$\\mathcal{M}_0$ 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 (i.e. we exclude the past and furture event horizons from these regions). $\\mathcal{R}_{\\mathrm{I}}$ and $\\mathcal{R}_{\\mathrm{III}}$ are asymtotically flat regions outside the event horizons; $\\mathcal{R}_{\\mathrm{II}}$ 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": [ "" ], "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 = M0.open_subset('R_I', r'\\mathcal{R}_{\\mathrm{I}}')\n", "regII = M0.open_subset('R_II', r'\\mathcal{R}_{\\mathrm{II}}')\n", "regIII = M0.open_subset('R_III', r'\\mathcal{R}_{\\mathrm{III}}')\n", "regIV = M0.open_subset('R_IV', r'\\mathcal{R}_{\\mathrm{IV}}')\n", "regI, regII, regIII, regIV" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The mass parameter $m$ of Schwarzschild spacetime is declared as a symbolic real-valued variable:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "m = var('m', domain='real') ; assume(m>=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Kruskal-Szekeres coordinates $(T,X,\\theta,\\varphi)$ cover $\\mathcal{M}_0$ and are subject to the restrictions $T^2<1+X^2$:
" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (M0, (T, X, th, ph))" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XKS.The Kruskal-Szekeres chart ploted in terms of itself:
" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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BMQIAADAoRgAAAAbFCAAAwKAYAQAAGBQjANYUCgXl83mVSqWwRwGAptj9OUZAFAS3TuD2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\nAAAABsUIAADAoBgBAAAY/x/M43rD6BnQQQAAAABJRU5ErkJggg==\n", "text/plain": [ "Graphics object consisting of 68 graphics primitives" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XKS.plot(XKS, ambient_coords=(X,T), max_range=3, nb_values=25)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinates $(\\tilde T, \\tilde X, \\theta, \\varphi)$ associated with the conformal compactification of the Schwarzschild spacetime are
" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (M0, (T1, X1, th, ph))" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XCP.The transition map from Kruskal-Szekeres coordinates to the compactified ones:
The Kruskal-Szekeres chart plotted in terms of the compactified coordinates:
The standard Schwarzschild-Droste coordinates (also called simply Schwarzschild coordinates) $(t,r,\\theta,\\varphi)$ are defined on $\\mathcal{R}_{\\mathrm{I}}\\cup \\mathcal{R}_{\\mathrm{II}}$:
" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_II = regI.union(regII)\n", "XSD.We naturally introduce two subcharts as the restrictions of the chart XSD to regions $\\mathcal{R}_{\\mathrm{I}}$ and $\\mathcal{R}_{\\mathrm{II}}$ respectively. Since, in terms of the Schwarzschild-Droste 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": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(Chart (R_I, (t, r, th, ph)), Chart (R_II, (t, r, th, ph)))" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XSDI = XSD.restrict(regI, r>2*m)\n", "XSDII = XSD.restrict(regII, r<2*m)\n", "XSDI, XSDII" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The metric tensor is defined by its components w.r.t. Schwarzschild-Droste coordinates:
" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "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": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.lorentzian_metric('g')\n", "M.set_default_chart(XSD)\n", "M.set_default_frame(XSD.frame())\n", "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\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
The transition map is obtained by composition of previously defined ones:
Plot of the Schwarzschild-Droste chart in region I in terms of the compactified coordinates:
Same thing for the Schwarzschild-Droste chart in region II:
We introduce a second patch $(t',r',\\theta,\\varphi)$ of Schwarzschild-Droste coordinates to cover $\\mathcal{R}_{\\mathrm{III}}\\cup \\mathcal{R}_{\\mathrm{IV}}$:
" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_III_union_R_IV, (tp, rp, th, ph))" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regIII_IV = regIII.union(regIV)\n", "XSDP.The transition maps to Kruskal-Szekeres coordinates and compactified coordinates are defined in a manner similar to above: