{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# From Boyer-Lindquist to Kerr coordinates\n", "\n", "This worksheet provides the primitives of the functions $r\\mapsto (r^2+a^2)/\\Delta$ and $r\\mapsto 1/\\Delta$, which appear in the relations between Boyer-Lindquist coordinates and Kerr ones.\n", "\n", "We set $m=1$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(r, a\\right)$ " ], "text/plain": [ "(r, a)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('r a', domain='real')" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "assume(a<1)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "assume(a>=0)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r}$ " ], "text/plain": [ "(a^2 + r^2)/(a^2 + r^2 - 2*r)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = (r^2 + a^2)/(r^2 - 2*r + a^2)\n", "f" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}r + \\frac{\\log\\left(\\frac{r - \\sqrt{-a^{2} + 1} - 1}{r + \\sqrt{-a^{2} + 1} - 1}\\right)}{\\sqrt{-a^{2} + 1}} + \\log\\left(a^{2} + r^{2} - 2 \\, r\\right)$ " ], "text/plain": [ "r + log((r - sqrt(-a^2 + 1) - 1)/(r + sqrt(-a^2 + 1) - 1))/sqrt(-a^2 + 1) + log(a^2 + r^2 - 2*r)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = integrate(f, r)\n", "s" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r}$ " ], "text/plain": [ "(a^2 + r^2)/(a^2 + r^2 - 2*r)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(s, r).simplify_full()" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [], "source": [ "rp = 1 + sqrt(1-a^2)\n", "rm = 1 - sqrt(1-a^2)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}r + \\frac{{\\left(\\sqrt{-a^{2} + 1} - 1\\right)} \\log\\left({\\left| \\frac{1}{2} \\, r + \\frac{1}{2} \\, \\sqrt{-a^{2} + 1} - \\frac{1}{2} \\right|}\\right) + {\\left(\\sqrt{-a^{2} + 1} + 1\\right)} \\log\\left({\\left| \\frac{1}{2} \\, r - \\frac{1}{2} \\, \\sqrt{-a^{2} + 1} - \\frac{1}{2} \\right|}\\right)}{\\sqrt{-a^{2} + 1}}$ " ], "text/plain": [ "r + ((sqrt(-a^2 + 1) - 1)*log(abs(1/2*r + 1/2*sqrt(-a^2 + 1) - 1/2)) + (sqrt(-a^2 + 1) + 1)*log(abs(1/2*r - 1/2*sqrt(-a^2 + 1) - 1/2)))/sqrt(-a^2 + 1)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "F = r + 1/sqrt(1-a^2)*(rp*ln(abs((r-rp)/2)) - rm*ln(abs((r-rm)/2)))\n", "F" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r}$ " ], "text/plain": [ "(a^2 + r^2)/(a^2 + r^2 - 2*r)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dFdr = diff(F,r).simplify_full()\n", "dFdr" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bool(dFdr == f)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{a^{2} + r^{2} - 2 \\, r}$ " ], "text/plain": [ "1/(a^2 + r^2 - 2*r)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = 1/(r^2 - 2*r + a^2)\n", "g" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\log\\left(\\frac{r - \\sqrt{-a^{2} + 1} - 1}{r + \\sqrt{-a^{2} + 1} - 1}\\right)}{2 \\, \\sqrt{-a^{2} + 1}}$ " ], "text/plain": [ "1/2*log((r - sqrt(-a^2 + 1) - 1)/(r + sqrt(-a^2 + 1) - 1))/sqrt(-a^2 + 1)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(g,r)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\log\\left({\\left| \\frac{r - \\sqrt{-a^{2} + 1} - 1}{r + \\sqrt{-a^{2} + 1} - 1} \\right|}\\right)}{2 \\, \\sqrt{-a^{2} + 1}}$ " ], "text/plain": [ "1/2*log(abs((r - sqrt(-a^2 + 1) - 1)/(r + sqrt(-a^2 + 1) - 1)))/sqrt(-a^2 + 1)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "G = 1/(2*sqrt(1-a^2))*ln(abs((r-rp)/(r-rm)))\n", "G" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{a^{2} + r^{2} - 2 \\, r}$ " ], "text/plain": [ "1/(a^2 + r^2 - 2*r)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dGdr = diff(G,r).simplify_full()\n", "dGdr" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bool(dGdr == g)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.0.beta1", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 2 }