{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Schwarzschild spacetime\n", "\n", "This worksheet demonstrates a few capabilities of SageManifolds (version 1.0, as included in SageMath 7.5) in computations regarding Schwarzschild spacetime.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.0/SM_Schwarzschild.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, via the command `sage -n jupyter`\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 7.5 is required to run this worksheet:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 7.5, Release Date: 2017-01-11'" ] }, "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": { "collapsed": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime manifold\n", "\n", "We declare the Schwarzschild spacetime as a 4-dimensional differentiable manifold:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/plain": [ "4-dimensional differentiable manifold M" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', r'\\mathcal{M}') ; M" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional differentiable 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": 5, "metadata": { "collapsed": false }, "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/plain": [ "(Open subset R_I of the 4-dimensional differentiable manifold M,\n", " Open subset R_II of the 4-dimensional differentiable manifold M,\n", " Open subset R_III of the 4-dimensional differentiable manifold M,\n", " Open subset R_IV of the 4-dimensional differentiable manifold M)" ] }, "execution_count": 5, "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": 6, "metadata": { "collapsed": false }, "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": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}$ " ], "text/plain": [ "Open subset R_I_union_R_II of the 4-dimensional differentiable manifold M" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_II = regI.union(regII) ; regI_II" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "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": 9, "metadata": { "collapsed": false }, "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/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 9, "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": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (R_I, (t, r, th, ph))" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_I = X.restrict(regI, r>2*m) ; X_I" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\phi})\\right)$ " ], "text/plain": [ "Chart (R_II, (t, r, th, ph))" ] }, "execution_count": 11, "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": 12, "metadata": { "collapsed": false }, "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/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": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "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/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 13, "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": 14, "metadata": { "collapsed": false }, "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/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": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "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": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}$ " ], "text/plain": [ "Open subset R_I_union_R_II of the 4-dimensional differentiable manifold M" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame().domain()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Metric tensor

\n", "

The metric tensor is defined as follows:

" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Lorentzian metric g on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "g = M.lorentzian_metric('g')\n", "print(g)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The metric tensor is set by its components in the coordinate frame associated with Schwarzschild coordinates, which is the current manifold's default frame:

" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "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": 19, "metadata": { "collapsed": false }, "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/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": 19, "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": 20, "metadata": { "collapsed": false }, "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/plain": [ "v = d/dt + (-2*m/r + 1) d/dr" ] }, "execution_count": 20, "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": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}$ " ], "text/plain": [ "Open subset R_I of the 4-dimensional differentiable manifold M" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.domain()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/plain": [ "4-dimensional differentiable manifold M" ] }, "execution_count": 22, "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": 23, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(v,v) on the Open subset R_I of the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "s = g(v,v) ; print(s)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "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/plain": [ "g(v,v): R_I --> R\n", " (t, r, th, ph) |--> 0" ] }, "execution_count": 24, "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": 25, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "nab = g.connection() ; 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": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab(g) == 0" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} g = 0$ " ], "text/plain": [ "nabla_g(g) = 0" ] }, "execution_count": 27, "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": 28, "metadata": { "collapsed": false }, "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/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": 28, "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": 29, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "R = g.riemann() ; print(R)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Weyl conformal tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field C(g) of type (1,3) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "C = g.weyl() ; print(C)" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "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/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": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "Ric = g.ricci() ; 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": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == 0" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = 0$ " ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 34, "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": 35, "metadata": { "collapsed": false }, "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/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": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R[:]" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "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/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": 36, "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": 37, "metadata": { "collapsed": false }, "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/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": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "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/plain": [ "[0 0 0 0]\n", "[0 0 0 0]\n", "[0 0 0 0]\n", "[0 0 0 0]" ] }, "execution_count": 38, "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": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 39, "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": 40, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(Riem(g)) of type (1,4) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "DR = nab(R) ; print(DR)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "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\n" ] } ], "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] ," ] }, { "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": 42, "metadata": { "collapsed": false }, "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\n" ] } ], "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] ," ] }, { "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": 43, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (0,4) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "dR = R.down(g) ; 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": 44, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (4,0) on the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "uR = R.up(g) ; print(uR)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}$ " ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 45, "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 contract() method, asking that the contraction takes place on all indices (positions 0, 1, 2, 3):

" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}$ " ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 46, "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": 47, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{48 \\, m^{2}}{r^{6}}$ " ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 47, "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": 48, "metadata": { "collapsed": false }, "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/plain": [ "Chart (R_I_union_R_II, (v, r, th, ph))" ] }, "execution_count": 48, "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": 49, "metadata": { "collapsed": false }, "outputs": [], "source": [ "ch_BL_EF_I = X_I.transition_map(X_EF, [t+r+2*m*ln(r/(2*m)-1), r, th, ph], restrictions2=r>2*m)" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "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/plain": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (v, r, th, ph))" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_I) ; ch_BL_EF_I" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "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/plain": [ "v = 2*m*log(1/2*r/m - 1) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_I.display()" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/plain": [ "Chart (R_I, (v, r, th, ph))" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF_I = X_EF.restrict(regI) ; X_EF_I" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [], "source": [ "ch_BL_EF_II = X_II.transition_map(X_EF, [t+r+2*m*ln(1-r/(2*m)), r, th, ph], restrictions2=r<2*m)" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "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/plain": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (v, r, th, ph))" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_II) ; ch_BL_EF_II" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "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/plain": [ "v = 2*m*log(-1/2*r/m + 1) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_II.display()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right)$ " ], "text/plain": [ "Chart (R_II, (v, r, th, ph))" ] }, "execution_count": 56, "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": 57, "metadata": { "collapsed": false }, "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/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": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default chart is 'BL':

" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "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/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 58, "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": 59, "metadata": { "collapsed": false }, "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() ; print(ch_EF_BL_I)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "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/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": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_BL_I.display()" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "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() ; print(ch_EF_BL_II)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "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/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": 62, "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": 63, "metadata": { "collapsed": false }, "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/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": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default frame is:

" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "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/plain": [ "Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph))" ] }, "execution_count": 64, "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": 65, "metadata": { "collapsed": false }, "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/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": 65, "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": 66, "metadata": { "collapsed": false }, "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/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": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "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/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": 67, "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": 68, "metadata": { "collapsed": false }, "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/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": 68, "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": 69, "metadata": { "collapsed": false }, "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/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": 69, "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": 70, "metadata": { "collapsed": false }, "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/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[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": 71, "metadata": { "collapsed": false }, "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/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": 71, "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": 72, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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DVL9+/XSfvEaBVasQmTZzppprZ/BgouPHiXbtIrr9dvlx//7rfik8+aTaXE+f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\neYAAAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 18 graphics primitives" ] }, "execution_count": 72, "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}, 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": 73, "metadata": { "collapsed": false }, "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/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": 73, "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": 74, "metadata": { "collapsed": false }, "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') ; 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": 75, "metadata": { "collapsed": false }, "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/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": 75, "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": 76, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field e_0 on the Open subset R_I of the 4-dimensional differentiable manifold M\n" ] } ], "source": [ "print(e[0])" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false }, "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/plain": [ "e_0 = 1/sqrt(-2*m/r + 1) d/dt" ] }, "execution_count": 77, "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": 78, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1$ " ], "text/plain": [ "-1" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(e[0],e[0]).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "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 g(e[i],e[j]).expr() , \n", " print \" \"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Another view of the above result:

" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false }, "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/plain": [ "[-1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[e,:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

or, equivalently,

" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false }, "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/plain": [ "g = -e^0*e^0 + e^1*e^1 + e^2*e^2 + e^3*e^3" ] }, "execution_count": 81, "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": 82, "metadata": { "collapsed": false }, "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/plain": [ "e_0 = sqrt(r)/sqrt(-2*m + r) d/dv" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_EF_I.frame())" ] }, { "cell_type": "code", "execution_count": 83, "metadata": { "collapsed": false }, "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/plain": [ "e_1 = sqrt(r)/sqrt(-2*m + r) d/dv + sqrt(-2*m + r)/sqrt(r) d/dr" ] }, "execution_count": 83, "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": 84, "metadata": { "collapsed": false }, "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/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": 84, "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": 85, "metadata": { "collapsed": false }, "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/plain": [ "m*sqrt(-2*m + r)/((2*m*r - r^2)*sqrt(r)) e_0" ] }, "execution_count": 85, "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": 86, "metadata": { "collapsed": false }, "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/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": 86, "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": 87, "metadata": { "collapsed": false }, "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/plain": [ "Open subset R_I_union_R_II_union_R_III_union_R_IV of the 4-dimensional differentiable manifold M" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M0 = regI.union(regII).union(regIII).union(regIV) ; M0" ] }, { "cell_type": "code", "execution_count": 88, "metadata": { "collapsed": false }, "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/plain": [ "Chart (R_I_union_R_II_union_R_III_union_R_IV, (U, V, th, ph))" ] }, "execution_count": 88, "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": 89, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/plain": [ "Chart (R_I, (U, V, th, ph))" ] }, "execution_count": 89, "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": 90, "metadata": { "collapsed": false }, "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/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": 90, "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)), th, ph])\n", "print(ch_BL_KS_I)\n", "ch_BL_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)$ " ], "text/plain": [ "Chart (R_II, (U, V, th, ph))" ] }, "execution_count": 91, "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": 92, "metadata": { "collapsed": false }, "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/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": 92, "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)), 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": 93, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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mvjd9uirKOHz4ywe4dCn06wcJCbBggapd9B/yTsc6lywu7PHeQy67XJjpzFh4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\nt2frgFWrWNj98cesgwoMtP3+KYxqiETEZqo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"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": 94, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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w9dcwfDiEhUHr1uZPb2+YPPnfBZurV832/GvXzLlEr7/+ZN2MiDDnG23e/GSv\nS2IvRSACuOR/iapTqnIr+BaCMLzucHp596JstrKsbLuS5aeX02ZhG7Knzo5fsB8RtggaFGxA0Yyr\n+a1eQndsAcwAnJK2I0qlQBqIlHpSC4C2QGSc629y6MZAano1J8IWAcDGjhsplqkYIkK3ld2YeWQm\nHpk8cHV0Ze+8vfHDUJkypvSGpyd8/70pzRERYbbZb9hgnjh7NrRs+f+befKk2VovYkpxFCv2ZN0U\ngfffh1mzIPzFLrSeoqfM/im3e258OvuQyTUTFiz0XNeTX2v/yt5re2k6pylNizRlUpNJXAu8hruz\nO85WZ1afXc1JvwZcuPcz8T+KBUAjICjpO6OUUuolNh54j7hh6GZQOXZc7kuVKY2JlmiibdGsbLuS\nCjkq4Oboxtxjc5l5ZCals5QmjVMa9s5NIAyVK2cCjKcn/PSTCUPR0Sb8rF9vpsyWLv13YWjnTlOK\nI3Vqc/r0k4YhMAuvp041o1EvuJcmEAHkSZsH386+ZHI1Bei+Xf8tv9b+lR1XdtBgZgMaFmrIrOaz\nuBNyBxerC26Obqw+u5qNF3JjAlDcbYPeQC3AL2k7opRS6iUkwC+YchyxJ2duBdXkSsAoak1rjLPV\nmdCoUJa0XkLVPGZdh9dBL7ou78qHZT7EzdGNnXN2Eu0dJwyVL28Kqw4caGqU9esHNhu0awfLl5ut\n8atXQ8OG/7+pq1aZkh7FisGWLZAz55N39/Zt+PVXE8zatXvy1yexlyoQgQlFPp19yJgqIzax0cu7\nF4NqDsL3oi9F/ixCzXw1mf/efALDA7FgIYNLBt7K9RZmHnc1ELdmy26gCnA5qbuilFLqpWEDegA/\nxnvklF8tMruto2jG1yiUvhB3Q+8yr8U86haoC8DsI7N5f9n7dHytI6fvnGb77O3YvGOq3g+ys6NX\nxYqmQOvgwWZU6KefYqar5s0zJTg2boRatf5/U7/5xkyv1aljRpXSpXu6LmfKZGqc9e37dK9PYi9d\nIALImzYvvl18yZgqI9ESzY+bfqTHGz3wD/enyJ9FqJyzMkvbLCUiOoLQqFDqzazHlYArQE1gE5Ax\nzh1PAm/+70+l1KO0bt2aJk2aMHv27ORuilIvkEigMzAi3iN9fWDusSqI2PHt+m85cusI05tNp2nR\npgDMPzaf9ovb08ajDRf9L7J11tb4YeiNN6BCBbNwevhwU19MxOwkmzbNFGfdujWm+OqjiJhiryNG\nwGuvwcL9tcMsAAAgAElEQVSFj69l9m/kzZtizil6aRZVJ+T8vfNUnVoVvxA/HOwc6FquK8N3DCeN\nUxpOfHaCk34n+X7999wIvoGrgyu+XXzJ7JoZOIU5ICvuqFBGzChS+aTuilIvNF1UrdSjhGLWC62I\n98jnqyGr6wB6v9Wb772/Z9iOYbT2aM3sd80vFItPLKbl/Ja8W/xdbgXdwnemb/wwVLkyeHjAuHEw\napTZXg9w+TKULGlCzvbtUKLE45v5YGptzhwoXBiOHgUHh0T6DFKGl3KE6IF86fLh09mHDC4ZiLRF\nMmHfBHq+0ZOA8ACKjS5GkQxF2P3RbjZ12oR/uD91ptfhbuhdoAiwjeuBcYcJ/YAawMYk74tSSqmU\n5j5Qj7hhKMpmod0iKJx+JD9W/ZEBWwYwbMcwAHKmzomIMP3QdFotaEXTok3xC/ZLOAy99RYULWrC\n0LhxMWHo/HmoVg3c3WHfvv8fhiIioGlTE4by5zfTXK9YGIKXPBAB5E+XH98uvqR3SU+kLZK/9v/1\nMBQVH1Ocy/6XKZi+IN4dvLkacJUGMxsQEB6ASE4+XF6YvdfifkRBQANgUTL0RimlVMpwA6gObIl1\nNTzKnmZzLNQvMI3PK33OsO3D+HnzzwD8XO1nBtceTPO5zem4pCO189cmICyATTM2xQ9DVatCvnzm\nxOhJk8z0GMCpU1CliinN4esLBQs+vplBQeaMoBUrzMLpvXuffZoshXrpAxH8LxR1NqEoPDqcifsn\n0uONHgSEB1BzWk1sYqNE5hKs67COk34naTy7MSGRIVTIXp9uKwqz4VzcjykCaAlMTIbeKKWUerGd\nA94CDsW6Ghhupd50CxWy/0yHUh0Yu2cs3643p0QPrj2YPtX68M7cd1hyagklMpUgyhbFhukbEO+Y\nlS2D7OzoVa2aKa46fTp4eZmF0wCHD0PVqpA2rQlDefI8vpm3b5uRpC1bzALoHTuefgF1SMjTve4F\n8koEIoAC6Qvg09mHdM7pCIsOY9KBSfR4owfn7p3js5WfISKUzVaW1e1Ws/faXprMbsLkA5O5GRxM\nn01lWHA87kdlAz7C1D97aZdhKaWUeiKHMZtw/o51VSQbw7Y3xsWxNp4+ngzwHcCnqz4FYFT9UfSs\n3JOms5uy/PRyimcsTs7UOfGe5h07DFks9KpRw4SXOXNMbbEOHcyDu3dD9epmlGfzZsiW7fHNPH/e\nnDF04oRZdO3j83Rb68Fs5S9QAM6effRzrsctTfLieWUCEUDB9AXx7eJLWue0hEWFMeXAFH6u/jPj\n9o3jyzVfIiJUzlWZZa2Xse3yNgpnKIyz1Zm/711h9O43mbAvoZXyvYBvMQFJKaXUq2sbUA0zXRZD\npACRto0c97Nn/d/r+bzi5/TZ1AeASY0n0a1CNxrNasSKMysolrEYedLmYd20dfHDUO3aZl3QokUm\nELVpYx709YXatc2ZQRs3Qsa4O6XjOHTI7Diz2czus82bn+7QRYDjx6FVK7PLLV++hJ+zaxeUK/d0\n909Cr1Qggv+Fos4mFIVEhjBy10h+qvoTo3aPoue6nogItfLX4rs3v2PD+Q3kcc9DVresHLp5lIUn\najNwS0J3HQ68T/wj2JVSSr0aVgC1MQupY4iUIjRqHU3nfM2yU8vo/VZvxu4dS6UclZjZfCYdS3fk\n7Vlvs/rsaopmKEr+dPlZM3VN/DBUt65Z27N8OSxYAC1amAcHDjRnC1WoYEpruLs/vpk+PmZaLXt2\n2LYNypaFMmWerst37kDjxmZr/cyZZt1SXFOmmPfLm/fp3iMJvXKBCKBQhkL4dPbB3dmdkMgQxuwZ\nww9v/cBvO3/jhw0/ICJ8XvFz8qbNi/d5b3KmyUnRjEXZfnkHe640ZfqhhP7j8cIc7pjy51GVUko9\nCS/gHSAs1tXrgQUJCF9K/Rmd2XJxC7/U/IVhO4bxdqG38e3iS9uSbemxtgfr/l5HkfRFKJi+IKum\nrIofhurVMyU31qyBxYvNjjCAAQNM8dYMGWDJEjP19TiLFpmyHhUrmlGhLFmevsuRkSaUBQTAsmWm\nvEfcx7/80qxv6tgRNm16+vdKKvIKO+V3SrIMzSJO/Z0k45CM8t367wRPpM/GPiIicjfkruT/Pb/g\niTSf01xqe9UWp/5OsuTEEhHxEhF7ESHOV2URuZNMPVIqefj7+wsg/v7+yd0UpZLYEIn/7wBy6X5p\nuR18ScqOLytpf00r4/eOF9eBrlJ3el0JiwwTEZH91/ZL2l/TSrE/i0mTWU3EUssimEWpAsggi0Wk\nQQORunVFnJ1F1qyJeVtPTxEQyZJF5N98340dK2KxiLRuLRIe/uzd7tZNxGoV8fGJ/9jt2yI1apjH\nR48Wsdme/f2SwCs5QvRA4QyFH44UBYYHMuXAFL554xv6+faj+dzmnL9/noOfHKRg+oIsOrmITK6Z\naFasGS4OLkBHYCkQd3vidqAqcDWpu6OUUirJ2DDrR7+L98il+zXJ5b6HXVcOcz3wOuMajuO79d9R\nNltZFrdajJPViSM3j1Bneh0KpCtA4fSFWT5pObIhzshQw4YQFmbW+axcaUZ3APr0McVbM2eG06fh\ncYehipjndusGn39uprYc49btfEJjxsDYsearatXYjx0+bKbvjh6FDRvg009TzEnVr/QI0QMnbp+Q\nzEMzi1N/J8k8NLN8vupzwRNxHuAsu67skuCIYCk8qrDgiXRa3CnOq7eJSFqJ/xtCbhE5maT9UCq5\n6AiRerVEiEhHSWhkaOg2O5m0f+LDZ+65ukcyDskoFSZUEP8w8/1x7NYxyTQkk5QaW0qazW4m1CL+\nyFDjxiJVqoi4usYehend24wMZcokcu/e45sZFSXStat5vqdn4ozU2GwinTuLfPFF/MfmzRNJlUqk\ndGmRCxee/b2SmAai/zlx+4RkGpJJnPo7SZahWeTDZR8KnohTfyfZenGrhEWGSZFRRcyU2qY+cV59\nRESyS/xvjowisjtpO6JUMtBApF4dwSLSUBIKQz3WWqT1gtYSHmWmpLZf2i5ZhmaRkmNKyp0Qs5Ti\n5O2TkmVoFvEY7SHN5zSPH4ZApEkTkTfeEEmdWmTr1pi37tHDhJuMGc201OOEhoo0by5iby8ycKBI\n5swiS5cmzkdgs5mw9UB0tMgPP5i2tW4tEhycOO+TxF7pKbN/KpqxKD6dfUjtlJr7YfdZfmo5XUp3\nITw6nJpeNdlxZQeHPznMO0XeoZ9PPyYfmPyPV3sA2wkIyxrnrg9KfaxPsn4opZR6Xu5i6lyujHU1\nygYdF0NwxMfMaDYDR3tHdl7eSdWpVQmODGZ9h/Wkd0nP2btnqTmtJuld0lMkYxEW/bUINsTcZxDQ\n65134MYNs519/XpzVpAI/PCDKdyaLZuZlnrc1vr798302urVMHkyTJhgzi6KO731tCyWmB1l/v7Q\npAkMGgSDB8OsWZAqVeK8TxLTQPQPxTIVw6ezD26ObtwPu8+as2voWKojEbYI6k6vi+8lXxa1WsQn\n5T7hw2UfMufonH+8Og9j9rblwHVrnLsGAw2BOSillEqprmDWh26PdTUi2krTOZArzQ+MaTgGezt7\ndl7eSZWpVQBY024NTlYnBm8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NCROgY0cThipXBgeHhJsZFgatWpmpsokTzXb6f64rslhM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88aXGv37d1Quijxx94/sPzicli7W6Opf1O9qQtDRCZGwtrdGk7eTlAICowJGYPHKY9x5+kdDFgz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\nIcug6J9/YNuwgVmiHj2ACRP45m7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"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": 95, "metadata": { "collapsed": false }, "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/plain": [ "Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (U, V, th, ph))" ] }, "execution_count": 95, "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": 96, "metadata": { "collapsed": false }, "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/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": 96, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 97, "metadata": { "collapsed": false }, "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/plain": [ "Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (U, V, th, ph))" ] }, "execution_count": 97, "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": 98, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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RTpRescK8YUgpGDVKC0Jr1z4VYQgkEKVzd3Bn15Bd1CldBxSExoYyZvsY/Hv6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\nf7i8+Q03swizAAAAAElFTkSuQmCC\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)}, 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": 99, "metadata": { "collapsed": false }, "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/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": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}9$ " ], "text/plain": [ "9" ] }, "execution_count": 100, "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": 101, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\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), \\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{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}},(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{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{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{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}},(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)\\right\\}$ " ], "text/plain": [ "{(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)),\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, (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_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, (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_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, (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_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))}" ] }, "execution_count": 101, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.coord_changes()" ] }, { "cell_type": "code", "execution_count": 102, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}8$ " ], "text/plain": [ "8" ] }, "execution_count": 102, "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": 103, "metadata": { "collapsed": false }, "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/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": 103, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 104, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}10$ " ], "text/plain": [ "10" ] }, "execution_count": 104, "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": 105, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}14$ " ], "text/plain": [ "14" ] }, "execution_count": 105, "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": 106, "metadata": { "collapsed": false }, "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/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": 106, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "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/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": 107, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_KS_I.frame(),:]" ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": false }, "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/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_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": 109, "metadata": { "collapsed": false }, "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/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": 109, "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": 110, "metadata": { "collapsed": false }, "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/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": 110, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.riemann().display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 111, "metadata": { "collapsed": false }, "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/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": 111, "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": 112, "metadata": { "collapsed": false }, "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 differentiable manifold M\n" ] } ], "source": [ "om = g.connection().curvature_form(0,1, X_KS_I.frame()) ; print(om)" ] }, { "cell_type": "code", "execution_count": 113, "metadata": { "collapsed": false }, "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/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": 113, "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": 114, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{III}}$ " ], "text/plain": [ "Open subset R_I_union_R_III of the 4-dimensional differentiable manifold M" ] }, "execution_count": 114, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_III = regI.union(regIII) ; regI_III" ] }, { "cell_type": "code", "execution_count": 115, "metadata": { "collapsed": false }, "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/plain": [ "Chart (R_I_union_R_III, (t, ri, th, ph))" ] }, "execution_count": 115, "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": 116, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{R}_{\\mathrm{I}},(t, {\\bar{r}}, {\\theta}, {\\varphi})\\right)$ " ], "text/plain": [ "Chart (R_I, (t, ri, th, ph))" ] }, "execution_count": 116, "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": 117, "metadata": { "collapsed": false }, "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/plain": [ "t = t\n", "r = 1/4*ri*(m/ri + 2)^2\n", "th = th\n", "ph = ph" ] }, "execution_count": 117, "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": 118, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " t == t\n", " ri == ri\n", " th == th\n", " ph == ph\n", " t == t\n", " r == r\n", " th == th\n", " ph == ph\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": 119, "metadata": { "collapsed": false }, "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/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": 119, "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": 120, "metadata": { "collapsed": false }, "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/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": 120, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 121, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}11$ " ], "text/plain": [ "11" ] }, "execution_count": 121, "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": 122, "metadata": { "collapsed": false }, "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/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": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 123, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}12$ " ], "text/plain": [ "12" ] }, "execution_count": 123, "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": 124, "metadata": { "collapsed": false }, "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/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": 124, "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": 125, "metadata": { "collapsed": false }, "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/plain": [ "-(m^2 - 4*m*ri + 4*ri^2)/(m^2 + 4*m*ri + 4*ri^2)" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_iso_I.frame(), 0,0, X_iso_I]" ] }, { "cell_type": "code", "execution_count": 126, "metadata": { "collapsed": false }, "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/plain": [ "-(m - 2*ri)^2/(m + 2*ri)^2" ] }, "execution_count": 126, "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 factorize the other components:

" ] }, { "cell_type": "code", "execution_count": 127, "metadata": { "collapsed": false }, "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": 128, "metadata": { "collapsed": false }, "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/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": 128, "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": 129, "metadata": { "collapsed": false }, "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/plain": [ "e_0 = -(m + 2*ri)/(m - 2*ri) d/dt" ] }, "execution_count": 129, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 130, "metadata": { "collapsed": false }, "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/plain": [ "e_1 = 4*ri^2/(m^2 + 4*m*ri + 4*ri^2) d/dri" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[1].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 131, "metadata": { "collapsed": false }, "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/plain": [ "e_2 = 4*ri/(m^2 + 4*m*ri + 4*ri^2) d/dth" ] }, "execution_count": 131, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[2].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 132, "metadata": { "collapsed": false }, "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/plain": [ "e_3 = 4*ri/((m^2 + 4*m*ri + 4*ri^2)*sin(th)) d/dph" ] }, "execution_count": 132, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[3].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.5", "language": "", "name": "sagemath" }, "language": "python", "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 0 }