{ "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/Worksheets/v1.3/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 worksheet:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.7.rc0, Release Date: 2019-03-19'" ] }, "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": [ "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.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": [ "The Levi-Civita connection $\\nabla$ associated with $g$:
\n", "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": [ "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": [ "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": [ "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": [ "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": [ "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.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], 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], 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
Note that their form is identical to that in Region I.
\n", "\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}, ambient_coords=(r,v), style={t:'--', r:'-'}, parameters={m:1})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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": [ "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)), 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)), th, ph])\n", "print(ch_BL_KS_II)\n", "ch_BL_KS_II.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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": [ "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": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAGGCAYAAAB2a4afAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAAPYQAAD2EBqD+naQAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4zLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvIxREBQAAIABJREFUeJzsnXd8k9Uexp8kHVAKpZRZqCxZMkUQUa96QUBAZKooIKgIF3FdVBBRTFml7L33LLMCZe89hbIKlJbVsgpdKd1NnvvHuQlNmpSkO+35fj75KG/ecU6Svs97flNBkpBIJBJJkUWZ3wOQSCQSSf4ihUAikUiKOFIIJBKJpIgjhUAikUiKOFIIJBKJpIgjhUAikUiKOFIIJBKJpIgjhUAikUiKOFIIJBKJpIgjhUAikUiKOFIIJBKJpIgjhUAikUiKOFIIJHbHkCFDULFiRSgUCigUCri4uKBRo0a4fft2hn0HDx4MJycnKBQKODo6okmTJkhOTs6HUUskBReFrD4qsUd0Oh1q1KiBu3fv4sKFC2jSpInFfcePH4+tW7di/fr1qFKlSh6OUiKxD+SKQGKXKJVK1KpVCwAQFRVlcb+kpCTs378f27dvlyIgkVhACoHEbqlRowYA4NatWxb3GTlyJH7++We4u7vn1bAkErtDCoHEbnmREJw5cwaRkZH44IMP8nJYEond4ZDfA5BIsopeCMw5iVNSUvD7779jw4YNeT0sicTukCsCid1SvXp1AOZXBKNHj8a3334rTUISiRVIIZDYLZZMQ4GBgbh16xa6deuWH8OSSOwOGT4qsWvc3d0RExMDjUaDkiVLIi0tDR988AHWrFmD8uXL5/fwJBK7QK4IJHaNqXlo/Pjx6NevnxQBicQGpBBI7Jr05qGgoCBcuHABvXv3zudRSST2hYwaktg1+hXBzZs3MXfuXCxZsiSfRySR2B9yRSCxa/QrgokTJ6JHjx4ye1giyQJSCCR2jV4IGjZsiG+++SafRyOR2CdSCCR2TZ06dVC+fHksXLgQCoUiv4cjkdglMnxUIpFIijhyRSCRSCRFHCkEEolEUsSRQiCRSCRFHCkEEolEUsSRQiCRSCRFHCkEEolEUsSRQiCRSCRFHCkEEolEUsSRQiCRSCRFHCkEEolEUsSRQiCRSCRFHCkEEolEUsSRQiCRSCRFHCkEkkIDSWg0GsiCuhKJbWRZCLQ6Lb7c8iWU3koovBVwG18KvseUABRmXsMByD9OSe4SFxcHNzc3xMXF5fdQJIWeVRCdfo3vdVtvKFDW1xWvzn8VTxOeou8vfaFQKAp8r4wsC4FKqcKiTovQt0lfKKCAJjkOi87XxO/7ze09HsB3AHRZvZxEIpEUEOYD+AKA1mir32Wgj39xVChZFe1fbo+F0xdixeQV+TJCW8lW83q9GADA8sDlCIkOwfhjCjxLIWa0N917DoA4AEuye1mJRCLJJyYCGJphq98VV4RpRqBBuW0IehqEFbNX4L7/fcP748ePz8Mx2k62fQSmKwOCmHkG+GoLoM2wAFgJ4BMAydm9rKQQMnfuXDRq1AilSpVCqVKl0LJlS+zcuTO/hyWRQJi2/4A5EQh60g6daj/Eu9Va4eqTqyh5pmQGERg2bFjeDTUL5IizWC8GPRv0NGxbGgh8tgnQ6kwv4Q+gE4D4nLi0pBBRpUoVjB8/HufOncO5c+fQqlUrdO7cGVevXs3voUmKNDoAPwIYm+Gd6afd0GzBYQzdOwxtVraB21k3hG0OM7xvEIFbt/JuuFkgR3sWp6SloMHcBrgZdRMAoFKo0P5lYsMnRDEH08u8CWA7gNI5dXlJIaRMmTKYOHEivv766xfuq9Fo4ObmhtjYWJQqVSoPRicp/KQB6A9geYZ3ftvngKmnlGhVvRXWdFuDrt93xeHFhw3vG60EdDpAWXCDNHN0ZE4OTrj67VXULlMbAKClFiFRtdFuFRGXwRp0AsC/AUTk5BAkhQStVgs/Pz/Ex8ejZcuWZvdJTk6GRqMxekkkOUcygJ4wFQEdge92OGHySaC5Z3P89c5fmD99vnkRCAoCdu8u0CIA5PCKQE+qNhWN5jbC9cjrAIDiDsVRv3widvUCPFxM964DYB+AKjk9DIkdcvnyZbRs2RJJSUlwdXXFmjVr0KFDB7P7qtVqeHt7Z9guVwSS7BMPoBuAPUZbU7XA4B0lkarrjtvRt3H2wVnUvFoTl9dcNuxjEIGbN4F33gEqVwbOnCnQYpArQgAAado0dFvfDQHBAeD/cwjqlwP29gEqlTTduyqEGLycG0OR2BEpKSm4d+8eYmJisGnTJixatAiHDx/GK6+8kmHf5ORkJCc/X2pqNBp4eXlJIZBkk1gAHQEcN9qanKZAmGYqKpT4CqfCT6GzX2dUuVQFN9ffNOxjEIHbt4UIuLoChw4BFSrk6QxsJdeEABBJZ/239seyi8sM22q4A6f7l0RZF9Okn4oA9gJokFvDkdgh77//PmrWrIn58+e/cF/pI5BknycAPgBw3mhrXDIw4fhb8D1+BpPbTsave39FtSvVcGPdDcM+BhEICxMioFIBR44Anp55O4UskKtrFZVShYWdFqJB+ec391vRQNP5zxCuMV0WPALwDoAzuTkkiZ1B0uipXyLJPcIh7kHGIhCdqED39W4Yc/Q4vmn6DZpXbo561+uZFwEASE0FqlYFDhywCxEA8qDWkIPKARcGXECjCo0M20o41sFrC57h/EPTvaMBtAZwILeHJSmA/P777zh69Cju3LmDy5cvY8SIETh06BB69eqV30OTFHpCALwN4LrR1kfPFOiyzh17b8WiZ/2emNJuCg6tOoTA1YGGfQwi8PQpEBsL1KghzEEvvZSnM8gOeeK9cFA54PyA82hSsQkA4HrkdShRAf9eDhy7Z7r3MwAdAGzJi6FJChCPHz9Gnz59UKdOHbRu3RqnT5/Grl270KZNm/wemqRQcwlCBO4abQ2LVeHrLbUQrimN7vW6Y+O1jej2fTcMHz7csI9BBKKjgTZtAEsPLdu25d7wc4Bc9RGYotVp0XpFaxy5e8TgQHZxBDZ9AnyQwU+sArAMQO+8Gp7EzpE+AontnIR48Iwx2vogzg1K7EUxx1rwv+6Pb7Z+g/rB9XFp9SXDPgYR0GiECISGipVAAzN+zhkzgB9+yNWZZIc8jWdSKVXY/8V+QzkKAEhIBT5aCxy4XcZkby2APgBm5+UQJRJJkWEfgPdhKgIXHirhe6wD2qz6Cn9f/xv9t/ZHw5sNzYtAfDzQsSNw4wawZ09GEYj5/7kLsAgA+dCPQF+OokXlFoZtaToF2q6MwvmHTc0c8R1EarcsYy2RSHIKf4gQ0QSjrUfvKtBnsydmnFmLAU0H4ErEFTQJaYLAVWZ8AgCwaxdw8aL4b1OT+1dICFC3LrByZe5OJQfIlwwHlVKFY18dM4gBQdRwr4XXFwZi8glzR+iLPUkxkEgk2WU5gB4AUoy27rgJfLOtCoKe3odPKx983+J7ePzjgfMrn0cRGURAb1Hv3h0IDgbeeMP4Eg8fAm3bAqVLAx98kLvTyQHyLdVNpVTh+FfH8UZl8QHejLqJ6qVr4Je9wAizQUOTAAyAaQ1wiWT27Nl45ZVX0Lx58/weiqTAMwNAP5j2RtlwVYWZp1shNPoh+jbui5GHRuL737/H77//btjHIAJpacAnnwDTpok3KlY0vkRMDNC+PZCSIspLlCuXqzPKEZjPpGnT+Nbit6hQKwg1qPRWEmpw8HaQNPf6mGRSPo5YUlCJjY0lAMbGxub3UCQFDh3Jv2junnIt4h0GRVxmmjaN6oNqQg227NeSECYIAuD48ePFadLSyF69SAcHcutW85f69lvS3Z28ciUP5pUz5GnUkCW0Oi36b+uP5YHLQRAqhQpaajHwNRfM6pAAhwzrljYANgNwzfvBSgosMmpIYh4dgJ8AzMzwzoqLlbDhalO8UaUlXBxdMGTPELx9920cW3rMsI9hJaDTAQMGAEuXAn5+wMcfm79cXJyIIGrSJHemkwsUiCpIegdy25ptAYiqpSqFCvP/ScDWG18gKc30iL0QYhCVxyOVSCT2RSqAvjAnAsP3A96HimPf7f1oUL4BZp+djffC3jMvAoAwBS1ZAixbllEESOCPP0ShuZIl7UoEgAIiBIAQg+2fb8e7Vd8FIMSgcsnK+GTDanzh72GmjPUpAO8CeJDHI5VIJPZBIkQF0VVGW3UE/hMArLtSHY/jH8P/U390rtsZvWJ74dDiQ4b9MnQW698f2LwZ6NMn46VGjADGjgXOncuVmeQ2BcI0lB6tTos2K9vg4J2DAIA6HnUQHBmM1ysrEPC5DmUzlLGuDrFCqJnHI5UUNKRpSPKcWAAfAThitDVVC/T2ByITWuPM/TP4+JWPce7hOfSI7IGRf4w07GckAlOnAh9+CNSqZf5S06YB//0vMHkyMGRI7kwnlykwKwI9KqUKe/vsRavqrQAANyJvwFHliNP3dWi93AHhGXqP3IZID79k+oZEIimSREA0vTIWgcRUBR7EzcOPLY5jS88t6NmgJ5YELoFnoKdlEZgwQdzcLfXOXr1aiMCwYZmLQAEvnFjghAAQYrCn9x70a9IPCiiQok2Bs8oZlyLS0Hq5M0KjFCZHPIIwE5lNQpBIJEWGewD+BeCC0daYJMD/2iD02+KHkk4loT6kxsLzC9HhUQfsmr/LsJ+RCMyaJW7wf/5pOTO4XDngu+8AHx/LQyKFWakAUyCFAHjuQO7ZoCcAIFmbDGeVM4KjkrHm8re4+Mh06DEQDuRdpqeSSCRFgmsA3gIQbLQ1MsERbVY44td9fyM4MhhBT4Iw+eRkdIrohB3zdhj2MxKBpUuB778XT/lmuuAhNFREEbVtC8ycCShMH07ToVAA776b/enlIgVWCAAhBiu7rkS7mu0ACDEoU7wMRh2Zj4kn2uPoXdMjEgB0ArA2j0cqkUjylzMQK4Fwo61hsQ54b7kK4XEecFA44Ei/I/i0waf4b9p/sW3O84qgGRzDnp7ATz8BkyZlvMkHBQHNm2e+CtBz65b4bwFfERQ4Z7E5tDotPlz7IXaFiKf92mVqIzQ6FOWKl8DiLhp0yODDUUCEiw3O45FK8hPpLC6q7APQBaLP8HOuP1Why9riSNaVhUqhQvPKzdGgXAOoTqjMl5IGgAsXgMaNLfcXDgsD3nwTcHcX3cdKl7Y8rFOngPfeE6uLzz7L1gxzmwK9ItCjUqoQ8FkAOtQSTcyDo4JRwqkEHiVo0O/v0lidwU9MiGJ13pD1iQo/ssREUWYjRBlpUxFwgxJHse7jY1jZZSVeq/Qa1l1Zh8CNgZZFYM8eUTNo3jzzl4qOFnWDVCpRZC4zEbhzB+jcWawcunXL1gzzhPxMa7aVNG0a+/n3M5SjKDG2BKEGy/qW4azTSpovSfEdSW0+jlqSV8gSE0WNeRTFEYz/5g/ddub+0C38Zus3TExNZK9Nvaj0VvLTHz81XzaCJA8fJosXJzt2JJOTzV9uzBjSw4O8fj3zYcXGkg0akDVqkE+e5MhMcxu7MA2lR1+OYlngMgBASaeSiEuJQ4963fGa5w789naimaM+g6g46JiHI5XkNdI0VFQgAB8AIzK88/f1YhiyywNPEzV4pdwraPdyO4w9MhafRH+CtdOe+w6NVgJnzgCtWwOvvw5s3w4UK2b+sjqdeNKvUSPz4Q0bBsyfD5w8CdSrl6UZ5jV2JwSAEINu67pha/BWAEDpYqURlxyH3976DdFJYzGzA6DM4MRvD2ADgBJ5PFpJXiGFoCigA/ArgCkZ3llywRnjjlbCg7jHaF65OQI+C4BSocRPf/yERRMXGfbL4BgePBgIDBSVQl1N6peRwC+/iOigdu2sG2JSEnD9ul2VmbALH4EpKqUKmz/djC51ugAAYpJiUN29OnxP+GL7zar4fBOQkqFa9U7I+kQSiT2TClFCOqMIjD2qwrRTtRCueYi3vN6CVykvhGnCMHPKTMsikPb/ImYzZ5oXAQAYMwaYMgW4f//Fw1u6VAhKsWJ2JQKAnQoBIMRg4ycb0bVuVwBASFQIyrmUw93Yuzh2rzI6rQXiU0yPOglzIWYSiaSgEw8RGZSx29eJsJ54pewGHPvyOCa2mQgogE3XNmHKxCmWHcOhocArrwBHj4oIIXMisGgRMHKkEIOvvsp8eLt3ixDRdeuyPsV8xG6FABBisOHjDejdsDcUUODhs4fwKO6B+3H3cS3CC61XALFJKpOjggC8CeB6PoxYIpHYThREb+EdRltTtcDsMy0RGtUBDkoHqJQq+F/3x4mwE+gd2xuLJy427GskAmFhwiegUAC1a5u/ZEAAMHAg8O23QLrmNGa5cQP49FMRUTRmTNanmY/YpY/AFFMHskdxD0QmRqJppaYopgrFhk/i4FlSZ3KUB8QP6/U8Hq0kt5A+gsJIOIB2EA9wz0lOc0C39Vo8iW+Gsw/O4ttm3+JO7B0cuXsEn8d8jgUTFhj2NRKBx4+Bd94RtX+OHgW8vMxfNjgYmD1bmIVUpg+T6YiOFiGnKpVwDru5ZW+6+YRdrwj06MtRfPyKqBEemRiJci7lEPgoEN3q/Yl3lqpw7YnpUZEAWgHYk8ejlUgk1nEdYvVuLAKxSSq8uywN0Ylv4OyDs/jh9R8ws8NMDH1zKPpo+lgWAQD44gtAowH27TMvAqGhorFM7drA9OmZiwAA3L4t/rt1q92KAFBIhAAQYrC2+1p8Uv8TAMCThCeo5lYNw/YNQzW3d/CvpcDpDP6eeAAdIUtSSCQFjdMQVYXDjLaGaxRouVgLV6fWOBl+Ej+1+Aluxdygow4n/U5i7vi5hn0ziAAgHMP79gEvv5zxkvfvA61aAYMGWTdEnQ5o2lSUnDB3PjuiUJiG0qPVadFrcy+suyqcNi+XeRkhUSGoUrIKopPCsekToJ3Z72wagB/zcqiSHEaahgoLuwD0gGm2cEiUA5JTt+LMg0d4r9p7WHtlLTYEbUBYbBi+fPYlJo2eZNjXSAQSE4XTd+RI0T3MHDExwmQUEwOcOAFUqZL5EJcsATZuBPz9AWfnrE+1gFBoVgR6VEoVVndbjZ4NekIBhRCBUlUQHheO0sVENNHf14ubOfInAMMhS1LYH7LERGFiJUThSGMROPfAAQ816xFw8xK61O0CF0cXrL2yFg/iHqC3prdlEUhJAbp3B+bMEU5dcyQnA127AuHhonTEi0Tg+HHgP/8RpiUnp6xPtQBR6FYEevQO5OWBy0EQFUtUxKP4R/As6YnktERMbQf0aRxt5si+ABZCZiHbH3JFYO9MgkgWM+afB+Xw72VP0dTzHRy5ewTLOi+Dz3EfxCbFomdMT0wdM9Wwb4Y8gc8+E/b77duB9983f9mtW0XUz969wNtvZz7Ee/dE/aC6dcX+hUQICt2KQI/egfxZA1H171H8I1R0rYgHcQ/wXfMf8N3ONIzYb66G+HKYq2QokUhyCx2An2FOBHaHeODdpTFoWKEljt07htXdVqN3497oUqcLPo/93LIIkMA33wjTzfr1lkUAAD76SDSdf5EIJCaKQnIuLsCmTbaJgL+/9fvmA4VWCAAhBiu6rkCvhr0AAI+ePcJLbi9hzNEx6NPoC4w/rsDXWxRIM40sxQ4ArQE8zeMRSyRFjRQAX8BctvCUk0D3dQloUKEpzj08h7kd56KsS1koFUq4nXPD5NGTDftmcAwrFCJhbOVKcfM2x+zZos8w8GJzECAyhnv1EiuIsmWtn+K9ey9OSMtnCq1pKD1anRb9/u6HVZdXAQBqe9TGzcibcHN2gyZFg461CL8ehEsGa1AdALsBVM3jEUuygjQN2RtxEE7hjCHcay83xZA9D6CEEoObD4ZnKU/8efBPlC5WGp/FfIYRvz8vOJdBBM6dA5o1y/zS/v7Cd/DTTyJX4EXcuAHUqWPdtNKTliZ6EoSFAXczdNIqMBTqFYEelVKFZV2WoXcjkYEcHBmMGu41EJMcgxKOJbDjphIdVjtBR3eTI28AaAkgQ8MDiUSSLR5DNJg3FoFULeB9uCZaV9+JX1r+AmcHZ8w5NwdD9w6Fq5MrOkV0ylwExo0TNvwLxj2LjTh5Evj8c6BHD9GB7EVs2CCqiB47ZtsUAeFYPncOWLPG9mPzkCKxItBj6kCu5FoJD589hIuDC1RKFdrUrILZHUJR0dW0SFEpAP4QCWiSgopcEdgLNwF8AOCW0dYUrRP6+rvg1P3SKO5QHE8SnmBOhznouaknVAoVBiYOxCyfWYb9M4jAzJmiybxaDfz1l/lLh4SITOB69YSz11LJaT2BgcBbbwk/wpo1mfcmtsSDB6L1ZQGmSKwI9OgdyH0a9QEAPHz2EBVdKyIhLQGd63TGoTuP8a8lSlyNMP2yNRA/XL+8HrJEUsg4A5EtbCwCUYkqdFxdDBcfV4JSoURsciwO9zuM1zxfQ6vqrVDidInMRWD5ciECP/8s8gUsUa6cMAlt2fJiEYiMFGGldeoAixfbJgJPnwITJwJabYEXAaCICQEgxGBJ5yX4ovEXAIQDuZJrJay9shbD3hyGexod3l5KHL9nmlqeCtHgxgp7okQiMcN2CHOQcRDG7WjgvWUOiEp6GXEpcdBRh7kd58KzpCdquNdA6/utEbM9xrD/+N9+MxYBrVbkCfTvL26+5m7Y8fGiqYybm2gaU6bMi4f7xx+i3IS/v4gUshYS+PJLYMIE4EmG2jYFkiJlGkqPVqfF11u/xvKLywEAVd2qIkwTBgUUUClUUChSsP5jR3xUJ9XM0UMATEQR1NECjTQNFWQWAxgIwLhRSOAjFVK1/jgRdgs9G/TEsH3D8FHtj9B3S1/0a9wPlS9VNi4l7e6OYaVLAwcPAlWrChFQqcQN28XFfG2gtDTxZB8UJBrGOFqZI6TRCFNS06a2TXX6dOGEDggAOna07dh8osjeyVRKFRZ/tBh9G/cFANyNvYtaZWpBRx1SdakgnNB1XSruxnxk5ugpAHoBSM7LIUssIDOLCzIEMApAf5iKwIHbDvjn/iQQFdC2ZltUcK2ALxp/gT5/90Fzz+Yod6Fcxn4CgYHiH//+t4jlb9JEhGeWLGleBEjgu++AnTtFuKg1IrBrlxCAUqVsF4ELF4ChQ4UQ2IkIALCv5vW5QZo2jf3+7keFWkGowfITyhv+v/iY4mw6/1VeePgZTRtki9d7JKPzcfSS9Mjm9QWNVJL9ae5vJ/BhIzqOAnus60HXca78eP3H3BG8g8XGFGO7le04aswoy43m794lK1cmFQqyRQsyIcHyEHx8SIBcvNi6IV++TJYoQX7zTdam/N//kq++SiYlZe34fKLImobSY9rPoKxLWUQmRKK2R208TXiKYg7F0LXuM0xpFwvHDA8d9SES0F7K20FLMiBNQwWJZwA+gWgRa8zGoNr4eEMwetbvCf/r/ni32rvY/MlmLA1cin239qHZrWb4c8Sfhv0zOIZv3ADefFPY/StWBI4cAV4y8/f36JGoCvrzz4C394uHHB0tQk9LlBCF50pkob85KQrXuZuGohdw8luJCgpp2jT28+9HqEGoQQ9fD6q8VZx0fBJdx7lSoVbw0w1lqEkytzLwJHkhP4cvoVwRFBwekmxK078TrQ5UHypLqMF+/v2o8lax27puvPTokuHIcePGWV4JkGRyMlmjBlm/PhkYSFavLl5375ofys2bpE734iFrtWSHDqS7OxkaavuU168nt2yx/bgCglwRpMN0ZVDOpRyiEqPgpHKCllpodVq8V80Nq7vFoIKraV2KkgA2AWiTx6OW6JErgoLAdQDtAdwx2pqYCiy+8G+0rTkPgY8CUc6lHNZdXYfXK7+Ob7Z9gx2f78CFjRcs9xhOz759onyEp6fwD7z3ntiudyCHhgqH7aRJ1tcDunJF1Bpatw5o1862KYeEAK++CnTpIkpa2CFSCEwwFYO6HnURHBUMHXVwUjmBJL5u+j7GtQqEe/GHJkc7AFgEUcFUktdIIchvjgH4CIBxVd+oRCWG7K6Jqe1O48jdI/iw9odQKVVY8M8CDAwYiP6v9keNqzXwe7rewBlEICoKWLgQ+PVX0Ww+PenFYPNmUUmUBE6dsi5MVE9MDFC6tE0zRmqqSDiLjgbOn7fc76CAU2SjhiyhTzrr16QfFFDgeuR1eBT3gFKhRKpWhJKeuHcf5FGcCDONQEgD0A8iSkLqq6QosQGiwbyxCDxLLod+f9fG39cj8POen9FlXRcEBAdgxukZGBgwEN81/+7FIvDsGdChg8gRuJ+hzaDwDxw6JG7+LVuK2P0dO6wTgWvXgAEDgIQE20UAECUtzp8HVq+2WxEApBCYRS8GfZv0hQIKPEl4AlcnVygUCjipnHBPcw/vLu+CL/zLY+NVcx/hXwC+gqisKLEWHx8fNG/eHCVLlkT58uXRpUsX3LDUTERSQCBETs0nMA2nvvjICa1XuODs/Ri0f7k9lgYuxeS2k9GhVgesvbIWv775KzwvemYuAsnJwuQSFATs3m252byXF9C4MZCUBBQvbp1JSKMR+QXHj4t8BFtJSREhrCNGAK+/bvvxBQgpBBZILwYAoEnWwMXRBUlpSfjjX3/gQdwDPI6PRa/NSkw/ZS42eRmADgBizLwnMcfhw4cxePBgnDp1Cnv37kVaWhratm2L+HjZG6JgkgbgWwBDM7yz86YKnda6IjrJCS2qtMC6q+uw4MMF+KLxF3BUOeLAFwfgfs7dSAR8fHyMRSAtTRSHO3YM2LYNeO01y0PR6YR/YNYs0TryvfeEycgSJNCvn4gs+vvvrD3NOzkBp0+LDGR7J19d1XaAaZ5BibElqPJW8fUFr1PlraKbjxsVagWH7XVhmtZcRFF9knfycwp2S0REBAHw8OHDVu0vo4bykjiSHWguR2B5oAsDH57j6kur+TjuMQduG0i/y34cumcoq0ypwtikWPr4+BhFB/n4+GS8REoK2acPuXVr5kN58sT433fvPo8mumPhb2/SJJFfkNVIn2nTyKCgrB1bAJErghdgaiaKT41HrTK18M/Df+Dq5Iq4lDiULlYa00/psC34S8RnsAZdBfAGgH/yfvB2TmxsLACgjAVbb3JyMjQajdFLkhfcB/AviPwZY8YeccONpz+iWumX4VXKC+Vdy2N6fSpFAAAgAElEQVROxzk4du8YJpyYgJ9b/ow5U+cYRQf5+Pjgt99+e34SUtTud3QEVqwAOnWyPJSDB4Fq1YR5R4/eZwCIDGRzfQCePgWGDRNVRW1l716RObx3r+3HFlTyW4nsBdOVQaVJlajyVrGsb1kq1ArWnF6TUQlRHHXoIz6MM7cycCFpv3HGeY1Op2OnTp349ttvW9znr7/+Mnqq1L/kiiA3CSRZhaa/75Q0FXfd/JoKtYJd/bry1XmvstKkSoxLiuPXW8T2+efmW7cSGDtWZPeGhWU+lKAg0s2NbNNGrB5MSb8y0OcZpM8psCa/wJSoKJHV3Lq1yD0oJEghsAFTMSg2phhV3ip6Tvb8fzmKpvx84+dsNr8sr0aYEwMFySkks/ADLGJ8++23rFq1KsMyuRkkJSUxNjbW8AoLC5NCkKtsJ+lK09/1k3hw+L53CTX4pf+XrD2zNitMrMCLjy7y4qOLLDG2BFcErrBOBObMESYbb+/Mh/LkyfPEspgYy/ulF4Nbt8j27a0vN2GOXr2E+Ny7l/VzFECkENiIqRg4j3amylvFiccn0sPXg1WnVqXzaGfWmlGWB26ZEwOQHERRh0Viju+++45VqlThrVu3bDpO+ghyk5kklTT9LQc/BT/b2IxQg0N2DWGVKVVYfVp1BkUEMU2bRpJ8/OxxhoxhsyKwZo2oH/Tjjy9+Wv/sM7JcOfL27RcPXS8GpUuL8x84YNPMDdy7R5YqRa5cmbXjCzBSCLKAqRg4jXaiylvFwdsHG8xGpXxKseLEMlx2wZIYtCMpb1jp0el0HDx4MD09PRkcHGzz8VIIcoM0kt/T3G/46F1w8vHhjE2M5aagTbwVdYttVrRhSGQI269qz0EBg0hmLBthVgTi40lPT/KLL6wzuTx8SJ49a/00li8XKw13d8vlKKwhIiJrJqUCjswsziKmbS9rl6mN0OhQNCzfEIGPA1HJtRIUCgVeq9QUjSoEYIzZLpcNAAQAqJq3gy+gfPvtt1izZg22bNmCOukahbu5uaF48eIvPF5mFuc0cRDNmLZneOfA7Yr440AlLOvih8iESNQrVw+li5XGs5Rn+GjtRzh9/zS29NyCs+vOZggRNXIMp+fuXaByZcDBwfKQ1q4FWrUCKlSwfhphYaIERMOGzx3Hhw6ZL1RnDhKYNg346ivR2KYwkt9KZM+YrgxqTKtBlbeKby56k1CDFSdWZERcBBf+s5CfbgATU82tDCqQPJmf0ygwwIzjFwCXLl1q1fFyRZCT3CXZiOZWAo/i/sNXZtdjGd8y9D3mS+fRzvx1z6+MSYzhm4vfZMlxJXn07lHrVgKBgWS3bqRG8+IhbdkiTDu+vrZN5fFjEYb69Kl1oaWmzJsnVhO7dtl2XTtCCkE2MRUDlbeKKm8V265oS6W3kk3mNuHukN10GePClovAiGfmxMCZ5Jr8nEahQApBTnGS4gHF+HealAr+vLsCK0+uzKpTq3LskbFUeivZY30PJqUmcdShUXQf784z4WescwyHhJAVKpBNm5Iv+s4uXBCRRN272xat8+xZxm3mooksERJCuriQAwdaf007RApBDmAqBkpvJVXeKo45PIYevh5sMKcB265oS4VawerTwCCzEUUgOZJk4QlJy2ukEOQEaygeTDJGBvX1r8lKkyqx8dzG/PPAn4QaHLhtIFPSROhmqjaVIZEh1onAgwci6qdWLfHEnhkPHpBVqgjBMHdjt8T69cLvEB6e8T1rxECnI997j6xWjYyLs/66dogUghzCkhhMOD6BDqMc6Objxt6behNq0M0HvPSoIs2LwSck4/NxJvaLFILsoCX5J839Jq89Afv5v01NkoYnw04yNimW009N5x/7/+DtqNtsNLcRj9w5QtJKx3BCAtmokbhJW2Oe2byZrFrV/A3dEiEhZMmS5CefWHbuvkgMjh8Xpqi9e62/rp0ihSAHMRUDryleVHmr2H9Lf0INuo515ZBdQ1jGtwwdR4F7Q+vQvBg0J3k/H2din0ghyCrxJD+mud9ieOwr7LD6TX664VMmpyUz4EaA4ajgp8H0muLF6tOq81bULetEQM/UqeSVK5kPK/0NPDHR+ukkJ5PNm5M1a77Y5PQin8HNm9ZfNzMuX86Z8+QSUghyGFMxqDOzDlXeKn63/Tsq1AoWH1OcZ8PPcvH5xVSoFfxuu4KpZmsUVSJ5Oj+nYndIIcgK90i+SvNO4R4cumcIoQZ/2vkTP1r7ER1GOTAkMoSXH19mhYkVWHdWXYbHhlsnAqmptj1dq9XkL7/YHq45dCjp4GB9eKmpGOh0wjGdlmbbdS2h04ns5wKMFIJcwFQM9KaiX3b/QqW3kjWm12BUQhRfmfUKoQY7rnZkdKIlJ3LhS17JaWbNmsV69eqxdu3aUghs4jjJ8jT93aVqwe93KNlqWSsq1Ar6HPHhO0vfYfExxbk9eDu1Oi0bzGnAxnMb8/Gzx9b5BLRasm9f0snJujj+TZtEpM7o0bZP6+RJctEi245JLwb67OY9e2y/tjkCAsT5CjAFe3R2jCUxGHtkLD18Pdh0flOuu7KOpXxKEWqw2YISDIlU0LypaChFYo8kM+SKwBaWknSi6W8tNknJHutL8efdP7PYmGJc9M8iNp7bmKXHl+axu8eo+//T+fUn1xmVEGWdCOh05M8/C3v7Giui4y5dEhFCH39s22ogOlqsOrLK3bvCF6FUku3aZf086UlJIevUIVu1ypnz5RJSCHIRS2Iw6fgkuo5zpdJbyfln57PipIqEGmy9vAl1ulY0LwYdSWZSU0UihcAqUkn+l+Z+YzcjlWy7oipDIkOo1WkZFBHEuOQ4dvXrysuPL3Nv6F62XdmWz5JF5I7VPgFfX/FEPHPmi4f39KmI0mnc2LYIIa1W3Ly7dbP+GHP06CEE66WXrM8zyIyzZ0VZiosXs3+uXESWoc5FTEtYA4CrkyuG7RuG4W8Ph0qhwsDtA/H727+jVplaOHD7Ir7f+TKO3Glk5mzbAbSAaA4ukWSFSAAfAJia4Z3Hz5pg/dUhuByRgjsxd3D+4XkoFAq4Orli86ebERoVio5rOkKlUEGpUMLHx8e6jOHkZMDPD/jzT+C77148RBcXoGNH0SymRAnrpzZliuhgNnCg9ceYcvIksHEjMH48oFKJEtaZNbexhmbNRHvNRub+pgsQ+a1ERQHTlUHdmXWp8lbR96jIyoQaHHdknMGBXH92fX6zFUxJM7cyKEnyBY06iihyRZAZgSSr0dxK4Mid5tx23Z8uY13YYmEL+l32Y4mxJfjJhk9IkqsurqLKW8Ue63swOS0540pg3Djzl9QnfsXHW2fiefAga1M7c0Y4h3/9NWvH69FqSX9/MVZbks4ssWVL5pVRCxBSCPIIUzFQqBVUqBX0PebL4mOKs+S4kox4FsER+0cYQk9bL3fik3hLfoNRlMlnxkghsIQfyeI0/Q0lp4H9/ga7rO1CpbeSXf26cvH5xXQY5cCOqzsyPiWelx9fpkKtYL+/+zFVm5rRJ2BJBE6eJBs2tP4mumiR8AvYWHGW8fEiTLR5cxE2mlVCQzNuy44YXL8uxGnSpKyPKQ+RQpCHmPMZ6MVA70A+e/8s+2zuY6hV1Gx+OYZGlaZ5MehCWcH0OVIITEmlCDTI+Nt5oFGwxULwr4N/scb0Gvxhxw+cfmq66Cnw95eGbGGS3HVzF7U6rfUiEBRElilD/utfInnsRRw/Tjo6kgMG2B4qqtMJEQkJse249Jw8KRzE5moJZVUMPvxQ+DpsyX/IR6QQ5DGWxGD80fH08PWgwygHtl3Zlt/v+J5Qg2V8yzD4SSDJXjQvBrVJXs2/CRUgpBCkJ4Kk+cCDK4/d2HBOWdafXZ+lfEpxT4gIk9x/az9H7B9BrVbLP/b/wbln5xrOZrVjOCyM9PIiGzQQ3bxeRFiYqDf09tu2P9Fn1ZSUnpQU0dymWTPLEUe2isGePcI5vn599seXR0ghyAdMxcBxlCNV3ipOOj6JpXxKUalWsun8pvx1z6+EGvxm6zcMjw3jiP3FmWY2+awESfv50eUWUgj0nKa5dpIkeC+mA688/oeN5zZmy0Ut2XJRSxYbU4zH7x0nSWp1Wv6w4wdCDU48PpGkDSKQlkY2aSIibqwtB9GmjRCOF9UbMuXOHRGNY2VlWotMnixWA+fPZ76ftWKg04mIp7fesqu+BVII8glLDuTRh0fTzceNDqMcWH1adfoc9aFCrWCP9T342YbP2Go5GJlgyW/wM4ty5zMpBDqS82kuPyA5DfxlT0kevnOYXlO86DnZk4fuHOK7S9+lQq1gibElePzecX7595dUqBWcd3YeSTMiYMkcpGfnTvLaNeuHHBpqe2ilVkv++99CQKKjbTs2PQ8eiHpEgwdbt7+1YnDo0IuFpYAhhSAfMRUD17GuhBpUH1TTfbw7i48pzv2h+7n4/GKDGamffz/WmenMCw8daF4M3iGZA0tmO0JmFpOiXlBfmvtNPIxzYIuF4rdTclxJNpnXhGfun2GjuY1Yenxp7gjewbcWv0Xn0c5UqpVcdXEVSVrvE0hLI5cts6089K5dL64DZIlp04TpZd++rB2vJyaG/O0360xYejKrTaTT2dUqID1SCPIZc/0MoAZH7B9hcCA/0Dzg0L1DDWLQY10PNp5bh2Gx7WheDMqT3J+Ps8ofiu6K4DrJBjT3WzgT7srq01zof82fqy6uYqc1nXgz8ia9pnixypQqvPJYFH7TJGnYYmELuox14enw09aLgE5HDhpEqlTkuXPWDffUKeEcVquzMNXrZLFi5A8/2H5serJzw7YkBqtXky1bikgmO0MKQQHAVAwcRjkQanDYnmH08PVglclVqFQrDct2hVrBT9d/SpK8H6tmktnOZ0qSo1mUQkyLphD4kXSlORE4e/89vjKrNiccm2AoDaHT6ajVaTnywEiGxYYxKiGKH6//mHdj7lKTpBErg3bO1lcRHTtWPJ1bW9vn8WOycmVxw8xKuGd8PDlqVPZutqmpwobv55f1c5iaiVJTRW+FDz/M+jnzESkEBQRTMXAa7WQoR+Hh68GyvmUJNdh5bWcq1Ap2XtuZWp2Wyy4sY/MF4J1oc2IAkh9QRJAUfoqWECSSHExz37kmScF7MZMZkxjDtivbUuWt4qVHl7jh6gajMtIP4x6y0dxGLONbhv88+Ick+deov6wXgWXLhAhY+2SfmioavZQvb1tvAT05lZw1fbooI2FtdVJLpBeDyZPFZ/HPPzkzxjxGCkEBwlQM6s2qR5W3in8d+Isevh70nORJqMG3Fr9FhVrBgdsGstfGXvxx548sN0HJ7cGWnMieJA/m38TyiKIjBDdINqG57/raE0fWmgHOPj2bdWfVZenxpbknZA8nn5hMhVrBr7d8TZK8HX2bL894mZ6TPQ3moQyOYWdn8rSFUug6Hfn552T//tabWfbsESahQ4dsn7K+Zs+pU7Yfm55Hj8R5BgzI3nn06MXAwSHnCtXlA1IIChimYlBlchVCDfbf2p8evh6sNq0af9z5o6EchedkTzp4O3DonqEs5+vB3/YpLISYKihaYRbeKqZFQwhWUoQLZ/yO/a+Voft4Z/oc9aH7eHfWnlmbVyOucvD2wYQaHL5vOLU6LZPTkllzek3WmF6Dt6JEJm8GEfjrL2E+KVUqoxjo4+21WturfWalXENCAlm3LvnaayLuPzv06yeS3Z4+zd550rNrlwhBrVw56+Uo8hkpBAWQDNFE40Q0Ue9NvQ0O5KiEKA4OGGyoTaTPCB15YCQ/31iZj+KUtBxVFJaPs8s9CrcQxJHsR3PfaVKqkj/uLEWvKV48d/8ck9OS+dve3xiVEMXvd3xPlbeK88/NNzrbrpu7+EAjosssOoY1GvLNN42fxG/dErbwo0etH/rVq+SECVl30P74I+nsLDKWs0Nysgg7nTv3xfvaSnBw9msT5SNSCAoopmKg71vQza8bPXw9+Oq8V1l9WnWWHl+aCrWC/1ryL6q8VVxzaQ3DY8NZf1ZZ7g6xZCpyJ7kxH2eXOxReIThD8mWa+y5vRbnwcdwB+h715cBtA3k63PjpPSQyhDtv7iQpMof/u+u/BscxaUWeQHox2LNH1NavWdP6BLC4OPE0X79+1hy8hw4J2/vUqbYfaw6dzrYw1xdx7NjzzyInCtXlE1IICjCmYuA+3p1Qw1CbqOGchmw0txGLjS4mIok2fMo0rTD9jD48mgo1+NteS60wQfIriifNwkHhE4I0kmNJms8ZWX/FjX6XljDiWQTfXfouHUY5cHngcl6NuMpOazoxJvG5c3Vz0GY6jXZiu5XtmJgq6t9YnTGs0YgoH5WKdHOzvo+vTkf26iWKyV2/nrWP4NYt0a4yuzfv9etzvgl9QgLp6SnMTXrsVAykEBRwTMWg9szawoF8UDiQm8xrwrYr2hrCSv+z7T/cHrydpXxKceT+kXQY5cCWi8CwGBXNi8HLJLPpgCsgFC4huE1hxjNnCnLmoAB3lvIpxSknpvClqS+x/MTyPHLnCPff2k83Hzc2nNOQ9zX3SZKLzy+m0ltpaEBPZiFjuG9fEWlTooT1DtuFC8XT/OrVWfkAcq5nsEZDlisn5pCTTJsmxDE42Hh7ZklnBRQpBHaAqRjUmVlHtLec30z4DOY15bA9wzjlxBRDZEjntZ0JNTho2yC6jHVh+QlO1CR1pHkxUJIcQTIbZXwLAIVDCHQkF1P0ncj4XUU8q8l6s5zZaG4jXn18lTWm12CzBc14L+YeF/6z0FC0UL8a2BG8g1CD/9n2H8Nq0WYRIMUT/YYNlh3IGaahI7t3z3p0zvnz5MsvZ7zJZoWRI0US2r172T+Xnvh4USzvyy/Nv29nYiCFwE4wFQN9KGm9WfWMHMhTTkwh1ODXW76m9yFvKtQKtl7emtWnVafXlCrcEfwZNUmWTEVNSF7Ox1lmjcJTYuIhyQ9p7rtJ04L7Qt9is/lN2Htzb4OjNygiiImpibz06BKhBr8N+Jap2ueRPKnaVK68uNLgF7A6Y1hPQICw8+vRaKwXA602a0ljqalk06ain0F2o4QePhSrmKFDs3ceU6ZPF6uBzPon3LljN2YiKQR2hKkYeE3xItRgtWnVWMa3DJvOb8qAGwGGrmd9Nvfhthvb+Nbit3j9yXXWmlGLTqOd+OrcUjz/oBjNi4ETyXG0x+J19rsi0JFcQ9KD5r6TezFKbg8eTpK8FXWL7y9/n21XtqVOp2NiaqLhJn/2/lnqdDqmalM5KGAQj941juyxeSWwbZsIi5w40Xi7uWgiw1R05PffZ88eP2GCuO6ZM1k/h55ffiHd3W2rJ2QNZ88KMXgR6cWgACOFwM4wFYOqU6sangT1K4Ojd48aHMs91vcwmASO3DnC8hPL03m0Mx28wfnnvCy0wwTJphTtDe0H+xSC+yQ/ovnvANxy3YMVJxVnF78uPBV2il5TvFh2Qlnuv7Wf4bHhfHXeq5xwbILhbPEp8ey0phMdRjlw7eW1hu1WO4b1nDtHuriQXbqYt9WnF4P0K4P584VfYM2arH0cN28KM86QIVk73pS4OPLEiZw5V1bRi0EBRgqBHWLOZ6DyVnHCsQl0H+/OpvObMigiiNWmVSPU4FdbvqJWp+Xv+34n1GDp8aUNfQ8Gb29Jsj7N34gcSP5JMinf5moL9iUEel+AG8199qlaD0489i/x/f39FWecmkHHUY58Y9EbvBdzj+fun6PnZE9WmVKFgQ+FYEclRPGtxW/RZawLdwTvMFzJ5pXA3btkxYrk669nHvJpKgYXLoh4/4EDs/6xnDpFtmpFPnuW9XPoyelVgJ4BA8j9NhZ1lKYhSW5grp+BUq1kiTEl6ObjxqbzmzI8Npy/7P7FEE2UkpbCyScmU+WtYrExxVh2Qln6XfYjmcjQqB4WMpJB0QXtQP5N1krsRwiCaCkiiAQP3S7H9qua0XWcK1dfEhE3009N5/c7vmdyWjL9Lvux+JjibL6gucFXQJJtV7ZlGd8yPBl20rAtS47hBQvEE+yjRy/eN73PwMtLNGUpCO0ZT50iixfP+do/J06IFc/WrTl73nxGCoEdYyoGtWbUMhSsK+VTik3nN2VkQqShn0HN6TUZlxzHY3ePseLEinQc5UivKV68GXmT7y9/n//Z9iqDIixlJIPkFyzIBewKvhAkkPyDpCPNfb6P4sB1Vz5mqjaVB28f5O6buznj1AyjM+h0OrZf1Z69NvViQopxP+BLjy4xKOJ59q3NjuH0xNmQX6LRCOeuUpn19owPH5LduonWldlFpyPfeUc4m3MqBFVP9+5k7do5m5RWAJBCYOeYyzNQqBVUqpV0HedqEIPBAaLeTNkJZXk35i4jnkVww5UNrDWjFqtMqcJxR8bRZawL686szqUXvDJJQnMnOYsF0ZlccIVAR/JvktVp2RdQiu8ubcDu67qTJFdfWk3Xca6sO6sunyU/Y1xynMEElJCSYHAQnw4/zS5+XTKIQpZE4Mcfs57Ba8lnYC1ffEF6eORMDaCAAPHUvn179s+VntBQIXa5UaIin5FCUAgw5zNQqBV8bf5rBgdyZEIk1QfVhBosPqa4IaIkPDacJceVpPNoZ667so4tFrag0lvJ3/Z+QK2uOS2vDhqRPJxPMzZPwRSC6yQtNRAC45LLstMaJVXeKlabWo17Q/byq7+/ItRgr029qEnS8Hb0bTac05A1ptdgStrzcMrtwdvpMtaFby5+k1EJz+3hNjuGSXLGDHHznDfPtumFhZHt2wsbeFbF4Phxce0FC2y7tjnS0sgGDch33835bmFDhwqxssPGMy9CCkEhwVIP5EnHJ7H0+NIGMZh6cqrhfa1OLG/9LvtR5a2i0lvJRf8s4tgjY/9vkkjjo7iRjEm0JAYg2Z2klSUHcpmCJQRPSP5AS+UhUrXglut12XtTV0Nnuvar2nPYnmF0GevCJeeXUKfTcf+t/Sw7oSxrTK9hKBdNkkvOL6HKW8XOazsbrQayJALbt4sn3f/+17YppqWJIm6VK5ORkWKbrWKQlka++irZrFnOmHEiI8lOnbK2KnkRCQnWd2GzM6QQFCIsiYHTaCc6j3bmq/NeZWRCJBecW0CFWsFBAYN4Muwkk1KTeOHBBbqMdRElr7f0N5gehu0dxkEBnbkpKGND9OcvR4qb3pP8mfj/KRhCkEjSl5aigUjw+D0llwf+zFRtKqeenMpNVzdxyfkldB7tzPar2vPiI9HMXV8a4v0V7/Np/HOTybn75wg1OGDrAKPksSw5hq9cEQ3cO3Wy/UY8frwoO3HAJJDAlqSz8+dJV9fs9xnIbbKSGGdHSCEoZJiNJvIWpgfHUY5sPLexwYGsUCvoPNqZLRa24H3Nfd6LucdyE8qxlE8phkaFUqfTcfi+4VR6K9lwTkP+uvt1XnyU2eqgJEXPgxzqJGUj+SsEySTnkqxMS5/Pwzglxx2tz/eWvkv1QTVJ8kn8E3b168qS40py7eW1dB7tzE5rOjE5LZnnH5zn8H3DDTf79FVD94XuM/p3lh3DUVHkd9/Z5hwmRUKVgwM5bJj5920Rg+ho265tib17hbM6p01CyclkjRrkkiU5e94ChBSCQoglMSg2uhhV3io2nNOQUQlRBjFwGevCChMr8NjdYwyPDWetGbXoNcWLfx38iwvOLeCZ8DNsMq8Jld5Kdl7TkcP2ujI60VIRO71DeSzzShDyt8REMsklJKvR0ueRpnWgVvcLp54cxZLjStJzsicDbgRwR/AOVpxUkR6+Htx4VZQF15t8Oq7uyKTU5/kb8Snx7Ly2Mxecy2hHz5IIJCdnL0Jn0ybRdjKzJ+UXicHmzbYLkCV0OhG6+v77OXO+9KxcKXwYV668eF87RQpBIcWSmajK5CqGpLP0YlBxUkWjfga1ZtQyNMTpsLoD70bfpe8xX/5n238YFBHE2jPKcdJxFZNSM1shuJH8naSVteuzSd6uCOJJTifpRcvzB9deVnLT1cl8Y9EbhhpQ0YnRnHZyGqEGP1j1gSEXIOBGAN3Hu7PSpEp0Gu1kWBlEPItgi4Ut6DLWhduDjSNhsuQT0OnIr74Scf/Zifm35snbks/g3DlhVsqpCBx9pNDBgzlzvvS89ppdt6G0BikEhZjMHMhlfMuw9szaRmLQdH5ThkSGkKRBDMpNKMeyE8rSfby7UcmC0YdH03m0MxvPcePqiwpqdZkJQjGSA5jbBe3yRgjCKSq1lmVmArD/Fjh0Tyveib5DkhyyawgP3zls6AUQ/DSYs07PMtQGGr5vOKEGO63pxKiEKO68uZPOo53Zankr1pxek+UnlufZ+8bN1rMkAqQIEQXI5cttn/6BA6JPcULCi/fVYyoGWq3ob9Cgge2tLs2h05FvvCFWHzltFrpwQXxWW7bk7HkLGFIICjmWxKD1staGgnXpxWBQwCBGJ0az4+qO3Be6j7Vm1GLlyZXZcXVHfrDqA4NdelPQJpadUJZQgx6+Hhy49W3uDS3NzG6O4vVvkuuZG2Urck8ItCT3k/yEZGYmMTAhpSkHB9Tjnwf+ZMVJFbk5aDNJEfv/y+5fWHdWXcanGIcfbg/eTpW3ir7HfA2RXCS58+ZOKr2VLDG2BK9FXDM6JkuOYfJ5f91ffrH9Y9BoyGrVRGimrQlV6c1Ef/0lbq5ZaWJvjgMHxPl27HjxvrYyf76IisoJwSrASCEoAlgSg2bzmxFqsNKkSoyMf+5A7r25NxvOacjiY4pzyokpBp/B1cdXSZK7Q3Zz1cVVjE2M5ddbvibUoOMoRx68fZDkeT6Jf+8FKwRQVNr8ieR5ioSr7JPzQhBKchQzSwTTv86EOzMlbSsP3znE5guaE2rw4/UfMyw2jEfuHGGtGbXoPNqZvsd8Dc7fG09vGK50/YlxB69nyaLWjt9lPyMHMpkNx3BiIlmpEtmhQ9ZCNQcOFCWdQ0NtP5Z83ulMoRBhpzlFYqJofpPTqwE9tqx+7BQpBEUES2LQbmU7w7FtdEcAACAASURBVFN9ejH4Zus3/NL/S0INdvXryprTa9JrihdDo0I5ZNcQQg22W9mOt6Nvc0fwDrqMdWHDOQ0Z8SyCG69uZO/Nzbj4PDKpbpr+9TKFL+EsxdN31sgZIQgmOZFkMyvGDW4PBj/bWIVrL6/l+qvrCTXYfEFzHr4jku1GHx5NqME3F79puNmnadPofcibSm8lA24EGF1dp9NxwrEJrDq1Kp/Ei3BcvZmo05pOHD12dNbLRpAiXDMmC0783bvFU/ecObYfmx6NRpiEXF1zJtY/t27+pCh7kd1+CHaCFIIihCUx+Hzj5ywxtkQGB/KggEFceXElS/mUot8VP8PKIDQqlAE3Aug1xYsuY104+cRkXnx0kRUmVmDtGbWFE3TlB5x3dh67r2tJ70OgJqk4rbmxkuVJ9iW5kuQdm+aXNSGIILmZ5Pe01CDe9KXTuXDNJTd28avNqSencuXFlSTJxNRErr+ynmnaNGqSNCSFA3j6qemGUuBhsWF8Z+k7VHorqT6oNsoDSElL4YCtAwg1OGL/iAxmIlUbVdZ8AlotOXs2mZQNc9zo0WSbNtmrsZOSIm7c2S1HkZ6uXcXYcoP27cmOHXPn3AUMKQRFjMwcyB6+Hnxp6ks8/+C8kRjok5nuxdxjpUmVDGKgSdLwhx0/sNaMWkxMTWRQRBArTKzAl6a+xJrTaxJqsPu67lx1cRUfP7tLcgVvR1eldYKgf3lRZC+PIrmVomSD+Rta5kKQSvIWyT0kJ5HsTbKeTWO5FqHiA82vJKN4Ouw0+2/tT8dRjvSc7GlwAl97co2tlrdix9UZbyAXH11kGd8yrDKlimHFoCcmMYZtV7alwygHLjmfMV49g0/AlpufWi3MMdm1yWf36Xj4cBF9k1NioHfkrlqVvXGZIzxc+FJyouyFHSCFoAhiSQzGHRlnSD7bEbzDSAy0Oi0P3j5IqEGXsS70nOzJ0ChhK9Y7P68/uc4PVn7AshPKssGcBpx6YiprTq/JH3b8YLju+KPj2WJhGY45At6NMV9+4cUvBUXi1qsk25LsSbIvY2P7/V8IPif5GckOJJtTxPhn7VqRCSrOOQs2XwC+u/Qd3nx6kwO3DaTDKAeWm1COvsd8GZccx5jEGA7dM5SOoxxZc3pN7ry50/B565/sk9OSOWzvMEYmRGb4Tk6GnWT5ieW5/1bGOvcZREClEpnA1jzhb9kibpZZfWo+eFDUIcputc3790VZ6BEjnm/LrNOZNXz5JfnSS7njyB03Toy3QJQryX2kEBRRLInBqIOj6DDKgQq1gnPOzMkgBntC9rDCxApUeivp4ethEAOS3H9rvyEGvsTYEnxl9it8qHloMJNMOj6Jry98nUvOL+H6K+vZxa8zX1+gpP+1l3n9SVYEwfgVG4v/C0H2zhOXXJJa3decefojvrPkTS78ZyHbrmhLhVrBJeeXcFDAIE46Psng0E1OSzaYybwPeRtWByR5/N5x1ptVj+cfnDf7PQQ+DDSYh/TnS4/Z6KCdO0UDmE6dMk/ounZNlI/o0iVrN/LERLJWLVHSObu2+AEDRME2U/+ELRnI6Xn6VHQys9VHYg06nZh3nz45f+4CihSCIowlMfA96mvoezzv7LwMYhCZEMlOazoRarDchHJGYqBJ0nD4vuF0Gu1EhVrBatOqMeKZ6GFw4NYBtl4uwlbL+JbhkF1DeO7+OaZp0zgoYBBrTge/3wHuDXXlE03eCUFCipInw1z5xwElm8wD/f6fL3E76jYnHJtgMHO1X9XeIAY6nY7+1/wZlywyY/2v+fO+5r7hc0hKTeJve3+j0lvJNxa9wZuRGQvzLbuwjE6jnTL0HNCTaXSQNWIwbhxZr5642WaFkSNJR0cyKOjF+2bGtWui0fuUKebfz4oYbNokntgjcqE/xqNHIkvZtIZSIUYKQRHHkhhMPD6RruNc2WRuE0YlRHHRP4uMxIAkV11cZXAgb7uxzei84bHh/HT9pyw3oRwbzmnII3eOGEomBD8N5q97fqWHr4ehpeLlR5f56YZPqVAr+PrC16n4GXxlNnjp0U8kf2DEs/qMTjTf0MUWIXiWrGJQhCsfajqRnM3f973P8hPd2dWvK2ednsWwWFF24cedPxpqMX228TOeDDtJrU7LAdsGUKFW0GuKF6EGl11YluEzvfbkGhvNbUTHUY4cd2SckUNY/5kP2zvMUOBPHxaaHqvyBKwRg6yaNq5dEyIwcmTWjk/P33+TTZpknsWcFZ9BbrWi1JObEUkFDCkEkhc6kOvPrs83Fr1Bn6M+hraXejEIjw1n1alVDeGkD+MeGp1b70B2HOXIihMrcvKJyYan6KTUJEM0jX6FUWFiBTGOf4Nz/eYa/A/qg2q6j3dnyXFgo7ngh2tAv0vtmJAwivei+nBVYEUuPuFBABy/T8Xxx5S8HzuY5AL23uTK1xeCHr6gy9jibLagGa9GiJyImMQYXn9yndNOTmObFW144p5odO5/zZ+L/lnEmMTnpowDtw6wwewGhBqEGhy2x7jgmj7Z7lHcI7Ze3poXHl7I8Fknpiayw+oOVHorOeXEFKPCcXpsyhg2JwZz5vyvvfMOj6ra+vBK6IQEIoTepUmRS/eKhSoWRJArF1EUQQUEVCwIqB+hExQUBZSiCIKCSpOAQuiCdEEgQAiQQGhJQEiB1Jnf98e6k0yfc86cSWYm632ePPdm5px99hnieWfvtfda7idIGz2ap0f0KjvpTjoKa65f99wGr7Q0XmJbxBARCABc1zMoNbkUKsyogHc3v2sjg4TbCaj6aVUETgxEyLQQLD6y2GLpY3RSNCpGVETojFAUm1gMoTNC8fH2jy2SqqVkpmD538vx9A9Po/ik4qBwQvcvugPgefSo81FITEtEUnoSdsfvxrJjy3Dw8kEAXKJxwOoB6Pd9PxAR+iztg0oRlVD5k8qITorGutPrsPncZlz45wLuZN/J61v4jnDU+7xeXnnPx75/DHsu7rH4XG5l3Mqb1tlybgu6L+uOXfG7MGzDsLxpIgCIOh+FBxY/YJEu2h5GoxEjN47E77G/231f02Yxcxns3MlZQd95x/V5zjAYOMDrLitWKKt9bEJJAPmpp3hTnCdYupRXWF2+7Jn2vRQRgZCHq9xEwdOCERgeiP4/97c7Mqg/p35eTYOIPREWbUcnRaPKJ1XQZG4TvLr+VXRY1CHv2/Cei3ssxJGZkwn6LyEgPADf/PUN3oh8I+9bePVZ1dHpu05YcnQJACApPQnLji3DyhMrseLQChAR1h1bh+Q7ybj/q/tRblo59F3ZF52/64zan9VGQHhA3mhgzv45eOu3t7D+zHqLQK3RaMShK4cwPHI4yk4ti8e+f8zmszIYDXkyMO0k7vRdJ1y8fdHuZxsZE5mXbsIRmtNGAPkyKF2a8+5oXeqZlKRf6oeYGG2J5ZyNDC5c4DYXL9anj9b07s3XLmKICAQLXE0TVf6kMspNLYdP9n5iVwYNv2iIyp9UzlslsyNuR14ZRZMMWsxvgcQ0zkh6Ovk0KJxQ+7PaeDPyTSzfvhx//fUXKIDw4NQHERAegBlbZiDmRgxWnVyF8VvHo/8v/bHs2DIAnKfHJAkayw/QmtNrAuBc/6WnlEbxScXR4/seGBs1FouOLEJiuuNsqH9f/xtN5jbJS70RviPcZrrLxHdHv8sbvbz262t2p3kMRgPCd4QjIDwAA1YPcHhdtwrNA5wyonVrXiravbv2zWNDhwKhoUC67Qom1QwZwikttEwvOZLBmDFAhQqeKRd55w4HoCMiXB/rZ4gIBBtcyaDF/Ba4efcm5h+cbxNANq9ncCb5DGrO5rTXn+79FBk5GRYySEpPgtFoxJ+X/sSwDcNQYWoFfqC/+r8HYgCBnuaH/OIj9r8BGo1GZOZkcm3fa3EgIsRdj8t73zQyME0TmZNjyMGei3swbus4fLTtIwAcM3hp7UvYfG5zXvzCHIPRkCeS/Qn7Mfq30Ri8frDFNJGJm3dv4skVTyIgPAATd060GPWY49ZIwMSNG7zyZto0ZUtL7fH337yJSmsBe3MuXeJg8yefaG/DepooI4OXoKotqakU056LM2dcH+tniAgEu7iSQesFrTFy40iUmVLGqQwOJBzA8MjhKDaxGOp8VgfL/16O6ERLGZjIMeRg24VtWHF8BQCe9ik+sTiqfFIlL5f/8evHHT5QHe0sTr6TjBbzWyBsZhiik6Jx9NpRdFnaBcHTgkHhhEozK+VtenNG1PkotF7QGm0WtLH49m8+TWQug2d+fAahM0ItNpdZozmVtD1MfVK6z8D63M6dgcaN9cmv8+abPLLQunTVhLkMfv6Z8xSdPet+/+yxYoVnCtv4AAEAQIJgB4PRQK9ueJWWHltKIFCTik0o9p9YiugWQdP3TKeaITWpRkgN2hS7iYiIhrYZSvOfmk+BAYF0JfUKdV7amTJzM2nnoJ2UY8ihsdvG0q2MW7Rz0E6KToqmrsu6UuWgyrTtpW0UFhRmc/2UzBT64cQPtC1uG205v4XSstOoZLGSlPFhBgUGBNLL616mjJwMCisbRqWKlyJkgj7v/TmlpKTQlstbaM3pNXQz4yZduHWB4m7FUYXSFahYYDH6rvd3tODwAupQowN1qdeF2lZvS8UCi9n9DADQ9rjtNPWPqbQjfgc9UPMBiugWQY/UecTiOCOMNGLjCFpwZAHNeXwOjeowii7cukCBAYFUt0Jdu21Pnz6dxo8fn//7tGk0dtw4df9It24RPfss0eefE7Vsmf/6778T9e5N9NhjRL/8QlSypPN21q/n4zduJHrySXV9sMfBg0RxcUT//a/7baWlET3+ONHJk0RbthB16OB+m4IlhSwiwctRMjKYsmsKAicG5hVUtzcysE5HseXcFtT+rDZCpoeg2bxmFiMDexiMBgxZPyQvgAwAIzeORNelXfGvr/+F++beh3oR9fJGBIuOLELn7zqjz8o+eOf3dzDv4DxEnY/KmyY6mei87KDpG//d7LsImxmG1gtaY+3ptXbjACZuZdxCgy8agMLJbklJc3SZDjIagWef5Tnzi3aC1OYjA1cxg9u3vTuvTnw870XQmo7CFVeu6LNKykcREQguUSKDzec24z+r/sMB5EjbALK5DAAOHD+76llQOK8OqjSzEg5fOey0HwajAcMjh1vIwBwl2UedxQxM70fsiUDjLxvnlZCMvxXvVAAAcOTqETT4ogHKTS2H7su6240ZmHA7MGziq694TnuNk9VISmSgNpbgjDt3gC5d9F+L/+WXvCy2fXvPyOC994CqVYvUJjJzRASCIpTI4Obd/HoGTec2zcsxdCX1il0ZAMC5m+cwaO2gvLrJSelJSEpPstjIZY7RaLSRgdri9eYyMI0MNp3dhP6/9EepyaVQanIpDFwzEHG34hR9NvMPzkeJSSXQekFrxN6MdRgzAHSUwOnTvFR0+HDXxzqTwY0bQPXqwJYt2vphzTff8PLOCxf0ac/Ev/8N9OzpfqI6exiNQIMGXIKziCIiEBSjVAavrOOCNhWmV8jb9OVMBgBw+MphVP6kMlrMb4E3It9Aqcml0OvHXvjq0Fc2D2RHIwM19QhOJ51G7c9qI2xmGE4mnsR/f/4v7v/qfszcMzOvIIxSNsRswJgtYyxSRdiTga6B4T/+4OCu0mWUjmQwZgwXidErZ0+bNvpv9oqL45HPCl5EoDlRnSOio7n9DRtcH+uniAgEVSiVwZRdU/KmfT7a9hFyDDkuZWBaWtr4y8aYsGMCHv72YRSbWAwUTpj9JycsS0hJwB8X/8C11GsYFjnMQgauRLD87+UYtmEYWn3dKm/vQb3P66HyJ5Vx5MoRxZ+B0WjEt399ixfXvOh0yshgNGDohqEICA9A31F99ZOAVqxlcO0ar5v/6CN92j940DMP1OnTuZ9pafmv6SmDadO4BGcRKEnpCBGBoBolMvjn7j9YeHhhXoI2U+I1pTJoPr953hTR6lOrcSLxBABg9p+z8x7ipSeXRoUZvPfgm7++Qfz1eBARui3shk7fdULz+c0ROiM075t6t2Xd0Hx+c7y89mUsO7YMV1OvuowZWHM55TKeXPEkKJzw0tqXLNJk2MNgNKD9S+31mQ4C+GHbs6f2hGvmMnjjDQ4037qlvT/mDBsG1KmjrR6yM2bNAkaNsn1dLxl8/DHw8svaz/cDRASCJpTKwDyF9eZzm/He5vcQeyNWkQys9xkAnPv/7+t/Y+3ptZj952yM2jgKD337EALCAzBz20wQEXp+2xPP//I8RmwcgYg9EXkrlRztP7AXM7DHoiOLEDI9BNVnVbfJtuoIm5hA376KzrPLnTtAo0ZA27burfU3yaByZa5ephd37wInTujXnhLMZeCJ1URFBBGBoBm1Mui0pBNKTS6Fup/XxdKjSzXLwBpTANmUYkJL8XpnMjBN/0zdPRWv//p6XsoMV9hIoH17DqRqzQz6/vv8AD99Wtv55phk0LOnPquGPJUNdN8+nsJyhjtlL5OTLaeciigiAsEt1MpgwOoB6L6se16Stnqf19NFBgajAUN+GgIiwtzdczXdi/U00fW063jt19cwfut4AHC5hNQcu4Fhg4GnT7TI4NAhTv+gR2zh4kUWwaZNyvcZOMNg4B2/8+a53zdrmjThkpSu0CqDDz7gcpdFHBGB4DZqZTAschjWnFqDR5c8mjdNVHN2TbdlcOv2LX7wjiW7+wyUkHwnGc3nN0fQ1CCUnVIWoTNC8dUhddkzna4O0iqDpUt5CaVe6R/uuYcTy6nZdOaIvXs5SLxjh/t9Myc+ntv95Rdlx2uRQceOQL9+2vvoJ4gIBF3QEjMwzdkfvHwQgRMDETwtGAcSbP8DNhW3cSUD06qhIT8NcbjpzBXJd5JRMaIiKJxQZkoZ7L20V9X5inYMa5WBuwXkAeDmTV4h8/HH+a+5K4NRo3gvgh79M2fhQh4FqQlmqwkgZ2QAJUsCX9gvFaornpo60wkRgaAbWmVwO+N2XmlICif0/7k/Ym7EWLStZGRgEsGt27ec7kC25lraNcz6c1be1M+c/XNw9NpRtJjfQvFqIkDlZjGDgVM+u5JBbCwwdqx+lcKmTuWHvnWxGK0yyM0FqlQB3n5bn/6Z85//8ChILUpHBn/8wSOOIliRzBoRgaAr7owMTiSeQMWIigicGIheP/YCwKuEMnL4IehKBub7COztQLbu52+xv6Hfz/1QcnJJBE0NwulkyyCs0tVEgMYdw65kYDQCTzzBSzL1yL+fmckP7ddft/++Fhn89Rd/a/fEip1XXwU+/VTbuUpksGQJT5EVxLd1GREIRQ13ZHA55TIafNEA1WdVx/l/zmPZsWUoN60c+v3cDz8c/wG743fbyMBRignrHchpWWl53/ofXfIoKJzQbF4zzPpzFm5l2J9+UCIDt3YMO5OBKT++s1xCarh7l+sDxMQ4PkaLDBITvTNHj5J0FHrEXJTQv3/BXEcjIgLBI6gOINupdFZrdi3sjNuJKbumoPWC1nkbyZ7/5XlU+aQKms5riqVHl2J/wn5c+OcCYi7HWIhgy7ktmP3nbDSd1zRvl7NpyikyJhIHLx9UtBLImQx0SRthTwZ37wJ16wI9ehT8Q1apDHJzPfdN9+xZ92sZAI5lYDTqH9NwRt26BXctDYgIBI/h7sjAep/B5ZTL+OH4D9h6fiuik6IROiM0v0xlONnsI6gxqwZKTS6F5vOao9GXjRAQHoA5++douhd7O5B1SSVtwloGK1ZwINPZt3c1HDoEvPWW8oerEhls2sSb0q5e1aeP5nTsyDECPbAngxMneFf133/rcw1nJCXxyM6L8e7eCT6P3jIw52TiSYTNDEPDLxrip5M/4ee/frYQwbW0a3nlJl2lsFaCuQze/vBt/SRgwloGcXHut2ni1Vd5vbya9A+uZDBoEFc003vEcvs2UKwYsGCBfm1axwy++orTWutRm9kVu3aJCATBkzIwDyCfv3re6c5ivWRQ5Zkq+ksgr5MG/ibszg5ka9LSOMPohAnqz3VU9tJo5ML0772nTx/NWbuWH5x6p7I2l8ETTwDt2unbviOWLBERCAJQMDJoOqupyxQT7srAZjpI72WTZ8/yt+HOnfWTgalGQHy8tvPtyeDkSX64bd7sfv+s+eADoEYN/dsF8mUQGAg884xnrmFNRgZw5kzBXEsjIgKhwPCkDE4lnULY5DAQEc5ftX3fHK0ysJFA1ao8Rx6tbJ+BIvr1A2rW5CkLJfsMlPDggxx0dgfraaKFC/l3T6RufvNN4L//1b9dE7dvswhKl9annoEfICIQChRPyuDA+QMgIjSd1VRRbiI1MrAbGE5OBu6/n2Vw0vk+A0WcOsUPftPcuNJNZ644fhw4orzegkPs1TPwRYxGTtzXrl3BZC0dNQr47jvPXsNNRARCgeMpGZg2lIVNDlOVtdSVDJyuDtJTBgMH8mjAPDCrlwz0Qo/cRM7Izi64ZZ3uZC1VQ1gYMGmS59rXARGBUCh4QgYmERw4f0BV1lJnMlC0Y9hcBu5ME332mf1vju7I4PXXgZ9/1t4ne0REcF+6d9e38D3ACebKleOcSJ5i1SpgyBD+/56WQWqqZZlNL0VEIBQa7sjAXqUz8xQTalNY25OBqn0CesnAYSc1yOD6dc+MJD78kB+c9lYTucu4cbwayZMMHw60aJH/uydlcOwYi2DfPn3b1RkRgVCo2JNB8UnFVckgdFIo7m17r02KCaVZSwFbGWjaMaxVBrdv8zJM60RwNp1UKYMFC3gFUnKy8r4ooXNnoE8fx0tL3aFHDy6W40k6dwaee87yNT1rIJuzZg2LIDFRvzY9gIhAKHT0kEGt2bVwLP6YzfJRLSMD6krKRwLWaIkZzJnDm5uU7NBVI4MnnuCHnp4Yjbwjd/Jk/l1PGRiNQKVK2vY7qKFaNcs03CY8MTKIieGay96Yi8kMEYHgFbgjA1PMoMa0Gnb3EaiRwdRpU93fLKZGBkYj785VUxxFiQyys7nuQESE8naVEBfH33A3bsx/Ta8A8sWL3Pb69W530yG3b/M1fvjB/vsFFUD2MkQEgtegRAY37950KIP6EfVBRDgWf8ymbSUysAkME/FmLC0olYEp/cDOnerad1XcJicH+P13fdNUACyYQ4dscxbpMTIwGjkvj161F+yRmsrpJS5edH6Mq6ylSlm92ieEIiIQvAp3RganE06DiFBjWg3VZS9tJDB1KgcVAwL0kYGjmMHQoVxvQMvUgTs1kD2BJ2IGhYVeMmjQAHj3Xf365SFEBILXoVUGplVD9SPqK8pNZJKBw8Cw0eh5Gfz1FxAZqa1twPE00dSp+tUxMOfjjzl3jiPckcGCBY6L5ujFgQO8fFQJesigZEmOAXk5IgLBK9EiA1Px+tMJp12mozDJYHz4eOergwwGlgERsHixtpsp6KWlublA+fL5AV09qVYNGD/e+TFaZfDyy9pKU6rh3XeBRo2UH++ODDIz+e9m2TJ15xUCIgLBa1ErgyE/DckLFivJTVTuyXLKAsN6yaBFC8uYwWefaW/PXh9NMpgyhfu6bZs+bZv45x9u98cfXR+rRQadOnk2xxDAsnnwQXXnaA0gm+oQrFun7npmxMXFoVWrVqhSJT/jbY0aNXD//ffj8OHDecdt2rQJtWvXzjsmJCQEnTp1UnwdEYHg1aiRgakwza3bXHbSmQxmzJhhKYEPP3TeEb1lcOIExwZGjdLWlqM+mmIGRPwg0pO//1a3OUrtaqL69YExY9zroyt69uT+qEWLDK5dA7p04eC6m+zatQtEhM4ulgMPHjwYgwcPRpbKaTkRgeD1KJXB3N1zQUQY8tMQp+kobCRQrhw/oF09OE0ycDdm0KIFULEiP1TdiQ846mPbtty23gHkyEhu9/Jl5ecolYHBAJQoAcyd634/nfHAA8Arr2g7txCXlk6ePBlEhFmzZjk9rk+fPojXkG5cRCD4BM5kEDQxCKXfLI17m93LD/ex5DA30bsfv2sbE4iOBqpUKVgZ1KjBD1Udvi3asHkz75LVezXRmTMcd1BT5QxQJoPsbO6rp/P2v/468OWX2s9XswM5O5vTdOuwmaxTp04gIvztpLSmwWBA+/btNbUvIhB8Blcjg5aftwQRYe7uuXb3GVR8uqLjwHBBy6B/f6BMGe/KTeRJPJ21tCBROjL49VeWvZvpujMyMlCqVClUrlwZRidSOXToEIYOHarpGiICwadwJoPQiaEgIsRfj7dZWjp52mRLCbzzjm3jamTg7tLSLVt4Gabeq4lyc4GffgKuXNFfBtu3u5c8zZkMzp/nYjee3ExmMHDAW48010pWE61YwSK4c8etS0VFRYGI0L9/f6fHzZgxAz/99JOma4gIBJ/DkQymbJkCIkLLz1tarCbqMKiDpQQqVgRq1eKHjzWFMU2kZ3Eb00oV0x4CPWXw6KM8knEHRzL48Ufud1qae+07IzHR8rNxF1cjg6++4qR/bk4NjRs3DkSERYsWOT2uR48euHHjhqZriAgEn8RuCuvxxUBECJ0YmhdA7vtmX0sJTJrEwc6GDQtPBleuAF9/Ddzi1U267jMw1RLes8eyj3rIoFEjfXbJ2lta+s033G+18Qc1nDrF1/jjD/3adCaDiAggNNTtS7Rv3x5EhDgn6UIyMzPRsWNHzdcQEQg+i7UMGs5sCCLClC1TUDGiIqr2rmopgfbt86cFClMGprnjK1fyX9NLBjt2cNtnz9r20V0Z1KypX2ZQaxl8+SX/7kn++IM/m1On9G3XkQw+/JCXCLtBSkoKihUrhvr16zs9bvv27Xj//fc1X0dEIPg0FjL43z6CYuOL4ZEhj1hIYNqzz/JDcNiwwpfB3LmcesB6rloPGaxaxQ+727ft99EdGVSsCCipy6AUcxlMnarLt2enREXxZ6NheaVL7MkgPd3tQHFkZCSICK+99prT48aPH4/Nmzdrvo6IQPB5TDIwiSD4sWBLCVSpwkHCb77xDhmM6RmnRwAAIABJREFUHQvUq2f/PXdlsGYN0KqV43lpd2TQqROwfLn6PjnDJINWrbjYjScxiUDvjKwmPLDPYNasWSAiLHGS3yknJwetW7dWvYnMHBGB4BfkGnIx4IcBFgIgIkx7+GH+Jtu6tXYZnDqlrwxGjABatnTcht4BZHt9LIpLS7Oy+LP1ZBzCXAbDhnG6DzeYM2cOiMjpt/0pU6ZgxowZbl1HRCD4DTM/mWkhganBwbxq49NPC0cGjtJRzJnjOrWEt8nAaPRsla21a/1rn0HHjlx1rl07t5o6efIk741xsON6zZo16NGjB3LdlFsAAJBKAFBaWpra0wRBV7KysigrK4uIiDIyMqhp06aUm5tLRERjiWgcEVHDhkTnzxNNmkQ0axZR7dpE69cTRUYSjRxJNHgwvx4YSHT1KtFTTxFlZRFt3EhUr57lBWNi+P2wMKING4gqVXLcOaOR6L33iL75hmjuXKKBA9Xf4M2bRE8/TZSYyP297z7X53z+OdFXX3FfXWE0Er3zDtGSJUTz5hG9+KLjY1NTiWrVIlq6lKh3b+X3oJQRI4gOHCC6eJGoSxeiZcuISpXS9xrR0UQTJxLNmUNUrZq+bVuTlkbUsiXRrVtEUVFEbdsSEVFwcDAFBASoaur999+nFStW0ObNm6lFixZERJSUlESzZ8+muLg4WrJkCZUtW9at7moSQWpqKpUvX96tCwuCIBQ1UlJSKCQkRPV5P/zwA82bN48yMzOpWLFiVLFiRRo6dCj11knKuo4IUlNTqVatWpSQkKDoZtu1a0eHDh3S7ThPHOupe1JzrCfaVHNfhfn5OzvOfERARLRu3ToaNWpU3u99a9emhZcuUXEiuyODLunptH30aB4ZDBlC9OmndkcG7fr1s7y+k5GBTV/tjQxGjiQ6dYrapaUp+vu7v1YtOn/ffVQsOdnlyOCr+vVpeEAA36sL8vqqYGTwSJs2tPvcOaLFi4mee851my6w+ft77z2iffuI9u7lb9ADBliMDHT5b2XvXqInnyQ6coSoQQPPPyveeYfvKTiY6NQposuXNY0ICoLiWk4KCAhw+vAICQlR9NAsVqyYrsd58li970nNsZ66JyJl91XYn7/S43r37k2jRo2iEiVKUE5ODq2+dIlK1K1L38fHU/HYWKImTYgmTCCKiCCaPp2W3rlDIQMGEJUpQ/Tqq0QlS/LDukkTot27iTp3Jnr6aapTsqTl9du1I9q1i9/v3Zto2zaWgqO+LlzIbY8axdeqUIEoK0vxfd0iouIbN1Jw795EvXoRbd9O1KyZ3WOLBwZSSE4OkdrPdfFi7uPIkdzHV16xODaneHEKCQggMhictq357y80lCgzk9vu25en73r3ZkH//LM+fyvlyvH/BgURhYR4/lnx/PNEDz9M9OyzRI8/rujfpNBwK8JghalUYEpKiqLjHQVAtB7niWM9dU9qjvVEm2ruqzA/fzXHJSQkgIjw3XffoWTJknlB4/516yKHiIO3TZrkBZDvBgXZBpCHD7cJIKeGhioOIDvsq/lqop49gZo11f/9KQggbxswgK+hIHhoc30nAeS5c+cC5coBLtIga/77mziRq5+ZY7aaaP7nnytq1+n1TRvKTp9W1Velxzr9byo1VfG1CoNCFYEv4I/3BPjnfZlEkJCQgMjISEUysLuayI4MdF9aGhDAaYpdYPPv5EoGN28CMTHaE6s5W0109qz9jWoasLmvlBTAXp4cd2ogW3PlCjB7Nn9GHsDmnk6fBtav98i19EZXEWRmZmLChAnI9PXlX2b44z0B/nlfSUlJICIk/e+BvGHDBgsZPF+vnnfIoGdP7kNEhMt7svvv5G1LSzWg6u9PTxl4EJt7mjYNqFSpcDulENlHIPgN9kY5XimDu3eBIUM8k7U0MZHr8rqbvM687KVJBhMnAi4yYGpm1y4WpKMHvR4ySEsDVq50O+2DYj79lDeW+QAiAsFvcDTd5ZUy8FQK6+Rkvq9fftHWpnUfzWXwyCPAgAHut2uPX37hfv/zj+Nj3JXB1at8jQ0btPdTDQWRSE8nRASC3+As7uF1Mhg7Fhg9Wn8ZGI1A+fL6JYczl0GHDsDDD+vTrjW//cb/FgkJro/TugM5O5uvofWzVsvChXw9T+7I1gkRgeA3uAqAe5UMRozga3piZNChA08P6YVJBkSeyxB64AC3/9dfro91Z2Rwzz2Am3l5FLN+PSfTy8kpmOu5gYhA8BuUrITyGhksXcptJiXpL4NevVgGemIwcIUyImDBAn3bBoDr17nttWuVHa9VBo0bA/bKlBZxdBXBhAkT0LhxY5QtWxYVKlRA165dsd9RPU8fIDs7G2PGjEHz5s1RtmxZVKtWDQMHDsQV84IiPsjq1avx2GOPoWJFLuZ+9OjRwu6SW8ydOxf33XcfGjVqpGhJrFfI4MoVvt4PP/DvdmSwa9cu9OzZE9WqVQMRYa2rh6RJBhUqAJMnK/no1GEabbixmmjatGlo27YtypUrh7CwMDzzzDM4c+YMT5/MnMlLX5WiRQaDBvFqHh2ZP38+WrRogeDgYAQHB+OBBx7Apk2b+E2DQZ8ayR5GVxGsWLECUVFROH/+PE6ePIkhQ4YgJCQkbzmfr3H79m1069YNq1atwpkzZ7Bv3z506NABbdq0KeyuucWyZcswceJELFq0yC9EYELN3givkEGLFpZTOFYy2LRpEz788EOsXr1amQgAfcte2sPNpaU9evTAkiVLcPLkSRw7dgxPPfUUateujfT0dG398YKlpb/++is2btyImJgYxMTEYPz48ShRogTORkYCJUroWxrTQ3h0asj0H+bWrVs9eZkC5eDBgyAiXLx4sbC74jZxcXFFVgQAy6BEiRKFJ4P9+wHrvyMH00SKRQDwEtI6dXg+XG8ZbNoE/P23bvsMTHs/du3aBezbB0RGqm9ErQwKQBihoaFY+sUX/Le0cqXHr+cuHhNBVlYWPvnkE5QvXx7JycmeukyBExUVhYCAAL/YkVvURQC4KQNP1TMwGm1koEoEAFC9OhAWpv/IoE4d4IMPdNt0FhsbCyLCiRMngNdf5+CqFpSuJpo0iT8bD5Gbm4sff/wRJUuWRPTJk5yW49NPPXY9vdBdBBs2bEBQUBACAgJQvXp1HDx4UO9LFBoZGRlo06YNXnjhhcLuii6ICJhCHRnMmAF89JHtsVYjA9UieP55oE0bvpaeMnjqKaBHj/w+uiEDo9GIp59+Gg899BC/MHUqj2K0omRkMHcuF4zReUnn8ePHERQUhGLFiqF8+fLYuHEjv9G4MS8T9nI0i2D58uUICgrK+9m9ezcAID09HbGxsdi3bx8GDx6MunXrIjExUbcOexJH9wRw4PiZZ55Bq1atfGo04OyeRAT5FJoMxo7lh5+9B5eZDF5RK4J583h++uLFfBnokY4iPJz7a3qQuiGDN954A3Xq1EGCae/A8uX8eaelae+fKxn89JPrjWsayMrKQmxsLA4dOoSxY8eiUqVKiI6OBrp2Bfr10/VankCzCFJTUxEbG5v3c/fuXbvHNWjQANN0jtJ7Ckf3lJ2djd69e+P+++/HDXuJsbwYZ/9OIgJLCkUGu3Zx+6tX2+/U/2RgIMKRESOU38zff3O7O3dyAFkvGWzcyO2a35cGGYwcORI1a9bEhQsX8l/cs4fbdrePzmTw55/K9yu4QdeuXfH666+ziD2U5E5PPL6P4N5778WECRM8fRmPYZJAs2bNfHb1kyNEBLYUigzatwe6dHHcKYMB84hgVLPPIDeXc/fs3cu/6yWDpCTgoYcA678ZhTIwGo0YMWIEqlevjrNnz1q+mZjIbR87pr1/JhzJ4NYt/rdcvtz9azihS5cueFnPTX0eRjcRpKenY9y4cdi3bx/i4+Nx5MgRDBkyBKVKlcJJT2RILABycnLQq1cv1KxZE8eOHcO1a9fyfrK8OAuiK27evImjR49i48aNICKsXLkSR48exbWCSsblIfRKrV3gMqhVi9s+d87i7bS0NBw9ehRHjx5FABGOPfggjAEBuDFzprYb03NkYA8FMhg+fDjKly+PnTt3Wvz35GhGwS0cySAuTlG9BqWMGzcOu3fvRlxcHI4fP47x48cjMDAQW7ZsAXbvBp591uv3EugmgoyMDPTp0wfVq1dHyZIlUa1aNfTq1cung8Wmb8z2fnbs2FHY3dPMkiVL7N6TL4/cAH1rLBS4DBo0sFlNtGPHDot/nwAizCOCQWm+nNxcnnoyF7y5DLQGkHNybKSVhwsZOPrvacmSJfltx8Vp65c9CmCfweDBg1GnTh2ULFkSYWFh6Nq1K0sA4DQTRAWX8VQjkmJC8HnU7ixWSqFME7lKuqYmHUVqKlCyJPDZZ5avuyuDTz4BgoIcf6t2ZzXRlCmcz0jPVT3WMvjxx4IL4B4+zH8vhw8XzPU0IiIQ/AZPVF0rMBlER/Mqn7AwfVNYP/EE0KmT7evuyMAU1HWWPkarDDZtsg1G64G5DObP53+3gijKZMqhtG6d56/lBiICwW/wVPnNApPB4MF8TtOm+sng66+5f/Y2dWqVQU4O5zMKD3fdR7UySEzkz/Wnn5T3RykmGXTsyNc4flz/a1hjMLDg583z/LXcQEQg+A2erMNcIDK4do2ncoKC9Ctuk5TEG6i++ML++1pzEz33nLIMp1pkULs2MGaM8r6owSSDAlg5lMenn+qzEsqDiAgEv8GTIgAKSAZvvw2ULctTRBrTUdgwapTzUYMWGXzzDd+zkukVtTJ49lneGe0pTEVwGjXy6hrIBYmIQPAbPC0CoABkkJzMD6jvvy/8speurqkGNTLIzlbXthY+/JBHXwWRtTQ+3jM1HHRERCD4DQUhAqAAZGB6X88ayOfOuc7sqWVkoGbXrNqRgY5r/e1SUCms163jvw0vxrt7JwgqKCgRAAUgg8uX+VukXjJ4/33OEeRqKkeNDGbM4OPUjA6UyMBgAJo3B2bNUt6uWhISONi9Zo3nZRAfLyIQhIKiIEUAeFgG337L7ezerY8MTp3i9lascH1jSmWwcye3uW+f6zat++hKBo89xj+ewvR5bNni+ZGB0cirrLwYEYHgNxS0CAAPysBg4FU5LVrwck09ZNCtG9C2rbLNWkpkkJsLVKvGwWi1uJLBrFlA6dKAJ1JPAPwZhIXlpwD3tAwefVT/NnVERCD4DYUhAsCDMjh0iI+ZM4ffj452TwamzVq7dim7MSUyeOcdfj8nR1mb1n10JIOTJ7mvv/+uvl2l9O0LPPJI/u9Ki9towcuL04gIBJ/HUykm1OAxGQwdynP7d+7w++7IwGjk6mJqCsS7ksGRI3xPp04pb9O6j/ZkYDQCNWtyvQZPMWcOP/gzMvJf86QMvBgRgeA3FNaIwIRHZHD4sO0OWHdHBmpxtrTUaHR/uafBwGU/rWVw6ZJns3aePs0jGuuVT56Qgc4V0fRGRCD4DYUtAsCDI4OsLMtv3e7IYOVKXvGjBlf7DJKS3JvPdySDwkJvGXj5xjURgeA3eIMIAA/J4KWXuOj67dv5F9Iqg2ee4UDs1avqbsyRDBITOY2FkhVJrvpoLoOcHC5U40kxXLgA/Pqr/feK0DSRiEDwG7xFBIAHZFC9OqeeeP11ywtplUGZMsBbb6m/MUcyePBB4Mkn1bdnr4/mMnjgAU454SkiIji3k6NgdxGRgYhA8Bu8SQSArQwG1K/vngxCQ/ncrVstL6RGBua5iYoX57bVYk8GCxYAgYG8ecpdzGXw3HM8evHUv+m+ffyZOiugVQRkICIQ/AZvEwHgpgyGDbOVQalSvHY/Pd3yQmpHBkOGcB969tR2Y9YySE/nTVN6ZQ01lwERsHSpPu1ak5XFIy1Xyzv9XAYiAsFv8EYRADqPDOrV4+McFbdRI4N+/fRdTTRmjO3UlTuYZEAE/Otf+rVrTdeuwFNPuT7Oj2UgIhD8Bm8VAeCBaaJatYC9e20vVJhLS0+c0NaGMwwG4IUXPLuaaOFCFo6SJZ5+KgMRgeA3eLMIAJ1lEBbGc/L28vyokcGGDdyOXjI4epRX4ei5bt7fl5Z6ASICwW/wdhEAOsrg+HE+tnRpIDbW9kJKZZCRwRXB6tbVRwYVKvC9/PGHtnYcMXUqr5zylAz++YcT/CnFz2QgIhB8Hm9IMaEG3WSwYgWfU6GCezGDlSu5naefdl8GLVpw/x9/XFsbjjDd64ABnpFBeDgQEqJu41dB1TMoAEQEgt/gCyMCE7rJYPhwPqdqVe0yMBp5H0CLFvlTMO7IoFo1+8tc3SEtjVf3TJnimWmi48e5z7/9pu48P5GBiEDwG3xJBIBOMsjO5m+zDRrYr4EMKJPBgQO8pDQlxf0A8oUL3PcyZZSVvVTK889zwRpPxAyMRuDee4HXXlN/rh/IQEQg+A2+JgJA5wBy3bruycCEHquJPv2U+6K0BrISIiP58zDVa9BbBu+9x/3VUiLTx2UgIhD8Bl8UAaCTDM6dA0qU4AeZOzL49ltg+nT9lpa2aKGfDHJzgStX8n/XWwb793Nsw5UoHeHDMhARCH6Dr4oA0CCDmzdtZTBsGMugdm3tMpg4kVNPnDzpvgwuXwb++1+gaVN9RwZZWfn1Gbx5aakPyUBEIPgNviwCQIeRwd27XIqyZk2e79Yig8xMoFEjzvppMLgng+RkTuj29tv5IwNnNZCVkJ0N1KljmUZbTxlkZQHbtrm3D8IHZSAiELyC1atX47HHHkPFihVBRDh69KjqNnxdBIAOMjh/npPT9etnWc/AGmcy2LaNr7d4Mf/ujgzeew8IDua9DnrJ4OWXWQbmc/l6yWDzZr53DX9/FviYDEQEglewbNkyTJw4EYsWLSrSIgB0kMHu3cC1a7bFbaw5dcqxDF58EXjiifzftcrg+nXe9BYebhkzcEcGhw7xZ7F+veXresggK4tF+sEH2vtnwodkICIQvIq4uDjFIsjMzERKSkreT0JCgl+IANApgJyYyCkktMggLc129YxWGbz1Fu9zyM7WTwbt2wPdu9u+rocMRo7M76+7mMvAixERCF6FGhFMmDAh70Fp/uMPIgB0kMFLL/F7+/drHxns3WuZz0iLDK5f570FJvSQwbJlQKtW+UFjc9yVwbFj/BmvW6etb9aYZODFiAgEr0JGBJa4JYNXXuG9Ba1a8fJStTIwGjlo3LixZT1irSODu3fzC8WbJ6rTIgODwXlA110ZDBhgO/XkDmp3LBcwIgKhwFm+fDmCgoLyfnabJftSIwJr/CVGYI1bMjBV+HrhBSAhQb0MoqOBkiWB99+3PFatDIxG7tvgwfmvuSsDgOMFaWn23/O2paVejIhAKHBSU1MRGxub93PX7NumiMA+bsmgWzd+/9gxbQHk6dPtp7xWK4Mvv+R2jh/Pf80dGVy9yvf12WeOj3FHBsePA1FR6s7xUUQEglchInCMWzJ48UX7xW2ULC3NyQHatQM6d7Y9Vo0MsrI4J5L5aiTAPRkMGsTpqTMyHB9jMABDh6qXwcCB3F89ayt4KSICwSu4efMmjh49io0bN4KIsHLlShw9ehTXrl1T3Ia/iwDQoQbyRx/xw1+tDC5cyJ/ft0aNDH75xX5mUq0yiInh6371lfPjtMhg507u686dyvvjo4gIBK9gyZIldlcATZgwQXEbRUEEgJsyCAnhlA8pKeplAHCuH/OpHRNKZWA08oYwew9XrTLo3583mLla7qlWBkYjjwgGDlTeFx9FRCD4DUVFBICbMihRAujZk/cJqJVBjx62q4hMGI361kBWKoMTJ4CxYx0Hjc1RK4Pp0znYfuuWsr74KCICwW8oSiIA3JQBUf7uWTUy2LOH18S/8479TimVwdmzvM8hPd32PT1WEzlDjQyuXuUNcYmJ+vfDixARCH5DURMB4KYMGjbkQDCgTgbh4dy+oxU1SmRw7hwLZdw4+++rlYHBAEyaBKxd6/pY0/FaAsh+iohA8BuKoggAjTJYvDh/B7IpD44aGTz8MFC/fr5IrFEigwkTeJrqzBn776uVweOP51cwU4JSGWRlAbNm8Z4FP0VEIPg8vla83hO4HUCOi+OGlMqgSRNgxw7nnXIlg7t3WSbdujleoqlGBnv28P0qHRUAymRgMPBn0rev8nZ9DBGB4DcU1RGBCU0ymD2b37/nnvyAqJqRQUIC5zJyhCsZREZyzYKYGMdtqJFBp05Amzbq1v4rkcHXX/P7584pb9eHEBEIfkNRFwGgUQaTJvH7tWurnyaqXBkoW5aDv45wtbQ0Odn1jSmVwdatfC9maUsU4UoGd+8CYWHAG2+oa9dHEBEIfoOIgNEkg3fe4fdbtVK3A7lyZc5F9K9/OV/H70oGKSk8OnCGuQwclb00Gm1TYSjFlQwmTgTKlAFu3NDWvhcjIhD8BhFBPppk8PrrlvUMAGUyuOcebvett5x3ypkMZs7kWsmnTjlvQ4kMABbC1avO23LUR0cyuHED2LhReTDahxARCH6DiMAStwLIffqok0G5ctzu3r3OO+VIBhkZvIu3SxfX8/tKZDBlCucgslevwBVKYgZ+ln9IRCD4DSICWzTJoHt3fr9fP+UyOH6cSzzaK25jjSMZ/PYbX3fZMtc35koG58/z0tSpU1235aiP9mSQk8PLVBcs0NaulyIiEPwGEYF9VMvg0iUOHBPx7l+1WUvr1uWayc5wJIMBA7gfStJFuJLBW2/x0lglwWhHfbQng759+XPwoykiEYHgN4gIHKNaBidP8jd8ImDIEOUyWL+ez6lSRdvIIDnZ9f4Ec5zJIDmZReAqduGqj9Yy2LdP/X4FL0dEIPgNIgLnqJbBnj28SkZtAPmNN7jN+vW1TxMZDJZ1jp3hTAaffw5ERChrx1kfTTIw9fGhh4COHd1r14sQEQh+g4jANaplEBubH0BWKoPcXKBDB65Gdt992mTw8cdA1aocs1CC0tVEWrEeGaxbxxvhLl3S/1qFgIhA8HkkxYQ6XMqgeHHbAHLfvupkcO0an1+2rLYA8pUrQPnyXIFMKY5kcPcuMGYMcPiw8rYc9dEkg8WLlUvKBxARCH6DjAiUo1oGH3zA76mRwenTXCfZuriNI6xlsHgxX/P335XfmD0Z5ORwMR4lS1NdYT0yuHULiI93r00vQEQg+A0iAnWolsHzz/PUkdqYwapV2paWLl4MdO3KK5iUrCIyYU8GGzboF+A1l8G999rWYPZBRASC3yAiUI8qGbRqxUndSpdWJ4NevXiaJyxMvQxmzAC++ILjDmqwloHRCDz5JEtFyyYze300yYCIA+s+jIhA8BtEBNpQJYOWLTkVxbx5ymVw4wbXFG7WTPs0EaBuVADYyiA2lovhrFyprh1nfRw6lD+nxo19erexiEDwG0QE2tEUQJ42TbkMDh/mh/Bzz7EMmjdXJ4OBA4F69Tg5nRqsZXDxorrzXWEw8E5jIuC99/RtuwAREQh+g4jAPVTLoEEDoFEj5TL45huWwK5d2kYGpUurW0Vkwt40UVSUft/gc3M5GO3DZS9FBILfICJwH1UyaNyYv+W3batcBqbiN+bFbdTIgAhYvVr9jZnL4JtvuJ1169S346yPPlwDWUQg+A0iAn1QJYP69fnh162bchkkJwOvvMKZStXIYNgw7kNQkOtcRvYwl0HHjhy30CNwbGL/fpajD8pARCD4DSIC/VAlg1q1+OEcEaFMBvHxfN7jj3PWUjUyGDSI+zB3rrYbM8mgYkW+h48/1taOPXbv5r517+5zMhARCD6P7Cz2DKpk0KKFZT0DVzLYsoWP++gj7dNEX3+t7cZMMggK4lTV9vY+aKVHD/5sTEV+fEQGIgLBb5ARgf6o3mfQrx8werQyGUyfzu38/LN6GfTpw+dqrTeQnMwrl4KDefezXhw8yP367jufihmICAS/QUTgGVTJoHx5zi/00UeuZWA0Ai+8wHsSAHUySEvjFNNEwMKF2m7MPGbgbh4ic3r3Btq186kAsohA8BtEBJ5DlQyCgriO8dSpymRgIjubaxYrlcHevfn5j6xrICslORmoWZNTZxw8qK0Nay5fzt/vYC4DL0ZEIPgNIgLPolgG99zDa/5r1uR6AEqmiWbM4KRw2dnqRgamkYc7Mjh0iM8vU0bfFNZXrwK3b+evePJiRASC3yAi8DyKZRAaClSrxukllASQd+7kc998k99XKoPsbOC117jEpTsymDSJ7yE0VB8ZZGby/Y8ezb97eVlLEYHgN4gICgbVO5B/+41X+LiSwfz53IZpPl1tAPm117TLIDeXg92lSulX3GbqVP4szpxxvy0PIyIQ/AYRQcGhapooMBDo2RNYtMi5DM6d44d5yZI8XQMol0FqKqe7eOQR7TI4cYJTbTdvro8MMjKAunWBp55yr50CQEQg+A0igoJFsQyCg/m1l1/mGgPOZHD6NDBhApCenn8hpTIYNoy/0ffr534AWa+ylz//rL64TiEgIhD8BhFBwaNYBuXK8WvvvKN801lMDJeZBJTJ4O5d/jbfpAnw6qvaZbBuHfDoo3wtd2VgNALvv8/992JEBILfICIoHBTLoGxZnvaJj3ctg+hoDrb275+/xFSJDKKjefXP4MG29QyUsn8/nzdhgn4jAy9HRCD4PJJiovBRFTNo3VrZaqK5c/n8SZPyL6REBitWABs32i9uo5TRo3kJ7IED+shAz93LHkBEIPgNMiIoXFTJoGxZrhTmSgajR/P5P/2UfyGlMQOjkYWjRQbp6VwI59FHgcRE92Vw/bq28woIEYHgN4gICh9FMpg5k6eIAgKATZtcy6BXL54mMsULAGUyGDmSayVkZGiTwdatfH5ysr4BZC9ERCD4DSIC70CRDGbM4P8NDAT++MO5DGrW5IeyNa5kcOgQZxcdPVr7NJH5RjBzGXh58FctIgLBbxAReA+KZDBlCuf4KV6cawm7GhmcOAGMHWtZTMaVDD77jK+5Zo12GRw5AowYwVNNfioDEYHgN4gIvAtFMggP530GrVu7rmdQtSraucJqAAAN5UlEQVSvCOrb1/KbujMZGI18fEgIy0aLDH77jftsypLqhzIQEQh+g4jA+1CVjqJ2beDCBecyqFSJz/3oI8sLOctampICzJmT35YWGQwfzhKKieHf/UwGIgKh0MnOzsaYMWPQvHlzlC1bFtWqVcPAgQNx5coVVe2ICLwTRTKYMIEfzEFBQEKCcxmUL8/nLl9ueSElAeSTJ3mUoFYG6el87fbtgZwcfs2PZCAiEAqd27dvo1u3bli1ahXOnDmDffv2oUOHDmjTpo2qdkQE3osiGbz5Jv8eHMzF6Z3JICgoP7OnOc5kcPIkB6dNhWzUymDfPl69dOpU/mt+IgMRgeCVHDx4EESEixcvKj5HRODdKJLBsGH8e4UK/JA1yWDYMMfpKDIzLS/kTAamfERHjvDvamWQkWH7mh/IQEQgeCVRUVEICAhw+lDPzMxESkpK3k9CQoKIwMtRJINBg3jZ57/+5TqAfM89wL338sYxcxzJIDOT9wbUrcttAzxVpEYGiYnAqFGWq5d8XAYiAsHryMjIQJs2bfDCCy84PW7ChAl5DxTzHxGBd6MqgHz//c5lULcuT/e0bat8ZBAXxwVohg7Nf03NyOD0aU4/MWqU5es+LAMRgVDgLF++HEFBQXk/u3fvznsvOzsbzzzzDFq1auXygS4jAt9FkQwmTuQHc40azmVQqxaf16uXZQ1kIF8GzZtbymDfvvwRgQk1Mpgzh6+5ZYvl6z4qAxGBUOCkpqYiNjY27+fu/1IHZGdno3fv3rj//vtxw3qorwCJEfgW1jJ4vl49Wxl0786/167tXAZVq9pfVgo4jxnExwN//pn/u1IZGAxA1675kjLHB2UgIhC8ApMEmjVrhiRXZQkdICLwPRSNDDp25N/r1XMtgxo18usZmONIBv36AWFhfL4JpTK4dIn3Naxda/uej8lARCAUOjk5OejVqxdq1qyJY8eO4dq1a3k/WVlZitsREfgmimTQujU/mFu2dB1Arl6dK4NZY08GSUmcy6hDB8sYg1IZpKY6fs+HEtWJCIRCJy4uzm7Ql4iwY8cOxe2ICHwXRTJ4/30OILtKRxEczK9HRdleyJ4MDh7kJaVDhljGGJTKIDsbmDyZRwjWmMvAixERCH6DiMC3USSDiAh+aDdp4lgG0dGc5rpYMX7IW2NPBkuWcAqJ2FjLY5XI4J9/OGD98MP5u47NMcnAixERCH6DiMD3USSD2rX5wdy0qWMZ/Pkni6BkSftz9PZkYB4nMEeJDHbv5mWsEybYfz85WdXnUNCICAS/QUTgHyiSQbVq/GBu1syxDDZs4Idz5crKA8iZmcC773K+I3OUyGDSJL6eiulMb0FEIPgNIgL/QZEMKlbkB6+zTWdnzuSnozh3zvZC1jIwBY/bt7dNJ+FKBrm5QKdOthvNfAARgeDzSPF6/0SRDLp3dx1AvnSJ6xFUqKBsZHDoEMchBg+23aDmSgbp6bbn+AAiAsFvkBGB/6E4HUX58s6Xlr7xBp9TsaIyGSxdalmMxhwl00SrVwMLFuj3QXgYEYHgN4gI/BNFMggK4v91lKjOYACeeYZfUxozePNN+6muAdcyePttDlT/9Ze+H4aHEBEIfoOIwH9xKQNT7eMSJYBWrezLIDMT+Pe/Oa5QvbprGVy/nv+6eWlM89ccySAzk/vRsKHzTWdegohA8BtEBP6NIhmYlow6ksGtW7wqyLyegTXWI4OVK4GHHrJfi8CZDM6eBcqVA156yTMfiI6ICAS/QUTg/yiSQcWKnGba1Q7kWrV4hZArGURFcdrpQYPsB4KdyWDZMs58evu2/h+GjogIBL9BRFA0UFXPwJEM0tO5qE3ZsspGBvPmcfuzZ9vvlDMZqMiXVViICAS/QURQdFAkgxo1OG2EIxls3swjiJAQZTIYNYrjC7//br9TzmRw+rT+H4KOiAgEv0FEULRwKYPAQP7/Zcs6lsGSJfnLSh3J4NSpfBm8+SZw9arjTjmSgZevHhIRCH6DiKDooVgGQUGOZTBpEtcyuPde5QHkxETg5k37nVJT6cxLEBEIfoOIoGiiWAYVKtiXQW4ucONGfj0DJTJo2RLo0oVTUNvDx2QgIhB8HkkxIShaTeQqgPzPP5xKulYt1zKoX5/bHD7ccad8SAYiAsFvkBFB0UaRDJ5+mgvX2JNBejqXxaxQAahb17UMatRwnIbChLkMvJgAACBB8ANSU1OpfPnylJKSQiEhIYXdHaEQiIyMpGeffZZycnKIiGhA/fq09MIFKk5E1KQJUUwMUWAgUVAQUYMGRFu3Eq1dS/Tqq0TDhhFNmkT0yCNEaWlEJUsS5eQQ7dxJVL++5YVOnSLq0oUoN5coPZ0oPp6oalX7nTIaiUaOJJo/34N37h6Bhd0BQRAEvejZsyetWbOGSpQoQUREP1y4QC/Xr0+5RERnzhA1asQP5owMovPnibp1I+rTh2jxYqKvvyb6v/8j2rSJxxGlSxOVKkXUqRPRhQuWF2ralGj7dqLixYlq1SIqVsxxpwIDiebO9dQt64KIQBAEv8KpDGJiiBo25G/6WVn8gLeWwcyZRBs3Er31Fo8GSpd2LoO0NB4d/N//Ed2+bb9TgV7+qC3suSlB0AuJEQjmOI0ZNGrE8/ZTpriuZ/DDD65XE4WF8eqkzp3t1y32crxcU4IgCNpwOjI4e5aocWOiCRM4PhAXZzsyGDmS6ORJohde4BGBs5HBzp1E5csT7djB5/kYEiwW/AYJFgv2cBlAPnOGqEIFooAAonr1bAPITZrwNNHHHxOtXEmUmek4gPzAAzxVNHs20ejRBX2rmhERCH6DiEBwhFMZNGhAdO4cUcWKHEi2J4Py5YlmzCD68kuiL75wLIPoaKJ27Xj0cOYMUeXKBX2rmpCpIUEQ/B6n00TnzhHdey/RzZtEJUrwUlDraaLbt4leeonol1+Itm1zPE3UrBnRgQO89LRbN6LExAK+U22ICARBKBI4lcH58/ztPimJqH9/oosXLWWwYAFRmTJEGzbwctHt2x3LoEULfv/yZaK6dYlOny7gO1WPiEDweebNm0dNmzaldu3aFXZXBC/HqQwuXOAA8oIFRGPH2spg4UKiDz7gB3uvXvyaswDyqlVE2dlEbdsSJSQU9K2qQmIEgt8gMQJBKYp2INevz1NCdepYxgwGDSLav5/ozh2i1auJBgxwHDNYuZLo+ec5xuBoj4EXICMCQRCKHC53INeqxdNFlSrZjgy++46/5RuNRK+8woJwNDLo359XEKWkFPAdqkNEIAhCkcSpDC5dYhnExBBVq5Yvg969WQbLlxM99BDRlSu8tNRZzGD0aKLw8AK+O3XI1JDgN8jUkKAFp9NENWty0Pc//+HNYtbTRH36EI0YwSkmrlwh6tzZ8TSRFyMjAkEQijRORwaXL/PDf+1aonHjbKeJ1q7lJaU5Obys1NnIwIsREQiCUORxKoOLFzlR3ZgxRM2b28rg6695xPDyy7zhbMcOn5OBiEAQBIEUBJArVybatcu+DDZsIOrYkTOXrlrlczIQEQiCIPwPpzK4fp2oShWWQevWtjL480+iVq2I3n2Xp4h8SAYiAkEQBDOcyiAxkSgsjCgqilcNWcvg2DHeh7B7N1GNGpYy8GJEBILPIzuLBb1xKoPkZJ4mioy0H0COieHKZUYjJ7IzycCLkeWjgt8gy0cFvXG5Azk2lmjgQI4RWC8t7dWL6OBBoi1biEJDeYTgpciIQBAEwQEuA8gVK/JO4+7dbUcGv/7KuYa6d+f/9WJEBIIgCE5wKoOkJM4jtHIl0ZNP2srgn3+I7t5lGXgxIgJBEAQXOJVBSgpRSAjR99/zdJC1DNLSiK5dK9T+u0JEIAiCoACnMkhNJSpXjmjpUvsB5IyMQu27K0QEgiAICnEqg/R0okaNuGbByJG2MvBiRASCVxAeHk5NmjShoKAgCg0NpW7dutGBAwcKu1uCYIPLAHLVqkQTJ3IdAnMZeDEiAsEraNSoEc2dO5dOnDhBe/bsobp169Jjjz1GycnJhd01QbDBqQyuXOGylnPnEr34Yr4MvBjZRyB4JaY9AVu3bqWuXbuqOkf2EQgFhdN9BqVLc0rq0aOJli0junGjUPvqDBkRCF5HdnY2LVy4kMqXL08tW7Z0eFxWVhalpqZa/AhCQeJ0ZJCZyTL4/HMOIHsxIgLBa4iMjKRy5cpR6dKl6bPPPqOoqCiqVKmSw+OnT59O5cuXz/upVatWAfZWEBiXMmjQgAPIXoxMDQkFzooVK2jo0KF5v//222/08MMP0507d+jatWt048YNWrRoEW3fvp0OHDhAlStXtttOVlYWZWVl5f2emppKtWrVkqkhoVBwOk1Uvz7XQPZSRARCgZOWlkaJiYl5v9eoUYPKlCljc1zDhg1p8ODBNE7hsFpiBEJh41QGXvyolakhocAJDg6mBg0a5P3YkwAREQCLb/yC4O04nSbyYkQEQqFz584dGj9+PO3fv58uXrxIf/31F7366qt0+fJleu655wq7e4KgCnsyeKlevULulXNEBEKhU6xYMTpz5gz17duXGjVqRD179qTk5GT6448/qFmzZoXdPUFQjbUMfoyLK+QeOUdiBILfIDECwdswjxl486NWRCD4DSICQdCGiEDwGwBQWloaBQcHU0BAQGF3RxB8BhGBIAhCEUeCxYIgCEUcEYEgCEIRR0QgCIJQxBERCIIgFHFEBIIgCEUcEYEgCEIRR0QgCIJQxBERCIIgFHFEBIIgCEUcEYEgCEIR5/8Bw/ZDbvweL+cAAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 54 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphI_EF = X_EF_I.plot(X_KS, ranges={v:(-12,12), r:(2.001,5)}, number_values={v:17, r:9},\n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V),\n", " style={v:'--', r:'-'}, parameters={m:1})\n", "graphII_EF = X_EF_II.plot(X_KS, ranges={v:(-12,12), r:(0.001,1.999)}, number_values={v:17, r:9},\n", " fixed_coords={th:pi/2,ph:0}, ambient_coords=(U,V),\n", " style={v:'--', r:'-'}, color='green', parameters={m:1})\n", "show(graphI_EF+graphII_EF+graph_r0+graph_r2, xmin=-3, xmax=3, ymin=-3, ymax=3,\n", " axes_labels=['$U$', '$V$'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are now 9 charts defined on the spacetime manifold:
" ] }, { "cell_type": "code", "execution_count": 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_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)),\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, (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, (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, (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_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_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_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))}" ] }, "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": [ "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.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\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": 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 factorize 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 8.7.rc0", "language": "", "name": "sagemath" }, "language": "python", "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.6" } }, "nbformat": 4, "nbformat_minor": 1 }