vdtti0P2uKfttgR0jZ7emIgYqT1aGAv8PucrzUZOBK4HzgK+BnwKuCPOV+nSMYBU4AHKr6PNviyTNrmr7bZ42MaFE9ZGtQGX8XRZtvyoC3+aYsdIW2zpyf8kdZxwDtJ92MxDHwnw7V2IGcD7AR2A4c7x18CPApM7fI9twMfznDdPHkOeAOd992NYbL5CdSbuinheDdfQnp/ulTp2yKpsuwN04z8r0veV6lBUH8d6kWDoFy9/iCwFXU87gKOCPY3pf60TTuHKa+exOU9lO+jg03321ZuI/EbrR8AVqY890PAe3q87knAGOC3zr6PAncC27y0n/bu6RPAYmBSj9dOYjZwB/AIcACYF5Hmv4F/zPCd/fgpDVG+hHh/JlGUbxcB9wFPAruAbwMn5HyNtJRd9pqS/0XWqyxUqUEQ7Te/HEBx/ipCg6A8vf4UcAXKx9cDfw5cHxxrSv2pk3ZC//pZVj1JynuoVmMOBt2vW7ktBL/R+mbgwZTnPgHs6eGahwNfRAU7ZAIwH/h8RPpTgHXO503ArymmQEwEfo4K3N6YNE+iyjgh5Xf26qc0RPkSkv2ZRFG+HQJuAU4F3gTsA35A9tGifqmi7DUl/4usV1moSoMg3m9+OYDi/FWEBkE5en0yamBdCKxG6+5uAs4Njjeh/tRNO6F//SyjnnTLe6hOYw4G3a9juS0Et9F6AvC4d3w2cC/wB5QJ6xjp4Q3TOfy9BlWsq9Gw9WPAZ7xrDALfBK4FfuLsPweNKvzY2fcC4NngHpYExzcHx+4A3p3GwIzciXorK4DnE9JtAf7a+Zy3nwg+9+JLiPbn11D+LnT2HQ88jYQmpAjfngUsRz/Gm4C5aNroNCfNAJqmeBD4E/AwcI1zfBBYhqY/nkH+Pt05npQH4flFlb225H9R9SotVWoQjPZbUjmAYvxVhAZB8XoNGmW7O7iPkF3Ay5zPda8/cbZVWX/S6GeV9QTS5T2M9pHpfidtKreF4DpjFmpxh4wFvgXcA5yIhvuXAfsTvu9i1As8Fa1tWQhcEBwbQBm5GviSd94ZaFHxAWffPrR2i+DakxmppOtQb2h8wr0UyTbgdcH/efspTZokX0K0PxcCXwE+FnweBG5DP45fddKV4dsXobLnrlu8GrgKCdZr0ZOR7tTH9cj+96OyugmtmZpM9zwosuy1Kf+rrldVahCM9ltSOYBq/dWvBkG+ej0OOB/4hpd2PPpBD6l7/YmyDepVf3z9rLqepM17GO0j0/30aZpebnPBbbQeiaadQl4MHIbWz2wDfomM35LwfZuBf0VPtv4X8EM03QfK/AuAv0GLie8HpgfHjgW2e9/1PCo0T6H1PDsZqaTbUUU5uruJhfAk8hfk76c0aZJ8CdH+3AF8NrjXY1FP7cXAAi9dGb69Ed3z2uDzROCfgH8B/gMt5F+LeqYALwQuA64Evot8+w+o972A7nlQZNlrU/5XXa+q1CAY7bekcgDV+qtfDYJ89XommqK8Do08hdutwK+cdHWvP1G2Qb3qj6+fVdeTtHkPnT4y3T+4ym0ujHX+H6Czlf471KpfCawKthXAbxK+b6P3eTvw8uD/e4iPCzseFUSfWcAGRvcewrVecT2DT6LFxUmciYbse+F55C/I309p0iT5EuL9+RBaZ/MRtB5mNqqgLkm+zcOvN6DpndMZ6ZVOQz3AVTHnTEWVyp362I8Ebhrd86DIstem/O9Wr4qmSg2CaL/FlQMovq4k0a8GQb56/Wo0tTrd238bnfW27vUnru5Af/Unr/IQpZ9V15O0eQ+dPjLdr3+5rR2uAx5DLXCXS9Cw993A21Cv6ayE73vO+3yA5MIfshuFa/CZCayP2B8uQN8V833L0LqNpM1/sCILk+gsIEX4qVdfQrw/QRXyciSiayOOJ/m2X7/+G1o78yY6p4H7IRSXrHkQkkfZa0v+d6tXRVOlBkG03+LKARRbV7rRrwZBvno9Kdi/1dn2IP+tcNLVvf4k1R3ovf7kUR6S9LPKepI27yE/jTHd76Socls7XGdsAo6JSLMBDfsPoV7g+wq4j/Wo5+RzIqN7HqDFzo8Q37PYjYbrk7an+7jfYyPuqww/pSXOn6DRmV+gwhtFkm/78euNjAjuL71jW9AifH+qJGQb6sm76wnHoPVH7oMxveRBXmWvDfnfrV4VTZUaBNF+iysHUFxdSUOVGhTlp92MrLUMWYR+IN0fybrXn6S6A73Xn37LQ5J+hlSV/2nzHjp9ZLqfH0WV27J5MwrptxOVje1oNPucMIFbyNbTuaZhClr/cCoSyDOBGXQWlrxYiXqaL/X2jwVeE9zXYc7+M0gfyzELE1FPaybyzTHB//4P6XTge8H/ZfopLXH+XICmBsYSv2C8CN/ejHqlF6E1QkcF28Tg+FNIlK8J0k1FC8MvC47/Ea2Pug4V3uODz0ei9U/95EG/Za9N+V9UvUpLlRoE0X6LKwdQjL+aoEFRflqNfLUYOA49TT4XuNQ7t+71J67uQHX1p5t+Vp3/afMeOn1kup8fdSy3WbkehXI7GUU0+Cxay3wE6hQAnY3WMFxCKI5Po1eE3Y4WBX8B+DIqQHmzCU2NXOjtXxzsc8NgHAq8A/hcAfdxMvrhXI/Wd3w8+P8TTpoJqPUfrgkp009pifLnNBQ0+WbglUTHeCzKt5ejnvgqtDA83K5w0ixCPrsK9cC/DrzCOX4lCt+xHC1AnwGcHXxPP3nQb9lrS/7nlffzkJYM9XBulRoE0X6LKgdQXF1pggZF+WkX8F4UK3Iz8BbgjXTG3G1C/YnTg7LqTxTd9LPq/E+T9xDtI9P9fKhjuc3C3wP/jPw4Fa29/Wiw/3U468EHli5d6p54KAotcQvlczbqdU0jOWzEAuDtwJwybiqCi5F47Kzo+mlx/TkW+CkSlPnox+40OmPqQfW+rYqmlL0sZM3/vGybh35gen3IqEoNgmaUhTpoUFo/uTSl/vi2DVJe/WkKbcj/JtT1LDS13A6iMGd7UcP62aTEY4aGhtzP+xgJ7v1MATeXxFY08ruD0bHdXGahKQI/CHkZjENP7WZ5TVpVuP5cgqYYz0ehSC5CUxr30enrKn1bJU0oe1nJmv952TYThWT5AnpqNStVahDUvyzURYPS+smlKfXHt+0zlFd/mkIb8r/udT0rTS23b0WN6uVoydNbgXeiWadD0Kj3/+OPtBrtYw5aFzLESOiQuahAb0YjYkZ7KTv/59HfSKth1AnTT6OJNKncfhzFpr0WOI/ON5qBojP8LUF0A2u0GoaRJ/OwRqthGIaRjlvRSyP2owb15Wj98hTUyJ4D/IjgOYm0cdkMwzB8HkIPT7nb8uDYDyOODZd+h4ZhGEaePMRobU/a/rPL94Xt0H0o3u09aDnDJvSQ2MPowb43QOcbsQzDMLKwjNFhoGaihf1Ra1rvL+GeDMMwjOLYRrbnDaJeL+uyJ/i7ntG/GU+jcFyXonBoa63RahhGryyL2DcPNVqHseUBhmEYbSPuZRC98qvg756Y478P/o6HkZHWqPdqG4ZhwMg77ovENMgwDKM+lKH7oPB9B1CorkNQdBSX8MGs/4GRtQQDttV2OwYtQt6C1nj8XQ3uyew5uLYyqNpG28rb2qIBbbGjrfbY1t9WFv8LfBuVvw95x+YAZ6FR2LvA1rQ2gX3AQrQe8CjgZ8Cd6BV3TaRt9hiGkY22aEBb7Ahpmz1Gc1iAYsbeAJyL1rdOQTG/96M4rk+ARQ9oAjsYeYBlJ7AbONxL8xLgUfT6syRuBz6c8frDwHcypF8D3JRwPE97XHqxzTCM/hkmm0b0owEfREHU96KRlyOcY2VrQDc76qTLUI02my4baXgYOAmVz1eiEdchNAJ7Gnq9L4DFaW0YJ6Gnsv3gu58GXgZc4u2bgYbWQ6aj6Z8ppH+LySQ0VRC3SNpnDfBz9OPSjSz2dKMX2wzD6J+sGuGSRQM+hV5h+37UoFqBAqeHaarUgCg74nTM1+YydBmq0WbTZSNXbKS1ORwOfBH4gLd/Aho6/7y3/xRgnbdvE/Br4D0ZrvsEvf0YdSOrPd3oxTbDMPqnV43IogEnA4uAC4HVwEY0KnOuk6YqDYiyI0nHfG2uky5DvtpsumzkijVa82MATYM8CPwJDXdfExwbROGBHkXxze4FTvfOnx3s/wMSpHWM9HIHgW+i15z9xDvvHPTkXfiqthcAzwbftyQ4ttlJfwfw7pTXhc5pqDXALcDVaKTjMfTGCr8cHdIlTRZ7AL6G3o280Nl3PIrhdmGCbYZh5EOSTgzTOVW9hu46kVUDrkCvc7zX2bcLjfq5+BqQpMvhfSRpczd9jLMjyoYkbY7Srrx9DtVos+mykRvWaM2Pq4GrkCC+FngX8Nvg2PXABWhaaxbqfd4FTA6OjwW+hd4EcSLweiSk+5HoDqPRhS9FXPcMtGA+DBm0j+DNEcH3TEZrQkLWoZ7++C7XjePi4BqnommmhYFtadNktYfg/K8AHws+DwK3oenBr8bYZhhGPuStE1k1YBxwPvANL914Rk85+xqQpMuQrM3d7E6yI0rHkrTZv+82abPpspEbtqY1HyainutC4N+9Yy9EwXHnoykXgDHAA6hyL0HTMY+jhcc/8s4/HY0wbHT2zUXiCuoVPwG8zzl+HhKSSYyOfzkD2AD8RXBfcdcNGUajGeeh3vwgI8IL8H0UsmJ+8Llbml7sATgOxWk7Dvn57UjMn4qxbVuMPYZhZCNJn6BTIyB/DfhL1PB5hs5G2zj0lPFfOftcDXiUeF2G7tp8Qxe7k+yI07E4bfa1K2+fp0lTlDabLhu5YSGv8mEaEoNVEcemInF1p1T2A2uD8wB+h0RoZfAdq1BP9Teop500Ij4eibPLLCQSUQHb9zrnbUu4bhwbvc/bgZdnSNOLPaDXu+0BPoLWWs2ms8EKnbYZhpEPSfoUR54a8Go0rT7dS3cbnboKnRqQpMvQXZu72Z1kR5yOxWmzr115+zxNmqK02XTZyA1bHlAtrnBdgqaA7gbehl5tdlbUSR67URgSl5loBCKKMITJrh6v+5z3+QCjy1GaNHFE2ROyAbgc+CT6YfHxbTMMIx+K0Ik4fA2YFOzb6mx7kM6t8M7NSwNCbc5TlyFem6Puuy3abLps5IY1WvNhC1rkH/VO3m1olMBdVzoGTdFs9tJuAK5DU0JrGD0NE8V6RkZsQ05kdI865ATgETp7zL1ctyii7AkZAH6BhDGKKNsMw8iHsnTC14DdwIvo/L1ahBpHfgPJ1YAkXYb02pyXLkO8NsdpVxu02XTZyA1rtObDU8CNaLH/JWja6RTgMvQ2kVuR8JyDnqy8FTgSPe0JimF3LVocfyxwJloH5Ddqo1gZfOdLnX1jgdcARwOHeenPCM7p97pFEWUP6I0Zs5FtcQ8juLYZhpEPZeuErwGrUb1fjNZNXoHWWl4aca6rAUm6DN21OW9dhnht9rWrTdpsumzkhjVa82MREr+rUA//68ArgmNXotAgy9EbR2YAZ6M3kIDCg7wKvT3kARTU+cvB93VjE3pIwQ39tDj47Id3ORR4B/C5HK5bFFH2TENBrW9Gb8uYEHGeb5thGPlQtk74GrALeC96WGgz8BbgjSiMlUuUBiTpMiRrc966DNHaHHXfbdFm02UjVyx6QDs4G40oTCM5JMoC9GTnnDJuqg9ce8YCP0U/VvPR6MlpdMZrhObYZhhGd9JqmkvdNKBtugzZtblJthkNYMzQ0FDV92D0z1Y0ar6D5FflzULTX4+XcVN94NqzBD28cD4Ksn0Rmi67j05bm2KbYRjdSatpLnXTgLbpMmTX5ibZZjQAG2k16swc4LvoIYQwLM1c9BaXzWidl2EYhlEups1GJVij1TAMwzAMw6g99iCWYRiGYRiGUXus0WoYhmEYhmHUHmu0GoZhGIZhGLXHGq2GYRiGYRhG7fk/fHm2D17St94AAAAASUVORK5CYII=\n", "text/latex": [ "$\\displaystyle \\frac{\\left(\\sin^{2}{\\left(t \\right)} - 1\\right) \\sinh^{2}{\\left(\\chi \\right)} - 2 \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}}{\\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}} + \\frac{\\left(\\sin^{2}{\\left(t \\right)} - 1\\right) \\sin^{2}{\\left(\\theta \\right)} \\sinh^{2}{\\left(\\chi \\right)} - 2 \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}}{\\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(t \\right)} \\sinh^{2}{\\left(\\chi \\right)}} - 6$" ], "text/plain": [ "⎛ 2 ⎞ 2 2 2 ⎛ 2 ⎞ 2 2 \n", "⎝sin (t) - 1⎠⋅sinh (χ) - 2⋅cos (t)⋅sinh (χ) ⎝sin (t) - 1⎠⋅sin (θ)⋅sinh (χ) -\n", "─────────────────────────────────────────── + ────────────────────────────────\n", " 2 2 2 2 \n", " cos (t)⋅sinh (χ) sin (θ)⋅cos (t\n", "\n", " 2 2 2 \n", " 2⋅sin (θ)⋅cos (t)⋅sinh (χ) \n", "─────────────────────────── - 6\n", " 2 \n", ")⋅sinh (χ) " ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "scalar_curvature_anti_de_sitter.expr" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On simplifying the expression we got above, we indeed obtain a constant" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle -12$" ], "text/plain": [ "-12" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(scalar_curvature_anti_de_sitter.expr)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Searching for a predefined metric\n", "find function returns a list of available functions" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['AntiDeSitter', 'AntiDeSitterStatic', 'DeSitter']" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "find(\"sitter\")" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.0" } }, "nbformat": 4, "nbformat_minor": 2 }