{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Schwarzschild spacetime\n", "\n", "This notebook illustrates a few capabilities of SageMath in computations regarding Schwarzschild spacetime. The corresponding tools have been developed within the [SageManifolds](http://sagemanifolds.obspm.fr) project.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Notebooks/SM_Schwarzschild.ipynb) to download the notebook file (ipynb format). To run it, you must start SageMath with the Jupyter interface, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 8.2 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.2.beta6, Release Date: 2020-07-25'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we initialize a time counter for benchmarking:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import time\n", "comput_time0 = time.perf_counter()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime manifold\n", "\n", "We declare the Schwarzschild spacetime as a 4-dimensional Lorentzian manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}\n", "\\end{math}" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', r'\\mathcal{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": [ "

The spacetime manifold $\\mathcal{M}$ can be split into 4 regions, corresponding to the 4 quadrants in the Kruskal diagram.Let us denote by $\\mathcal{R}_{\\mathrm{I}}$ to $\\mathcal{R}_{\\mathrm{IV}}$ the interiors of these 4 regions. $\\mathcal{R}_{\\mathrm{I}}$ and $\\mathcal{R}_{\\mathrm{III}}$ are asymtotically flat regions outside the event horizon; $\\mathcal{R}_{\\mathrm{II}}$ is inside the future event horizon and $\\mathcal{R}_{\\mathrm{IV}}$ is inside the past event horizon.

" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}, \\mathcal{R}_{\\mathrm{II}}, \\mathcal{R}_{\\mathrm{III}}, \\mathcal{R}_{\\mathrm{IV}}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}, \\mathcal{R}_{\\mathrm{II}}, \\mathcal{R}_{\\mathrm{III}}, \\mathcal{R}_{\\mathrm{IV}}\\right)\n", "\\end{math}" ], "text/plain": [ "(Open subset R_I of the 4-dimensional Lorentzian manifold M,\n", " Open subset R_II of the 4-dimensional Lorentzian manifold M,\n", " Open subset R_III of the 4-dimensional Lorentzian manifold M,\n", " Open subset R_IV of the 4-dimensional Lorentzian manifold M)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI = M.open_subset('R_I', r'\\mathcal{R}_{\\mathrm{I}}')\n", "regII = M.open_subset('R_II', r'\\mathcal{R}_{\\mathrm{II}}')\n", "regIII = M.open_subset('R_III', r'\\mathcal{R}_{\\mathrm{III}}')\n", "regIV = M.open_subset('R_IV', r'\\mathcal{R}_{\\mathrm{IV}}')\n", "regI, regII, regIII, regIV" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The parameter $m$ of the Schwarzschild spacetime is declared as a symbolic variable:

" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "m = var('m') ; assume(m>=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Boyer-Lindquist coordinates

\n", "

The standard Boyer-Lindquist coordinates (also called Schwarzschild coordinates) are defined on $\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}$

" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\n", "\\end{math}" ], "text/plain": [ "Open subset R_I_union_R_II of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_II = regI.union(regII) ; regI_II" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R_I_union_R_II, (t, r, th, ph))\n" ] } ], "source": [ "X. = regI_II.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "print(X)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We naturally introduce two subcharts as the restrictions of the chart $X$ to regions $\\mathcal{R}_{\\mathrm{I}}$ and $\\mathcal{R}_{\\mathrm{II}}$ respectively. Since, in terms of the Boyer-Lindquist coordinates, $\\mathcal{R}_{\\mathrm{I}}$ (resp. $\\mathcal{R}_{\\mathrm{II}}$) is defined by $r>2m$ (resp. $r<2m$), we set

" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I, (t, r, th, ph))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_I = X.restrict(regI, r>2*m) ; X_I" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_II, (t, r, th, ph))" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_II = X.restrict(regII, r<2*m) ; X_II" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

At this stage, the manifold's atlas has 3 charts:

" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\phi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\phi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (R_I_union_R_II, (t, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph))]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Three vector frames have been defined on the manifold: the three coordinate frames:

" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\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 (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph))]" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph))\n" ] } ], "source": [ "print(M.default_frame())" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\n", "\\end{math}" ], "text/plain": [ "Open subset R_I_union_R_II of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame().domain()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Metric tensor\n", "\n", "The metric tensor is obtained as follows:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Lorentzian metric g on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "g = M.metric()\n", "print(g)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One has to set its components in the coordinate frame associated with Schwarzschild coordinates, which is the current manifold's default frame:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "g[0,0], g[1,1] = -(1-2*m/r), 1/(1-2*m/r)\n", "g[2,2], g[3,3] = r^2, (r*sin(th))^2" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "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": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

As an example, let us consider a vector field defined only on $\\mathcal{R}_{\\mathrm{I}}$:

" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\frac{\\partial}{\\partial t } + \\left( -\\frac{2 \\, m}{r} + 1 \\right) \\frac{\\partial}{\\partial r }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\frac{\\partial}{\\partial t } + \\left( -\\frac{2 \\, m}{r} + 1 \\right) \\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "v = d/dt + (-2*m/r + 1) d/dr" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v = regI.vector_field('v')\n", "v[0] = 1\n", "v[1] = 1 - 2*m/r\n", "# unset components are zero \n", "v.display()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\n", "\\end{math}" ], "text/plain": [ "Open subset R_I of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.domain()" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}\n", "\\end{math}" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.domain()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Since $\\mathcal{R}_{\\mathrm{I}}\\subset \\mathcal{M}$, it is possible to apply $g$ to $v$:

" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(v,v) on the Open subset R_I of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "s = g(v,v)\n", "print(s)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(v,v\\right):& \\mathcal{R}_{\\mathrm{I}} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(v,v\\right):& \\mathcal{R}_{\\mathrm{I}} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}\n", "\\end{math}" ], "text/plain": [ "g(v,v): R_I --> R\n", " (t, r, th, ph) |--> 0" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display() # v is indeed a null vector" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", "

Levi-Civita Connection

\n", "

The Levi-Civita connection $\\nabla$ associated with $g$:

\n", "

" ] }, { "cell_type": "code", "execution_count": 26, "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": [ "nab = g.connection()\n", "print(nab)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us verify that the covariant derivative of $g$ with respect to $\\nabla$ vanishes identically:

" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab(g) == 0" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} g = 0$ " ], "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": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab(g).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The nonzero Christoffel symbols of $g$ with respect to Schwarzschild coordinates, skipping those that can be deduced by symmetry:

" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, m^{2} - m r}{r^{3}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & 2 \\, m - r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & {\\left(2 \\, m - r\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$ " ], "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{\\, 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}} \\, {\\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)} \\end{array}\n", "\\end{math}" ], "text/plain": [ "Gam^t_t,r = -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_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_r,ph = 1/r \n", "Gam^ph_th,ph = cos(th)/sin(th) " ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Curvature

\n", "\n", "

The Riemann curvature tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 30, "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": [ "R = g.riemann()\n", "print(R)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Weyl conformal tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field C(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "C = g.weyl()\n", "print(C)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{C}\\left(g\\right) = \\left( -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r + \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r -\\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + \\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{C}\\left(g\\right) = \\left( -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r + \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r -\\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + \\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\n", "\\end{math}" ], "text/plain": [ "C(g) = -2*m/(2*m*r^2 - r^3) d/dt*dr*dt*dr + 2*m/(2*m*r^2 - r^3) d/dt*dr*dr*dt - m/r d/dt*dth*dt*dth + m/r d/dt*dth*dth*dt - m*sin(th)^2/r d/dt*dph*dt*dph + m*sin(th)^2/r d/dt*dph*dph*dt - 2*(2*m^2 - m*r)/r^4 d/dr*dt*dt*dr + 2*(2*m^2 - m*r)/r^4 d/dr*dt*dr*dt - m/r d/dr*dth*dr*dth + m/r d/dr*dth*dth*dr - m*sin(th)^2/r d/dr*dph*dr*dph + m*sin(th)^2/r d/dr*dph*dph*dr + (2*m^2 - m*r)/r^4 d/dth*dt*dt*dth - (2*m^2 - m*r)/r^4 d/dth*dt*dth*dt - m/(2*m*r^2 - r^3) d/dth*dr*dr*dth + m/(2*m*r^2 - r^3) d/dth*dr*dth*dr + 2*m*sin(th)^2/r d/dth*dph*dth*dph - 2*m*sin(th)^2/r d/dth*dph*dph*dth + (2*m^2 - m*r)/r^4 d/dph*dt*dt*dph - (2*m^2 - m*r)/r^4 d/dph*dt*dph*dt - m/(2*m*r^2 - r^3) d/dph*dr*dr*dph + m/(2*m*r^2 - r^3) d/dph*dr*dph*dr - 2*m/r d/dph*dth*dth*dph + 2*m/r d/dph*dth*dph*dth" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 33, "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": [ "

Einstein equation

\n", "

Let us check that the Schwarzschild metric is a solution of the vacuum Einstein equation:

" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == 0" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0$ " ], "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": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display() # another view of the above" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Contrary to the Ricci tensor, the Riemann tensor does not vanish:

" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}, 0, 0\\right], \\left[\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, -\\frac{m}{r}, 0\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{m}{r}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}, 0, 0, 0\\right]\\right]\\right], \\left[\\left[\\left[0, -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}}, 0, 0\\right], \\left[\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, -\\frac{m}{r}, 0\\right], \\left[0, \\frac{m}{r}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}\\right], \\left[0, 0, 0, 0\\right], \\left[0, \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}, 0, 0\\right]\\right]\\right], \\left[\\left[\\left[0, 0, \\frac{2 \\, m^{2} - m r}{r^{4}}, 0\\right], \\left[0, 0, 0, 0\\right], \\left[-\\frac{2 \\, m^{2} - m r}{r^{4}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, -\\frac{m}{2 \\, m r^{2} - r^{3}}, 0\\right], \\left[0, \\frac{m}{2 \\, m r^{2} - r^{3}}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r}\\right], \\left[0, 0, -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r}, 0\\right]\\right]\\right], \\left[\\left[\\left[0, 0, 0, \\frac{2 \\, m^{2} - m r}{r^{4}}\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[-\\frac{2 \\, m^{2} - m r}{r^{4}}, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{m}{2 \\, m r^{2} - r^{3}}\\right], \\left[0, 0, 0, 0\\right], \\left[0, \\frac{m}{2 \\, m r^{2} - r^{3}}, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{2 \\, m}{r}\\right], \\left[0, 0, \\frac{2 \\, m}{r}, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right]\\right]\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}, 0, 0\\right], \\left[\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, -\\frac{m}{r}, 0\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{m}{r}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}, 0, 0, 0\\right]\\right]\\right], \\left[\\left[\\left[0, -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}}, 0, 0\\right], \\left[\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, -\\frac{m}{r}, 0\\right], \\left[0, \\frac{m}{r}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}\\right], \\left[0, 0, 0, 0\\right], \\left[0, \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r}, 0, 0\\right]\\right]\\right], \\left[\\left[\\left[0, 0, \\frac{2 \\, m^{2} - m r}{r^{4}}, 0\\right], \\left[0, 0, 0, 0\\right], \\left[-\\frac{2 \\, m^{2} - m r}{r^{4}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, -\\frac{m}{2 \\, m r^{2} - r^{3}}, 0\\right], \\left[0, \\frac{m}{2 \\, m r^{2} - r^{3}}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r}\\right], \\left[0, 0, -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r}, 0\\right]\\right]\\right], \\left[\\left[\\left[0, 0, 0, \\frac{2 \\, m^{2} - m r}{r^{4}}\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[-\\frac{2 \\, m^{2} - m r}{r^{4}}, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{m}{2 \\, m r^{2} - r^{3}}\\right], \\left[0, 0, 0, 0\\right], \\left[0, \\frac{m}{2 \\, m r^{2} - r^{3}}, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{2 \\, m}{r}\\right], \\left[0, 0, \\frac{2 \\, m}{r}, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right]\\right]\\right]\n", "\\end{math}" ], "text/plain": [ "[[[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, -2*m/(2*m*r^2 - r^3), 0, 0],\n", " [2*m/(2*m*r^2 - r^3), 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, -m/r, 0], [0, 0, 0, 0], [m/r, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, -m*sin(th)^2/r],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [m*sin(th)^2/r, 0, 0, 0]]],\n", " [[[0, -2*(2*m^2 - m*r)/r^4, 0, 0],\n", " [2*(2*m^2 - m*r)/r^4, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, -m/r, 0], [0, m/r, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, -m*sin(th)^2/r],\n", " [0, 0, 0, 0],\n", " [0, m*sin(th)^2/r, 0, 0]]],\n", " [[[0, 0, (2*m^2 - m*r)/r^4, 0],\n", " [0, 0, 0, 0],\n", " [-(2*m^2 - m*r)/r^4, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, -m/(2*m*r^2 - r^3), 0],\n", " [0, m/(2*m*r^2 - r^3), 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 2*m*sin(th)^2/r],\n", " [0, 0, -2*m*sin(th)^2/r, 0]]],\n", " [[[0, 0, 0, (2*m^2 - m*r)/r^4],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [-(2*m^2 - m*r)/r^4, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, -m/(2*m*r^2 - r^3)],\n", " [0, 0, 0, 0],\n", " [0, m/(2*m*r^2 - r^3), 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -2*m/r], [0, 0, 2*m/r, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]]]" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R[:]" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = \\left( -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r + \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r -\\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + \\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = \\left( -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r + \\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} r\\otimes \\mathrm{d} t -\\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r + \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, m^{2} - m r}{r^{4}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} t\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{m}{2 \\, m r^{2} - r^{3}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r -\\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + \\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\n", "\\end{math}" ], "text/plain": [ "Riem(g) = -2*m/(2*m*r^2 - r^3) d/dt*dr*dt*dr + 2*m/(2*m*r^2 - r^3) d/dt*dr*dr*dt - m/r d/dt*dth*dt*dth + m/r d/dt*dth*dth*dt - m*sin(th)^2/r d/dt*dph*dt*dph + m*sin(th)^2/r d/dt*dph*dph*dt - 2*(2*m^2 - m*r)/r^4 d/dr*dt*dt*dr + 2*(2*m^2 - m*r)/r^4 d/dr*dt*dr*dt - m/r d/dr*dth*dr*dth + m/r d/dr*dth*dth*dr - m*sin(th)^2/r d/dr*dph*dr*dph + m*sin(th)^2/r d/dr*dph*dph*dr + (2*m^2 - m*r)/r^4 d/dth*dt*dt*dth - (2*m^2 - m*r)/r^4 d/dth*dt*dth*dt - m/(2*m*r^2 - r^3) d/dth*dr*dr*dth + m/(2*m*r^2 - r^3) d/dth*dr*dth*dr + 2*m*sin(th)^2/r d/dth*dph*dth*dph - 2*m*sin(th)^2/r d/dth*dph*dph*dth + (2*m^2 - m*r)/r^4 d/dph*dt*dt*dph - (2*m^2 - m*r)/r^4 d/dph*dt*dph*dt - m/(2*m*r^2 - r^3) d/dph*dr*dr*dph + m/(2*m*r^2 - r^3) d/dph*dr*dph*dr - 2*m/r d/dph*dth*dth*dph + 2*m/r d/dph*dth*dph*dth" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The nonzero components of the Riemann tensor, skipping those that can be deduced by antisymmetry:

" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\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}$ " ], "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": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[0 0 0 0]\n", "[0 0 0 0]\n", "[0 0 0 0]\n", "[0 0 0 0]" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Since the Ricci tensor is zero, the Weyl tensor is of course equal to the Riemann tensor:

" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C == R" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Bianchi identity

\n", "\n", "

Let us check the Bianchi identity $\\nabla_p R^i_{\\ \\, j kl} + \\nabla_k R^i_{\\ \\, jlp} + \\nabla_l R^i_{\\ \\, jpk} = 0$:

" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(Riem(g)) of type (1,4) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "DR = nab(R)\n", "print(DR)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 " ] } ], "source": [ "from __future__ import print_function # because the print below is valid only in Python3\n", "for i in M.irange():\n", " for j in M.irange():\n", " for k in M.irange():\n", " for l in M.irange():\n", " for p in M.irange():\n", " print(DR[i,j,k,l,p] + DR[i,j,l,p,k] + DR[i,j,p,k,l], end=' ')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us check that if we turn the first $+$ sign into a $-$ one, the Bianchi identity does no longer hold:

" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12*m/(2*m*r^3 - r^4) 0 0 12*m/(2*m*r^3 - r^4) 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 0 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 0 0 0 6*m/r^2 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12*(2*m^2 - m*r)/r^5 0 0 12*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*(4*m^3 - 4*m^2*r + m*r^2)/r^4 0 0 0 0 0 6*(4*m^3 - 4*m^2*r + m*r^2)/r^4 0 0 0 0 0 0 0 0 0 0 -6*(4*m^3 - 4*m^2*r + m*r^2)*sin(th)^2/r^4 0 0 0 0 0 0 0 0 6*(4*m^3 - 4*m^2*r + m*r^2)*sin(th)^2/r^4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)*sin(th)^2/r 0 0 -6*(2*m^2 - m*r)*sin(th)^2/r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*(2*m^2 - m*r)*sin(th)^2/r 0 0 6*(2*m^2 - m*r)*sin(th)^2/r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 0 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/(2*m*r^3 - r^4) 0 0 6*m/(2*m*r^3 - r^4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12*m*sin(th)^2/r^2 0 0 -12*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/(2*m*r^3 - r^4) 0 0 0 0 0 6*m/(2*m*r^3 - r^4) 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12*m/r^2 0 0 12*m/r^2 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 0 0 0 6*m/r^2 0 0 0 0 0 0 0 0 6*m/r^2 0 0 -6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 " ] } ], "source": [ "for i in M.irange():\n", " for j in M.irange():\n", " for k in M.irange():\n", " for l in M.irange():\n", " for p in M.irange():\n", " print(DR[i,j,k,l,p] - DR[i,j,l,p,k] + DR[i,j,p,k,l], end=' ')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Kretschmann scalar

\n", "

Let us first introduce tensor $R^\\flat$, of components $R_{ijkl} = g_{ip} R^p_{\\ \\, jkl}$:

" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (0,4) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "dR = R.down(g)\n", "print(dR)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

and tensor $R^\\sharp$, of components $R^{ijkl} = g^{jp} g^{kq} g^{lr} R^i_{\\ \\, pqr}$:

" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (4,0) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "uR = R.up(g)\n", "print(uR)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Kretschmann scalar is $K := R^{ijkl} R_{ijkl}$:

" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}$ " ], "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": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Kr = 0\n", "for i in M.irange():\n", " for j in M.irange():\n", " for k in M.irange():\n", " for l in M.irange():\n", " Kr += uR[i,j,k,l]*dR[i,j,k,l]\n", "Kr" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of the above loops, the Kretschmann scalar can also be computed by means of the method `contract()`, asking that the contraction takes place on all indices (positions 0, 1, 2, 3):" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}$ " ], "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": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Kr = uR.contract(0, 1, 2, 3, dR, 0, 1, 2, 3)\n", "Kr.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The contraction can also be performed by means of index notations:

" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}$ " ], "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": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Kr = uR['^{ijkl}']*dR['_{ijkl}']\n", "Kr.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Eddington-Finkelstein coordinates

\n", "

Let us introduce new coordinates on the spacetime manifold: the ingoing Eddington-Finkelstein ones:

" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R_I_union_R_II, (v, r, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I_union_R_II, (v, r, th, ph))" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF. = regI_II.chart(r'v r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\varphi')\n", "print(X_EF); X_EF" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The change from Schwarzschild (Boyer-Lindquist) coordinates to the ingoing Eddington-Finkelstein ones:

" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [], "source": [ "ch_BL_EF_I = X_I.transition_map(X_EF, [t+r+2*m*ln(r/(2*m)-1), r, th, ph], \n", " restrictions2=r>2*m)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (v, r, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (v, r, th, ph))" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_I) ; ch_BL_EF_I" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} v & = & 2 \\, m \\log\\left(\\frac{r}{2 \\, m} - 1\\right) + r + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} v & = & 2 \\, m \\log\\left(\\frac{r}{2 \\, m} - 1\\right) + r + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "v = 2*m*log(1/2*r/m - 1) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_I.display()" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I, (v, r, th, ph))" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF_I = X_EF.restrict(regI) ; X_EF_I" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [], "source": [ "ch_BL_EF_II = X_II.transition_map(X_EF, [t+r+2*m*ln(1-r/(2*m)), r, th, ph], \n", " restrictions2=r<2*m)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (v, r, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (v, r, th, ph))" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_II) ; ch_BL_EF_II" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} v & = & 2 \\, m \\log\\left(-\\frac{r}{2 \\, m} + 1\\right) + r + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} v & = & 2 \\, m \\log\\left(-\\frac{r}{2 \\, m} + 1\\right) + r + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "v = 2*m*log(-1/2*r/m + 1) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_II.display()" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_II, (v, r, th, ph))" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF_II = X_EF.restrict(regII) ; X_EF_II" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The manifold's atlas has now 6 charts:

" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (R_I_union_R_II, (t, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph)),\n", " Chart (R_I_union_R_II, (v, r, th, ph)),\n", " Chart (R_I, (v, r, th, ph)),\n", " Chart (R_II, (v, r, th, ph))]" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default chart is 'BL':

" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The change from Eddington-Finkelstein coordinates to the Schwarzschild (Boyer-Lindquist) ones, computed as the inverse of `ch_BL_EF`:" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (t, r, th, ph))\n" ] } ], "source": [ "ch_EF_BL_I = ch_BL_EF_I.inverse()\n", "print(ch_EF_BL_I)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & 2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left(-2 \\, m + r\\right) - r + v \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "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(-2 \\, m + r\\right) - r + v \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "t = 2*m*log(2) + 2*m*log(m) - 2*m*log(-2*m + r) - r + v\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_BL_I.display()" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (t, r, th, ph))\n" ] } ], "source": [ "ch_EF_BL_II = ch_BL_EF_II.inverse()\n", "print(ch_EF_BL_II)" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & 2 \\, m \\log\\left(2\\right) - 2 \\, m \\log\\left(2 \\, m - r\\right) + 2 \\, m \\log\\left(m\\right) - r + v \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "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(2 \\, m - r\\right) + 2 \\, m \\log\\left(m\\right) - r + v \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "t = 2*m*log(2) - 2*m*log(2*m - r) + 2*m*log(m) - r + v\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_BL_II.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

At this stage, 6 vector frames have been defined on the manifold: the 6 coordinate frames associated with the various charts:

" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dv,d/dr,d/dth,d/dph))]" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default frame is:

" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\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 (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph))" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The coframes are the duals of the defined vector frames:

" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} v,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\varphi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\mathrm{d} v,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\varphi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} v,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\varphi}\\right)\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} v,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\varphi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\mathrm{d} v,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\varphi}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\mathrm{d} v,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\varphi}\\right)\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Coordinate coframe (R_I_union_R_II, (dt,dr,dth,dph)),\n", " Coordinate coframe (R_I, (dt,dr,dth,dph)),\n", " Coordinate coframe (R_II, (dt,dr,dth,dph)),\n", " Coordinate coframe (R_I_union_R_II, (dv,dr,dth,dph)),\n", " Coordinate coframe (R_I, (dv,dr,dth,dph)),\n", " Coordinate coframe (R_II, (dv,dr,dth,dph))]" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.coframes()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

If not specified, tensor components are assumed to refer to the manifold's default frame. For instance, for the metric tensor:

" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$ " ], "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & 0 & 0 & 0 \\\\\n", "0 & -\\frac{1}{\\frac{2 \\, m}{r} - 1} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$ " ], "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": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The tensor components in the frame associated with Eddington-Finkelstein coordinates in Region I are obtained by providing the frame to the function `display()`:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m - r}{r} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r +\\mathrm{d} r\\otimes \\mathrm{d} v + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m - r}{r} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r +\\mathrm{d} r\\otimes \\mathrm{d} v + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\n", "\\end{math}" ], "text/plain": [ "g = (2*m - r)/r dv*dv + dv*dr + dr*dv + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_EF_I.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

They are also returned by the method comp(), with the frame as argument:

" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m - r}{r} & 1 & 0 & 0 \\\\\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m - r}{r} & 1 & 0 & 0 \\\\\n", "1 & 0 & 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)/r 1 0 0]\n", "[ 1 0 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.comp(X_EF_I.frame())[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

or, as a schortcut,

" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m - r}{r} & 1 & 0 & 0 \\\\\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m - r}{r} & 1 & 0 & 0 \\\\\n", "1 & 0 & 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)/r 1 0 0]\n", "[ 1 0 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_EF_I.frame(),:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Similarly, the metric components in the frame associated with Eddington-Finkelstein coordinates in Region II are obtained by

" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m - r}{r} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r +\\mathrm{d} r\\otimes \\mathrm{d} v + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{2 \\, m - r}{r} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v +\\mathrm{d} v\\otimes \\mathrm{d} r +\\mathrm{d} r\\otimes \\mathrm{d} v + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\n", "\\end{math}" ], "text/plain": [ "g = (2*m - r)/r dv*dv + dv*dr + dr*dv + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_EF_II.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Note that their form is identical to that in Region I.

\n", "\n", "

Plot of the Boyer-Lindquist coordinates in terms of the Eddington-Finkelstein ones

\n", "

Let us perform the plot in Region I:

" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 18 graphics primitives" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_I.plot(X_EF_I, ranges={t:(0,8), r:(2.1,10)}, fixed_coords={th:pi/2,ph:0}, \n", " ambient_coords=(r,v), style={t:'--', r:'-'}, parameters={m:1})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Tetrad of the static observer

\n", "

Let us introduce the orthonormal tetrad $(e_\\alpha)$ associated with the static observers in Schwarzschild spacetime, i.e. the observers whose worldlines are parallel to the timelike Killing vector in the Region I.

\n", "

The orthonormal tetrad is defined via a tangent-space automorphism that relates it to the Boyer-Lindquist coordinate frame in Region I:

" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} & 0 & 0 & 0 \\\\\n", "0 & \\sqrt{-\\frac{2 \\, m}{r} + 1} & 0 & 0 \\\\\n", "0 & 0 & \\frac{1}{r} & 0 \\\\\n", "0 & 0 & 0 & \\frac{1}{r \\sin\\left({\\theta}\\right)}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} & 0 & 0 & 0 \\\\\n", "0 & \\sqrt{-\\frac{2 \\, m}{r} + 1} & 0 & 0 \\\\\n", "0 & 0 & \\frac{1}{r} & 0 \\\\\n", "0 & 0 & 0 & \\frac{1}{r \\sin\\left({\\theta}\\right)}\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[1/sqrt(-2*m/r + 1) 0 0 0]\n", "[ 0 sqrt(-2*m/r + 1) 0 0]\n", "[ 0 0 1/r 0]\n", "[ 0 0 0 1/(r*sin(th))]" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_to_stat = regI.automorphism_field()\n", "ch_to_stat[0,0], ch_to_stat[1,1] = 1/sqrt(1-2*m/r), sqrt(1-2*m/r)\n", "ch_to_stat[2,2], ch_to_stat[3,3] = 1/r, 1/(r*sin(th))\n", "ch_to_stat[:]" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector frame (R_I, (e_0,e_1,e_2,e_3))\n" ] } ], "source": [ "e = X_I.frame().new_frame(ch_to_stat, 'e')\n", "print(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

At this stage, 7 vector frames have been defined on the manifold $\\mathcal{M}$:

" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(e_{0},e_{1},e_{2},e_{3}\\right)\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(e_{0},e_{1},e_{2},e_{3}\\right)\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Vector frame (R_I, (e_0,e_1,e_2,e_3))]" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The first vector of the tetrad is the static observer 4-velocity:

" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field e_0 on the Open subset R_I of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(e[0])" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial t }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial t }\n", "\\end{math}" ], "text/plain": [ "e_0 = 1/sqrt(-2*m/r + 1) d/dt" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

As any 4-velocity, it is a unit timelike vector:

" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1\n", "\\end{math}" ], "text/plain": [ "-1" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(e[0],e[0]).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Equivalently, we may use the method `dot` to get the scalar product (since $g$ is the default metric on $\\mathcal{M}$):" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1\n", "\\end{math}" ], "text/plain": [ "-1" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].dot(e[0]).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Let us check that the tetrad $(e_\\alpha)$ is orthonormal:

" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-1 0 0 0 \n", "0 1 0 0 \n", "0 0 1 0 \n", "0 0 0 1 \n" ] } ], "source": [ "for i in M.irange():\n", " for j in M.irange():\n", " print(e[i].dot(e[j]).expr(), end=' ') \n", " print(\"\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Another view of the above result:

" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "-1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1\n", "\\end{array}\\right)\n", "\\end{math}" ], "text/plain": [ "[-1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[e,:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

or, equivalently,

" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -e^{0}\\otimes e^{0}+e^{1}\\otimes e^{1}+e^{2}\\otimes e^{2}+e^{3}\\otimes e^{3}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -e^{0}\\otimes e^{0}+e^{1}\\otimes e^{1}+e^{2}\\otimes e^{2}+e^{3}\\otimes e^{3}\n", "\\end{math}" ], "text/plain": [ "g = -e^0*e^0 + e^1*e^1 + e^2*e^2 + e^3*e^3" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The expression of the 4-velocity $e_0$ and the vector $e_1$ in terms of the frame associated with Eddington-Finkelstein coordinates:

" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\left( \\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\right) \\frac{\\partial}{\\partial v }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\left( \\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\right) \\frac{\\partial}{\\partial v }\n", "\\end{math}" ], "text/plain": [ "e_0 = sqrt(r)/sqrt(-2*m + r) d/dv" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_EF_I.frame())" ] }, { "cell_type": "code", "execution_count": 85, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1} = \\left( \\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{\\sqrt{-2 \\, m + r}}{\\sqrt{r}} \\right) \\frac{\\partial}{\\partial r }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1} = \\left( \\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\right) \\frac{\\partial}{\\partial v } + \\left( \\frac{\\sqrt{-2 \\, m + r}}{\\sqrt{r}} \\right) \\frac{\\partial}{\\partial r }\n", "\\end{math}" ], "text/plain": [ "e_1 = sqrt(r)/sqrt(-2*m + r) d/dv + sqrt(-2*m + r)/sqrt(r) d/dr" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[1].display(X_EF_I.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Contrary to vectors of a coordinate frame, the vectors of the tetrad $e$ do not commute: their structure coefficients are not identically zero:

" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, -\\frac{m \\sqrt{-2 \\, m + r}}{{\\left(2 \\, m r - r^{2}\\right)} \\sqrt{r}}, 0, 0\\right], \\left[\\frac{m \\sqrt{-2 \\, m + r}}{{\\left(2 \\, m r - r^{2}\\right)} \\sqrt{r}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, -\\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}, 0\\right], \\left[0, \\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}\\right], \\left[0, 0, 0, -\\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)}\\right], \\left[0, \\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}, \\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)}, 0\\right]\\right]\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, -\\frac{m \\sqrt{-2 \\, m + r}}{{\\left(2 \\, m r - r^{2}\\right)} \\sqrt{r}}, 0, 0\\right], \\left[\\frac{m \\sqrt{-2 \\, m + r}}{{\\left(2 \\, m r - r^{2}\\right)} \\sqrt{r}}, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, -\\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}, 0\\right], \\left[0, \\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, 0, -\\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}\\right], \\left[0, 0, 0, -\\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)}\\right], \\left[0, \\frac{\\sqrt{-2 \\, m + r}}{r^{\\frac{3}{2}}}, \\frac{\\cos\\left({\\theta}\\right)}{r \\sin\\left({\\theta}\\right)}, 0\\right]\\right]\\right]\n", "\\end{math}" ], "text/plain": [ "[[[0, -m*sqrt(-2*m + r)/((2*m*r - r^2)*sqrt(r)), 0, 0],\n", " [m*sqrt(-2*m + r)/((2*m*r - r^2)*sqrt(r)), 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, -sqrt(-2*m + r)/r^(3/2), 0],\n", " [0, sqrt(-2*m + r)/r^(3/2), 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, -sqrt(-2*m + r)/r^(3/2)],\n", " [0, 0, 0, -cos(th)/(r*sin(th))],\n", " [0, sqrt(-2*m + r)/r^(3/2), cos(th)/(r*sin(th)), 0]]]" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e.structure_coeff()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Equivalently, the Lie derivative of one vector along another one is not necessarily zero:

" ] }, { "cell_type": "code", "execution_count": 87, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{m \\sqrt{-2 \\, m + r}}{{\\left(2 \\, m r - r^{2}\\right)} \\sqrt{r}} \\right) e_{0}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( \\frac{m \\sqrt{-2 \\, m + r}}{{\\left(2 \\, m r - r^{2}\\right)} \\sqrt{r}} \\right) e_{0}\n", "\\end{math}" ], "text/plain": [ "m*sqrt(-2*m + r)/((2*m*r - r^2)*sqrt(r)) e_0" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].lie_der(e[1]).display(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The curvature 2-form $\\Omega^0_{\\ \\, 1}$ associated with the tetrad $(e_\\alpha)$:

" ] }, { "cell_type": "code", "execution_count": 88, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Omega^0_{\\ \\, 1} = \\frac{2 \\, m}{r^{3}} \\mathrm{d} t\\wedge \\mathrm{d} r$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Omega^0_{\\ \\, 1} = \\frac{2 \\, m}{r^{3}} \\mathrm{d} t\\wedge \\mathrm{d} r\n", "\\end{math}" ], "text/plain": [ "curvature (0,1) of connection nabla_g w.r.t. Vector frame (R_I, (e_0,e_1,e_2,e_3)) = 2*m/r^3 dt/\\dr" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.connection().curvature_form(0,1,e).display(X_I.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Kruskal-Szekeres coordinates

\n", "

Let us now introduce the Kruskal-Szekeres coordinates $(U,V,\\theta,\\varphi)$ on the spacetime manifold, via the standard transformation expressing them in terms of the Boyer-Lindquist coordinates $(t,r,\\theta,\\varphi)$:

" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}\n", "\\end{math}" ], "text/plain": [ "Open subset R_I_union_R_II_union_R_III_union_R_IV of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M0 = regI.union(regII).union(regIII).union(regIV) ; M0" ] }, { "cell_type": "code", "execution_count": 90, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}},(U, V, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I_union_R_II_union_R_III_union_R_IV, (U, V, th, ph))" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_KS. = M0.chart(r'U V th:(0,pi):\\theta ph:(0,2*pi):\\varphi')\n", "X_KS.add_restrictions(V^2 < 1 + U^2)\n", "X_KS" ] }, { "cell_type": "code", "execution_count": 91, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I, (U, V, th, ph))" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_KS_I = X_KS.restrict(regI, [U>0, V-U]) ; X_KS_I" ] }, { "cell_type": "code", "execution_count": 92, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (U, V, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} U & = & \\sqrt{\\frac{r}{2 \\, m} - 1} \\cosh\\left(\\frac{t}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)} \\\\ V & = & \\sqrt{\\frac{r}{2 \\, m} - 1} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{t}{4 \\, m}\\right) \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} U & = & \\sqrt{\\frac{r}{2 \\, m} - 1} \\cosh\\left(\\frac{t}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)} \\\\ V & = & \\sqrt{\\frac{r}{2 \\, m} - 1} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{t}{4 \\, m}\\right) \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "U = sqrt(1/2*r/m - 1)*cosh(1/4*t/m)*e^(1/4*r/m)\n", "V = sqrt(1/2*r/m - 1)*e^(1/4*r/m)*sinh(1/4*t/m)\n", "th = th\n", "ph = ph" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_KS_I = X_I.transition_map(X_KS_I, [sqrt(r/(2*m)-1)*exp(r/(4*m))*cosh(t/(4*m)), \n", " sqrt(r/(2*m)-1)*exp(r/(4*m))*sinh(t/(4*m)), \n", " th, ph])\n", "print(ch_BL_KS_I)\n", "ch_BL_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 93, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_II, (U, V, th, ph))" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_KS_II = X_KS.restrict(regII, [V>0, V>U, V>-U]) ; X_KS_II" ] }, { "cell_type": "code", "execution_count": 94, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (U, V, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} U & = & \\sqrt{-\\frac{r}{2 \\, m} + 1} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{t}{4 \\, m}\\right) \\\\ V & = & \\sqrt{-\\frac{r}{2 \\, m} + 1} \\cosh\\left(\\frac{t}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} U & = & \\sqrt{-\\frac{r}{2 \\, m} + 1} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{t}{4 \\, m}\\right) \\\\ V & = & \\sqrt{-\\frac{r}{2 \\, m} + 1} \\cosh\\left(\\frac{t}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "U = sqrt(-1/2*r/m + 1)*e^(1/4*r/m)*sinh(1/4*t/m)\n", "V = sqrt(-1/2*r/m + 1)*cosh(1/4*t/m)*e^(1/4*r/m)\n", "th = th\n", "ph = ph" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_KS_II = X_II.transition_map(X_KS_II, [sqrt(1-r/(2*m))*exp(r/(4*m))*sinh(t/(4*m)), \n", " sqrt(1-r/(2*m))*exp(r/(4*m))*cosh(t/(4*m)), \n", " th, ph])\n", "print(ch_BL_KS_II)\n", "ch_BL_KS_II.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Plot of the Boyer-Lindquist coordinates in terms of the Kruskal ones

\n", "

We draw the Boyer-Lindquist chart in Region I (red) and Region II (green), with lines of constant $r$ being dashed:

" ] }, { "cell_type": "code", "execution_count": 95, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 52 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphI = X_I.plot(X_KS, ranges={t:(-12,12), r:(2.001,5)}, number_values={t:17, r:9}, \n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V), \n", " style={t:'--', r:'-'}, parameters={m:1})\n", "graphII = X_II.plot(X_KS, ranges={t:(-12,12), r:(0.001,1.999)}, number_values={t:17, r:9}, \n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V), \n", " style={t:'--', r:'-'}, color='green', parameters={m:1})\n", "show(graphI+graphII, xmin=-3, xmax=3, ymin=-3, ymax=3, axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

We may add to the graph the singularity $r=0$ as a Boyer-Lindquist chart plot with $(r,\\theta,\\phi)$ fixed at $(0, \\pi/2, 0)$. Similarly, we add the event horizon $r=2m$ as a Boyer-Lindquist chart plot with $(r,\\theta,\\phi)$ fixed at $(2.00001m, \\pi/2, 0)$:

" ] }, { "cell_type": "code", "execution_count": 96, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 54 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_r0 = X_II.plot(X_KS, fixed_coords={r:0, th:pi/2, ph:0}, ambient_coords=(U,V),\n", " color='yellow', thickness=3, parameters={m:1})\n", "graph_r2 = X_I.plot(X_KS, ranges={t:(-40,40)}, fixed_coords={r:2.00001, th:pi/2, ph:0},\n", " ambient_coords=(U,V), color='black', thickness=2, parameters={m:1})\n", "show(graphI+graphII+graph_r0+graph_r2, xmin=-3, xmax=3, ymin=-3, ymax=3, \n", " axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Plot of the Eddington-Finkelstein coordinates in terms of the Kruskal ones

\n", "

We first get the change of coordinates $(v,r,\\theta,\\phi) \\mapsto (U,V,\\theta,\\phi)$ by composing the change $(v,r,\\theta,\\phi) \\mapsto (t,r,\\theta,\\phi)$ with $(t,r,\\theta,\\phi) \\mapsto (U,V,\\theta,\\phi)$:

" ] }, { "cell_type": "code", "execution_count": 97, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (U, V, th, ph))" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_KS_I = ch_BL_KS_I * ch_EF_BL_I\n", "ch_EF_KS_I" ] }, { "cell_type": "code", "execution_count": 98, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} U & = & \\frac{\\sqrt{2} \\sqrt{-2 \\, m + r} \\cosh\\left(\\frac{2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left(-2 \\, m + r\\right) - r + v}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)}}{2 \\, \\sqrt{m}} \\\\ V & = & \\frac{\\sqrt{2} \\sqrt{-2 \\, m + r} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left(-2 \\, m + r\\right) - r + v}{4 \\, m}\\right)}{2 \\, \\sqrt{m}} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} U & = & \\frac{\\sqrt{2} \\sqrt{-2 \\, m + r} \\cosh\\left(\\frac{2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left(-2 \\, m + r\\right) - r + v}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)}}{2 \\, \\sqrt{m}} \\\\ V & = & \\frac{\\sqrt{2} \\sqrt{-2 \\, m + r} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left(-2 \\, m + r\\right) - r + v}{4 \\, m}\\right)}{2 \\, \\sqrt{m}} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "U = 1/2*sqrt(2)*sqrt(-2*m + r)*cosh(1/4*(2*m*log(2) + 2*m*log(m) - 2*m*log(-2*m + r) - r + v)/m)*e^(1/4*r/m)/sqrt(m)\n", "V = 1/2*sqrt(2)*sqrt(-2*m + r)*e^(1/4*r/m)*sinh(1/4*(2*m*log(2) + 2*m*log(m) - 2*m*log(-2*m + r) - r + v)/m)/sqrt(m)\n", "th = th\n", "ph = ph" ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 99, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (U, V, th, ph))" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_KS_II = ch_BL_KS_II * ch_EF_BL_II\n", "ch_EF_KS_II" ] }, { "cell_type": "code", "execution_count": 100, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 54 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphI_EF = X_EF_I.plot(X_KS, ranges={v:(-12,12), r:(2.001,5)}, number_values={v:17, r:9},\n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V),\n", " style={v:'--', r:'-'}, parameters={m:1})\n", "graphII_EF = X_EF_II.plot(X_KS, ranges={v:(-12,12), r:(0.001,1.999)}, \n", " number_values={v:17, r:9},\n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V),\n", " style={v:'--', r:'-'}, color='green', parameters={m:1})\n", "show(graphI_EF+graphII_EF+graph_r0+graph_r2, xmin=-3, xmax=3, ymin=-3, ymax=3,\n", " axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

There are now 9 charts defined on the spacetime manifold:

" ] }, { "cell_type": "code", "execution_count": 101, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (R_I_union_R_II, (t, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph)),\n", " Chart (R_I_union_R_II, (v, r, th, ph)),\n", " Chart (R_I, (v, r, th, ph)),\n", " Chart (R_II, (v, r, th, ph)),\n", " Chart (R_I_union_R_II_union_R_III_union_R_IV, (U, V, th, ph)),\n", " Chart (R_I, (U, V, th, ph)),\n", " Chart (R_II, (U, V, th, ph))]" ] }, "execution_count": 101, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 102, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}9$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}9\n", "\\end{math}" ], "text/plain": [ "9" ] }, "execution_count": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(M.atlas())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

There are 8 explicit coordinate changes (the coordinate change KS $\\rightarrow$ BL is not known in explicit form):

" ] }, { "cell_type": "code", "execution_count": 103, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right\\}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right\\}\n", "\\end{math}" ], "text/plain": [ "{(Chart (R_I, (t, r, th, ph)),\n", " Chart (R_I, (v, r, th, ph))): Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (v, r, th, ph)),\n", " (Chart (R_II, (t, r, th, ph)),\n", " Chart (R_II, (v, r, th, ph))): Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (v, r, th, ph)),\n", " (Chart (R_I, (v, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph))): Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (t, r, th, ph)),\n", " (Chart (R_II, (v, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph))): Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (t, r, th, ph)),\n", " (Chart (R_I, (t, r, th, ph)),\n", " Chart (R_I, (U, V, th, ph))): Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (U, V, th, ph)),\n", " (Chart (R_II, (t, r, th, ph)),\n", " Chart (R_II, (U, V, th, ph))): Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (U, V, th, ph)),\n", " (Chart (R_I, (v, r, th, ph)),\n", " Chart (R_I, (U, V, th, ph))): Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (U, V, th, ph)),\n", " (Chart (R_II, (v, r, th, ph)),\n", " Chart (R_II, (U, V, th, ph))): Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (U, V, th, ph))}" ] }, "execution_count": 103, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.coord_changes()" ] }, { "cell_type": "code", "execution_count": 104, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}8$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}8\n", "\\end{math}" ], "text/plain": [ "8" ] }, "execution_count": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(M.coord_changes())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

There are 10 vector frames (among which 9 coordinate frames):

" ] }, { "cell_type": "code", "execution_count": 105, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(e_{0},e_{1},e_{2},e_{3}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(e_{0},e_{1},e_{2},e_{3}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Vector frame (R_I, (e_0,e_1,e_2,e_3)),\n", " Coordinate frame (R_I_union_R_II_union_R_III_union_R_IV, (d/dU,d/dV,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dU,d/dV,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dU,d/dV,d/dth,d/dph))]" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 106, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}10$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}10\n", "\\end{math}" ], "text/plain": [ "10" ] }, "execution_count": 106, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(M.frames())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

There are 14 fields of tangent space automorphisms expressing the changes of coordinate bases and tetrad:

" ] }, { "cell_type": "code", "execution_count": 107, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}14$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}14\n", "\\end{math}" ], "text/plain": [ "14" ] }, "execution_count": 107, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(M.changes_of_frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thanks to these changes of frames, the components of the metric tensor with respect to the Kruskal-Szekeres can be computed by the method `display()` and are found to be:" ] }, { "cell_type": "code", "execution_count": 108, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} U\\otimes \\mathrm{d} U -\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} V\\otimes \\mathrm{d} V + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} U\\otimes \\mathrm{d} U -\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} V\\otimes \\mathrm{d} V + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\n", "\\end{math}" ], "text/plain": [ "g = 32*m^3*e^(-1/2*r/m)/r dU*dU - 32*m^3*e^(-1/2*r/m)/r dV*dV + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 109, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} & 0 & 0 & 0 \\\\\n", "0 & -\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} & 0 & 0 & 0 \\\\\n", "0 & -\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{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": [ "[ 32*m^3*e^(-1/2*r/m)/r 0 0 0]\n", "[ 0 -32*m^3*e^(-1/2*r/m)/r 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 109, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_KS_I.frame(),:]" ] }, { "cell_type": "code", "execution_count": 110, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} U\\otimes \\mathrm{d} U -\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} V\\otimes \\mathrm{d} V + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} U\\otimes \\mathrm{d} U -\\frac{32 \\, m^{3} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r} \\mathrm{d} V\\otimes \\mathrm{d} V + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\n", "\\end{math}" ], "text/plain": [ "g = 32*m^3*e^(-1/2*r/m)/r dU*dU - 32*m^3*e^(-1/2*r/m)/r dV*dV + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 110, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_KS_II.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The first vector of the orthonormal tetrad $e$ expressed on the Kruskal-Szekeres frame:

" ] }, { "cell_type": "code", "execution_count": 111, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\frac{\\sqrt{2} \\sqrt{r} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{t}{4 \\, m}\\right)}{8 \\, m^{\\frac{3}{2}}} \\frac{\\partial}{\\partial U } + \\frac{\\sqrt{2} \\sqrt{r} \\cosh\\left(\\frac{t}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)}}{8 \\, m^{\\frac{3}{2}}} \\frac{\\partial}{\\partial V }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\frac{\\sqrt{2} \\sqrt{r} e^{\\left(\\frac{r}{4 \\, m}\\right)} \\sinh\\left(\\frac{t}{4 \\, m}\\right)}{8 \\, m^{\\frac{3}{2}}} \\frac{\\partial}{\\partial U } + \\frac{\\sqrt{2} \\sqrt{r} \\cosh\\left(\\frac{t}{4 \\, m}\\right) e^{\\left(\\frac{r}{4 \\, m}\\right)}}{8 \\, m^{\\frac{3}{2}}} \\frac{\\partial}{\\partial V }\n", "\\end{math}" ], "text/plain": [ "e_0 = 1/8*sqrt(2)*sqrt(r)*e^(1/4*r/m)*sinh(1/4*t/m)/m^(3/2) d/dU + 1/8*sqrt(2)*sqrt(r)*cosh(1/4*t/m)*e^(1/4*r/m)/m^(3/2) d/dV" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_KS_I.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Riemann curvature tensor in terms of the Kruskal-Szekeres coordinates:

" ] }, { "cell_type": "code", "execution_count": 112, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} U\\otimes \\mathrm{d} V + \\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} V\\otimes \\mathrm{d} U -\\frac{m}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} U -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\varphi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} U -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} U\\otimes \\mathrm{d} V + \\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} V\\otimes \\mathrm{d} U -\\frac{m}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} V -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\varphi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} V + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\theta} -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} U -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\theta} + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} V + \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\varphi} -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\theta} + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\varphi} -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} U -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\varphi} + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} V -\\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\varphi} + \\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\theta}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} U\\otimes \\mathrm{d} V + \\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} V\\otimes \\mathrm{d} U -\\frac{m}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} U -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\varphi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial U }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} U -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} U\\otimes \\mathrm{d} V + \\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} V\\otimes \\mathrm{d} U -\\frac{m}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\theta} + \\frac{m}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} V -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\varphi} + \\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial V }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} V + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\theta} -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} U -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\theta} + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} V + \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\varphi} -\\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\theta} + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\varphi} -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} U\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} U -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\varphi} + \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} V\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} V -\\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\varphi} + \\frac{2 \\, m}{r} \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\theta}\n", "\\end{math}" ], "text/plain": [ "Riem(g) = -64*m^4*e^(-1/2*r/m)/r^4 d/dU*dV*dU*dV + 64*m^4*e^(-1/2*r/m)/r^4 d/dU*dV*dV*dU - m/r d/dU*dth*dU*dth + m/r d/dU*dth*dth*dU - m*sin(th)^2/r d/dU*dph*dU*dph + m*sin(th)^2/r d/dU*dph*dph*dU - 64*m^4*e^(-1/2*r/m)/r^4 d/dV*dU*dU*dV + 64*m^4*e^(-1/2*r/m)/r^4 d/dV*dU*dV*dU - m/r d/dV*dth*dV*dth + m/r d/dV*dth*dth*dV - m*sin(th)^2/r d/dV*dph*dV*dph + m*sin(th)^2/r d/dV*dph*dph*dV + 32*m^4*e^(-1/2*r/m)/r^4 d/dth*dU*dU*dth - 32*m^4*e^(-1/2*r/m)/r^4 d/dth*dU*dth*dU - 32*m^4*e^(-1/2*r/m)/r^4 d/dth*dV*dV*dth + 32*m^4*e^(-1/2*r/m)/r^4 d/dth*dV*dth*dV + 2*m*sin(th)^2/r d/dth*dph*dth*dph - 2*m*sin(th)^2/r d/dth*dph*dph*dth + 32*m^4*e^(-1/2*r/m)/r^4 d/dph*dU*dU*dph - 32*m^4*e^(-1/2*r/m)/r^4 d/dph*dU*dph*dU - 32*m^4*e^(-1/2*r/m)/r^4 d/dph*dV*dV*dph + 32*m^4*e^(-1/2*r/m)/r^4 d/dph*dV*dph*dV - 2*m/r d/dph*dth*dth*dph + 2*m/r d/dph*dth*dph*dth" ] }, "execution_count": 112, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.riemann().display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 113, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, U} \\, V \\, U \\, V }^{ \\, U \\phantom{\\, V} \\phantom{\\, U} \\phantom{\\, V} } & = & -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, U} \\, {\\theta} \\, U \\, {\\theta} }^{ \\, U \\phantom{\\, {\\theta}} \\phantom{\\, U} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, U} \\, {\\varphi} \\, U \\, {\\varphi} }^{ \\, U \\phantom{\\, {\\varphi}} \\phantom{\\, U} \\phantom{\\, {\\varphi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, V} \\, U \\, U \\, V }^{ \\, V \\phantom{\\, U} \\phantom{\\, U} \\phantom{\\, V} } & = & -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, V} \\, {\\theta} \\, V \\, {\\theta} }^{ \\, V \\phantom{\\, {\\theta}} \\phantom{\\, V} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, V} \\, {\\varphi} \\, V \\, {\\varphi} }^{ \\, V \\phantom{\\, {\\varphi}} \\phantom{\\, V} \\phantom{\\, {\\varphi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, U \\, U \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, U} \\phantom{\\, U} \\phantom{\\, {\\theta}} } & = & \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, V \\, V \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, V} \\phantom{\\, V} \\phantom{\\, {\\theta}} } & = & -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\varphi} \\, {\\theta} \\, {\\varphi} }^{ \\, {\\theta} \\phantom{\\, {\\varphi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\varphi}} \\, U \\, U \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, U} \\phantom{\\, U} \\phantom{\\, {\\varphi}} } & = & \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\varphi}} \\, V \\, V \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, V} \\phantom{\\, V} \\phantom{\\, {\\varphi}} } & = & -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\varphi}} \\, {\\theta} \\, {\\theta} \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\varphi}} } & = & -\\frac{2 \\, m}{r} \\end{array}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, U} \\, V \\, U \\, V }^{ \\, U \\phantom{\\, V} \\phantom{\\, U} \\phantom{\\, V} } & = & -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, U} \\, {\\theta} \\, U \\, {\\theta} }^{ \\, U \\phantom{\\, {\\theta}} \\phantom{\\, U} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, U} \\, {\\varphi} \\, U \\, {\\varphi} }^{ \\, U \\phantom{\\, {\\varphi}} \\phantom{\\, U} \\phantom{\\, {\\varphi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, V} \\, U \\, U \\, V }^{ \\, V \\phantom{\\, U} \\phantom{\\, U} \\phantom{\\, V} } & = & -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, V} \\, {\\theta} \\, V \\, {\\theta} }^{ \\, V \\phantom{\\, {\\theta}} \\phantom{\\, V} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, V} \\, {\\varphi} \\, V \\, {\\varphi} }^{ \\, V \\phantom{\\, {\\varphi}} \\phantom{\\, V} \\phantom{\\, {\\varphi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, U \\, U \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, U} \\phantom{\\, U} \\phantom{\\, {\\theta}} } & = & \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, V \\, V \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, V} \\phantom{\\, V} \\phantom{\\, {\\theta}} } & = & -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\varphi} \\, {\\theta} \\, {\\varphi} }^{ \\, {\\theta} \\phantom{\\, {\\varphi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\varphi}} \\, U \\, U \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, U} \\phantom{\\, U} \\phantom{\\, {\\varphi}} } & = & \\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\varphi}} \\, V \\, V \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, V} \\phantom{\\, V} \\phantom{\\, {\\varphi}} } & = & -\\frac{32 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\varphi}} \\, {\\theta} \\, {\\theta} \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\varphi}} } & = & -\\frac{2 \\, m}{r} \\end{array}\n", "\\end{math}" ], "text/plain": [ "Riem(g)^U_V,U,V = -64*m^4*e^(-1/2*r/m)/r^4 \n", "Riem(g)^U_th,U,th = -m/r \n", "Riem(g)^U_ph,U,ph = -m*sin(th)^2/r \n", "Riem(g)^V_U,U,V = -64*m^4*e^(-1/2*r/m)/r^4 \n", "Riem(g)^V_th,V,th = -m/r \n", "Riem(g)^V_ph,V,ph = -m*sin(th)^2/r \n", "Riem(g)^th_U,U,th = 32*m^4*e^(-1/2*r/m)/r^4 \n", "Riem(g)^th_V,V,th = -32*m^4*e^(-1/2*r/m)/r^4 \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_U,U,ph = 32*m^4*e^(-1/2*r/m)/r^4 \n", "Riem(g)^ph_V,V,ph = -32*m^4*e^(-1/2*r/m)/r^4 \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 113, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.riemann().display_comp(X_KS_I.frame(), only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The curvature 2-form $\\Omega^0_{\\ \\, 1}$ associated to the Kruskal-Szekeres coordinate frame:

" ] }, { "cell_type": "code", "execution_count": 114, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-form curvature (0,1) of connection nabla_g w.r.t. Coordinate frame (R_I, (d/dU,d/dV,d/dth,d/dph)) on the Open subset R_I of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "om = g.connection().curvature_form(0,1, X_KS_I.frame())\n", "print(om)" ] }, { "cell_type": "code", "execution_count": 115, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Omega^0_{\\ \\, 1} = -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\mathrm{d} U\\wedge \\mathrm{d} V$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Omega^0_{\\ \\, 1} = -\\frac{64 \\, m^{4} e^{\\left(-\\frac{r}{2 \\, m}\\right)}}{r^{4}} \\mathrm{d} U\\wedge \\mathrm{d} V\n", "\\end{math}" ], "text/plain": [ "curvature (0,1) of connection nabla_g w.r.t. Coordinate frame (R_I, (d/dU,d/dV,d/dth,d/dph)) = -64*m^4*e^(-1/2*r/m)/r^4 dU/\\dV" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "om.display(X_KS_I.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Isotropic coordinates

\n", "

Let us now introduce isotropic coordinates $(t,\\bar{r},\\theta,\\varphi)$ on the spacetime manifold:

" ] }, { "cell_type": "code", "execution_count": 116, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}}\n", "\\end{math}" ], "text/plain": [ "Open subset R_I_union_R_III of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 116, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_III = regI.union(regIII) ; regI_III" ] }, { "cell_type": "code", "execution_count": 117, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R_I_union_R_III, (t, ri, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I_union_R_III, (t, ri, th, ph))" ] }, "execution_count": 117, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_iso. = regI_III.chart(r't ri:(0,+oo):\\bar{r} th:(0,pi):\\theta ph:(0,2*pi):\\varphi')\n", "print(X_iso) ; X_iso" ] }, { "cell_type": "code", "execution_count": 118, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "text/plain": [ "Chart (R_I, (t, ri, th, ph))" ] }, "execution_count": 118, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_iso_I = X_iso.restrict(regI, ri>m/2) ; X_iso_I" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The transformation from the isotropic coordinates to the Boyer-Lindquist ones:

" ] }, { "cell_type": "code", "execution_count": 119, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (t, ri, th, ph)) to Chart (R_I, (t, r, th, ph))\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ r & = & \\frac{1}{4} \\, {\\bar{r}} {\\left(\\frac{m}{{\\bar{r}}} + 2\\right)}^{2} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ r & = & \\frac{1}{4} \\, {\\bar{r}} {\\left(\\frac{m}{{\\bar{r}}} + 2\\right)}^{2} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "t = t\n", "r = 1/4*ri*(m/ri + 2)^2\n", "th = th\n", "ph = ph" ] }, "execution_count": 119, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assume(2*ri>m) # we consider only region I\n", "ch_iso_BL_I = X_iso_I.transition_map(X_I, [t, ri*(1+m/(2*ri))^2, th, ph])\n", "print(ch_iso_BL_I)\n", "ch_iso_BL_I.display()" ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " t == t *passed*\n", " ri == ri *passed*\n", " th == th *passed*\n", " ph == ph *passed*\n", " t == t *passed*\n", " r == r *passed*\n", " th == th *passed*\n", " ph == ph *passed*\n" ] } ], "source": [ "assume(r>2*m) # we consider only region I\n", "ch_iso_BL_I.set_inverse(t, (r-m+sqrt(r*(r-2*m)))/2, th, ph, verbose=True)" ] }, { "cell_type": "code", "execution_count": 121, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ {\\bar{r}} & = & -\\frac{1}{2} \\, m + \\frac{1}{2} \\, r + \\frac{1}{2} \\, \\sqrt{-{\\left(2 \\, m - r\\right)} r} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ {\\bar{r}} & = & -\\frac{1}{2} \\, m + \\frac{1}{2} \\, r + \\frac{1}{2} \\, \\sqrt{-{\\left(2 \\, m - r\\right)} r} \\\\ {\\theta} & = & {\\theta} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "text/plain": [ "t = t\n", "ri = -1/2*m + 1/2*r + 1/2*sqrt(-(2*m - r)*r)\n", "th = th\n", "ph = ph" ] }, "execution_count": 121, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_iso_BL_I.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

At this stage, 11 charts have been defined on the manifold $\\mathcal{M}$:

" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Chart (R_I_union_R_II, (t, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph)),\n", " Chart (R_I_union_R_II, (v, r, th, ph)),\n", " Chart (R_I, (v, r, th, ph)),\n", " Chart (R_II, (v, r, th, ph)),\n", " Chart (R_I_union_R_II_union_R_III_union_R_IV, (U, V, th, ph)),\n", " Chart (R_I, (U, V, th, ph)),\n", " Chart (R_II, (U, V, th, ph)),\n", " Chart (R_I_union_R_III, (t, ri, th, ph)),\n", " Chart (R_I, (t, ri, th, ph))]" ] }, "execution_count": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 123, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}11$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}11\n", "\\end{math}" ], "text/plain": [ "11" ] }, "execution_count": 123, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(M.atlas())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

12 vector frames have been defined on $\\mathcal{M}$: 11 coordinate bases and the tetrad $(e_\\alpha)$:

" ] }, { "cell_type": "code", "execution_count": 124, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(e_{0},e_{1},e_{2},e_{3}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial {\\bar{r}} },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial {\\bar{r}} },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right)\\right]$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial v },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(e_{0},e_{1},e_{2},e_{3}\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}\\cup \\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{II}}, \\left(\\frac{\\partial}{\\partial U },\\frac{\\partial}{\\partial V },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial {\\bar{r}} },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right), \\left(\\mathcal{R}_{\\mathrm{I}}, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial {\\bar{r}} },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\varphi} }\\right)\\right)\\right]\n", "\\end{math}" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Vector frame (R_I, (e_0,e_1,e_2,e_3)),\n", " Coordinate frame (R_I_union_R_II_union_R_III_union_R_IV, (d/dU,d/dV,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dU,d/dV,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dU,d/dV,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_III, (d/dt,d/dri,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dri,d/dth,d/dph))]" ] }, "execution_count": 124, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 125, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}12$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}12\n", "\\end{math}" ], "text/plain": [ "12" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(M.frames())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The components of the metric tensor in terms of the isotropic coordinates are given by

" ] }, { "cell_type": "code", "execution_count": 126, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -\\frac{m^{2} - 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{4} + 8 \\, m^{3} {\\bar{r}} + 24 \\, m^{2} {\\bar{r}}^{2} + 32 \\, m {\\bar{r}}^{3} + 16 \\, {\\bar{r}}^{4}}{16 \\, {\\bar{r}}^{4}} \\right) \\mathrm{d} {\\bar{r}}\\otimes \\mathrm{d} {\\bar{r}} + \\left( \\frac{m^{4} + 8 \\, m^{3} {\\bar{r}} + 24 \\, m^{2} {\\bar{r}}^{2} + 32 \\, m {\\bar{r}}^{3} + 16 \\, {\\bar{r}}^{4}}{16 \\, {\\bar{r}}^{2}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{{\\left(m^{4} + 8 \\, m^{3} {\\bar{r}} + 24 \\, m^{2} {\\bar{r}}^{2} + 32 \\, m {\\bar{r}}^{3} + 16 \\, {\\bar{r}}^{4}\\right)} \\sin\\left({\\theta}\\right)^{2}}{16 \\, {\\bar{r}}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -\\frac{m^{2} - 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{m^{4} + 8 \\, m^{3} {\\bar{r}} + 24 \\, m^{2} {\\bar{r}}^{2} + 32 \\, m {\\bar{r}}^{3} + 16 \\, {\\bar{r}}^{4}}{16 \\, {\\bar{r}}^{4}} \\right) \\mathrm{d} {\\bar{r}}\\otimes \\mathrm{d} {\\bar{r}} + \\left( \\frac{m^{4} + 8 \\, m^{3} {\\bar{r}} + 24 \\, m^{2} {\\bar{r}}^{2} + 32 \\, m {\\bar{r}}^{3} + 16 \\, {\\bar{r}}^{4}}{16 \\, {\\bar{r}}^{2}} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{{\\left(m^{4} + 8 \\, m^{3} {\\bar{r}} + 24 \\, m^{2} {\\bar{r}}^{2} + 32 \\, m {\\bar{r}}^{3} + 16 \\, {\\bar{r}}^{4}\\right)} \\sin\\left({\\theta}\\right)^{2}}{16 \\, {\\bar{r}}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\n", "\\end{math}" ], "text/plain": [ "g = -(m^2 - 4*m*ri + 4*ri^2)/(m^2 + 4*m*ri + 4*ri^2) dt*dt + 1/16*(m^4 + 8*m^3*ri + 24*m^2*ri^2 + 32*m*ri^3 + 16*ri^4)/ri^4 dri*dri + 1/16*(m^4 + 8*m^3*ri + 24*m^2*ri^2 + 32*m*ri^3 + 16*ri^4)/ri^2 dth*dth + 1/16*(m^4 + 8*m^3*ri + 24*m^2*ri^2 + 32*m*ri^3 + 16*ri^4)*sin(th)^2/ri^2 dph*dph" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The $g_{00}$ component can be factorized:

" ] }, { "cell_type": "code", "execution_count": 127, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{m^{2} - 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{m^{2} - 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}}\n", "\\end{math}" ], "text/plain": [ "-(m^2 - 4*m*ri + 4*ri^2)/(m^2 + 4*m*ri + 4*ri^2)" ] }, "execution_count": 127, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_iso_I.frame(), 0,0, X_iso_I]" ] }, { "cell_type": "code", "execution_count": 128, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(m - 2 \\, {\\bar{r}}\\right)}^{2}}{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{2}}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(m - 2 \\, {\\bar{r}}\\right)}^{2}}{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{2}}\n", "\\end{math}" ], "text/plain": [ "-(m - 2*ri)^2/(m + 2*ri)^2" ] }, "execution_count": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_iso_I.frame(), 0,0, X_iso_I].factor()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us also factor the other components:" ] }, { "cell_type": "code", "execution_count": 129, "metadata": {}, "outputs": [], "source": [ "for i in range(1,4):\n", " g[X_iso_I.frame(), i,i, X_iso_I].factor()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The output of the `display()` command looks nicer:" ] }, { "cell_type": "code", "execution_count": 130, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -\\frac{{\\left(m - 2 \\, {\\bar{r}}\\right)}^{2}}{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{4}}{16 \\, {\\bar{r}}^{4}} \\mathrm{d} {\\bar{r}}\\otimes \\mathrm{d} {\\bar{r}} + \\frac{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{4}}{16 \\, {\\bar{r}}^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{4} \\sin\\left({\\theta}\\right)^{2}}{16 \\, {\\bar{r}}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = -\\frac{{\\left(m - 2 \\, {\\bar{r}}\\right)}^{2}}{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{2}} \\mathrm{d} t\\otimes \\mathrm{d} t + \\frac{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{4}}{16 \\, {\\bar{r}}^{4}} \\mathrm{d} {\\bar{r}}\\otimes \\mathrm{d} {\\bar{r}} + \\frac{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{4}}{16 \\, {\\bar{r}}^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{{\\left(m + 2 \\, {\\bar{r}}\\right)}^{4} \\sin\\left({\\theta}\\right)^{2}}{16 \\, {\\bar{r}}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\n", "\\end{math}" ], "text/plain": [ "g = -(m - 2*ri)^2/(m + 2*ri)^2 dt*dt + 1/16*(m + 2*ri)^4/ri^4 dri*dri + 1/16*(m + 2*ri)^4/ri^2 dth*dth + 1/16*(m + 2*ri)^4*sin(th)^2/ri^2 dph*dph" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Expression of the tetrad associated with the static observer in terms of the isotropic coordinate basis:

" ] }, { "cell_type": "code", "execution_count": 131, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\left( -\\frac{m + 2 \\, {\\bar{r}}}{m - 2 \\, {\\bar{r}}} \\right) \\frac{\\partial}{\\partial t }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{0} = \\left( -\\frac{m + 2 \\, {\\bar{r}}}{m - 2 \\, {\\bar{r}}} \\right) \\frac{\\partial}{\\partial t }\n", "\\end{math}" ], "text/plain": [ "e_0 = -(m + 2*ri)/(m - 2*ri) d/dt" ] }, "execution_count": 131, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 132, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1} = \\left( \\frac{4 \\, {\\bar{r}}^{2}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\frac{\\partial}{\\partial {\\bar{r}} }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1} = \\left( \\frac{4 \\, {\\bar{r}}^{2}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\frac{\\partial}{\\partial {\\bar{r}} }\n", "\\end{math}" ], "text/plain": [ "e_1 = 4*ri^2/(m^2 + 4*m*ri + 4*ri^2) d/dri" ] }, "execution_count": 132, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[1].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 133, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{2} = \\left( \\frac{4 \\, {\\bar{r}}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\frac{\\partial}{\\partial {\\theta} }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{2} = \\left( \\frac{4 \\, {\\bar{r}}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\frac{\\partial}{\\partial {\\theta} }\n", "\\end{math}" ], "text/plain": [ "e_2 = 4*ri/(m^2 + 4*m*ri + 4*ri^2) d/dth" ] }, "execution_count": 133, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[2].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 134, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{3} = \\frac{4 \\, {\\bar{r}}}{{\\left(m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}\\right)} \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\varphi} }$ " ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{3} = \\frac{4 \\, {\\bar{r}}}{{\\left(m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}\\right)} \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\varphi} }\n", "\\end{math}" ], "text/plain": [ "e_3 = 4*ri/((m^2 + 4*m*ri + 4*ri^2)*sin(th)) d/dph" ] }, "execution_count": 134, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[3].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 135, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Total elapsed time: 253.3007331409999 s\n" ] } ], "source": [ "print(\"Total elapsed time: {} s\".format(time.perf_counter() - comput_time0))" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2.beta6", "language": "sage", "name": "sagemath" }, "language": "python", "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 1 }