{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Carter time machine" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(a, r\\right)$ " ], "text/plain": [ "(a, r)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('a r')" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\theta}$ " ], "text/plain": [ "th" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('th', latex_name=r'\\theta')" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( r, a, {\\theta} \\right) \\ {\\mapsto} \\ 2 \\, a^{2} r \\sin\\left({\\theta}\\right)^{2} + {\\left(a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}\\right)} {\\left(a^{2} + r^{2}\\right)}$ " ], "text/plain": [ "(r, a, th) |--> 2*a^2*r*sin(th)^2 + (a^2*cos(th)^2 + r^2)*(a^2 + r^2)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(r,a,th) = (r^2+a^2)*(r^2+a^2*cos(th)^2) + 2*a^2*r*sin(th)^2\n", "f" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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ys7MZOHAgCxcuJDIyksGDB7N+/Xo6depkdjQyMzPx8/MjJCQEgOeff56DBw+y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T17qmqM9fXrBzfdBA3rljz4oLplX1+vXo2HG4FKi9qOhg4dWhs0OBwOMjMzCYVC2Gw2PvywLmCw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\njuS2227jW9/6VrybI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"text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g0 = plot(f(r,0.9,0), (r,-1.5,1.5), legend_label=r'$\\theta=0$', \n", " thickness=2, linestyle=':', color='red')\n", "g1 = plot(f(r,0.9,pi/4), (r,-1.5,1.5), legend_label=r'$\\theta=\\pi/4$', \n", " thickness=2, linestyle='-.', color='grey')\n", "g2 = plot(f(r,0.9,pi/3), (r,-1.5,1.5), legend_label=r'$\\theta=\\pi/3$', \n", " thickness=2, linestyle='--', color='blue')\n", "g3 = plot(f(r,0.9,pi/2), (r,-1.5,1.5), legend_label=r'$\\theta=\\pi/2$', \n", " thickness=2, color='violet')\n", "graph = g0+g1+g2+g3\n", "graph.axes_labels([r'$r/m$', r'$\\rho^2 (r^2+a^2) + 2 a^2 m r \\, \\sin^2\\theta$'])\n", "graph.set_legend_options(loc='upper right')\n", "graph" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": true }, "outputs": [], "source": [ "graph.save('ker_sign_gpp.pdf')" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": true }, "outputs": [], "source": [ "rp(a) = 1 + sqrt(1-a^2)\n", "rm(a) = 1 - sqrt(1-a^2)" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\ {\\mapsto}\\ \\sqrt{-a^{2} + 1} + 1$ " ], "text/plain": [ "a |--> sqrt(-a^2 + 1) + 1" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rp" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\ {\\mapsto}\\ -\\sqrt{-a^{2} + 1} + 1$ " ], "text/plain": [ "a |--> -sqrt(-a^2 + 1) + 1" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rm" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.43588989435407$ " ], "text/plain": [ "1.43588989435407" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rp(0.9)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.564110105645933$ " ], "text/plain": [ "0.564110105645933" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rm(0.9)" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, a^{2} r + 4 \\, r^{3} - 2 \\, {\\left(a^{2} r - a^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}$ " ], "text/plain": [ "4*a^2*r + 4*r^3 - 2*(a^2*r - a^2)*sin(th)^2" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = diff(f(r,a,th), r).simplify_full()\n", "df" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left\\{r : -\\frac{{\\left(\\sin\\left({\\theta}\\right)^{2} - 2\\right)} a^{2} {\\left(-i \\, \\sqrt{3} + 1\\right)}}{12 \\, {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}}} - \\frac{1}{2} \\, {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}} {\\left(i \\, \\sqrt{3} + 1\\right)}\\right\\}, \\left\\{r : -\\frac{{\\left(\\sin\\left({\\theta}\\right)^{2} - 2\\right)} a^{2} {\\left(i \\, \\sqrt{3} + 1\\right)}}{12 \\, {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}}} - \\frac{1}{2} \\, {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}} {\\left(-i \\, \\sqrt{3} + 1\\right)}\\right\\}, \\left\\{r : \\frac{{\\left(\\sin\\left({\\theta}\\right)^{2} - 2\\right)} a^{2}}{6 \\, {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}}} + {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}}\\right\\}\\right]$ " ], "text/plain": [ "[{r: -1/12*(sin(th)^2 - 2)*a^2*(-I*sqrt(3) + 1)/(-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3) - 1/2*(-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3)*(I*sqrt(3) + 1)},\n", " {r: -1/12*(sin(th)^2 - 2)*a^2*(I*sqrt(3) + 1)/(-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3) - 1/2*(-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3)*(-I*sqrt(3) + 1)},\n", " {r: 1/6*(sin(th)^2 - 2)*a^2/(-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3) + (-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3)}]" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = solve(df==0, r, solution_dict=True)\n", "s" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sin\\left({\\theta}\\right)^{2} - 2\\right)} a^{2}}{6 \\, {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}}} + {\\left(-\\frac{1}{4} \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{1}{12} \\, \\sqrt{-\\frac{2}{3} \\, a^{2} \\sin\\left({\\theta}\\right)^{6} + {\\left(4 \\, a^{2} + 9\\right)} \\sin\\left({\\theta}\\right)^{4} - 8 \\, a^{2} \\sin\\left({\\theta}\\right)^{2} + \\frac{16}{3} \\, a^{2}} a^{2}\\right)}^{\\frac{1}{3}}$ " ], "text/plain": [ "1/6*(sin(th)^2 - 2)*a^2/(-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3) + (-1/4*a^2*sin(th)^2 + 1/12*sqrt(-2/3*a^2*sin(th)^6 + (4*a^2 + 9)*sin(th)^4 - 8*a^2*sin(th)^2 + 16/3*a^2)*a^2)^(1/3)" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rmin = s[2][r]\n", "rmin" ] }, { "cell_type": "code", "execution_count": 97, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/plain": [ "0" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.subs(r=rmin).simplify_full()" ] }, { "cell_type": "code", "execution_count": 105, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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qQtIVhylTppCVlRU6hgI699yoLJx7LmzcCPfcY3mQJEmqbElXHCSIJg/Vq0cb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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(rmin.subs(th=pi/2), (a,0.01, 0.9))" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{8} + 4 \\, {\\left(\\frac{1}{12} \\, \\sqrt{\\frac{2}{3} \\, a^{2} + 9} a^{2} - \\frac{1}{4} \\, a^{2}\\right)}^{\\frac{2}{3}} a^{6} - 18 \\, \\sqrt{\\frac{2}{3} \\, a^{2} + 9} a^{6} + 54 \\, a^{6} - 72 \\, {\\left(\\frac{1}{12} \\, \\sqrt{\\frac{2}{3} \\, a^{2} + 9} a^{2} - \\frac{1}{4} \\, a^{2}\\right)}^{\\frac{4}{3}} a^{4} + 864 \\, {\\left(\\frac{1}{12} \\, \\sqrt{\\frac{2}{3} \\, a^{2} + 9} a^{2} - \\frac{1}{4} \\, a^{2}\\right)}^{\\frac{5}{3}} a^{2} + 432 \\, {\\left(\\frac{1}{12} \\, \\sqrt{\\frac{2}{3} \\, a^{2} + 9} a^{2} - \\frac{1}{4} \\, a^{2}\\right)}^{\\frac{8}{3}}}{432 \\, {\\left(\\frac{1}{12} \\, \\sqrt{\\frac{2}{3} \\, a^{2} + 9} a^{2} - \\frac{1}{4} \\, a^{2}\\right)}^{\\frac{4}{3}}}$ " ], "text/plain": [ "1/432*(a^8 + 4*(1/12*sqrt(2/3*a^2 + 9)*a^2 - 1/4*a^2)^(2/3)*a^6 - 18*sqrt(2/3*a^2 + 9)*a^6 + 54*a^6 - 72*(1/12*sqrt(2/3*a^2 + 9)*a^2 - 1/4*a^2)^(4/3)*a^4 + 864*(1/12*sqrt(2/3*a^2 + 9)*a^2 - 1/4*a^2)^(5/3)*a^2 + 432*(1/12*sqrt(2/3*a^2 + 9)*a^2 - 1/4*a^2)^(8/3))/(1/12*sqrt(2/3*a^2 + 9)*a^2 - 1/4*a^2)^(4/3)" ] }, "execution_count": 107, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fmin = f(rmin.subs(th=pi/2), a, pi/2).simplify_full()\n", "fmin" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.1", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", 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