{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Geodesics in Kerr spacetime\n", "## Computation with `kerrgeodesic_gw`\n", "\n", "This Jupyter notebook requires [SageMath](http://www.sagemath.org/) (version $\\geq$ 8.2), with the package [kerrgeodesic_gw](https://github.com/BlackHolePerturbationToolkit/kerrgeodesic_gw) (version $\\geq$ 0.3). To install the latter, simply run \n", "```\n", "sage -pip install kerrgeodesic_gw\n", "```\n", "in a terminal." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.4, Release Date: 2021-08-22'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, we set up the notebook to use LaTeX-formatted display:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we ask for CPU demanding computations to be performed in parallel on 8 processes:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A Kerr black bole is entirely defined by two parameters $(m, a)$, where $m$ is the black hole mass and $a$ is the black hole angular momentum divided by $m$.\n", "In this notebook, we shall set $m=1$ and we denote the angular momentum parameter $a$ by the symbolic variable `a`, using `a0` for a specific numerical value:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "a = var('a')\n", "a0 = 0.998" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The spacetime object is created as an instance of the class `KerrBH`:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Kerr spacetime M\n" ] } ], "source": [ "from kerrgeodesic_gw import KerrBH\n", "M = KerrBH(a)\n", "print(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The object `M` is endowed with many methods, which can be discovered via the TAB key:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# M." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One of them returns the Boyer-Lindquist coordinate $r$ of the event horizon:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{-a^{2} + 1} + 1\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{-a^{2} + 1} + 1$$" ], "text/plain": [ "sqrt(-a^2 + 1) + 1" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rH = M.event_horizon_radius()\n", "rH" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.06321392251712\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.06321392251712$$" ], "text/plain": [ "1.06321392251712" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rH0 = rH.subs({a: a0})\n", "rH0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another one returns the chart of Boyer-Lindquist coordinates and allows the user to instanciate the Python variables `(t, r, th, ph)` to the coordinates $(t,r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)$$" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL. = M.boyer_lindquist_coordinates()\n", "BL" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The metric tensor is naturally returned by the method `metric()`:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} - 2 \\, r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, a^{2} r \\sin\\left({\\theta}\\right)^{4} + {\\left(a^{2} r^{2} + r^{4} + {\\left(a^{4} + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} - 2 \\, r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, a^{2} r \\sin\\left({\\theta}\\right)^{4} + {\\left(a^{2} r^{2} + r^{4} + {\\left(a^{4} + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$$" ], "text/plain": [ "g = -(a^2*cos(th)^2 + r^2 - 2*r)/(a^2*cos(th)^2 + r^2) dt⊗dt - 2*a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt⊗dph + (a^2*cos(th)^2 + r^2)/(a^2 + r^2 - 2*r) dr⊗dr + (a^2*cos(th)^2 + r^2) dth⊗dth - 2*a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph⊗dt + (2*a^2*r*sin(th)^4 + (a^2*r^2 + r^4 + (a^4 + a^2*r^2)*cos(th)^2)*sin(th)^2)/(a^2*cos(th)^2 + r^2) dph⊗dph" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.metric()\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Again many methods are available on metric objects and one can discover them via the TAB key:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "# g." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For instance `christoffel_symbols_display()` computes the Christofell symbols with respect to the default chart (Boyer-Lindquist) and displays them:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{a^{4} - r^{4} - {\\left(a^{4} + a^{2} r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a^{2} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, r \\, {\\phi} }^{ \\, t \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} - {\\left(a^{5} - a^{3} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, {\\theta} \\, {\\phi} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{5} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{5} - {\\left(a^{5} r + a^{3} r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{3}\\right)}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & \\frac{a^{2} r^{2} + r^{4} - 2 \\, r^{3} - {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, {\\phi} }^{ \\, r \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{3} r^{2} + a r^{4} - 2 \\, a r^{3} - {\\left(a^{5} + a^{3} r^{2} - 2 \\, a^{3} r\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{{\\left(a^{2} r - a^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} + a^{2} - r^{2}}{a^{2} r^{2} + r^{4} - 2 \\, r^{3} + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} r + r^{3} - 2 \\, r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3} - {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} r^{5} + r^{7} - 2 \\, r^{6} + {\\left(a^{6} r + a^{4} r^{3} - 2 \\, a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{3} + a^{2} r^{5} - 2 \\, a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, t \\, t }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, a^{2} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, t \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, {\\left(a^{3} r + a r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, r }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} r^{2} + r^{4} - 2 \\, r^{3} + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left({\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{4} r + 4 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, r }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{a^{3} \\cos\\left({\\theta}\\right)^{2} - a r^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a r \\cos\\left({\\theta}\\right)}{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}\\right)} \\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{r^{5} + {\\left(a^{4} r - a^{4}\\right)} \\cos\\left({\\theta}\\right)^{4} - a^{2} r^{2} - 2 \\, r^{4} + {\\left(2 \\, a^{2} r^{3} + a^{4} - a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{a^{4} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{2} r^{2} - a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(r^{4} + 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)}{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}\\right)} \\sin\\left({\\theta}\\right)} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{a^{4} - r^{4} - {\\left(a^{4} + a^{2} r^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a^{2} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, r \\, {\\phi} }^{ \\, t \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} - {\\left(a^{5} - a^{3} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, {\\theta} \\, {\\phi} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{5} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{5} - {\\left(a^{5} r + a^{3} r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)^{3}\\right)}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & \\frac{a^{2} r^{2} + r^{4} - 2 \\, r^{3} - {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, {\\phi} }^{ \\, r \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left(a^{3} r^{2} + a r^{4} - 2 \\, a r^{3} - {\\left(a^{5} + a^{3} r^{2} - 2 \\, a^{3} r\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{{\\left(a^{2} r - a^{2}\\right)} \\sin\\left({\\theta}\\right)^{2} + a^{2} - r^{2}}{a^{2} r^{2} + r^{4} - 2 \\, r^{3} + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} r + r^{3} - 2 \\, r^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3} - {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{4} - {\\left(a^{2} r^{5} + r^{7} - 2 \\, r^{6} + {\\left(a^{6} r + a^{4} r^{3} - 2 \\, a^{4} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{3} + a^{2} r^{5} - 2 \\, a^{2} r^{4}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, t \\, t }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, a^{2} r \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, t \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, {\\left(a^{3} r + a r^{3}\\right)} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, r }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} r^{2} + r^{4} - 2 \\, r^{3} + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -\\frac{a^{2} \\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\frac{{\\left({\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{4} r + 4 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{6} \\cos\\left({\\theta}\\right)^{6} + 3 \\, a^{4} r^{2} \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{4} \\cos\\left({\\theta}\\right)^{2} + r^{6}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, r }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{a^{3} \\cos\\left({\\theta}\\right)^{2} - a r^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, t \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{2 \\, a r \\cos\\left({\\theta}\\right)}{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}\\right)} \\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{r^{5} + {\\left(a^{4} r - a^{4}\\right)} \\cos\\left({\\theta}\\right)^{4} - a^{2} r^{2} - 2 \\, r^{4} + {\\left(2 \\, a^{2} r^{3} + a^{4} - a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \\, r^{5} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} \\cos\\left({\\theta}\\right)^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} - 2 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{a^{4} \\cos\\left({\\theta}\\right)^{5} + 2 \\, {\\left(a^{2} r^{2} - a^{2} r\\right)} \\cos\\left({\\theta}\\right)^{3} + {\\left(r^{4} + 2 \\, a^{2} r\\right)} \\cos\\left({\\theta}\\right)}{{\\left(a^{4} \\cos\\left({\\theta}\\right)^{4} + 2 \\, a^{2} r^{2} \\cos\\left({\\theta}\\right)^{2} + r^{4}\\right)} \\sin\\left({\\theta}\\right)} \\end{array}$$" ], "text/plain": [ "Gam^t_t,r = -(a^4 - r^4 - (a^4 + a^2*r^2)*sin(th)^2)/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^t_t,th = -2*a^2*r*cos(th)*sin(th)/(a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4) \n", "Gam^t_r,ph = -(a^3*r^2 + 3*a*r^4 - (a^5 - a^3*r^2)*cos(th)^2)*sin(th)^2/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^t_th,ph = -2*(a^5*r*cos(th)*sin(th)^5 - (a^5*r + a^3*r^3)*cos(th)*sin(th)^3)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^r_t,t = (a^2*r^2 + r^4 - 2*r^3 - (a^4 + a^2*r^2 - 2*a^2*r)*cos(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^r_t,ph = -(a^3*r^2 + a*r^4 - 2*a*r^3 - (a^5 + a^3*r^2 - 2*a^3*r)*cos(th)^2)*sin(th)^2/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^r_r,r = ((a^2*r - a^2)*sin(th)^2 + a^2 - r^2)/(a^2*r^2 + r^4 - 2*r^3 + (a^4 + a^2*r^2 - 2*a^2*r)*cos(th)^2) \n", "Gam^r_r,th = -a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) \n", "Gam^r_th,th = -(a^2*r + r^3 - 2*r^2)/(a^2*cos(th)^2 + r^2) \n", "Gam^r_ph,ph = ((a^4*r^2 + a^2*r^4 - 2*a^2*r^3 - (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^2)*sin(th)^4 - (a^2*r^5 + r^7 - 2*r^6 + (a^6*r + a^4*r^3 - 2*a^4*r^2)*cos(th)^4 + 2*(a^4*r^3 + a^2*r^5 - 2*a^2*r^4)*cos(th)^2)*sin(th)^2)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^th_t,t = -2*a^2*r*cos(th)*sin(th)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^th_t,ph = 2*(a^3*r + a*r^3)*cos(th)*sin(th)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^th_r,r = a^2*cos(th)*sin(th)/(a^2*r^2 + r^4 - 2*r^3 + (a^4 + a^2*r^2 - 2*a^2*r)*cos(th)^2) \n", "Gam^th_r,th = r/(a^2*cos(th)^2 + r^2) \n", "Gam^th_th,th = -a^2*cos(th)*sin(th)/(a^2*cos(th)^2 + r^2) \n", "Gam^th_ph,ph = -((a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^5 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^3 + (a^2*r^4 + r^6 + 2*a^4*r + 4*a^2*r^3)*cos(th))*sin(th)/(a^6*cos(th)^6 + 3*a^4*r^2*cos(th)^4 + 3*a^2*r^4*cos(th)^2 + r^6) \n", "Gam^ph_t,r = -(a^3*cos(th)^2 - a*r^2)/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^ph_t,th = -2*a*r*cos(th)/((a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)*sin(th)) \n", "Gam^ph_r,ph = (r^5 + (a^4*r - a^4)*cos(th)^4 - a^2*r^2 - 2*r^4 + (2*a^2*r^3 + a^4 - a^2*r^2)*cos(th)^2)/(a^2*r^4 + r^6 - 2*r^5 + (a^6 + a^4*r^2 - 2*a^4*r)*cos(th)^4 + 2*(a^4*r^2 + a^2*r^4 - 2*a^2*r^3)*cos(th)^2) \n", "Gam^ph_th,ph = (a^4*cos(th)^5 + 2*(a^2*r^2 - a^2*r)*cos(th)^3 + (r^4 + 2*a^2*r)*cos(th))/((a^4*cos(th)^4 + 2*a^2*r^2*cos(th)^2 + r^4)*sin(th)) " ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Riemann curvature tensor is naturally returned by the method `riemann()`" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the Kerr spacetime M\n" ] } ], "source": [ "Riem = g.riemann()\n", "print(Riem)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The $R^t_{rtr}$ and $R^t_{r\\theta\\phi}$ components:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3 \\, a^{4} r \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{3} + 2 \\, r^{5} - {\\left(9 \\, a^{4} r + 7 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{6} + r^{8} - 2 \\, r^{7} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{6} r^{2} + a^{4} r^{4} - 2 \\, a^{4} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{4} r^{4} + a^{2} r^{6} - 2 \\, a^{2} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{2}}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{3 \\, a^{4} r \\cos\\left({\\theta}\\right)^{4} + 3 \\, a^{2} r^{3} + 2 \\, r^{5} - {\\left(9 \\, a^{4} r + 7 \\, a^{2} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{2}}{a^{2} r^{6} + r^{8} - 2 \\, r^{7} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{6} r^{2} + a^{4} r^{4} - 2 \\, a^{4} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{4} r^{4} + a^{2} r^{6} - 2 \\, a^{2} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{2}}$$" ], "text/plain": [ "(3*a^4*r*cos(th)^4 + 3*a^2*r^3 + 2*r^5 - (9*a^4*r + 7*a^2*r^3)*cos(th)^2)/(a^2*r^6 + r^8 - 2*r^7 + (a^8 + a^6*r^2 - 2*a^6*r)*cos(th)^6 + 3*(a^6*r^2 + a^4*r^4 - 2*a^4*r^3)*cos(th)^4 + 3*(a^4*r^4 + a^2*r^6 - 2*a^2*r^5)*cos(th)^2)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem[0,1,0,1]" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left(a^{7} + a^{5} r^{2} - 2 \\, a^{5} r\\right)} \\cos\\left({\\theta}\\right)^{5} - {\\left(3 \\, a^{7} + 8 \\, a^{5} r^{2} + 5 \\, a^{3} r^{4} - 2 \\, a^{5} r - 6 \\, a^{3} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{3} + 3 \\, {\\left(3 \\, a^{5} r^{2} + 5 \\, a^{3} r^{4} + 2 \\, a r^{6} - 2 \\, a^{3} r^{3}\\right)} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{2} r^{6} + r^{8} - 2 \\, r^{7} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{6} r^{2} + a^{4} r^{4} - 2 \\, a^{4} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{4} r^{4} + a^{2} r^{6} - 2 \\, a^{2} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{2}}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left(a^{7} + a^{5} r^{2} - 2 \\, a^{5} r\\right)} \\cos\\left({\\theta}\\right)^{5} - {\\left(3 \\, a^{7} + 8 \\, a^{5} r^{2} + 5 \\, a^{3} r^{4} - 2 \\, a^{5} r - 6 \\, a^{3} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{3} + 3 \\, {\\left(3 \\, a^{5} r^{2} + 5 \\, a^{3} r^{4} + 2 \\, a r^{6} - 2 \\, a^{3} r^{3}\\right)} \\cos\\left({\\theta}\\right)\\right)} \\sin\\left({\\theta}\\right)}{a^{2} r^{6} + r^{8} - 2 \\, r^{7} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} \\cos\\left({\\theta}\\right)^{6} + 3 \\, {\\left(a^{6} r^{2} + a^{4} r^{4} - 2 \\, a^{4} r^{3}\\right)} \\cos\\left({\\theta}\\right)^{4} + 3 \\, {\\left(a^{4} r^{4} + a^{2} r^{6} - 2 \\, a^{2} r^{5}\\right)} \\cos\\left({\\theta}\\right)^{2}}$$" ], "text/plain": [ "-((a^7 + a^5*r^2 - 2*a^5*r)*cos(th)^5 - (3*a^7 + 8*a^5*r^2 + 5*a^3*r^4 - 2*a^5*r - 6*a^3*r^3)*cos(th)^3 + 3*(3*a^5*r^2 + 5*a^3*r^4 + 2*a*r^6 - 2*a^3*r^3)*cos(th))*sin(th)/(a^2*r^6 + r^8 - 2*r^7 + (a^8 + a^6*r^2 - 2*a^6*r)*cos(th)^6 + 3*(a^6*r^2 + a^4*r^4 - 2*a^4*r^3)*cos(th)^4 + 3*(a^4*r^4 + a^2*r^6 - 2*a^2*r^5)*cos(th)^2)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem[0,1,2,3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course, since the Kerr metric is a solution of the **vacuum Einstein equation**, the Ricci tensor identically vanishes:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0$$" ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.ricci().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Bound timelike geodesic" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us choose the initial point $P$ for the geodesic:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point P on the Kerr spacetime M\n" ] } ], "source": [ "P = M.point((0, 6, pi/2, 0), name='P')\n", "print(P)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A geodesic is constructed by providing the range $[\\lambda_{\\rm min},\\lambda_{\\rm max}]$ of the affine parameter $\\lambda$, the initial point and either \n", " - (i) the Boyer-Lindquist components $(p^t_0, p^r_0, p^\\theta_0, p^\\phi_0)$ of the initial 4-momentum vector\n", " $p_0 = \\left. \\frac{\\mathrm{d}x}{\\mathrm{d}\\lambda}\\right| _{\\lambda_{\\rm min}}$,\n", " - (ii) the four integral of motions $(\\mu, E, L, Q)$\n", " - or (iii) some of the components of $p_0$ along with with some integrals of motion. \n", "We shall also specify some numerical value for the Kerr spin parameter $a$.\n", "\n", "Examples of (i) and (iii) are provided below. Here, we choose $\\lambda\\in[0, 300\\, m]$, the option (ii) and \n", "$a=0.998 \\,m$, where $m$ in the black hole mass::\n" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Initial tangent vector: \n" ] }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 1.29225788954106 \\frac{\\partial}{\\partial t } + 0.00438084990626460 \\frac{\\partial}{\\partial r } + 0.0189826106258554 \\frac{\\partial}{\\partial {\\theta} } + 0.0646134478134985 \\frac{\\partial}{\\partial {\\phi} }\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 1.29225788954106 \\frac{\\partial}{\\partial t } + 0.00438084990626460 \\frac{\\partial}{\\partial r } + 0.0189826106258554 \\frac{\\partial}{\\partial {\\theta} } + 0.0646134478134985 \\frac{\\partial}{\\partial {\\phi} }$$" ], "text/plain": [ "p = 1.29225788954106 ∂/∂t + 0.00438084990626460 ∂/∂r + 0.0189826106258554 ∂/∂th + 0.0646134478134985 ∂/∂ph" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The curve was correctly set.\n", "Parameters appearing in the differential system defining the curve are [a].\n" ] } ], "source": [ "Li = M.geodesic([0, 300], P, mu=1, E=0.883, L=1.982, Q=0.467, a_num=0.998,\n", " name='Li', latex_name=r'\\mathcal{L}', verbose=True)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Geodesic Li of the Kerr spacetime M\n" ] } ], "source": [ "print(Li)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The numerical integration of the geodesic equation is performed via `integrate()`, by providing the integration step $\\delta\\lambda$ in units of $m$:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "Li.integrate(step=0.005)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can then plot the geodesic:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.plot()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Actually, many options can be passed to the method `plot()`. For instance to a get a 3D spacetime diagram:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.plot(coordinates='txy')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "or to get the trace of the geodesic in the $(x,y)$ plane:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.plot(coordinates='xy', plot_points=2000)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "or else to get the trace in the $(x,z)$ plane:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.plot(coordinates='xz')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a curve, the geodesic $\\mathcal{L}$ is a map from an interval of $\\mathbb{R}$ to the spacetime $M$:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathcal{L}:& \\left(0, 300\\right) & \\longrightarrow & M \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathcal{L}:& \\left(0, 300\\right) & \\longrightarrow & M \\end{array}$$" ], "text/plain": [ "Li: (0, 300) → M" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.display()" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0, 300\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0, 300\\right)$$" ], "text/plain": [ "Real interval (0, 300)" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.domain()" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}M\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}M$$" ], "text/plain": [ "Kerr spacetime M" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.codomain()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It maps values of $\\lambda$ to spacetime points:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mbox{Point on the Kerr spacetime M}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mbox{Point on the Kerr spacetime M}$$" ], "text/plain": [ "Point on the Kerr spacetime M" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li(0)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0.0, 6.0, 1.5707963267948966, 0.0\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0.0, 6.0, 1.5707963267948966, 0.0\\right)$$" ], "text/plain": [ "(0.0, 6.0, 1.5707963267948966, 0.0)" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li(0).coordinates() # coordinates in the default chart (BL)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0.0, 6.0, 1.5707963267948966, 0.0\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0.0, 6.0, 1.5707963267948966, 0.0\\right)$$" ], "text/plain": [ "(0.0, 6.0, 1.5707963267948966, 0.0)" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "BL(Li(0)) # equivalent to above" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(553.4637326813786, 3.703552505462962, 1.6613834863942039, 84.62814710987239\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(553.4637326813786, 3.703552505462962, 1.6613834863942039, 84.62814710987239\\right)$$" ], "text/plain": [ "(553.4637326813786, 3.703552505462962, 1.6613834863942039, 84.62814710987239)" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li(300).coordinates()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The initial 4-momentum vector $p_0$ is returned by the method `initial_tangent_vector()`:" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector p at Point P on the Kerr spacetime M\n" ] } ], "source": [ "p0 = Li.initial_tangent_vector()\n", "print(p0)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0 in M.tangent_space(P)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 1.29225788954106 \\frac{\\partial}{\\partial t } + 0.00438084990626460 \\frac{\\partial}{\\partial r } + 0.0189826106258554 \\frac{\\partial}{\\partial {\\theta} } + 0.0646134478134985 \\frac{\\partial}{\\partial {\\phi} }\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 1.29225788954106 \\frac{\\partial}{\\partial t } + 0.00438084990626460 \\frac{\\partial}{\\partial r } + 0.0189826106258554 \\frac{\\partial}{\\partial {\\theta} } + 0.0646134478134985 \\frac{\\partial}{\\partial {\\phi} }$$" ], "text/plain": [ "p = 1.29225788954106 ∂/∂t + 0.00438084990626460 ∂/∂r + 0.0189826106258554 ∂/∂th + 0.0646134478134985 ∂/∂ph" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0.display()" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]$$" ], "text/plain": [ "[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985]" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For instance, the components $p^t_0$ and $p^\\phi_0$ are recovered by" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1.29225788954106, 0.0646134478134985\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1.29225788954106, 0.0646134478134985\\right)$$" ], "text/plain": [ "(1.29225788954106, 0.0646134478134985)" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0[0], p0[3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that the scalar square of $p_0$ is $-1$, i.e. is consistent with the mass parameter $\\mu = 1$ used in the construction of the geodesic:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1.00000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1.00000000000000$$" ], "text/plain": [ "-1.00000000000000" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.metric()\n", "g.at(P)(p0, p0).subs(a=0.998)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The 4-momentum vector $p$ at any value of the affine parameter $\\lambda$ is obtained by\n", "the method `evaluate_tangent_vector()`; for instance, for $\\lambda=200\\,m$: " ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.316592599498746 \\frac{\\partial}{\\partial t } -0.07370434215844164 \\frac{\\partial}{\\partial r } -0.01091195426423706 \\frac{\\partial}{\\partial {\\theta} } + 0.07600209768075264 \\frac{\\partial}{\\partial {\\phi} }\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.316592599498746 \\frac{\\partial}{\\partial t } -0.07370434215844164 \\frac{\\partial}{\\partial r } -0.01091195426423706 \\frac{\\partial}{\\partial {\\theta} } + 0.07600209768075264 \\frac{\\partial}{\\partial {\\phi} }$$" ], "text/plain": [ "1.316592599498746 ∂/∂t - 0.07370434215844164 ∂/∂r - 0.01091195426423706 ∂/∂th + 0.07600209768075264 ∂/∂ph" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = Li.evaluate_tangent_vector(200)\n", "p.display()" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p in M.tangent_space(Li(200))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The particle mass $\\mu$ computed at a given value of $\\lambda$ is returned by the method `evaluate_mu()`:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.00000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.00000000000000$$" ], "text/plain": [ "1.00000000000000" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.evaluate_mu(0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course, it should be conserved along $\\mathcal{L}$; actually it is, up to the numerical accuracy::" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.0000117978600134\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.0000117978600134$$" ], "text/plain": [ "1.0000117978600134" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.evaluate_mu(300)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, the conserved energy $E$, conserved angular momentum $L$ and Carter constant $Q$ are computed at any value of $\\lambda$ by respectively `evaluate_E()`, `evaluate_L()` and `evaluate_Q()`:" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.883000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.883000000000000$$" ], "text/plain": [ "0.883000000000000" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.evaluate_E(0)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.98200000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.98200000000000$$" ], "text/plain": [ "1.98200000000000" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.evaluate_L(0)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.467000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.467000000000000$$" ], "text/plain": [ "0.467000000000000" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.evaluate_Q(0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that the values of $\\mu$, $E$, $L$ and $Q$ evaluated at $\\lambda=300 \\, m$ are equal to those at $\\lambda=0$ up to the numerical accuracy of the integration scheme:" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "

\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "

quantity	value	initial value	diff.	relative diff.
\$\\mu^2\$	\$1.0000235958592163\$	\$1.00000000000000\$	\$0.00002360\$	\$0.00002360\$
\$E\$	\$0.883067996080701\$	\$0.883000000000000\$	\$0.00006800\$	\$0.00007701\$
\$L\$	\$1.98248080818931\$	\$1.98200000000000\$	\$0.0004808\$	\$0.0002426\$
\$Q\$	\$0.467214137649741\$	\$0.467000000000000\$	\$0.0002141\$	\$0.0004585\$

\n", "

" ], "text/plain": [ " quantity value initial value diff. relative diff.\n", " $\\mu^2$ 1.0000235958592163 1.00000000000000 0.00002360 0.00002360\n", " $E$ 0.883067996080701 0.883000000000000 0.00006800 0.00007701\n", " $L$ 1.98248080818931 1.98200000000000 0.0004808 0.0002426\n", " $Q$ 0.467214137649741 0.467000000000000 0.0002141 0.0004585" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.check_integrals_of_motion(300)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Decreasing the integration step leads to smaller errors:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "

quantity	value	initial value	diff.	relative diff.
\$\\mu^2\$	\$1.0000047183936422\$	\$1.00000000000000\$	\$4.718 \\times 10^{-6}\$	\$4.718 \\times 10^{-6}\$
\$E\$	\$0.883013604456676\$	\$0.883000000000000\$	\$0.00001360\$	\$0.00001541\$
\$L\$	\$1.98209626120918\$	\$1.98200000000000\$	\$0.00009626\$	\$0.00004857\$
\$Q\$	\$0.467042771975860\$	\$0.467000000000000\$	\$0.00004277\$	\$0.00009159\$

\n", "

" ], "text/plain": [ " quantity value initial value diff. relative diff.\n", " $\\mu^2$ 1.0000047183936422 1.00000000000000 4.718e-6 4.718e-6\n", " $E$ 0.883013604456676 0.883000000000000 0.00001360 0.00001541\n", " $L$ 1.98209626120918 1.98200000000000 0.00009626 0.00004857\n", " $Q$ 0.467042771975860 0.467000000000000 0.00004277 0.00009159" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.integrate(step=0.001)\n", "Li.check_integrals_of_motion(300)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Various ways to initialize a geodesic" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of providing the integral of motions, as for ``Li`` above, one can initialize a geodesic by providing the Boyer-Lindquist components $(p^t_0, p^r_0, p^\\theta_0, p^\\phi_0)$ of the initial 4-momentum vector $p_0$. For instance:" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector p at Point P on the Kerr spacetime M\n" ] } ], "source": [ "print(p0)" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]$$" ], "text/plain": [ "[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985]" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0[:]" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li2 = M.geodesic([0, 300], P, pt0=p0[0], pr0=p0[1], pth0=p0[2], pph0=p0[3], \n", " a_num=0.998)\n", "Li2.initial_tangent_vector() == p0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a check, we recover the same values of $(\\mu, E, L, Q)$ as those that were used to initialize ``Li``:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.00000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.00000000000000$$" ], "text/plain": [ "1.00000000000000" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li2.evaluate_mu(0)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.883000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.883000000000000$$" ], "text/plain": [ "0.883000000000000" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li2.evaluate_E(0)" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.98200000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.98200000000000$$" ], "text/plain": [ "1.98200000000000" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li2.evaluate_L(0)" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.467000000000000\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.467000000000000$$" ], "text/plain": [ "0.467000000000000" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li2.evaluate_Q(0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may also initialize a geodesic by providing the mass $\\mu$ and the three spatial components $(p^r_0, p^\\theta_0, p^\\phi_0)$ of the initial 4-momentum vector:" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [], "source": [ "Li3 = M.geodesic([0, 300], P, mu=1, pr0=p0[1], pth0=p0[2], pph0=p0[3], \n", " a_num=0.998)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The component $p^t_0$ is then automatically computed:" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]$$" ], "text/plain": [ "[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985]" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li3.initial_tangent_vector()[:] " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we check the identity with the initial vector of ``Li``, up to numerical errors:" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2.22044604925031 \\times 10^{-16}, 0, 0, 0\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2.22044604925031 \\times 10^{-16}, 0, 0, 0\\right]$$" ], "text/plain": [ "[2.22044604925031e-16, 0, 0, 0]" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(Li3.initial_tangent_vector() - p0)[:] " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another way to initialize a geodesic is to provide the conserved energy $E$, the conserved angular momentum $L$ and the two components $(p^r_0, p^\\theta_0)$ of the initial 4-momentum vector:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985\\right]$$" ], "text/plain": [ "[1.29225788954106, 0.00438084990626460, 0.0189826106258554, 0.0646134478134985]" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li4 = M.geodesic([0, 300], P, E=0.8830, L=1.982, pr0=p0[1], pth0=p0[2], \n", " a_num=0.998)\n", "Li4.initial_tangent_vector()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Again, we get a geodesic equivalent to ``Li``:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0, 0, 0, 0\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0, 0, 0, 0\\right]$$" ], "text/plain": [ "[0, 0, 0, 0]" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(Li4.initial_tangent_vector() - p0)[:] " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Ingoing null geodesic with negative angular momentum" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We choose a ingoing null geodesic in the equatorial plane with $L = -6 E < 0$, starting at the point of Boyer-Lindquist coordinates $(t,r,\\theta,\\phi) = (0, 12, \\pi/2, 0)$:" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Initial tangent vector: \n" ] }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 1.20797381595070 \\frac{\\partial}{\\partial t } -0.901996260873419 \\frac{\\partial}{\\partial r } -0.0399489777089388 \\frac{\\partial}{\\partial {\\phi} }\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 1.20797381595070 \\frac{\\partial}{\\partial t } -0.901996260873419 \\frac{\\partial}{\\partial r } -0.0399489777089388 \\frac{\\partial}{\\partial {\\phi} }$$" ], "text/plain": [ "p = 1.20797381595070 ∂/∂t - 0.901996260873419 ∂/∂r - 0.0399489777089388 ∂/∂ph" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The curve was correctly set.\n", "Parameters appearing in the differential system defining the curve are [a].\n" ] } ], "source": [ "lambda_max = 13.063\n", "Li = M.geodesic([0, lambda_max], M((0,12,pi/2,0)), mu=0, E=1, L=-6, Q=0,\n", " r_increase=False, a_num=a0, name='Li', latex_name=r'\\mathcal{L}', \n", " verbose=True)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [], "source": [ "Li.integrate(step=0.00002)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We plot the trajectory of the geodesic in the equatorial plane with the tangent vector at 6 points:" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 8 graphics primitives" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.plot(coordinates='xy', color='green', prange=[0, lambda_max], plot_points=40000, \n", " display_tangent=True, scale=0.002, color_tangent='brown',\n", " plot_points_tangent=6, axes_labels=[r'$x/m$', r'$y/m$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note the turning point in $\\phi$ and the final winding in the direction of the black\n", "rotation (counterclockwise in the figure)." ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "

quantity	value	initial value	diff.	relative diff.
\$\\mu^2\$	\$0.000211815000511706\$	\$0.000000000000000\$	\$0.0002118\$	-
\$E\$	\$1.00000186057173\$	\$1.00000000000000\$	\$1.861 \\times 10^{-6}\$	\$1.861 \\times 10^{-6}\$
\$L\$	\$-5.99999596865291\$	\$-6.00000000000000\$	\$4.031 \\times 10^{-6}\$	\$-6.719 \\times 10^{-7}\$
\$Q\$	\$1.31244559310830 \\times 10^{-31}\$	\$0\$	\$1.312 \\times 10^{-31}\$	-

\n", "

" ], "text/plain": [ " quantity value initial value diff. relative diff.\n", " $\\mu^2$ 0.000211815000511706 0.000000000000000 0.0002118 -\n", " $E$ 1.00000186057173 1.00000000000000 1.861e-6 1.861e-6\n", " $L$ -5.99999596865291 -6.00000000000000 4.031e-6 -6.719e-7\n", " $Q$ 1.31244559310830e-31 0 1.312e-31 -" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Li.check_integrals_of_motion(0.99999*lambda_max)" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.4", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 2 }

quantity	value	initial value	diff.	relative diff.
\\(\\mu^2\\)	\\(1.0000235958592163\\)	\\(1.00000000000000\\)	\\(0.00002360\\)	\\(0.00002360\\)
\\(E\\)	\\(0.883067996080701\\)	\\(0.883000000000000\\)	\\(0.00006800\\)	\\(0.00007701\\)
\\(L\\)	\\(1.98248080818931\\)	\\(1.98200000000000\\)	\\(0.0004808\\)	\\(0.0002426\\)
\\(Q\\)	\\(0.467214137649741\\)	\\(0.467000000000000\\)	\\(0.0002141\\)	\\(0.0004585\\)

quantity	value	initial value	diff.	relative diff.
\\(\\mu^2\\)	\\(1.0000047183936422\\)	\\(1.00000000000000\\)	\\(4.718 \\times 10^{-6}\\)	\\(4.718 \\times 10^{-6}\\)
\\(E\\)	\\(0.883013604456676\\)	\\(0.883000000000000\\)	\\(0.00001360\\)	\\(0.00001541\\)
\\(L\\)	\\(1.98209626120918\\)	\\(1.98200000000000\\)	\\(0.00009626\\)	\\(0.00004857\\)
\\(Q\\)	\\(0.467042771975860\\)	\\(0.467000000000000\\)	\\(0.00004277\\)	\\(0.00009159\\)

quantity	value	initial value	diff.	relative diff.
\\(\\mu^2\\)	\\(0.000211815000511706\\)	\\(0.000000000000000\\)	\\(0.0002118\\)	-
\\(E\\)	\\(1.00000186057173\\)	\\(1.00000000000000\\)	\\(1.861 \\times 10^{-6}\\)	\\(1.861 \\times 10^{-6}\\)
\\(L\\)	\\(-5.99999596865291\\)	\\(-6.00000000000000\\)	\\(4.031 \\times 10^{-6}\\)	\\(-6.719 \\times 10^{-7}\\)
\\(Q\\)	\\(1.31244559310830 \\times 10^{-31}\\)	\\(0\\)	\\(1.312 \\times 10^{-31}\\)	-