{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Schwarzschild spacetime\n", "\n", "This notebook illustrates a few capabilities of SageMath in computations regarding Schwarzschild spacetime. The corresponding tools have been developed within the [SageManifolds](http://sagemanifolds.obspm.fr) project.\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Notebooks/SM_Schwarzschild.ipynb) to download the notebook file (ipynb format). To run it, you must start SageMath with the Jupyter interface, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 8.2 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.9, Release Date: 2019-09-29'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spacetime manifold\n", "\n", "We declare the Schwarzschild spacetime as a 4-dimensional Lorentzian manifold:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', r'\\mathcal{M}', structure='Lorentzian')\n", "M" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(Open subset R_I of the 4-dimensional Lorentzian manifold M,\n", " Open subset R_II of the 4-dimensional Lorentzian manifold M,\n", " Open subset R_III of the 4-dimensional Lorentzian manifold M,\n", " Open subset R_IV of the 4-dimensional Lorentzian manifold M)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI = M.open_subset('R_I', r'\\mathcal{R}_{\\mathrm{I}}')\n", "regII = M.open_subset('R_II', r'\\mathcal{R}_{\\mathrm{II}}')\n", "regIII = M.open_subset('R_III', r'\\mathcal{R}_{\\mathrm{III}}')\n", "regIV = M.open_subset('R_IV', r'\\mathcal{R}_{\\mathrm{IV}}')\n", "regI, regII, regIII, regIV" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "m = var('m') ; assume(m>=0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Boyer-Lindquist coordinates

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset R_I_union_R_II of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_II = regI.union(regII) ; regI_II" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R_I_union_R_II, (t, r, th, ph))\n" ] } ], "source": [ "X. = regI_II.chart(r't r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n", "print(X)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I, (t, r, th, ph))" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_I = X.restrict(regI, r>2*m) ; X_I" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_II, (t, r, th, ph))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_II = X.restrict(regII, r<2*m) ; X_II" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Chart (R_I_union_R_II, (t, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph))]" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph))]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph))\n" ] } ], "source": [ "print(M.default_frame())" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset R_I_union_R_II of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame().domain()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Metric tensor\n", "\n", "The metric tensor is obtained as follows:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Lorentzian metric g on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "g = M.metric()\n", "print(g)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One has to set its components in the coordinate frame associated with Schwarzschild coordinates, which is the current manifold's default frame:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "g[0,0], g[1,1] = -(1-2*m/r), 1/(1-2*m/r)\n", "g[2,2], g[3,3] = r^2, (r*sin(th))^2" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "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": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "v = d/dt + (-2*m/r + 1) d/dr" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v = regI.vector_field('v')\n", "v[0] = 1\n", "v[1] = 1 - 2*m/r\n", "# unset components are zero \n", "v.display()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset R_I of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.domain()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.domain()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(v,v) on the Open subset R_I of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "s = g(v,v)\n", "print(s)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g(v,v): R_I --> R\n", " (t, r, th, ph) |--> 0" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display() # v is indeed a null vector" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "

Levi-Civita Connection

\n", "
\n", "
 
\n", "
\n", "

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

\n", "
" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "nab = g.connection()\n", "print(nab)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab(g) == 0" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "nabla_g(g) = 0" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab(g).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Gam^t_t,r = -m/(2*m*r - r^2) \n", "Gam^r_t,t = -(2*m^2 - m*r)/r^3 \n", "Gam^r_r,r = m/(2*m*r - r^2) \n", "Gam^r_th,th = 2*m - r \n", "Gam^r_ph,ph = (2*m - r)*sin(th)^2 \n", "Gam^th_r,th = 1/r \n", "Gam^th_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_r,ph = 1/r \n", "Gam^ph_th,ph = cos(th)/sin(th) " ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.christoffel_symbols_display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Curvature

\n", "\n", "

The Riemann curvature tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "R = g.riemann()\n", "print(R)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Weyl conformal tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field C(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "C = g.weyl()\n", "print(C)" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "C(g) = -2*m/(2*m*r^2 - r^3) d/dt*dr*dt*dr + 2*m/(2*m*r^2 - r^3) d/dt*dr*dr*dt - m/r d/dt*dth*dt*dth + m/r d/dt*dth*dth*dt - m*sin(th)^2/r d/dt*dph*dt*dph + m*sin(th)^2/r d/dt*dph*dph*dt - 2*(2*m^2 - m*r)/r^4 d/dr*dt*dt*dr + 2*(2*m^2 - m*r)/r^4 d/dr*dt*dr*dt - m/r d/dr*dth*dr*dth + m/r d/dr*dth*dth*dr - m*sin(th)^2/r d/dr*dph*dr*dph + m*sin(th)^2/r d/dr*dph*dph*dr + (2*m^2 - m*r)/r^4 d/dth*dt*dt*dth - (2*m^2 - m*r)/r^4 d/dth*dt*dth*dt - m/(2*m*r^2 - r^3) d/dth*dr*dr*dth + m/(2*m*r^2 - r^3) d/dth*dr*dth*dr + 2*m*sin(th)^2/r d/dth*dph*dth*dph - 2*m*sin(th)^2/r d/dth*dph*dph*dth + (2*m^2 - m*r)/r^4 d/dph*dt*dt*dph - (2*m^2 - m*r)/r^4 d/dph*dt*dph*dt - m/(2*m*r^2 - r^3) d/dph*dr*dr*dph + m/(2*m*r^2 - r^3) d/dph*dr*dph*dr - 2*m/r d/dph*dth*dth*dph + 2*m/r d/dph*dth*dph*dth" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci tensor associated with $g$:

" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Ric = g.ricci()\n", "print(Ric)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Einstein equation

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == 0" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display() # another view of the above" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[[[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, -2*m/(2*m*r^2 - r^3), 0, 0],\n", " [2*m/(2*m*r^2 - r^3), 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, -m/r, 0], [0, 0, 0, 0], [m/r, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, -m*sin(th)^2/r],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [m*sin(th)^2/r, 0, 0, 0]]],\n", " [[[0, -2*(2*m^2 - m*r)/r^4, 0, 0],\n", " [2*(2*m^2 - m*r)/r^4, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, -m/r, 0], [0, m/r, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, -m*sin(th)^2/r],\n", " [0, 0, 0, 0],\n", " [0, m*sin(th)^2/r, 0, 0]]],\n", " [[[0, 0, (2*m^2 - m*r)/r^4, 0],\n", " [0, 0, 0, 0],\n", " [-(2*m^2 - m*r)/r^4, 0, 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, -m/(2*m*r^2 - r^3), 0],\n", " [0, m/(2*m*r^2 - r^3), 0, 0],\n", " [0, 0, 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 2*m*sin(th)^2/r],\n", " [0, 0, -2*m*sin(th)^2/r, 0]]],\n", " [[[0, 0, 0, (2*m^2 - m*r)/r^4],\n", " [0, 0, 0, 0],\n", " [0, 0, 0, 0],\n", " [-(2*m^2 - m*r)/r^4, 0, 0, 0]],\n", " [[0, 0, 0, 0],\n", " [0, 0, 0, -m/(2*m*r^2 - r^3)],\n", " [0, 0, 0, 0],\n", " [0, m/(2*m*r^2 - r^3), 0, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -2*m/r], [0, 0, 2*m/r, 0]],\n", " [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]]]" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R[:]" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Riem(g) = -2*m/(2*m*r^2 - r^3) d/dt*dr*dt*dr + 2*m/(2*m*r^2 - r^3) d/dt*dr*dr*dt - m/r d/dt*dth*dt*dth + m/r d/dt*dth*dth*dt - m*sin(th)^2/r d/dt*dph*dt*dph + m*sin(th)^2/r d/dt*dph*dph*dt - 2*(2*m^2 - m*r)/r^4 d/dr*dt*dt*dr + 2*(2*m^2 - m*r)/r^4 d/dr*dt*dr*dt - m/r d/dr*dth*dr*dth + m/r d/dr*dth*dth*dr - m*sin(th)^2/r d/dr*dph*dr*dph + m*sin(th)^2/r d/dr*dph*dph*dr + (2*m^2 - m*r)/r^4 d/dth*dt*dt*dth - (2*m^2 - m*r)/r^4 d/dth*dt*dth*dt - m/(2*m*r^2 - r^3) d/dth*dr*dr*dth + m/(2*m*r^2 - r^3) d/dth*dr*dth*dr + 2*m*sin(th)^2/r d/dth*dph*dth*dph - 2*m*sin(th)^2/r d/dth*dph*dph*dth + (2*m^2 - m*r)/r^4 d/dph*dt*dt*dph - (2*m^2 - m*r)/r^4 d/dph*dt*dph*dt - m/(2*m*r^2 - r^3) d/dph*dr*dr*dph + m/(2*m*r^2 - r^3) d/dph*dr*dph*dr - 2*m/r d/dph*dth*dth*dph + 2*m/r d/dph*dth*dph*dth" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Riem(g)^t_r,t,r = -2*m/(2*m*r^2 - r^3) \n", "Riem(g)^t_th,t,th = -m/r \n", "Riem(g)^t_ph,t,ph = -m*sin(th)^2/r \n", "Riem(g)^r_t,t,r = -2*(2*m^2 - m*r)/r^4 \n", "Riem(g)^r_th,r,th = -m/r \n", "Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n", "Riem(g)^th_t,t,th = (2*m^2 - m*r)/r^4 \n", "Riem(g)^th_r,r,th = -m/(2*m*r^2 - r^3) \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_t,t,ph = (2*m^2 - m*r)/r^4 \n", "Riem(g)^ph_r,r,ph = -m/(2*m*r^2 - r^3) \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[0 0 0 0]\n", "[0 0 0 0]\n", "[0 0 0 0]\n", "[0 0 0 0]" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "True" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C == R" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Bianchi identity

\n", "\n", "

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

" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(Riem(g)) of type (1,4) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "DR = nab(R)\n", "print(DR)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 " ] } ], "source": [ "from __future__ import print_function # because the print below is valid only in Python3\n", "for i in M.irange():\n", " for j in M.irange():\n", " for k in M.irange():\n", " for l in M.irange():\n", " for p in M.irange():\n", " print(DR[i,j,k,l,p] + DR[i,j,l,p,k] + DR[i,j,p,k,l], end=' ')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12*m/(2*m*r^3 - r^4) 0 0 12*m/(2*m*r^3 - r^4) 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 0 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 0 0 0 6*m/r^2 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12*(2*m^2 - m*r)/r^5 0 0 12*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*(4*m^3 - 4*m^2*r + m*r^2)/r^4 0 0 0 0 0 6*(4*m^3 - 4*m^2*r + m*r^2)/r^4 0 0 0 0 0 0 0 0 0 0 -6*(4*m^3 - 4*m^2*r + m*r^2)*sin(th)^2/r^4 0 0 0 0 0 0 0 0 6*(4*m^3 - 4*m^2*r + m*r^2)*sin(th)^2/r^4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)*sin(th)^2/r 0 0 -6*(2*m^2 - m*r)*sin(th)^2/r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*(2*m^2 - m*r)*sin(th)^2/r 0 0 6*(2*m^2 - m*r)*sin(th)^2/r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 0 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/(2*m*r^3 - r^4) 0 0 6*m/(2*m*r^3 - r^4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12*m*sin(th)^2/r^2 0 0 -12*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 -6*m*sin(th)^2/r^2 0 0 6*m*sin(th)^2/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6*(2*m^2 - m*r)/r^5 0 0 -6*(2*m^2 - m*r)/r^5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/(2*m*r^3 - r^4) 0 0 0 0 0 6*m/(2*m*r^3 - r^4) 0 0 0 0 0 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12*m/r^2 0 0 12*m/r^2 0 0 0 0 0 0 0 0 -6*m/r^2 0 0 0 0 0 6*m/r^2 0 0 0 0 0 0 0 0 6*m/r^2 0 0 -6*m/r^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 " ] } ], "source": [ "for i in M.irange():\n", " for j in M.irange():\n", " for k in M.irange():\n", " for l in M.irange():\n", " for p in M.irange():\n", " print(DR[i,j,k,l,p] - DR[i,j,l,p,k] + DR[i,j,p,k,l], end=' ')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Kretschmann scalar

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (0,4) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "dR = R.down(g)\n", "print(dR)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

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

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

" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Kr = 0\n", "for i in M.irange():\n", " for j in M.irange():\n", " for k in M.irange():\n", " for l in M.irange():\n", " Kr += uR[i,j,k,l]*dR[i,j,k,l]\n", "Kr" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of the above loops, the Kretschmann scalar can also be computed by means of the method `contract()`, asking that the contraction takes place on all indices (positions 0, 1, 2, 3):" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Kr = uR.contract(0, 1, 2, 3, dR, 0, 1, 2, 3)\n", "Kr.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Kr = uR['^{ijkl}']*dR['_{ijkl}']\n", "Kr.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Eddington-Finkelstein coordinates

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R_I_union_R_II, (v, r, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_II, (v, r, th, ph))" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF. = regI_II.chart(r'v r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\varphi')\n", "print(X_EF); X_EF" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [], "source": [ "ch_BL_EF_I = X_I.transition_map(X_EF, [t+r+2*m*ln(r/(2*m)-1), r, th, ph], \n", " restrictions2=r>2*m)" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (v, r, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (v, r, th, ph))" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_I) ; ch_BL_EF_I" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "v = 2*m*log(1/2*r/m - 1) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_I.display()" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I, (v, r, th, ph))" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF_I = X_EF.restrict(regI) ; X_EF_I" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [], "source": [ "ch_BL_EF_II = X_II.transition_map(X_EF, [t+r+2*m*ln(1-r/(2*m)), r, th, ph], \n", " restrictions2=r<2*m)" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (v, r, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (v, r, th, ph))" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(ch_BL_EF_II) ; ch_BL_EF_II" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "v = 2*m*log(-1/2*r/m + 1) + r + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_EF_II.display()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_II, (v, r, th, ph))" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_EF_II = X_EF.restrict(regII) ; X_EF_II" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The manifold's atlas has now 6 charts:

" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Chart (R_I_union_R_II, (t, r, th, ph)),\n", " Chart (R_I, (t, r, th, ph)),\n", " Chart (R_II, (t, r, th, ph)),\n", " Chart (R_I_union_R_II, (v, r, th, ph)),\n", " Chart (R_I, (v, r, th, ph)),\n", " Chart (R_II, (v, r, th, ph))]" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default chart is 'BL':

" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_II, (t, r, th, ph))" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The change from Eddington-Finkelstein coordinates to the Schwarzschild (Boyer-Lindquist) ones, computed as the inverse of `ch_BL_EF`:" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (t, r, th, ph))\n" ] } ], "source": [ "ch_EF_BL_I = ch_BL_EF_I.inverse()\n", "print(ch_EF_BL_I)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "t = 2*m*log(2) + 2*m*log(m) - 2*m*log(-2*m + r) - r + v\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_BL_I.display()" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (t, r, th, ph))\n" ] } ], "source": [ "ch_EF_BL_II = ch_BL_EF_II.inverse()\n", "print(ch_EF_BL_II)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "t = 2*m*log(2) - 2*m*log(2*m - r) + 2*m*log(m) - r + v\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_BL_II.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dv,d/dr,d/dth,d/dph))]" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The default frame is:

" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph))" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The coframes are the duals of the defined vector frames:

" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Coordinate coframe (R_I_union_R_II, (dt,dr,dth,dph)),\n", " Coordinate coframe (R_I, (dt,dr,dth,dph)),\n", " Coordinate coframe (R_II, (dt,dr,dth,dph)),\n", " Coordinate coframe (R_I_union_R_II, (dv,dr,dth,dph)),\n", " Coordinate coframe (R_I, (dv,dr,dth,dph)),\n", " Coordinate coframe (R_II, (dv,dr,dth,dph))]" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.coframes()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "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": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[ 2*m/r - 1 0 0 0]\n", "[ 0 -1/(2*m/r - 1) 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The tensor components in the frame associated with Eddington-Finkelstein coordinates in Region I are obtained by providing the frame to the function `display()`:" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = (2*m - r)/r dv*dv + dv*dr + dr*dv + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_EF_I.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[ (2*m - r)/r 1 0 0]\n", "[ 1 0 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.comp(X_EF_I.frame())[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

or, as a schortcut,

" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[ (2*m - r)/r 1 0 0]\n", "[ 1 0 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_EF_I.frame(),:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = (2*m - r)/r dv*dv + dv*dr + dr*dv + r^2 dth*dth + r^2*sin(th)^2 dph*dph" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_EF_II.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

\n", "\n", "

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

\n", "

Let us perform the plot in Region I:

" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 18 graphics primitives" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_I.plot(X_EF_I, ranges={t:(0,8), r:(2.1,10)}, fixed_coords={th:pi/2,ph:0}, \n", " ambient_coords=(r,v), style={t:'--', r:'-'}, parameters={m:1})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Tetrad of the static observer

\n", "

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

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[1/sqrt(-2*m/r + 1) 0 0 0]\n", "[ 0 sqrt(-2*m/r + 1) 0 0]\n", "[ 0 0 1/r 0]\n", "[ 0 0 0 1/(r*sin(th))]" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_to_stat = regI.automorphism_field()\n", "ch_to_stat[0,0], ch_to_stat[1,1] = 1/sqrt(1-2*m/r), sqrt(1-2*m/r)\n", "ch_to_stat[2,2], ch_to_stat[3,3] = 1/r, 1/(r*sin(th))\n", "ch_to_stat[:]" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector frame (R_I, (e_0,e_1,e_2,e_3))\n" ] } ], "source": [ "e = X_I.frame().new_frame(ch_to_stat, 'e')\n", "print(e)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[Coordinate frame (R_I_union_R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dt,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I_union_R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_I, (d/dv,d/dr,d/dth,d/dph)),\n", " Coordinate frame (R_II, (d/dv,d/dr,d/dth,d/dph)),\n", " Vector frame (R_I, (e_0,e_1,e_2,e_3))]" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field e_0 on the Open subset R_I of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(e[0])" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_0 = 1/sqrt(-2*m/r + 1) d/dt" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-1" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(e[0],e[0]).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Equivalently, we may use the method `dot` to get the scalar product (since $g$ is the default metric on $\\mathcal{M}$):" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-1" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].dot(e[0]).expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-1 0 0 0 \n", "0 1 0 0 \n", "0 0 1 0 \n", "0 0 0 1 \n" ] } ], "source": [ "for i in M.irange():\n", " for j in M.irange():\n", " print(e[i].dot(e[j]).expr(), end=' ') \n", " print(\"\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Another view of the above result:

" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[-1 0 0 0]\n", "[ 0 1 0 0]\n", "[ 0 0 1 0]\n", "[ 0 0 0 1]" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[e,:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

or, equivalently,

" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "g = -e^0*e^0 + e^1*e^1 + e^2*e^2 + e^3*e^3" ] }, "execution_count": 82, "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": 83, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_0 = sqrt(r)/sqrt(-2*m + r) d/dv" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_EF_I.frame())" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_1 = sqrt(r)/sqrt(-2*m + r) d/dv + sqrt(-2*m + r)/sqrt(r) d/dr" ] }, "execution_count": 84, "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": 85, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e.structure_coeff()[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "m*sqrt(-2*m + r)/((2*m*r - r^2)*sqrt(r)) e_0" ] }, "execution_count": 86, "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": 87, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 87, "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": 88, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset R_I_union_R_II_union_R_III_union_R_IV of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M0 = regI.union(regII).union(regIII).union(regIV) ; M0" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_II_union_R_III_union_R_IV, (U, V, th, ph))" ] }, "execution_count": 89, "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": 90, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I, (U, V, th, ph))" ] }, "execution_count": 90, "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": 91, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (t, r, th, ph)) to Chart (R_I, (U, V, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "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": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_KS_I = X_I.transition_map(X_KS_I, [sqrt(r/(2*m)-1)*exp(r/(4*m))*cosh(t/(4*m)), \n", " sqrt(r/(2*m)-1)*exp(r/(4*m))*sinh(t/(4*m)), \n", " th, ph])\n", "print(ch_BL_KS_I)\n", "ch_BL_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 92, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_II, (U, V, th, ph))" ] }, "execution_count": 92, "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": 93, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_II, (t, r, th, ph)) to Chart (R_II, (U, V, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "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": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_BL_KS_II = X_II.transition_map(X_KS_II, [sqrt(1-r/(2*m))*exp(r/(4*m))*sinh(t/(4*m)), \n", " sqrt(1-r/(2*m))*exp(r/(4*m))*cosh(t/(4*m)), \n", " th, ph])\n", "print(ch_BL_KS_II)\n", "ch_BL_KS_II.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 94, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 52 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphI = X_I.plot(X_KS, ranges={t:(-12,12), r:(2.001,5)}, number_values={t:17, r:9}, \n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V), \n", " style={t:'--', r:'-'}, parameters={m:1})\n", "graphII = X_II.plot(X_KS, ranges={t:(-12,12), r:(0.001,1.999)}, number_values={t:17, r:9}, \n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V), \n", " style={t:'--', r:'-'}, color='green', parameters={m:1})\n", "show(graphI+graphII, xmin=-3, xmax=3, ymin=-3, ymax=3, axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 95, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 54 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_r0 = X_II.plot(X_KS, fixed_coords={r:0, th:pi/2, ph:0}, ambient_coords=(U,V),\n", " color='yellow', thickness=3, parameters={m:1})\n", "graph_r2 = X_I.plot(X_KS, ranges={t:(-40,40)}, fixed_coords={r:2.00001, th:pi/2, ph:0},\n", " ambient_coords=(U,V), color='black', thickness=2, parameters={m:1})\n", "show(graphI+graphII+graph_r0+graph_r2, xmin=-3, xmax=3, ymin=-3, ymax=3, \n", " axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 96, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (R_I, (v, r, th, ph)) to Chart (R_I, (U, V, th, ph))" ] }, "execution_count": 96, "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": 97, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ch_EF_KS_I.display()" ] }, { "cell_type": "code", "execution_count": 98, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Change of coordinates from Chart (R_II, (v, r, th, ph)) to Chart (R_II, (U, V, th, ph))" ] }, "execution_count": 98, "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": 99, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 54 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphI_EF = X_EF_I.plot(X_KS, ranges={v:(-12,12), r:(2.001,5)}, number_values={v:17, r:9},\n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V),\n", " style={v:'--', r:'-'}, parameters={m:1})\n", "graphII_EF = X_EF_II.plot(X_KS, ranges={v:(-12,12), r:(0.001,1.999)}, \n", " number_values={v:17, r:9},\n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V),\n", " style={v:'--', r:'-'}, color='green', parameters={m:1})\n", "show(graphI_EF+graphII_EF+graph_r0+graph_r2, xmin=-3, xmax=3, ymin=-3, ymax=3,\n", " axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

There are now 9 charts defined on the spacetime manifold:

" ] }, { "cell_type": "code", "execution_count": 100, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 101, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "9" ] }, "execution_count": 101, "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": 102, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.coord_changes()" ] }, { "cell_type": "code", "execution_count": 103, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "8" ] }, "execution_count": 103, "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": 104, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 105, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "10" ] }, "execution_count": 105, "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": 106, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "14" ] }, "execution_count": 106, "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": 107, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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_I.frame())" ] }, { "cell_type": "code", "execution_count": 108, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_KS_I.frame(),:]" ] }, { "cell_type": "code", "execution_count": 109, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 109, "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": 110, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 110, "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": 111, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.riemann().display(X_KS_I.frame())" ] }, { "cell_type": "code", "execution_count": 112, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 112, "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": 113, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-form curvature (0,1) of connection nabla_g w.r.t. Coordinate frame (R_I, (d/dU,d/dV,d/dth,d/dph)) on the Open subset R_I of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "om = g.connection().curvature_form(0,1, X_KS_I.frame())\n", "print(om)" ] }, { "cell_type": "code", "execution_count": 114, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 114, "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": 115, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Open subset R_I_union_R_III of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "regI_III = regI.union(regIII) ; regI_III" ] }, { "cell_type": "code", "execution_count": 116, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (R_I_union_R_III, (t, ri, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I_union_R_III, (t, ri, th, ph))" ] }, "execution_count": 116, "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": 117, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Chart (R_I, (t, ri, th, ph))" ] }, "execution_count": 117, "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": 118, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change of coordinates from Chart (R_I, (t, ri, th, ph)) to Chart (R_I, (t, r, th, ph))\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "t = t\n", "r = 1/4*ri*(m/ri + 2)^2\n", "th = th\n", "ph = ph" ] }, "execution_count": 118, "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": 119, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " t == t *passed*\n", " ri == ri *passed*\n", " th == th *passed*\n", " ph == ph *passed*\n", " t == t *passed*\n", " r == r *passed*\n", " th == th *passed*\n", " ph == ph *passed*\n" ] } ], "source": [ "assume(r>2*m) # we consider only region I\n", "ch_iso_BL_I.set_inverse(t, (r-m+sqrt(r*(r-2*m)))/2, th, ph, verbose=True)" ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 120, "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": 121, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 121, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "11" ] }, "execution_count": 122, "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": 123, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 123, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "code", "execution_count": 124, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "12" ] }, "execution_count": 124, "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": 125, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 125, "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": 126, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-(m^2 - 4*m*ri + 4*ri^2)/(m^2 + 4*m*ri + 4*ri^2)" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_iso_I.frame(), 0,0, X_iso_I]" ] }, { "cell_type": "code", "execution_count": 127, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "-(m - 2*ri)^2/(m + 2*ri)^2" ] }, "execution_count": 127, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[X_iso_I.frame(), 0,0, X_iso_I].factor()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us also factor the other components:" ] }, { "cell_type": "code", "execution_count": 128, "metadata": {}, "outputs": [], "source": [ "for i in range(1,4):\n", " g[X_iso_I.frame(), i,i, X_iso_I].factor()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The output of the `display()` command looks nicer:" ] }, { "cell_type": "code", "execution_count": 129, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "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": 129, "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": 130, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_0 = -(m + 2*ri)/(m - 2*ri) d/dt" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[0].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 131, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_1 = 4*ri^2/(m^2 + 4*m*ri + 4*ri^2) d/dri" ] }, "execution_count": 131, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[1].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 132, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_2 = 4*ri/(m^2 + 4*m*ri + 4*ri^2) d/dth" ] }, "execution_count": 132, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[2].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": 133, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "e_3 = 4*ri/((m^2 + 4*m*ri + 4*ri^2)*sin(th)) d/dph" ] }, "execution_count": 133, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e[3].display(X_iso_I.frame(), X_iso_I)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.9", "language": "sage", "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": "2.7.15" } }, "nbformat": 4, "nbformat_minor": 1 }