{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Pseudo-Riemannian manifolds in SageMath\n", "## The Schwarzschild spacetime example" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook demonstrates some SageMath tools for pseudo-Riemannian geometry, developed through the [SageManifolds project](https://sagemanifolds.obspm.fr/), by these [authors](https://sagemanifolds.obspm.fr/authors.html). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook requires a version of SageMath at least equal to 9.0:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.2, Release Date: 2020-10-24'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display outputs via LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since some computations are quite heavy, we ask for running them in parallel on 8 \n", "threads:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the **Schwarzschild spacetime**, which is the spacetime of a **static black hole** in general relativity, as a **4-dimensional Lorentzian manifold** $M$:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}M\n", "\\end{math}" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', structure='Lorentzian')\n", "M" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$M$ is in the category of smooth manifolds over the real field:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}\n", "\\end{math}" ], "text/plain": [ "Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.category()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Bold{R}\n", "\\end{math}" ], "text/plain": [ "Real Field with 53 bits of precision" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.base_field()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At the moment, the real field is modeled by 53-bit floating-point approximations, but this plays no role in the manifold implementation:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Real Field with 53 bits of precision\n" ] } ], "source": [ "print(M.base_field())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Coordinate charts" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The function `Manifold` generates a manifold with no-predefined coordinate chart, so that the manifold (user) atlas is empty:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]\n", "\\end{math}" ], "text/plain": [ "[]" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the standard **Schwarzchild-Droste coordinates** $(t,r,\\theta,\\phi)$ on $M$, via the method `chart`:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "SD. = M.chart(r\"t r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi:periodic\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the argument of `chart()` is a raw string (hence the prefix `r` in front of it), which defines the range of each coordinate, if different from $(-\\infty, +\\infty)$, as well as its LaTeX symbol, if different from the Python symbol to denote the coordinate. The Python variables for each coordinate are declared within the `<...>` operator on the left-hand side, `SD` denoting the Python variable chosen for the coordinate chart." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (M, (t, r, th, ph))\n" ] } ], "source": [ "print(SD)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}\n", "\\end{math}" ], "text/plain": [ "t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thanks to the SageMath operator `<...>` used in the chart declaration, the coordinates are immediately available:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\theta}\n", "\\end{math}" ], "text/plain": [ "th" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "th" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "They are SageMath's symbolic expressions:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\text{SR}\n", "\\end{math}" ], "text/plain": [ "Symbolic Ring" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "th.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "They are also accessible as items of the chart:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, {\\phi}\\right)\n", "\\end{math}" ], "text/plain": [ "(t, ph)" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD[0], SD[3]" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right)\n", "\\end{math}" ], "text/plain": [ "(t, r, th, ph)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The manifold (user) atlas is no longer empty: " ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (M, (t, r, th, ph))]" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce a second chart on the manifold, that of **Eddington-Finkelstein coordinates** $(T,r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(T, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (M, (T, r, th, ph))" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF. = M.chart(r\"T r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi:periodic\")\n", "EF" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The transition map from Schwarzschild-Droste coordinates (chart `SD`) to Eddington-Finkelstein ones (chart `EF`) depends on a parameter $m$, the mass of the Schwarzschild black hole:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "m = var('m')\n", "assume(m > 0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We provide the explicit coordinate transformation via the method `transition_map`:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} T & = & 2 \\, m \\log\\left({\\left| \\frac{r}{2 \\, m} - 1 \\right|}\\right) + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "T = 2*m*log(abs(1/2*r/m - 1)) + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD_to_EF = SD.transition_map(EF, [t +2*m*ln(abs(r/(2*m)-1)), r, th, ph])\n", "SD_to_EF.display()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & 2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left({\\left| -2 \\, m + r \\right|}\\right) + T \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "t = 2*m*log(2) + 2*m*log(m) - 2*m*log(abs(-2*m + r)) + T\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD_to_EF.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are now two charts in the manifold atlas:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right), \\left(M,(T, r, {\\theta}, {\\phi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (M, (t, r, th, ph)), Chart (M, (T, r, th, ph))]" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One of them is the so-called *default chart*: it is the chart used by any function that requires a chart as argument and none is provided by the user. At this stage, the default chart is the first chart defined on the manifold, but this can be changed by the manifold method `set_default_chart`. " ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One can plot the `SD` chart in terms of `EF` one:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot1 = SD.plot(EF, ranges={t:(0, 8), r:(2.1, 10)}, fixed_coords={th:pi/2, ph:0}, \n", " ambient_coords=(r,T), style={t:'--', r:'-'}, parameters={m: 1}) \\\n", " + SD.plot(EF, ranges={t:(0, 8), r:(0.1, 1.9)}, fixed_coords={th:pi/2, ph:0}, \n", " ambient_coords=(r,T), number_values={t: 9, r: 3},\n", " style={t:'--', r:'-'}, parameters={m: 1})\n", "plot1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Manifold points" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To create a point on $M$, we use SageMath's *parent/element* syntax, i.e. the call operator `M(...)` acting on the parent `M`, with the point's coordinates in some chart as argument: " ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "p = M((m, 8*m, pi/2, 0), name='p')\n", "print(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since the chart has not been specified, the default chart (i.e. `SD`) is meant:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, 8 \\, m, \\frac{1}{2} \\, \\pi, 0\\right)\n", "\\end{math}" ], "text/plain": [ "(m, 8*m, 1/2*pi, 0)" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thanks to the transition map declared above, the coordinates of $p$ in the Eddington-Finkelstein chart can computed:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m {\\left(2 \\, \\log\\left(3\\right) + 1\\right)}, 8 \\, m, \\frac{1}{2} \\, \\pi, 0\\right)\n", "\\end{math}" ], "text/plain": [ "(m*(2*log(3) + 1), 8*m, 1/2*pi, 0)" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Manifold points have a `plot` method:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 32 graphics primitives" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot1 += p.plot(EF, color='blue', ambient_coords=(r,T), \n", " parameters={m: 1}, label_offset=0.4, fontsize=14)\n", "plot1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vector fields" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "When a chart is declared, the manifold is automatically endowed with some vector fields, those of the **coordinate vector frame**:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\n", "\\end{math}" ], "text/plain": [ "Coordinate frame (M, (d/dt,d/dr,d/dth,d/dph))" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.frame()" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M, \\left(\\frac{\\partial}{\\partial T },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\n", "\\end{math}" ], "text/plain": [ "Coordinate frame (M, (d/dT,d/dr,d/dth,d/dph))" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF.frame()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(M, \\left(\\frac{\\partial}{\\partial T },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Coordinate frame (M, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (M, (d/dT,d/dr,d/dth,d/dph))]" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As for charts, there is a *default frame*, i.e. a vector frame that is used by default in functions having a vector frame in their arguments. The default frame can be changed by the method `set_default_frame`." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\n", "\\end{math}" ], "text/plain": [ "Coordinate frame (M, (d/dt,d/dr,d/dth,d/dph))" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first vector of the Schwarzschild-Droste coordinate frame:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial t }\n", "\\end{math}" ], "text/plain": [ "Vector field d/dt on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt = SD.frame()[0]\n", "vt" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial t } = \\frac{\\partial}{\\partial t }\n", "\\end{math}" ], "text/plain": [ "d/dt = d/dt" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt.display()" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial t } = \\frac{\\partial}{\\partial T }\n", "\\end{math}" ], "text/plain": [ "d/dt = d/dT" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt.display(EF.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The second vector of the Schwarzschild-Droste coordinate frame:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "Vector field d/dr on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vr = SD.frame()[1]\n", "vr" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial r } = \\left( -\\frac{2 \\, m}{2 \\, m - r} \\right) \\frac{\\partial}{\\partial T } +\\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "d/dr = -2*m/(2*m - r) d/dT + d/dr" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vr.display(EF.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Creating a vector field from scratch, by providing its components with rest to a given vector frame:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\left( -\\frac{r}{2 \\, m - r} \\right) \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "k = -r/(2*m - r) d/dt - d/dr" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k = M.vector_field(1, -1, 0, 0, frame=EF.frame(), name='k')\n", "k.display()" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\frac{\\partial}{\\partial T }-\\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "k = d/dT - d/dr" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.display(EF.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot with respect to Schwarzschild-Droste coordinates (default chart):" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 81 graphics primitives" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.plot(ambient_coords=(r,t), fixed_coords={th: pi/2, ph: 0}, \n", " ranges={t: (0, 6), r: (0.1, 6)}, number_values=9,\n", " parameters={m: 1}, color='green', scale=0.3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot with respect to Eddington-Finkelstein coordinates:" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 25 graphics primitives" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.plot(chart=EF, ambient_coords=(r,T), chart_domain=EF,\n", " fixed_coords={th: pi/2, ph: 0}, ranges={T: (0, 8), r: (0.1, 10)}, \n", " number_values=5, parameters={m: 1}, color='green')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vector fields as sections of the tangent bundle" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent bundle TM over the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "TM = M.tangent_bundle()\n", "print(TM)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "k1 = TM.section({EF.frame(): [1, -1, 0, 0]})\n", "print(k1)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k1 == k" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The set of all vector fields on $M$ as a $C^\\infty(M)$-module:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Free module X(M) of vector fields on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathfrak{X}\\left(M\\right)\n", "\\end{math}" ], "text/plain": [ "Free module X(M) of vector fields on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XM = M.vector_field_module()\n", "print(XM)\n", "XM" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}C^{\\infty}\\left(M\\right)\n", "\\end{math}" ], "text/plain": [ "Algebra of differentiable scalar fields on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XM.base_ring()" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Algebra of differentiable scalar fields on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(XM.base_ring())" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}\n", "\\end{math}" ], "text/plain": [ "M --> R\n", "(t, r, th, ph) |--> 2\n", "(T, r, th, ph) |--> 2" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XM.base_ring().an_element().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vectors at a point" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The value of a vector field, and more generally of any tensor field, at a given point is obtained by the method `at`: " ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector k at Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "kp = k.at(p)\n", "print(kp)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}k = \\frac{4}{3} \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "k = 4/3 d/dt - d/dr" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The parent of `kp` is the tangent space at `p`:" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}T_{p}\\,M\n", "\\end{math}" ], "text/plain": [ "Tangent space at Point p on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.parent()" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent space at Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(kp.parent())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is accessible from the manifold via the method `tangent_space`:" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.parent() is M.tangent_space(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$T_p M$ is the fiber over $p$ in the tangent bundle $TM$:" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.parent() is TM.fiber(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Tangent vectors have a method `plot`:" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 34 graphics primitives" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot1 += kp.plot(EF, color='green', ambient_coords=(r,T), \n", " parameters={m: 1}, scale=2, label_offset=0.5, \n", " fontsize=16)\n", "plot1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vector field defined on an open subset" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce the exterior $E$ of the black hole as an open subset of $M$:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [], "source": [ "E = M.open_subset('E', coord_def = {SD: r>2*m})" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 2 \\, m , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}\n", "\\end{math}" ], "text/plain": [ "t: (-oo, +oo); r: (2*m, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.restrict(E).coord_range()" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p in E" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial t }\n", "\\end{math}" ], "text/plain": [ "u = 1/sqrt(-2*m/r + 1) d/dt" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = E.vector_field(name='u')\n", "u[0] = 1/sqrt(1-2*m/r)\n", "u.display()" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(E,(t, r, {\\theta}, {\\phi})\\right) \\rightarrow \\left(E,(T, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Change of coordinates from Chart (E, (t, r, th, ph)) to Chart (E, (T, r, th, ph))" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD_to_EF.restrict(E)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(E,(t, r, {\\theta}, {\\phi})\\right), \\left(E,(T, r, {\\theta}, {\\phi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (E, (t, r, th, ph)), Chart (E, (T, r, th, ph))]" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.atlas()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial T }\n", "\\end{math}" ], "text/plain": [ "u = 1/sqrt(-2*m/r + 1) d/dT" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.display(EF.frame().restrict(E))" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 25 graphics primitives" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.plot(ambient_coords=(r, t), fixed_coords={th: pi/2, ph: 0}, \n", " ranges={t: (0, 6), r: (2.1, 6)}, number_values=5, \n", " parameters={m: 1}, scale=0.3)" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = \\frac{2}{3} \\, \\sqrt{3} \\frac{\\partial}{\\partial t }\n", "\\end{math}" ], "text/plain": [ "u = 2/3*sqrt(3) d/dt" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.at(p).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Metric tensor\n", "\n", "We define next the **metric tensor** $g$ from its non-vanishing components in the manifold's default frame, namely the coordinate frame associated to Schwarzschild-Droste coordinate:" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [], "source": [ "g = M.metric()\n", "g[0, 0] = - (1 - 2*m/r)\n", "g[1, 1] = 1/(1 - 2*m/r)\n", "g[2, 2] = r^2\n", "g[3, 3] = r^2*sin(th)^2" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\n", "\\end{math}" ], "text/plain": [ "g = (2*m/r - 1) dt*dt - 1/(2*m/r - 1) dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right)\n", "\\end{math}" ], "text/plain": [ "Coordinate coframe (M, (dt,dr,dth,dph))" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.coframe()" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & 0 & 0 & 0 \\\\\n", "0 & -\\frac{1}{\\frac{2 \\, m}{r} - 1} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 2*m/r - 1 0 0 0]\n", "[ 0 -1/(2*m/r - 1) 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{\\frac{2 \\, m}{r} - 1}\n", "\\end{math}" ], "text/plain": [ "-1/(2*m/r - 1)" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[1,1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$g_{rr}$ is diverging at $r=2m$: this is a singularity of the Schwarszchild-Droste coordinates." ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, m}{r} - 1 \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & r^{2} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & r^{2} \\sin\\left({\\theta}\\right)^{2} \\end{array}\n", "\\end{math}" ], "text/plain": [ "g_t,t = 2*m/r - 1 \n", "g_r,r = -1/(2*m/r - 1) \n", "g_th,th = r^2 \n", "g_ph,ph = r^2*sin(th)^2 " ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The components of $g$ with respect to the Eddington-Finkelstein frame are evaluated via the methods `comp` or `display`:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, m}{r} \\mathrm{d} T\\otimes \\mathrm{d} r + \\frac{2 \\, m}{r} \\mathrm{d} r\\otimes \\mathrm{d} T + \\left( \\frac{2 \\, m + r}{r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\n", "\\end{math}" ], "text/plain": [ "g = (2*m/r - 1) dT*dT + 2*m/r dT*dr + 2*m/r dr*dT + (2*m + r)/r dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(EF.frame())" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & \\frac{2 \\, m}{r} & 0 & 0 \\\\\n", "\\frac{2 \\, m}{r} & \\frac{2 \\, m + r}{r} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[ 2*m/r - 1 2*m/r 0 0]\n", "[ 2*m/r (2*m + r)/r 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[EF.frame(),:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that these components are regular at $r=2m$, contrary to the components in Schwarzschild-Droste coordinates." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The metric tensor is a twice-covariant tensor (actually a field of symmetric bilinear forms):" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0, 2\\right)\n", "\\end{math}" ], "text/plain": [ "(0, 2)" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.tensor_type()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It can be thus be applied to a pair of vector fields, for instance $u$ and $k$:" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g\\left(u,k\\right)\n", "\\end{math}" ], "text/plain": [ "Scalar field g(u,k) on the Open subset E of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = g(u, k)\n", "s" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(u,k) on the Open subset E of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(s)" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(u,k\\right):& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\end{array}\n", "\\end{math}" ], "text/plain": [ "g(u,k): E --> R\n", " (t, r, th, ph) |--> -sqrt(r)/sqrt(-2*m + r)" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display()" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}}\n", "\\end{math}" ], "text/plain": [ "-sqrt(r)/sqrt(-2*m + r)" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr()" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(k,k\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}\n", "\\end{math}" ], "text/plain": [ "g(k,k): M --> R\n", " (t, r, th, ph) |--> 0\n", " (T, r, th, ph) |--> 0" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(k, k).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$u$ is a unit timelike vector:" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(u,u\\right):& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}\n", "\\end{math}" ], "text/plain": [ "g(u,u): E --> R\n", " (t, r, th, ph) |--> -1" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u, u).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Scalar products returned by the method `dot` are actually those formed with $g$:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{u}\\cdot{u}\n", "\\end{math}" ], "text/plain": [ "Scalar field u.u on the Open subset E of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.dot(u)" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} {u}\\cdot{u}:& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}\n", "\\end{math}" ], "text/plain": [ "u.u: E --> R\n", " (t, r, th, ph) |--> -1" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.dot(u).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Levi-Civita connection" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "nabla = g.connection()\n", "print(nabla)" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, r \\, t }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, m^{2} - m r}{r^{3}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & 2 \\, m - r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & {\\left(2 \\, m - r\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\n", "\\end{math}" ], "text/plain": [ "Gam^t_t,r = -m/(2*m*r - r^2) \n", "Gam^t_r,t = -m/(2*m*r - r^2) \n", "Gam^r_t,t = -(2*m^2 - m*r)/r^3 \n", "Gam^r_r,r = m/(2*m*r - r^2) \n", "Gam^r_th,th = 2*m - r \n", "Gam^r_ph,ph = (2*m - r)*sin(th)^2 \n", "Gam^th_r,th = 1/r \n", "Gam^th_th,r = 1/r \n", "Gam^th_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_r,ph = 1/r \n", "Gam^ph_th,ph = cos(th)/sin(th) \n", "Gam^ph_ph,r = 1/r \n", "Gam^ph_ph,th = cos(th)/sin(th) " ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla.display()" ] }, { "cell_type": "code", "execution_count": 85, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=1)" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} k\n", "\\end{math}" ], "text/plain": [ "Tensor field nabla_g(k) of type (1,1) on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dk = nabla(k)\n", "Dk" ] }, { "cell_type": "code", "execution_count": 87, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(k) of type (1,1) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(Dk)" ] }, { "cell_type": "code", "execution_count": 88, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} k = \\left( \\frac{m}{2 \\, m r - r^{2}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} t + \\left( -\\frac{m}{4 \\, m^{2} - 4 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r + \\frac{m}{r^{2}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r - r^{2}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} r -\\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\theta} -\\frac{1}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\phi}\n", "\\end{math}" ], "text/plain": [ "nabla_g(k) = m/(2*m*r - r^2) d/dt*dt - m/(4*m^2 - 4*m*r + r^2) d/dt*dr + m/r^2 d/dr*dt - m/(2*m*r - r^2) d/dr*dr - 1/r d/dth*dth - 1/r d/dph*dph" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dk.display()" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\nabla_{g} k_{ \\phantom{\\, t} \\, t }^{ \\, t \\phantom{\\, t} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, t} \\, r }^{ \\, t \\phantom{\\, r} } & = & -\\frac{m}{4 \\, m^{2} - 4 \\, m r + r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, r} \\, t }^{ \\, r \\phantom{\\, t} } & = & \\frac{m}{r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, r} \\, r }^{ \\, r \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, {\\theta}} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} } & = & -\\frac{1}{r} \\\\ \\nabla_{g} k_{ \\phantom{\\, {\\phi}} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} } & = & -\\frac{1}{r} \\end{array}\n", "\\end{math}" ], "text/plain": [ "nabla_g(k)^t_t = m/(2*m*r - r^2) \n", "nabla_g(k)^t_r = -m/(4*m^2 - 4*m*r + r^2) \n", "nabla_g(k)^r_t = m/r^2 \n", "nabla_g(k)^r_r = -m/(2*m*r - r^2) \n", "nabla_g(k)^th_th = -1/r \n", "nabla_g(k)^ph_ph = -1/r " ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dk.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The acceleration $\\nabla_k k$ of the vector field $k$:" ] }, { "cell_type": "code", "execution_count": 90, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Ak = Dk.contract(k)\n", "print(Ak)" ] }, { "cell_type": "code", "execution_count": 91, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}0\n", "\\end{math}" ], "text/plain": [ "0" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ak.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$k$ is thus geodesic vector. The field lines of $k$ are actually the ingoing radial null geodesics of Schwarzschild spacetime." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of the method contract, one can use index notations to compute $\\nabla_k k$ as $k^b \\nabla_b k^a$:" ] }, { "cell_type": "code", "execution_count": 92, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ak == Dk['^a_b']*k['^b']" ] }, { "cell_type": "code", "execution_count": 93, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The acceleration $\\nabla_u u$ of the vector field $u$:" ] }, { "cell_type": "code", "execution_count": 94, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{m}{r^{2}} \\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "m/r^2 d/dr" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Au = nabla(u).contract(u)\n", "Au.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\nabla g$ is identically zero, since the connection $\\nabla$ is compatible with $g$:" ] }, { "cell_type": "code", "execution_count": 95, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(g) of type (0,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Dg = nabla(g)\n", "print(Dg)" ] }, { "cell_type": "code", "execution_count": 96, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} g = 0\n", "\\end{math}" ], "text/plain": [ "nabla_g(g) = 0" ] }, "execution_count": 96, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dg.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Curvature" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Riemann curvature tensor** is computed as" ] }, { "cell_type": "code", "execution_count": 97, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Riem = g.riemann()\n", "print(Riem)" ] }, { "cell_type": "code", "execution_count": 98, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, r \\, t \\, r }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\theta} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\phi} \\, t \\, {\\phi} }^{ \\, t \\phantom{\\, {\\phi}} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, t \\, t \\, r }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, t \\, t \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, t \\, t \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}\n", "\\end{math}" ], "text/plain": [ "Riem(g)^t_r,t,r = -2*m/(2*m*r^2 - r^3) \n", "Riem(g)^t_th,t,th = -m/r \n", "Riem(g)^t_ph,t,ph = -m*sin(th)^2/r \n", "Riem(g)^r_t,t,r = -2*(2*m^2 - m*r)/r^4 \n", "Riem(g)^r_th,r,th = -m/r \n", "Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n", "Riem(g)^th_t,t,th = (2*m^2 - m*r)/r^4 \n", "Riem(g)^th_r,r,th = -m/(2*m*r^2 - r^3) \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_t,t,ph = (2*m^2 - m*r)/r^4 \n", "Riem(g)^ph_r,r,ph = -m/(2*m*r^2 - r^3) \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The component $\\mathrm{Riem}(g)^t_{\\ \\, rtr} = \\mathrm{Riem}(g)^0_{\\ \\, 101}$ is returned by " ] }, { "cell_type": "code", "execution_count": 99, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}\n", "\\end{math}" ], "text/plain": [ "-2*m/(2*m*r^2 - r^3)" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem[0,1,0,1]" ] }, { "cell_type": "code", "execution_count": 100, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, T \\, T \\, r }^{ \\, T \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, r} } & = & \\frac{4 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, r \\, T \\, r }^{ \\, T \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, r} } & = & \\frac{2 \\, {\\left(2 \\, m^{2} + m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, {\\theta} \\, T \\, {\\theta} }^{ \\, T \\phantom{\\, {\\theta}} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, {\\phi} \\, T \\, {\\phi} }^{ \\, T \\phantom{\\, {\\phi}} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, T \\, T \\, r }^{ \\, r \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, r \\, T \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, r} } & = & -\\frac{4 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, T \\, T \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, T \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, T} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, T \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} + m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, T \\, T \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, T \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, T} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, T \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} + m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}\n", "\\end{math}" ], "text/plain": [ "Riem(g)^T_T,T,r = 4*m^2/r^4 \n", "Riem(g)^T_r,T,r = 2*(2*m^2 + m*r)/r^4 \n", "Riem(g)^T_th,T,th = -m/r \n", "Riem(g)^T_ph,T,ph = -m*sin(th)^2/r \n", "Riem(g)^r_T,T,r = -2*(2*m^2 - m*r)/r^4 \n", "Riem(g)^r_r,T,r = -4*m^2/r^4 \n", "Riem(g)^r_th,r,th = -m/r \n", "Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n", "Riem(g)^th_T,T,th = (2*m^2 - m*r)/r^4 \n", "Riem(g)^th_T,r,th = 2*m^2/r^4 \n", "Riem(g)^th_r,T,th = 2*m^2/r^4 \n", "Riem(g)^th_r,r,th = (2*m^2 + m*r)/r^4 \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_T,T,ph = (2*m^2 - m*r)/r^4 \n", "Riem(g)^ph_T,r,ph = 2*m^2/r^4 \n", "Riem(g)^ph_r,T,ph = 2*m^2/r^4 \n", "Riem(g)^ph_r,r,ph = (2*m^2 + m*r)/r^4 \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(EF.frame(), only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Kretschmann scalar** is the \"square\" of the Riemann tensor defined by \n", "$$K = R_{abcd} \\, R^{abcd}, \\qquad R := \\mathrm{Riem}(g)$$\n", "To compute it, we must first form the tensor fields whose components are $R_{abcd}$ and \n", "$R^{abcd}$. They are obtained by respectively lowering and raising the indices of the components $R^a_{\\ \\, bcd}$ of the Riemann tensor, via the metric $g$. These two operations are performed by the methods `down()` and `up()`. The contraction is performed by summation on repeated indices, using LaTeX notations:" ] }, { "cell_type": "code", "execution_count": 101, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\end{array}\n", "\\end{math}" ], "text/plain": [ "M --> R\n", "(t, r, th, ph) |--> 48*m^2/r^6\n", "(T, r, th, ph) |--> 48*m^2/r^6" ] }, "execution_count": 101, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K = Riem.down(g)['_{abcd}'] * Riem.up(g)['^{abcd}']\n", "print(K)\n", "K.display()" ] }, { "cell_type": "code", "execution_count": 102, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}\n", "\\end{math}" ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $\\lim_{r\\to 0} K = +\\infty$, we may say that $r=0$ is a **curvature singularity** of Schwarzschild spacetime." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ricci tensor" ] }, { "cell_type": "code", "execution_count": 103, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Ric = g.ricci()\n", "print(Ric)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We check that the Schwarzschild metric is a solution of the vacuum **Einstein equation**:" ] }, { "cell_type": "code", "execution_count": 104, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0\n", "\\end{math}" ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Geodesics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, for graphical purposes, we introduce the **Euclidean space** $\\mathbb{E}^3$ and some map $M\\to \\mathbb{E}^3$:" ] }, { "cell_type": "code", "execution_count": 105, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{E}^{3} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\end{array}\n", "\\end{math}" ], "text/plain": [ "M --> E^3\n", " (t, r, th, ph) |--> (x, y, z) = (r*cos(ph)*sin(th), r*sin(ph)*sin(th), r*cos(th))\n", " (T, r, th, ph) |--> (x, y, z) = (r*cos(ph)*sin(th), r*sin(ph)*sin(th), r*cos(th))" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E3. = EuclideanSpace()\n", "X3 = E3.cartesian_coordinates()\n", "to_E3 = M.diff_map(E3, {(SD, X3): \n", " [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)]})\n", "to_E3.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A timelike geodesic" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us consider the geodesic starting at point $p$ and having the following tangent vector at $p$ (note that the tangent vector $v_0$ is constructed by means of the call operator `()` acting of the parent, which is the tangent space to $M$ at $p$): " ] }, { "cell_type": "code", "execution_count": 106, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}v_0 = 1.30000000000000 \\frac{\\partial}{\\partial t } + \\frac{0.0640000000000000}{m} \\frac{\\partial}{\\partial {\\phi} }\n", "\\end{math}" ], "text/plain": [ "v_0 = 1.30000000000000 d/dt + 0.0640000000000000/m d/dph" ] }, "execution_count": 106, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = M.tangent_space(p)((1.3, 0, 0, 0.064/m), name='v_0')\n", "v0.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We declare the geodesic with such initial conditions, denoting by $s$ the affine parameter (proper time), with $(s_{\\rm min}, s_{\\rm max})=(0, 1500\\,m)$:" ] }, { "cell_type": "code", "execution_count": 107, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mbox{Integrated geodesic in the 4-dimensional Lorentzian manifold M}\n", "\\end{math}" ], "text/plain": [ "Integrated geodesic in the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 107, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = var('s')\n", "geod = M.integrated_geodesic(g, (s, 0, 2000), v0)\n", "geod" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the initial point `p` is not explicitely passed in the argument list of `integrated_geodesic`, because this piece of information is contained in `v0`:" ] }, { "cell_type": "code", "execution_count": 108, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p is v0.parent().base_point()" ] }, { "cell_type": "code", "execution_count": 109, "metadata": {}, "outputs": [], "source": [ "sol = geod.solve(parameters_values={m: 1}) # numerical integration\n", "interp = geod.interpolate() # interpolation of the solution for the plot" ] }, { "cell_type": "code", "execution_count": 110, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 110, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod.plot_integrated(chart=X3, mapping=to_E3, plot_points=1000, \n", " thickness=2, label_axes=False) \\\n", "+ p.plot(chart=X3, mapping=to_E3, size=4, parameters={m: 1}) \\\n", "+ sphere(size=2, color='grey')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A 2D view by suppressing $z$ from the ambient coordinates:" ] }, { "cell_type": "code", "execution_count": 111, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bh_plot = circle((0, 0), 2, edgecolor='black', fill=True, facecolor='grey', alpha=0.5)\n", "geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), plot_points=1000, \n", " thickness=2) \\\n", "+ p.plot(chart=X3, mapping=to_E3, ambient_coords=(x,y), size=4, parameters={m: 1}) \\\n", "+ bh_plot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Null geodesics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us consider a null geodesique $\\mathscr{L}$ in the equatorial plane ($\\theta = \\pi/2$). \n", "The null vector $v$ tangent to $\\mathscr{L}$ and associated to some affine parameter $\\lambda$ is given by the following first integrals of the geodesic equation:\n", "$$\n", " v^t = \\frac{\\mathrm{d}t}{\\mathrm{d}\\lambda} = \\left(1 - \\frac{2m}{r} \\right)^{-1}\n", "$$\n", "$$\n", " v^r = \\frac{\\mathrm{d}r}{\\mathrm{d}\\lambda} = \\pm \\sqrt{1 - \\frac{b^2}{r^2} \\left(1 - \\frac{2m}{r} \\right) }\n", "$$\n", "$$\n", " v^\\theta = \\frac{\\mathrm{d}\\theta}{\\mathrm{d}\\lambda} = 0\n", "$$\n", "$$\n", " v^\\varphi = \\frac{\\mathrm{d}\\varphi}{\\mathrm{d}\\lambda} = \\frac{b}{r^2}\n", "$$\n", "where the constant $b$ is related to the conserved energy $E$ and conserved angular momentum $L$ along the geodesic by \n", "$$\n", " b = \\frac{L}{E} . \n", "$$\n", "For a geodesic arising from infinity, $b$ is the **impact parameter**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To set up the initial vector for the computation of a null geodesic, let us define a function that takes $b$ and some initial radius $r_0$ as input and returns the initial vector $v_0$. We take advantage that SageMath is built atop Python to construct this function as a pure Python function:" ] }, { "cell_type": "code", "execution_count": 112, "metadata": {}, "outputs": [], "source": [ "def initial_vector(r0, b, phi0=0, inward=True):\n", " r\"\"\"\n", " Evaluate the initial tangent vector along a null geodesic. \n", " \n", " INPUT:\n", " \n", " - r0: radial SD coordinate of the initial point\n", " - b: impact parameter\n", " - phi0: azimuthal SD coordinate of the initial point (default: 0)\n", " - inward: determines whether the geodesic has initially v^r < 0 (default: True)\n", " \n", " \"\"\"\n", " vt0 = 1/(1 - 2*m/r0)\n", " vr0 = sqrt(1 - b^2/r0^2*(1 - 2*m/r0))\n", " if inward:\n", " vr0 = - vr0\n", " vth0 = 0\n", " vph0 = b / r0^2\n", " p0 = M((0, r0, pi/2, phi0), name='p_0') # initial point\n", " return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use this function to construct the initial vector for $r_0 = 10m$ and $b=7m$:" ] }, { "cell_type": "code", "execution_count": 113, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}v_0 = \\frac{5}{4} \\frac{\\partial}{\\partial t } -\\frac{2}{5} \\, \\sqrt{\\frac{19}{5}} \\frac{\\partial}{\\partial r } + \\frac{7}{100 \\, m} \\frac{\\partial}{\\partial {\\phi} }\n", "\\end{math}" ], "text/plain": [ "v_0 = 5/4 d/dt - 2/5*sqrt(19/5) d/dr + 7/100/m d/dph" ] }, "execution_count": 113, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = initial_vector(10*m, 7*m)\n", "v0.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $v_0$ is a null vector:" ] }, { "cell_type": "code", "execution_count": 114, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}0\n", "\\end{math}" ], "text/plain": [ "0" ] }, "execution_count": 114, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0 = v0.parent().base_point()\n", "g.at(p0)(v0, v0)" ] }, { "cell_type": "code", "execution_count": 115, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 6 graphics primitives" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 40), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 = geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='green', thickness=1.5, display_tangent=True, \n", " color_tangent='green', plot_points_tangent=4, scale=1) \n", "plot2 += bh_plot\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A null geodesic plunging into the black hole:" ] }, { "cell_type": "code", "execution_count": 116, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}v_0 = \\frac{5}{4} \\frac{\\partial}{\\partial t } -2 \\, \\sqrt{\\frac{1}{5}} \\frac{\\partial}{\\partial r } + \\frac{1}{20 \\, m} \\frac{\\partial}{\\partial {\\phi} }\n", "\\end{math}" ], "text/plain": [ "v_0 = 5/4 d/dt - 2*sqrt(1/5) d/dr + 1/20/m d/dph" ] }, "execution_count": 116, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = initial_vector(10*m, 5*m)\n", "v0.display()" ] }, { "cell_type": "code", "execution_count": 117, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 10 graphics primitives" ] }, "execution_count": 117, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 13), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 += geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='green', thickness=1.5, display_tangent=True, \n", " color_tangent='green', plot_points_tangent=3, scale=0.2)\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The photon orbit" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The photon orbit corresponds to $r_0=3m$ with the following critical value of the impact parameter: \n", "$$\n", " b_{\\rm c} = 3\\sqrt{3}\\, m\n", "$$" ] }, { "cell_type": "code", "execution_count": 118, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}5.19615242270663\n", "\\end{math}" ], "text/plain": [ "5.19615242270663" ] }, "execution_count": 118, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bc = 3*sqrt(3)*m\n", "n(bc/m)" ] }, { "cell_type": "code", "execution_count": 119, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}v_0 = 3 \\frac{\\partial}{\\partial t } + \\frac{\\sqrt{3}}{3 \\, m} \\frac{\\partial}{\\partial {\\phi} }\n", "\\end{math}" ], "text/plain": [ "v_0 = 3 d/dt + 1/3*sqrt(3)/m d/dph" ] }, "execution_count": 119, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = initial_vector(3*m, bc)\n", "v0.display()" ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 11 graphics primitives" ] }, "execution_count": 120, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 100), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 += geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='green', thickness=1.5)\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A geodesic with $b$ close to $b_{\\rm c}$ wraps around the circular orbit: " ] }, { "cell_type": "code", "execution_count": 121, "metadata": {}, "outputs": [], "source": [ "v0 = initial_vector(10*m, 5.2025*m)" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 16 graphics primitives" ] }, "execution_count": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 40), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 += geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='orange', thickness=1.5,\n", " display_tangent=True, color_tangent='orange', \n", " plot_points_tangent=4, scale=1)\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using SymPy as the symbolic backend\n", "\n", "By default, the symbolic backend used in tensor calculus is SageMath's one (Pynac + Maxima), implemented via the symbolic ring `SR`. We can choose to use SymPy instead:" ] }, { "cell_type": "code", "execution_count": 123, "metadata": {}, "outputs": [], "source": [ "M.set_calculus_method('sympy')" ] }, { "cell_type": "code", "execution_count": 124, "metadata": {}, "outputs": [], "source": [ "v = 2*k" ] }, { "cell_type": "code", "execution_count": 125, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{2 r}{- 2 m + r} \\right) \\frac{\\partial}{\\partial t } -2 \\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "2*r/(-2*m + r) d/dt - 2 d/dr" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display()" ] }, { "cell_type": "code", "execution_count": 126, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 r}{- 2 m + r}\n", "\\end{math}" ], "text/plain": [ "2*r/(-2*m + r)" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v[0]" ] }, { "cell_type": "code", "execution_count": 127, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|2*r/(-2*m|\\phantom{\\verb!x!}\\verb|+|\\phantom{\\verb!x!}\\verb|r)|\n", "\\end{math}" ], "text/plain": [ "2*r/(-2*m + r)" ] }, "execution_count": 127, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v[0].expr()" ] }, { "cell_type": "code", "execution_count": 128, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb||\n", "\\end{math}" ], "text/plain": [ "" ] }, "execution_count": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(v[0].expr())" ] }, { "cell_type": "code", "execution_count": 129, "metadata": {}, "outputs": [], "source": [ "M.set_calculus_method('SR')" ] }, { "cell_type": "code", "execution_count": 130, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb||\n", "\\end{math}" ], "text/plain": [ "" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(v[0].expr())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Going further\n", "\n", "\n", "Visit the [SageManifolds examples](http://sagemanifolds.obspm.fr/examples.html). " ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2", "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }