{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Schwarzschild spacetime\n", "\n", "This worksheet demonstrates a few capabilities of SageManifolds (version 0.9.1) in computations regarding Schwarzschild spacetime.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v0.9.1/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": [ "First we set up the notebook to display mathematical objects using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 1, "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": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/plain": [ "4-dimensional differentiable manifold M" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', r'\\mathcal{M}') ; M" ] }, { "cell_type": "code", "execution_count": 3, "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": 4, "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": 4, "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": 5, "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": 6, "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": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_II = regI.union(regII) ; regI_II" ] }, { "cell_type": "code", "execution_count": 7, "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": 8, "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": 8, "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": 9, "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": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_I = X.restrict(regI, r>2*m) ; X_I" ] }, { "cell_type": "code", "execution_count": 10, "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": 10, "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": 11, "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": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 12, "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": 12, "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": 13, "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": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 14, "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": 15, "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": 15, "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": 16, "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": 17, "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": 18, "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": 18, "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": 19, "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": 19, "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": 20, "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": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.domain()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}$ " ], "text/plain": [ "4-dimensional differentiable manifold M" ] }, "execution_count": 21, "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": 22, "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": 23, "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": 23, "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": 24, "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": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab(g) == 0" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g} g = 0$ " ], "text/plain": [ "nabla_g(g) = 0" ] }, "execution_count": 26, "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": 27, "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": 27, "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": 28, "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": 29, "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": 30, "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": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 31, "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": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == 0" ] }, { "cell_type": "code", "execution_count": 33, "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": 33, "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": 34, "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": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R[:]" ] }, { "cell_type": "code", "execution_count": 35, "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": 35, "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": 36, "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": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 37, "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": 37, "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": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "execution_count": 38, "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": 39, "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": 40, "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": 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 -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": 42, "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": 43, "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": 44, "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": 44, "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": 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 = 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": 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['^{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": 47, "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": 47, "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": 48, "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": 49, "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": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_I) ; ch_BL_EF_I" ] }, { "cell_type": "code", "execution_count": 50, "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": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_I.display()" ] }, { "cell_type": "code", "execution_count": 51, "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": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF_I = X_EF.restrict(regI) ; X_EF_I" ] }, { "cell_type": "code", "execution_count": 52, "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": 53, "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": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_II) ; ch_BL_EF_II" ] }, { "cell_type": "code", "execution_count": 54, "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": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_II.display()" ] }, { "cell_type": "code", "execution_count": 55, "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": 55, "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": 56, "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": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default chart is 'BL':

" ] }, { "cell_type": "code", "execution_count": 57, "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": 57, "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": 58, "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": 59, "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": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_BL_I.display()" ] }, { "cell_type": "code", "execution_count": 60, "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": 61, "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": 61, "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": 62, "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": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default frame is:

" ] }, { "cell_type": "code", "execution_count": 63, "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": 63, "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": 64, "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": 64, "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": 65, "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": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 66, "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": 66, "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": 67, "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": 67, "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": 68, "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": 68, "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": 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[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": 70, "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": 70, "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": 71, "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": 71, "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": 72, "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": 72, "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": 73, "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": 74, "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": 74, "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": 75, "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": 76, "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": 76, "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": 77, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1$ " ], "text/plain": [ "-1" ] }, "execution_count": 77, "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": 78, "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": 79, "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": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[e,:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

or, equivalently,

" ] }, { "cell_type": "code", "execution_count": 80, "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": 80, "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": 81, "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": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_EF_I.frame())" ] }, { "cell_type": "code", "execution_count": 82, "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": 82, "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": 83, "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": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e.structure_coef()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 84, "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": 84, "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": 85, "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": 85, "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": 86, "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": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M0 = regI.union(regII).union(regIII).union(regIV) ; M0" ] }, { "cell_type": "code", "execution_count": 87, "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": 87, "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": 88, "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": 88, "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": 89, "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": 89, "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": 90, "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": 90, "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": 91, "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": 91, "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": 92, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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+N2uWKso4ZszzB7hiBQweDImJsHixql30H/JGxzqXbC7s77OfPHZ5MNOZseTU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\neHigSJEicV4XFhaGwMDAWK9ZtWqViqpF0ppMmYBt24B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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)}, nb_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)}, nb_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": 93, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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mfMb//R98+SX88guEh0PbtuZPb2+YOvXfBZvr1832/Bs3zLlE//nPk3UzMtKc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IAAAg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\n3n6b88b69k3+Z728gF6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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": 94, "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": 94, "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": 95, "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": 95, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 96, "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": 96, "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": 97, "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)}, nb_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)}, nb_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": 98, "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": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}9$ " ], "text/plain": [ "9" ] }, "execution_count": 99, "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": 100, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{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), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{II}},(t, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{II}},(U, V, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right)\\right) : \\left(\\mathcal{R}_{\\mathrm{I}},(v, r, {\\theta}, {\\varphi})\\right) \\rightarrow \\left(\\mathcal{R}_{\\mathrm{I}},(t, r, {\\theta}, {\\varphi})\\right), \\left(\\left(\\mathcal{R}_{\\mathrm{II}},(v, r, {\\theta}, {\\varphi})\\right), \\left(\\mathcal{R}_{\\mathrm{II}},(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)\\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": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.coord_changes()" ] }, { "cell_type": "code", "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}8$ " ], "text/plain": [ "8" ] }, "execution_count": 101, "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": 102, "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": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 103, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}10$ " ], "text/plain": [ "10" ] }, "execution_count": 103, "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": 104, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}14$ " ], "text/plain": [ "14" ] }, "execution_count": 104, "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": 105, "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": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 106, "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": 106, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_KS_I.frame(),:]" ] }, { "cell_type": "code", "execution_count": 107, "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": 107, "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": 108, "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": 108, "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": 109, "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": 109, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.riemann().display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 110, "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": 110, "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": 111, "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": 112, "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": 112, "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": 113, "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": 113, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_III = regI.union(regIII) ; regI_III" ] }, { "cell_type": "code", "execution_count": 114, "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": 114, "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": 115, "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": 115, "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": 116, "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": 116, "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": 132, "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": 118, "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": 118, "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": 119, "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": 119, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 120, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}11$ " ], "text/plain": [ "11" ] }, "execution_count": 120, "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": 121, "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": 121, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 122, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}12$ " ], "text/plain": [ "12" ] }, "execution_count": 122, "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": 123, "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": 123, "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": 124, "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": 124, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_iso_I.frame(), 0,0, X_iso_I]" ] }, { "cell_type": "code", "execution_count": 125, "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": 125, "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": 126, "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": 127, "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": 127, "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": 128, "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": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 129, "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": 129, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[1].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 130, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_2 = \\left( \\frac{4 \\, {\\bar{r}}}{m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}} \\right) \\frac{\\partial}{\\partial {\\theta} }$ " ], "text/plain": [ "e_2 = 4*ri/(m^2 + 4*m*ri + 4*ri^2) d/dth" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[2].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 131, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_3 = \\frac{4 \\, {\\bar{r}}}{{\\left(m^{2} + 4 \\, m {\\bar{r}} + 4 \\, {\\bar{r}}^{2}\\right)} \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\varphi} }$ " ], "text/plain": [ "e_3 = 4*ri/((m^2 + 4*m*ri + 4*ri^2)*sin(th)) d/dph" ] }, "execution_count": 131, "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 7.4.beta1", "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": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }