{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"# Timelike geodesic in Schwarzschild spacetime\n",
"\n",
"This Jupyter/SageMath worksheet presents the numerical computation of a timelike geodesic in Schwarzschild spacetime, given an initial point and tangent vector. It uses the `integrated_geodesic` functionality introduced by Karim Van Aelst in **SageMath 8.1**, in the framework of the [SageManifolds](http://sagemanifolds.obspm.fr) project.\n",
"\n",
"Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.1/SM_simple_goed_Schwarz.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath within the Jupyter notebook, via the command `sage -n jupyter`"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"A version of SageMath at least equal to 8.1 is required to run this worksheet:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/plain": [
"'SageMath version 8.1, Release Date: 2017-12-07'"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"version()"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"%display latex # LaTeX rendering turned on"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"We define first the spacetime manifold $M$ and the standard Schwarzschild-Droste coordinates on it:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"Chart (M, (t, r, th, ph))"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"M = Manifold(4, 'M')\n",
"X. = M.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:\\phi')\n",
"X"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"For graphical purposes, we introduce $\\mathbb{R}^3$ and some coordinate map $M\\to \\mathbb{R}^3$:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"M --> R^3\n",
" (t, r, th, ph) |--> (x, y, z) = (r*cos(ph)*sin(th), r*sin(ph)*sin(th), r*cos(th))"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"R3 = Manifold(3, 'R^3', latex_name=r'\\mathbb{R}^3')\n",
"X3. = R3.chart()\n",
"to_R3 = M.diff_map(R3, {(X, X3): [r*sin(th)*cos(ph), \n",
" r*sin(th)*sin(ph), r*cos(th)]})\n",
"to_R3.display()"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"Then, we define the Schwarzschild metric:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"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": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g = M.lorentzian_metric('g')\n",
"m = var('m'); assume(m >= 0)\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": {
"deletable": true,
"editable": true
},
"source": [
"We pick an initial point and an initial tangent vector:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"v_0 = 1.29751300000000 d/dt + 0.0640625000000000/m d/dph"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p0 = M.point((0, 8*m, pi/2, 1e-12), name='p_0')\n",
"v0 = M.tangent_space(p0)((1.297513, 0, 0, 0.0640625/m), name='v_0')\n",
"v0.display()"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"We declare a geodesic with such initial conditions, denoting by $s$ the affine parameter (proper time), with $(s_{\\rm min}, s_{\\rm max})=(0, 1500\\,m)$:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"Integrated geodesic in the 4-dimensional differentiable manifold M"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s = var('s')\n",
"geod = M.integrated_geodesic(g, (s, 0, 1500), v0); geod"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"We ask for the numerical integration of the geodesic, providing some numerical value for the parameter $m$, and then plot it in terms of the Cartesian chart `X3` of $\\mathbb{R}^3$:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sol = geod.solve(parameters_values={m: 1}) # numerical integration\n",
"interp = geod.interpolate() # interpolation of the solution for the plot\n",
"graph = geod.plot_integrated(chart=X3, mapping=to_R3, plot_points=500, \n",
" thickness=2, label_axes=False) # the geodesic\n",
"graph += p0.plot(chart=X3, mapping=to_R3, size=4, parameters={m: 1}) # the starting point\n",
"graph += sphere(size=2, color='grey') # the event horizon\n",
"show(graph, viewer='threejs', online=True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"A Tachyon view of the geodesic:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
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Iak9ZZG+//Hf48OH3ve99X/ziF9vsftU7fPhwFt/d+kqS6hncJbWPbG5JNtux/Zou8jj7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\nc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hBW0gHYE3AFLsF5+NoFxfes9A64T5NK5EpXV1tX1+/s3LkR2sME1YbQS1MHX4T7gFC+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\nYII=\n",
"text/plain": [
"Graphics3d Object"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(graph, viewer='tachyon', aspect_ratio=1, figsize=10)"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"Some details about the system solved to get the geodesic:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Geodesic in the 4-dimensional differentiable manifold M equipped with Lorentzian metric g on the 4-dimensional differentiable manifold M, and integrated over the Real interval (0, 1500) as a solution to the following geodesic equations, written with respect to Chart (M, (t, r, th, ph)):\n",
"\n",
"Initial point: Point p_0 on the 4-dimensional differentiable manifold M with coordinates [0, 8*m, 1/2*pi, 1.00000000000000e-12] with respect to Chart (M, (t, r, th, ph))\n",
"Initial tangent vector: Tangent vector v_0 at Point p_0 on the 4-dimensional differentiable manifold M with components [1.29751300000000, 0, 0, 0.0640625000000000/m] with respect to Chart (M, (t, r, th, ph))\n",
"\n",
"d(t)/ds = Dt\n",
"d(r)/ds = Dr\n",
"d(th)/ds = Dth\n",
"d(ph)/ds = Dph\n",
"d(Dt)/ds = 2*Dr*Dt*m/(2*m*r - r^2)\n",
"d(Dr)/ds = -(4*Dth^2*m^2*r^3 - 4*Dth^2*m*r^4 + Dth^2*r^5 - 4*Dt^2*m^3 + 4*Dt^2*m^2*r + (Dr^2 - Dt^2)*m*r^2 + (4*Dph^2*m^2*r^3 - 4*Dph^2*m*r^4 + Dph^2*r^5)*sin(th)^2)/(2*m*r^3 - r^4)\n",
"d(Dth)/ds = (Dph^2*r*cos(th)*sin(th) - 2*Dr*Dth)/r\n",
"d(Dph)/ds = -2*(Dph*Dth*r*cos(th) + Dph*Dr*sin(th))/(r*sin(th))\n",
"\n"
]
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"[[2*Dr*Dt*m/(2*m*r - r^2),\n",
" -(4*Dth^2*m^2*r^3 - 4*Dth^2*m*r^4 + Dth^2*r^5 - 4*Dt^2*m^3 + 4*Dt^2*m^2*r + (Dr^2 - Dt^2)*m*r^2 + (4*Dph^2*m^2*r^3 - 4*Dph^2*m*r^4 + Dph^2*r^5)*sin(th)^2)/(2*m*r^3 - r^4),\n",
" (Dph^2*r*cos(th)*sin(th) - 2*Dr*Dth)/r,\n",
" -2*(Dph*Dth*r*cos(th) + Dph*Dr*sin(th))/(r*sin(th))],\n",
" Tangent vector v_0 at Point p_0 on the 4-dimensional differentiable manifold M,\n",
" Chart (M, (t, r, th, ph))]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"geod.system(verbose=True)"
]
},
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"We recognize in the above list the Christoffel symbols of the metric $g$:"
]
},
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""
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"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": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.christoffel_symbols_display()"
]
},
{
"cell_type": "code",
"execution_count": 12,
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"Riem(g)^t_r,t,r = -2*m/(2*m*r^2 - r^3) \n",
"Riem(g)^t_r,r,t = 2*m/(2*m*r^2 - r^3) \n",
"Riem(g)^t_th,t,th = -m/r \n",
"Riem(g)^t_th,th,t = m/r \n",
"Riem(g)^t_ph,t,ph = -m*sin(th)^2/r \n",
"Riem(g)^t_ph,ph,t = m*sin(th)^2/r \n",
"Riem(g)^r_t,t,r = -2*(2*m^2 - m*r)/r^4 \n",
"Riem(g)^r_t,r,t = 2*(2*m^2 - m*r)/r^4 \n",
"Riem(g)^r_th,r,th = -m/r \n",
"Riem(g)^r_th,th,r = m/r \n",
"Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n",
"Riem(g)^r_ph,ph,r = m*sin(th)^2/r \n",
"Riem(g)^th_t,t,th = (2*m^2 - m*r)/r^4 \n",
"Riem(g)^th_t,th,t = -(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_r,th,r = m/(2*m*r^2 - r^3) \n",
"Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n",
"Riem(g)^th_ph,ph,th = -2*m*sin(th)^2/r \n",
"Riem(g)^ph_t,t,ph = (2*m^2 - m*r)/r^4 \n",
"Riem(g)^ph_t,ph,t = -(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_r,ph,r = m/(2*m*r^2 - r^3) \n",
"Riem(g)^ph_th,th,ph = -2*m/r \n",
"Riem(g)^ph_th,ph,th = 2*m/r "
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.riemann().display_comp()"
]
},
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