{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Allgemeine Relativitätstheorie mit dem Computer \n",
"# General Theory of Relativity on the Computer\n",
"\n",
"## Vorlesung gehalten an der J.W.Goethe-Universität in Frankfurt am Main (Wintersemester 2026/27)\n",
"\n",
"### von Dr.phil.nat. Dr.rer.pol. Matthias Hanauske\n",
"\n",
"### Frankfurt am Main 26.07.2026\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In diesem Python Jupyter Notebook werden die grundlegenden Größen der allgemeinen Relativitätstheorie (z.B. die Metrik der Raumzeit, Christoffel Symbole, Ricci- und Einstein-Tensor) am Beispiel einer allgemeinen statischen und isotropen Raumzeit in Python berechnet. Die oft komplizierten und zeitaufwendigen Berechnungen der tensoriellen Gleichungen der Allgemeinen Relativitätstheorie (ART) können mithilfe von Computeralgebra-Systemen erleichtert werden. Diverse Anwendungen der Einstein- und Geodätengleichung sind in schon vordefinierten Python Modulen implementiert, und analytische Berechnungen können durchgeführt und entsprechende Lösungen berechnet und visualisiert werden.\n",
"\n",
"Mittlerweile gibt es in Python mehrere leistungsstarke Bibliotheken/Module, die Berechnungen der ART durchführen können. In vergangenen Vorlesungen verwendeten wir das Python-Modul Gravipy (siehe Vorlesung im Sommersemester 2020) bzw. das Python-Modul EinsteinPy (siehe Vorlesung im Sommersemester 2023). Beide Module basieren auf dem Modul SymPy welches symbolische Berechnungen mit Python vereinfacht. In der aktuellen Vorlesung werden wir zusätzlich zu EinsteinPy das Python-Modul OGRePy: An Object-Oriented General Relativity Package for Python verwenden, welches ebenfalls auf der Bibliothek SymPy basiert.\n",
"\n",
"In diesem Jupyter Notebook stellen wir die Verwendung des Moduls \"EinsteinPy\" dar, wohingegen das Modul \"OGRePy\" in dem Jupyter Notebook Allgemeinen Relativitätstheorie mit Python und OGRePy vorgestellt wird (Download Notebook OGRePy.ipynb)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Allgemeinen Relativitätstheorie mit Python und EinsteinPy"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Im Folgenden werden einige grundlegende Größen der allgemeinen Relativitätstheorie mit dem Python Modul \"EinsteinPy\" berechnet. Wieder erläutern wir dies am Beispiel der allgemeinen statischen, isotropen Metrik. Zunächst wird das Python-Modul \"EinsteinPy\" eingebunden, welches auf dem Modul SymPy basiert und symbolische Berechnungen in der Allgemeinen Relativitätstheorie relativ einfach möglich macht. EinsteinPy kann einfach mit \"pip install einsteinpy\" in einem Terminal installiert werden und stellt zusammen mit dem Modul SymPy ein wichtiges Tool der Computer Algebra Systeme im Bereich der allgemeinen Relativitätstheorie dar. Weiteres über das Modul finden Sie unter EinsteinPy - Making Einstein possible in Python."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import sympy as sp\n",
"sp.init_printing()\n",
"import einsteinpy.symbolic as ep"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Koordinatensystem und Metrik\n",
"\n",
"Um symbolische Rechnungen in einer flachen oder gekrümmten Raumzeit machen zu können, müssen wir zunächst das verwendete Koordinatensystem und die Metrik der Raumzeit definieren. \n",
"\n",
"Wir verwenden im Folgenden sphärische Raumzeitkoordinaten $(t,r,\\theta,\\phi)$ und beschreiben Ereignisse in der Raumzeit mit dem Vierer-Ortsvektor $x^\\mu=(t,r,\\theta,\\phi)$ (Lichtgeschwindigkeit $c=1$). Des Weiteren nehmen wir eine allgemeine statische und isotrope Metrik an: \n",
"\n",
"\\begin{equation}\n",
"g_{\\mu\\nu}=\\left( \\begin{array}{ccc} A(r) & 0 & 0 & 0\\\\ 0& -B(r)& 0&0 \\\\ 0& 0& -r^2& 0\\\\ 0& 0& 0& -r^2 \\hbox{sin}^2(\\theta)\\\\ \\end{array} \\right)\n",
"\\end{equation}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Bei der Definition des sphärischen Koordinatensystems verwenden wir die SymPy-Variablen $t, r, \\phi$ und $\\theta$, wobei wir die Koordinate $r$ als positives SymPy-Symbol definieren. Die Funktionen $A(r)$ und $B(r)$ werden als SymPy-Funktionen. Den metrischen Tensor definieren wir schließlich als ein EinsteinPy-Objekt."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"t, theta, phi = sp.symbols('t, theta, phi')\n",
"r = sp.symbols('r', nonnegative=True)\n",
"A = sp.Function('A')(r)\n",
"B = sp.Function('B')(r)\n",
"Metric = sp.diag(A, -B, -r**2, -r**2*sp.sin(theta)**2).tolist()\n",
"g = ep.MetricTensor(Metric, [t, r, theta, phi])"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}A{\\left(r \\right)} & 0 & 0 & 0\\\\0 & - B{\\left(r \\right)} & 0 & 0\\\\0 & 0 & - r^{2} & 0\\\\0 & 0 & 0 & - r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡A(r) 0 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢ 0 -B(r) 0 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 2 ⎥\n",
"⎢ 0 0 -r 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 2 2 ⎥\n",
"⎣ 0 0 0 -r ⋅sin (θ)⎦"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Durch den Befehl \".config\" kann man herausfinden ob der Tensor kovariante (l) oder kontravariante (u) Index-Komponenten hat."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'ll'"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.config"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kontravariante Form der Metrik: \n",
"\n",
"$$ g^{\\mu\\nu}=\\left( \\begin{array}{ccc} \\frac{1}{A(r)} & 0 & 0 & 0\\\\ 0& -\\frac{1}{B(r)}& 0&0 \\\\ 0& 0& -\\frac{1}{r^2}& 0\\\\ 0& 0& 0& -\\frac{1}{r^2 \\hbox{sin}^2(\\theta)}\\\\ \\end{array} \\right)$$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{1}{A{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & - \\frac{1}{B{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & - \\frac{1}{r^{2}} & 0\\\\0 & 0 & 0 & - \\frac{1}{r^{2} \\sin^{2}{\\left(\\theta \\right)}}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ 1 ⎤\n",
"⎢──── 0 0 0 ⎥\n",
"⎢A(r) ⎥\n",
"⎢ ⎥\n",
"⎢ -1 ⎥\n",
"⎢ 0 ──── 0 0 ⎥\n",
"⎢ B(r) ⎥\n",
"⎢ ⎥\n",
"⎢ -1 ⎥\n",
"⎢ 0 0 ─── 0 ⎥\n",
"⎢ 2 ⎥\n",
"⎢ r ⎥\n",
"⎢ ⎥\n",
"⎢ -1 ⎥\n",
"⎢ 0 0 0 ──────────⎥\n",
"⎢ 2 2 ⎥\n",
"⎣ r ⋅sin (θ)⎦"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gInv = g.inv()\n",
"gInv.tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wir überprüfen dies, indem wir diese kontravariante Form der Metrik zweifach mit der ursprünglichen Metrik kontrahieren:$$g^{\\alpha\\beta} \\, g_{\\alpha\\mu} \\, g_{\\beta\\nu} = g_{\\mu\\nu}$$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}A{\\left(r \\right)} & 0 & 0 & 0\\\\0 & - B{\\left(r \\right)} & 0 & 0\\\\0 & 0 & - r^{2} & 0\\\\0 & 0 & 0 & - r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡A(r) 0 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢ 0 -B(r) 0 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 2 ⎥\n",
"⎢ 0 0 -r 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 2 2 ⎥\n",
"⎣ 0 0 0 -r ⋅sin (θ)⎦"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tens1 = sp.tensorproduct(gInv.tensor(),g.tensor(),g.tensor())\n",
"sp.tensorcontraction(tens1, (0, 2),(1,4))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Man kann die Art der Indices auch einfach mit dem Befehl \"change_config\" verändern:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'uu'"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gInv.config"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAT4AAABkCAYAAAAfWlwVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAATE0lEQVR4Ae2dX67cthXG5xr3uXBioAtwgC7g2l5B7B0kNtD3ODtI4CfnLUh2kOS9gBPvwMkKnHgBBZIFFLBjdAFtv59GZypxSF1RGvGOxHMADSX+50fqm8M/Ii+eP39+tdvtftMVk5dfffXVpzEHt3MEHAFH4FwREG/9rrzdjeVPbhe3Og7f6v7z4Pqu437SWyUezVQqkVz/qXjc3hFwBKpA4BuVMuSzl1byS7uR+Z3I5Y/Oc9ZtS0w/ybx3XUD5+UJ+SCsnvbsK94kuCNrFEXAEHIEkAuKJ70NH2WH1CT9d4uN5jvykwHSbBwXykoc7uQQm/z/rutL1VNdRoQYTXYGjysQ/FPJW10e6vpFdzh8DYTcjjkeZqqwV525XdzLSAg8yG0N6t+Xvmfx/OSUxhWu64zKJZzOi8jDG+hpc2jKCzyvdZw0HbAUQx6NMTdaM80mIT9X0SFfTf77mZUWrmTtuSHjTjnS7bhFeT1WC2zIP4w+6fy87nuditTpwHI8yVVY7zrOJTwBCQlzWLRvSxh7L/6xuahueeIbSKdN6TpMKs+ZvIlG9lt3DDZUzUsSoleMRheXkllXjPIv49FLeVXW8lQnpMTaFYHck8kN32MjxyD3TgngeZ4Y5V+8PlbF3kcwZVrjXJI5HmdquGufLmRgzJsWUMWIv6of7x6NfusM/h7YKz9jgE11MXDzSBXEyxkV373c9xzRE4iG+mJus1yEq2xitNYXnOgqZkUvHIwOsGV4d591ussYn8PjHeNXB34iPGcmY3JcliwpDeaK4mskOmYxpfS7TyDQ1lpdcnBhGfubPRmqQfErGkGMq7NrsHY8yNVY9znM0vk87BEV12csb7erKnRe416VTePwaGVIZrNUz4sR/SqMjnlQ6ctqU3NlUaeYXxvGYj+GYGDaN8yTiEzmxABmS6s462r9ISkPB3cjRgMfvj+3DlUzT9HaK+3Bvnjsm2mUqnY63s7/t/REEuTU8bew0cN7ko+NRplqrxzmb+ERIDeHIZIytJ7L7ryxGa2Ly/4YIZNJtRowE90/p3xiJpn2fqYvK/V4XuYuRuNnZEMKZluJ02XI8ToflUEyO87QvN/iiIKWN8ZKaphJiz7+MvcyhG1Prf1AhoUPimXiG/rUSwc7Smoma2J+F4Xg0IXSWpThdphyP02E5FFPVON8aQiZ0EzGhmfGVQUogrhS5QYqxF5y4iPewgBeLawRS2IomxKd+TPyEwjfPbzL+DMLwa312PMrUXNU4jyY+vYCQFpsQpCYcqK5GC2v9htVHt/ZBaNn6Je7uDHHoLXxuSCG0XONzi+c7maxzbET3/HmwTvGzvU09v45HmbquHefL62BuX0L+HdDKdnpG4/tYJtpdI7rnpaX72/iRCUGyqUCzTKXxtNu9wL697xqQHt3cnC4d6aS6292413IPkTOEwB8DkxmYYMyfRY3ieJSp9WpxvuhsRPqRXrRFu4+Kn6UrLIOZ/EIrbKMdyrRlL2WaiKfiCDgCq0ZAnMF38Wy/d3GrcElYkDxXU0OLTC1sLlwcT84RcATWiEBR4hPTMj7I+j+0tmxpwxF+aJwxO14P4Ag4AnUhUJT4WmhZutJd+JyDePNJW04A9+sIOAKOQIhAceKTtsakCN/j8vXHaGn9E27RccjRGXKPjoAjsFoErp3VXaJkLXllnZ2hMFn+l8i3x+kIOALbQKC4xrcN2LwUjoAjsGYEnPjWXHued0fAEZiEgBPfJNg8kCPgCKwZASe+Fdaexjvv5mZ7SpjcNNy/I7AWBJz41lJTbT5FYMyGX03INusfs2bSJ6ThQRyBVSBQbFZXL519bcG3qNUdmK3y87kMaxjte+ZwNxq0ODZ54ByT6Cd9sueb6OzD2BVmp7CrOpBd+a26vVBnJaRWnIt8qytw2djga5nNyy6T3Uew43Chqtblqbx/qty/yoxt5MomDpDbPbn3yE/PYPaLTD4snywKD+69TSYmR7ZQwDaP3l4WwteirQ1nlbfct7ptYn5gtlqbsECrg8BSW3DZFy2x75nRgMxdt5OFOEybmhzJUgG9vSyFbD/e2nEuMcZH966nvbRV8FrmQ1UARFCLWDc3tQUXxIjYAUz7p/3v7MPYiUZ4853zOR/I7u1lX99L/1aNcwni42Vn7CoU6+IaGYTuW3xuurcin9gfAeVF03uvqzf+J/90fw0v3c4W4mKj03MUby9laqVqnC+XxFgv7Bhtzs6WWDIr5xI3je1I22tx+kFu4MX4W0hyEOZRONnt5JcZ3ie6Vn8ge4sDxRqSmtrLEA6T3RznaYcN5QBujRQtJiVjyDEVdjX2LUE1ZdV9d1nJAxWCLi4bJNL9iMl9WabG95oD2RX2lS78cHIbmzkwjscV28KLrjQkfG7i7aVMjVSP86Ia38g6vDPS39q9GdEc7TAjkqIr+4PMna4YUUGYR8MF8gth2nggjZm1erYzNWFiccm6icvGE3lek9TSXm66TjaN89LEd/SydmrT/nVY13eWIhKBPH7R1WhqIzOZ2lqf7iraWNiN3cnupS40P7Q+lrqEY4Bg9T6SPvmys4ivdH+YDVYch/tIOPKQU6ZIFItYrbq9LILIMpFWjzPEBwj/0PWfU2Osl2/VB2aTf2Eya91cB1M0vt6kRceNW9Pc8BcSX+B1/2gEKZMwiJHg/in9myLSdIgCLmtvLwUgOkkSFeP8TwEI1+2Y1eUl+Ht7L+PkwqB8rFtFukh00H7vtI1fNTS0MSS1fg83W9AcIz3+nIY0NMYGt3Ige/XthcZQQGrE+W/CFa5riG9pjPkagcH5UNCkajkw2zSyKMm3xMg4H3jE/NA1jf15GKbXaZPmz0z+dI663OZ4w6a3lzIVUDXOaHyLil5kBthrPzCb5Sap8T1Ii3FENL2PdcUEN8YAj0T4QohcQ9pkGK750wktz+HZ20uZWqgd58syMDfjZNUdmK3Gxb8qXdSmq9s+dyGHsOjGfia3ofG/F/JDXDEhjq0dyA4xV9deYpW7sF21OBfZpGDhyqsiehEjkx+pGePRGCieRjuUacteRod1j47AmhFQmy+3ScGagTqzvLMYeWiJytjs+oHsY5Fyf5tF4NZmS7axgunfirHSyYexA4fiQNvzA9kBw6VqBJz41lX9LFtJfbo2piSEPYXWOCYt9+MInC0CTnxnWzXHGZPGxoJqPnnrfut77DFi04Y5+lwu4tWtHIHNI1BqVnfzQJYqoAiM9XfZh6srXHaYUmXydByB0gi4xlcacU/PEXAEbhwBiO8vbS7MvPFMeQYcAUfAEVgAgb9anHR1/90+mGlubjoCq0BA3XgWiT9rM3u3NVkUzpioiyNgCPzLbnyMz5Bwc80I8JXHYbZa98xec5qcL9Jec60umHcf41sQXI+6GAJPRXYPO6mx2Jv1is2ngh17v3UEGgSKaXxqhDRGhI1H+SfmX/pcdwghn0XF8ZkFN9rer7NiqDRwre2uiMYncOl2vJb5pS6WVfDZFGdE2HhMpc1uX2zHZ171C7/vdXXH8yBCNm5gVxuXBAI1t7vFiU/g8mHwbZmH3Ud0TyPlec5XCInqXJe143Pa+hKedG/Z25CdR1wSCNTe7hYnPuHOZ1axf97Xsn+oCmBGrmZxfE5U+2pL9CAYUrmn+64GeKIUNhVN1e2uBPEx6Pwu0mRsfK87KB3xtnkrx+cEVdySHkMpjyA9nlu7E8S+ySiqbneXS1apGt4Ybe7DJfNwznE7PsO1I3zotl57WHpLcAybQHw2k8s4H2PJLgEC3u6WP3PDSG2o2zGGHIOq28yj4zNclc1h6XjRywqxscmCrdezVQI4M3mGBoNpF0tchtqdvFYr1be7RTW+kc3qzkh/tXqrEh+RFuN1duQmLypdV1uQzJ8l+xM2IvsP7N7NkyGw6Xa3NPHFxvasZuxf561ZVGhuGh8REgTFQUo5Wr1tr0+YH9s2cSXTNL2d4j3ct+5u5CGw6XY3BopFiU8NdNUHio8BcI6freND+YTPpGUlCvsGbGXShUWMBPdP/jsZAepFF+Fjf0hmZ5OPk9M554AlZnVrPLg4p84dn2G0WHaRc1j6cGzuaghU3e5KEF/VBxdbKxswHZ8BcOSExndY/D7s1V0zEKi63S1OfFKp/UDxgdbo+KTBETZMcHDlHJaejtBdDgjU3u4WHeM7oLwf5/EDojuABLeMgzk+ASh6hPRyD0s/jsVtUghU2+78QPFUk3B7R8AR2BQC0nL9QPFN1agXxhFwBLIQWHyMLys37tkRcAQcgQIIOPEVANmTcAQcgfNCwInvvOrDc+MIOAIFEHDiKwCyJ+EIOALnhYAT33nVh+fGEXAECiBQah1fgaJ4Eo6AI1ASAS0P4bveZ22ad1tzFecZO/GVbCmeliMwAYEzJpjVnmfsXd0JDdGDOAKFEYBg2F2ai00b2FaKDVdvWtjslW+pTVZznrETn1WZm47A+SJwrgTDvoizzzMWeVo3OasGpoYjESe+LKjdsyNwIwichGBOnXMRz+zzjBXHF8oXG81OEXblJny2FBvjUwbtjAR2XGYLcdT3TW92mFMbjk8fLcfj/3gIi8M2+60tRDjrwHTFiZZFd5nucxh/m8x4Q3FAXlnnGSsM/u/I/DaVktwgNtsG/4Wemw1q8a/7n3Vd6UIjzipDEY1PmQLg1zIBmUJy+tUr3U9ScRV2U+L49KvT8ejj0X0SNtkE0w3fuefdY1Z20g7ZnXh2yhNxZZ1nrDDNjLDM5El4cmM7MnaLxs/XuthDsCdyg084hMp2ju65px4WJz5liB0Rbss8bCap+/ey45mTs6oWx6df/Y5HH4/uk7DJJphu+O694mIH5g9koj1OljZPKDS55xlDlMn3X/E1PUSZjSYnE86gaxvrFhOP9ShHlWVx4lMuqj6xfUQtOD59kByPPh7Nk154SG8KwURi21u1ZJJ0v86hzROk8x2ExKV7tLOhw4ws2sfyH+2etvHQxY2RGTj0pI2H+EZrfSWIj+nuGBA2vtedDu8VqJIHx6df0Y5HH4+dXmhe9qkEE8R20keGsKgvTLsYb0M7S4rcGduz9z/m7xnu8odW2ojujdQ+NLvAJL7HgV3y8TLpcgKHTmaHYksVZCjMJtwcn3411oaHyouG9EQX2hJdxUar0zPE8bueTSOCVHjxMQ8i92Q3VW7EDSkRl5HGE9nf00U6jJdhMkGAlr2TSZgfdGHPqXZoXJAU8kAX4/SHiQjdTz3P+JHiOpAakZsoTvJKmuHY3/3WT4owiY94DbPWe9xYlPiUpJEa4KfEKiXlvmV7x6dfu7XhARHRfWWiD42OgXwG6iEcLhvfyiIYheedYtUERHAQ4uZBJuQBAYZEyowp9hzkTl1AyA3RybzS828yX+pKkY+8jBJIjPLGhDkBJCRG0kdSaZNniH6ULE18YzJhU9Vj/Nbox/Hp1/om8BB5oFXxsiKQDAP3LPNCIK5Rmkvj+/gHYrmv+JhU7CodIdnEhqCIDXJ5qLCNJoiF7t/o4hZymZM34qB8qbTRgJFnbXr7p70WyB9DiviID0xHydLElyocmaOykbd7o8pfx6df7avCQy8hL/AvujDHyqcK90aeCUN3EkGbOXRb5X64b1wzfxSe7itY/sm9TJaFoKkduqkjokx9kZFT1lQyvPvvE45gceh+40f5Jk26v1/znBAIcXTeFiU+ZRiGJp+xDJldisEJt2lxfPrVuzY8yK9KMGkdnMJCfjuZaFCIkeD+af4v+WKSAMJous5Kiy8tZpHq/GylY1DejBPC40Rt0mJI0xwi06NEbx3ZnN6Cf5yYCkpGkbAvv7et59fx6dd1bXjQnWQGExI9iSiu5n2TyfjhR7ouFDGE91T3aFQ3LWijRnKxvDR/Ch0HJjrCz+M6zs0t8RHvKClBfFWf2D6iFhyfPki14YHGd1jc34di8hPkZpMETSQiPLQl0mH876aFXt6RMqQ8HpG/7NBYUZLCWd6wDPgZ3XtcnPhawN+1BWgyq3vYGfX1s8ai4h/Hp1/5NeGhsvLyc4Vduz4o056YHAi1Kp7DHlboh9QgkSUFjY7lMTGBnJs8tfmnm/6x7t/HPHfs6NqHmmLHuX+76BhfJykyxfQ6hWUyA5PCjM6o/G9ZHJ9+7daCB6TXW6jbh2HyEySBhkfX1iK5o5tmYxDZoREy/tdof3pGy0YJIT/Y4874PPZMKGBvY4MQ6gNddNGnygsFJO6YkA/ySZrMcrO+cYwmh+ZsedTtsFw8f/6cQrKeh7GAMQkMx+iujoAj4Ahcg4C4hqU8NsN9je9h55YkWQtpy4GiAeRO95/P6y5uRX24pSPgCDgCyyJAF3a0hnZNVhj/I77R4sQ3Gir36Ag4AqdCQFoXXXEWbdOlnSxteOIZWupyFL8T3xEkbuEIOAKFEGCcMPyaJDdpwmdrjk58uTC7f0fAETgJAtLSmITh22S2oMqWNhzhs+cmupMbsYT5zGXO7E0sTrdzBBwBR2BRBMRbTJ5Eu9Fyu2A5C2yZUhWzmXTR0njkjoAj4AiMQ2BwsuN/Mtvwh1VO/CkAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}A{\\left(r \\right)} & 0 & 0 & 0\\\\0 & - B{\\left(r \\right)} & 0 & 0\\\\0 & 0 & - r^{2} & 0\\\\0 & 0 & 0 & - r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡A(r) 0 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢ 0 -B(r) 0 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 2 ⎥\n",
"⎢ 0 0 -r 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 2 2 ⎥\n",
"⎣ 0 0 0 -r ⋅sin (θ)⎦"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gInv.change_config('ll').tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Das kontrahierte Tensorprodukt der kovarianten Metrik mit der kontravarianten Metrik ergibt die vierdimensionale Einheitsmatrix:\n",
"\n",
"$$g^{\\mu\\alpha} \\, g_{\\alpha\\nu} = \\eta^{\\mu}\\,_{\\!\\!\\nu} = \\delta^{\\mu}\\,_{\\!\\!\\nu} = \\left( \\begin{array}{ccc} 1 & 0 & 0 & 0\\\\ 0& 1& 0&0 \\\\ 0& 0& 1& 0\\\\ 0& 0& 0& 1\\\\ \\end{array} \\right)$$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$"
],
"text/plain": [
"⎡1 0 0 0⎤\n",
"⎢ ⎥\n",
"⎢0 1 0 0⎥\n",
"⎢ ⎥\n",
"⎢0 0 1 0⎥\n",
"⎢ ⎥\n",
"⎣0 0 0 1⎦"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eta = sp.tensorproduct(g.tensor(),gInv.tensor())\n",
"sp.tensorcontraction(eta, (1, 2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Die Christoffel Symbole"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die Christoffel Symbole (zweiter Art): \n",
"$$ \\Gamma^{\\sigma}_{\\mu \\nu} = \\frac{1}{2}g^{\\sigma \\alpha} \\left( g_{\\nu \\alpha| \\mu} + g_{\\mu \\alpha| \\nu} - g_{\\mu \\nu| \\alpha} \\right)$$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"llu\n"
]
},
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & - \\frac{\\frac{d}{d r} A{\\left(r \\right)}}{2 B{\\left(r \\right)}} & 0 & 0\\\\\\frac{\\frac{d}{d r} A{\\left(r \\right)}}{2 A{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}- \\frac{\\frac{d}{d r} A{\\left(r \\right)}}{2 A{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & \\frac{\\frac{d}{d r} B{\\left(r \\right)}}{2 B{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & - \\frac{1}{r} & 0\\\\0 & 0 & 0 & - \\frac{1}{r}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{r}{B{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & 0 & - \\frac{1}{\\tan{\\left(\\theta \\right)}}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{1}{\\tan{\\left(\\theta \\right)}}\\\\0 & \\frac{r \\sin^{2}{\\left(\\theta \\right)}}{B{\\left(r \\right)}} & \\frac{\\sin{\\left(2 \\theta \\right)}}{2} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ ⎡ d ⎤ ↪\n",
"⎢ ⎢-──(A(r)) ⎥ ↪\n",
"⎢⎡ d ⎤ ⎢ dr ⎥ ↪\n",
"⎢⎢ -──(A(r)) ⎥ ⎢────────── 0 0 0 ⎥ ⎡0 0 0 ↪\n",
"⎢⎢ dr ⎥ ⎢ 2⋅A(r) ⎥ ⎢ ↪\n",
"⎢⎢ 0 ────────── 0 0⎥ ⎢ ⎥ ⎢ 1 ↪\n",
"⎢⎢ 2⋅B(r) ⎥ ⎢ d ⎥ ⎢0 0 ─ ↪\n",
"⎢⎢ ⎥ ⎢ ──(B(r)) ⎥ ⎢ r ↪\n",
"⎢⎢d ⎥ ⎢ dr ⎥ ⎢ ↪\n",
"⎢⎢──(A(r)) ⎥ ⎢ 0 ──────── 0 0 ⎥ ⎢ r ↪\n",
"⎢⎢dr ⎥ ⎢ 2⋅B(r) ⎥ ⎢0 ──── 0 ↪\n",
"⎢⎢──────── 0 0 0⎥ ⎢ ⎥ ⎢ B(r) ↪\n",
"⎢⎢ 2⋅A(r) ⎥ ⎢ -1 ⎥ ⎢ ↪\n",
"⎢⎢ ⎥ ⎢ 0 0 ─── 0 ⎥ ⎢ ↪\n",
"⎢⎢ 0 0 0 0⎥ ⎢ r ⎥ ⎢0 0 0 ↪\n",
"⎢⎢ ⎥ ⎢ ⎥ ⎣ ↪\n",
"⎢⎣ 0 0 0 0⎦ ⎢ -1 ⎥ ↪\n",
"⎢ ⎢ 0 0 0 ───⎥ ↪\n",
"⎣ ⎣ r ⎦ ↪\n",
"\n",
"↪ ⎤\n",
"↪ ⎥\n",
"↪ ⎡0 0 0 0 ⎤⎥\n",
"↪ 0 ⎤ ⎢ ⎥⎥\n",
"↪ ⎥ ⎢ 1 ⎥⎥\n",
"↪ ⎥ ⎢0 0 0 ─ ⎥⎥\n",
"↪ 0 ⎥ ⎢ r ⎥⎥\n",
"↪ ⎥ ⎢ ⎥⎥\n",
"↪ ⎥ ⎢ 1 ⎥⎥\n",
"↪ ⎥ ⎢0 0 0 ──────⎥⎥\n",
"↪ 0 ⎥ ⎢ tan(θ)⎥⎥\n",
"↪ ⎥ ⎢ ⎥⎥\n",
"↪ ⎥ ⎢ 2 ⎥⎥\n",
"↪ -1 ⎥ ⎢ r⋅sin (θ) sin(2⋅θ) ⎥⎥\n",
"↪ ──────⎥ ⎢0 ───────── ──────── 0 ⎥⎥\n",
"↪ tan(θ)⎦ ⎣ B(r) 2 ⎦⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎦"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"chr = ep.ChristoffelSymbols.from_metric(g)\n",
"chr_llu = chr.change_config('llu')\n",
"print(chr_llu.config)\n",
"chr_llu.tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Speziell die $\\Gamma^{1}_{1 1} = \\Gamma^{r}_{r r}$ Komponente: "
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{\\frac{d}{d r} B{\\left(r \\right)}}{2 B{\\left(r \\right)}}$"
],
"text/plain": [
"d \n",
"──(B(r))\n",
"dr \n",
"────────\n",
" 2⋅B(r) "
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"chr_llu.tensor()[1,1,1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die Christoffel Symbole in kovarianter Form (erster Art): \n",
"$$ \\Gamma_{\\mu \\nu \\sigma} = \\Gamma^{\\rho}_{\\mu \\nu} g_{\\rho \\sigma} = \\frac{1}{2} \\left( g_{\\nu \\sigma| \\mu} + g_{\\mu \\sigma| \\nu} - g_{\\mu \\nu| \\sigma} \\right)$$"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"lll\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAABN4AAAByCAYAAACMYBygAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4Ae2dW9LdNLqG/6Ry3QWkag+gmUESRkCYQSAjSHoGULnbdymYAfQIAswg9AgImUHY911FOtUTYL+Pf8t4efl8kmS/qvKSrbOeT0uWPkv2nT///PPGxgRMwARyIPC///u/36qcj2U/zKG8LqMJmIAJmIAJmIAJmIAJmIAJmMC5Cdw9d/WPVXuUEjp+O1atXBsTuCDwWldvLlx8YQImYAImYAImYAImYAImYAImYAKJErgnRc0Dla1LWfOz/L9sK/vceG1p2W01AiglPlotNSe0GQH9f94p8b+3ZSC/O23ue7ol/P/+Qhxo5zYmYAImMJpA6n3u6Io4oAmYQNYEEh5fZc3VhTcBEzCBtQjMHTMOxbtXK+B3OkcZUDe/1y86zufG60jOzgsILFJKqLF8pONDV/7yeyK/t7LHtIsiGYX9WgdtxOaSAFsmmwb5wTglE/3/rfbzvASCUhk+3y8BpPRWbedu40uk4bgmsBuBXPrc3YA4IxMwgagEoo+votbemZuACZhAugTmjhl749UVb99rAjlaoVLjNDdeLQmfziWwllICZYTK8H+yP9fxtlmeMp/fZU9qIwr/nQ7ayD+aaZ75Wjx+aNZfbjilpniL+v8WE5Run8r+BjiyX+iY1AaJF4zirt7OlabbeABs2wQSJaD/aS59bqIEXSwTMIGVCUQdX61cFydnAiZgAochMHfMOBTP73jLuIlIuEEp8YPOeXL2iey5SokXio9S4mr7o9LE/QvZv8ieY35S3DYN8Jy0HOdcBFDYFivc1IbYFr/0/W5btXO38XO1S9fWBEzABEzABEzABEzABEzABEYRqK94GxXBgZIigFKieAffEqWE4j5WOq90oMi7UrzJDaXZ7O19Sv8XHXz4oXeLn/KwMYEmAZS+70tH2ulrtaMnOn5uBhy6VpzN2rnSdhsfEoD9TcAETMAETMAERhPQ2CI8tP5DkT7VwVh67gP20fnmFNCMcpJWd1ktx242uflYlt0Ss+Ktm00OPmspJR7oT8J2OW7mn7VU/LH8rraKyo1tkeG9cp/onPJ0DQxYLYfiY7LCRHFszkuALaZsL/21RED7+jATx9bt3G18pmAczQRMwARMwARM4C8CGvfw4buXsotxs2zG2L/JZgeKlW+CIQ5m9FeTyfbMcsxWdFcFtyyvkFw4WPF2gSO7i8VKCf1B6h8/YBvfozoF+bMCrkvRgbLtp/L4RmFR3nETZEtgc1CA4gQlnRVvgmAzjoDaE+1lcZtROnu0c7fxcWJ1KBMwgRYC6qe4d3IPbTOdX5lvC2w3EzgjAf2Hkv5i/FiZqB7sQGGXSDX+0fmH8podKIynT23M6BjitxyPIUdqYVkOy9KKt2FGyYZQA1+klFB8VqDV39vGgOWrRoVRvIWtfpWX4jJBCCt8+OhCeHH1M51ffZxBYVHetW1jrdL0iQlsQWDHdu42voUAnaYJnI/A4NcO1a8luwUt5bL1NaVcy91Xp7X9MmAU/hf1qqf4xfh6+drOeY1M21iaB3w8SPSrW25ftWNGba0nLze39bzk1Vday7KPjvyseBsAlIq3brJ/LimL4t9pic8fhCdoT0s/FGPczOs3dJa2X614U5jiZiebAc2rMv5NcA/XNft97dynJnBFQG1nizZOPnu1c7fxK6nawQRMYAaB3q8dqq9MdntVymXrk0Ou5e6r09p+OTBSGcND4Kr6cuM8tS/GV+XrOOHB+FVd5BZ2k+BfrYbrSOPozmZ0DAlbjseQI7WwLAdkeW/A396JENDA4UpxJjee7PH+tYdTi6k4PDG7eG+brlG8MTjBDk+RuMmjfOsy/MnGfHiBNMOAoSstu5+YgNrfqm0clEpzz3Z+yjYuxmGFwR9C3vWOx8O27LPXv/yfnboN7Nm41d6S3YKWctn6ZJRrufvqtLafGa1NtDs9se4bc4eIvOrltMaMjiF6y/EYcqQWluU4Wd4dFyztUBI2X/nhYJLNk2ImwGcwr1XJN1MrKj4oy9pWsQXFWMVPYVHAVdf1vErOrI6rb1etB6mfkwZbWXczKle0dhEz790A75PRrDZO0SSDvdv57m18HxF05yLGrLz5VXbxjked895Jvjzb2md0p5Snz9nrj9TMYPe227eV47HkMWbSvlWhUy5bX51zLXdfndb2M6O1iXanF5RqV+P0WpSY//NaMaKdmlE09KtmbDmuijNqYpblCPx3R4RJOsjJB/1s80QxMdqIF6vT+CDCxYtZGayXfqTFVyRRWgTDO9zaJtLc+McudSe/sWFDvrNtlTeaQiBm3rOBpRtxchunKpJBjHa+axuPLTIxbl15o3LxPx+zCjZ2FRblf/b6A88MFjWhuZG5N7dtaw8Pzur37rl5zI2Xctn66pRrufvqtLafGa1NdFl695dFP0VsMzqGmC3HY8iRWpxelvdylmXXoF/uYeJ3oVzKua6h7GWduUTpxbbQaoJb+j2UG0qnYFCo8aLmwuic7aUXW0zxkDtP1lr95M42IvxYzVIZxWE1HE9Be43CFU/mZIeJQW/4pZ7Kp1UhIPfN20XMvJdySyV+yZDizGrjRFQarW1Z7pu0c6W7axunjgmYvhUQZ3j589nrTxM0gx3/iKGfGcgyPHUeCLaud8pl66tpruXuq9Pafma0NtHB9NoU6yFS+H/zaoczGzM6hvQtx2PIkVpYliNkmfuKt75B/+ORg4URmNIIovqgUPpU9g86UKZ9IruuzKLRs52zCCObLaB1f11ON8qDdFDgta16G5NgUNyNCbtGmJjtImbea7CLmobaWJQ2TqUXtvO923hUOZWZn30FxNnrTzMwg33/iWHSzQOELlM8BOjy3NA95bL1VTvXcvfVaW0/M1qbaE96GouE/3fbfzm4LR7b9xQheS8zSl5EowpoOY7ClEUgy3KcmHJXvJ1t0M8qnmKFmxr4A52/qYtZbqzoYpXfK9x1zRZR3BYbpVOsIJIdbvqj0lR4FCk/yd5zkBCzXcTMe5RMEg8UrY3DRe10cjuP1MajilF1HtMPhMla1LJukfnZ6w9TM9iiZa2SZspbOVIuWx/8XMvdV6e1/cxoXaI88G572B3uq2PerbxuidJLzYzSk8mcElmOc6ilGceyHJBLtoq3kw76meyGpZwoeHiJefMT6Y/k9nZA7rO8le7FVtORifyoeLsNEJRXNIVAzLxHyiKHYFHbOIAkx6ntfNc2nogQw+D/Q095xvwXe6In7XX2+iMcM9i/iYb7f1vOQR6xtqClXLY2XsEt13KH8u9hm9EelC/z4F3Mjy6diquH+n2rcUrfvbcl2iGdzOgYYrUcjyFHamFZDsgyW8Wb6hUGmX03n6NN/FAI8OGDoGz7tMYBhQH13VTJNfVmPzW8yr/UxGwXMfNeyi2V+NHbOCCmtNspYVOBvFM5zr4C4uz1p5mZwYp/tlpf0za2CW57ri6vapdy2apCtpzkWu6WqmzmZEaboe1MWMx/kOd72WG8f6Nz/uNf6XjWGfFEHmZ0DGFbjseQI7WwLIdleW84SNYhDjXoV4Nm22jn1lH5f5A/7xiz6ScQs13EzLufykRftTcGgf/SESZ8Y1L4UvHe1gPq+s/6ddu5wlQfEcF/TJy2dPZ0Uxnv7JnfznmdfQXE2etPczODnf90ZXYpb+VIuWx90sq13H11WtvPjNYmOpzeQwX5VmOJz2SzkhX7c11fjKHkdmZjRseQvuV4DDlSC8uyR5Y5K94OP+jPQbnQ07Y6vTZWSMRsFzHz7uS9lYfkiKKXDnaR2bg9LCqbI7cTQPY68GxTuga3KCtv2ku8ruvZ6w9NM1i3TU1Ija0cfMylaeiLY29BS7lsTV7161zLXa/D1udmtDXhRvr0sXLivbM2HQTMqANMZs6WY2YC6ymuZdkDR17ZKt4QrA5qFyZ5nAcT3LKe+Kl+R14xE2S1qh2zXcTMe1WITswExhE4+wqIs9efVmIG4/4rq4XSfYavmn+j44mOYgW8bMY8bEH7fLWMZiSkciRbtr7q5Fruvjqt7WdGaxN1eiZgAiZgAmcjkK3irRSUB/1na7Hj6huzXcTMexydk4UqJ6UvymqHr4Q9kztPk23mEzj7Coiz15+WYwbz/z9LYqa8lSPlsvUxz7XcfXVa28+M1ibq9EzABEzABE5DIHfFmwf9C5rqgRUSMdtFzLwXtIZDR+UdKdV2DZ3zvrjfdPBxEpuZBMQxy9UtM6t7Fe3s9QeIGVw1i10cxJ2HBlWftkumIzNJuWx9Vci13H11WtvPjNYm6vRMwARMwATORCDnr5reaBDgr/4sa60oJNiywsFHGXhHGQqJrE3MdhEz76yFtm3hn0suj2tZ8H6kv8vtQc3Np/MIsALiC7GkL/la5//UcaaXP5+9/rQaM4CCjQmYgAmYgAmYgAmYgAl0EMh9xRvV8tL3DuGOcEYh8ZMOtkdiUEjg9kDH28Il35+Y7SJm3vlKbLuSszLkzXbJnzdl9RPJrrzZQypnrz+MzWCPluY8TMAETMAETMAETMAEciaQveLNg/5Fze+wComY7SJm3otaw0EjSx6sjK0b2v3vcs9duVyvk89NwARMwARMwARMwARMwARMIAsCmovxcajTvIc7e8VbFq0q0UJaIZGoYFyszQiozbO99IkOViXamIAJmIAJmIAJmIAJmIAJmIAJ7E/gVO/hzvodb/u3jePmaIXEcWXrmt0SUBvni6Zsp36oc3/R1A3DBEzABEzABEzABEzABEzABOIQONV7uK14i9PIksrVComkxOHCbECgbON8RIQPAXzgunTbIDcnaQImYAImYAImYAImYAImYAIm0ENglddezZnTzYnTU49RXt5qOgrTcQOVja5QSFDL0Ahl/37cWrtmZyJQtunvVWfaefiSKR39N2fi4LqagAmYgAmYgAmYgAmYgAmYQAoENC9b/B5upfG16oLeYqrugkUYT3R8txcLK972Ip1gPmpobL2zQiJB2bhIqxL4Tanx8k7syqj9o3yzMQETMAETMAETMIHkCGicwusxMH/o+FQH70OaOrkk/mGNGR1DtJZjXnKUvNAhMK9iUUNTeTarMkpn8nu4FYf3dt+X3ak8kx+KuftloV7puvi4nuxfdDzQwXbXVeowVPGkFG+qNDeYx7L94vMhya3jb4XEOhydSsIE1J98nHDxsiqa++isxOXCmoAJmIAJZEpA91vG6C9l/0wVZBcPEGXzygwr326ZmBGNI3Oj9mw55idDFG/0SavobNQGSG/Se7gVh/xfyO4sg/xeK8xPsr8rw9PWeIhRmNL9N9k/6vgQ3Ley726V8Mx0gfNmZlxHm0hADexjHXeax8RkHNwETOA8BNxHn0fWrqkJmIAJmEAEAhqXP1e2H8kulG4UQedMCrlmp8rpjRkdowlYjnnKUXL7RSVHj7B495DSQOk25z3cKOo6+0OlW6wYll2sZpNNH8r20vDaIV0WhjSKsOX1ZlZSK95Uyy90MLGzMQETMAETSI/ALn20boosHX8re/CpvsJ8raNziXl6CF0iE1hOQG0+DBL/UGpJbUFLuWzLyZ87Bct2N/l/qZyK7VCNHH/VNfc8lHKbr85o5J3apRmlJpF55bEc53GLHmuNPkhpoHSb+9qrrxS/VfEnd5RrbDFl3tI05Fn1rwr7g47/6ED5t2m/Gl3xpgryVAfzkQ4mW52aSwINGaXXezOS/+gJHXkpvCd1Q9DtbwImcFgC6gNn9dGKx00P5cBjHTxtCjcz+vrwdKu68cmtMGV+v8seVLoRQeFYPv69jtab722q/jWB4xBQW092W07KZTtOC4hTE8t2V+7hvtnMNNwX8a9WwzUDneTajI4haMvxGHKcWwvGM8wLsCszNKaXP/qc0B9W8WonL/BXOFbmFUbn5IP55Na6+CWtr3Rs+q63exdZ7nwhAEzoPpVdfF1QNvt0+yD2llBxAfp/sj/XsXhCR2ZKx5O6Xur2NAETOCoB9X+z+2j6YB0/iQ3Lui+UYrrmKdS/dFy8f0/u9OG8v4YnoFMM72/gpdP+Uu0Uag6bHQG1cf6TV1vQ5B62oLU93d2lnimXbRcAB87Est1PuGIdJod9mbZNHPvCH8rPjI4hTstxHzmKMw/Cn+rgQwKMsYuH37r+oOOdrq+UTWUclKKECX3SU7k/LOMX43v58YGCYsxexvmn3Ej/Rx08fEdBhvlMx68Kc7FDRdcX84Ai5LgfxjqVUq0eRWlSXvJtzgkeleHadE2kRZpXLMo4q1h3V0llfiJMxooVbqWw3sxPqoiJdhPYCPzClEKgsbUK6SLw9UUxqbt2tosJmIAJHJrA0j6668ZYv5HXAXKTnrzquezXH5f9fD09n5vA0Qj0bcuJ/R9IuWxHawd718ey3Y94UKpxn+wyzHXObMzoGNK3HPeRIwqzsMiJMfY/dB0eiIfXVlQlKcfSPMxm8RHbMLFRmBX9js5ZScYHDS4UWHLjgTvu73UgWxR9Ie5LXZPmlY5G7nMMSrR3HRHDTp2mzudBGf6i3KUbaa1Vto5i3dygePtb6RvszsANjxA+2A3vUZcI8H0ZEq3qawkkaEZHJRACKR7xX+ngRtUGbtaEjvSVNoKLPaClKDbHJvA/CVUv/K+DnVDRXJQdCSzto4t+vV5e9aekyQAg3PTr3vSzFzdKXT/RwVZSbD75zfZ/rpv9fNFP1xPzuQkMEEipzx0oauXNfyqMmypHnYSBJP6xTMpli8XkKPlatutJMoyrgj0n5ftzIp0sjhkdQ+CW4wI5lmPloKBCGcY4O6wEYzzetsILpdYjhcO/bpoPxtvGIoRnPEI+1XZ4nb8tE1prjFKfn5RJV9bT8oydlCyeKg65oQv6oOswXqoi6IS6NOcVdX/O544Zq3j3minufI3ggcLLQjG8IPhDcTb9J2hVgclyxqahAVxN9OSGoo9VGXzUgQaJINteVBwmdVUjUjgbEzABEzgygdl9tPpWbmD0p2w1DU+feBJGP/ul3MJNWJc3N2X4tv6f8Cxp5+DFpzw9410QD3TUb57cR+jL3UcLgs3xCKjd838aMvxfdjcpl20NGKof/Q2D+dFbddbIN4U0ji7bFBg3ytA1mSVY+H//0YhztkszOobELccRciz7YF7PMmYMEFIM42zisO0Tw32s0oUo3eq88C1/5M72UWTDBwfQf6Aj+VnnF9tEy+BdVtcuxil16Eobd/rCDx0BqGe1BZYwKjv5ovN5yXWLYT6xVtlakr91QvH239I32J2BGx4hfLAb3sOXgsAEafEkSenUP4CAoNHUVkb+TAC7hPOJ/Dypq2j5JCKBf0fMu5l1+F8Hu+nv6xMQWNhH81SL5ejVjVrn3NR4/wM3++YNn376YhCm8A/kFh56kFZ4MvdM52/lVzf08UNPq+rhfW4CKfW5Y6TBeAXTNZ7Bj/9YDJNy2dbgUWzVUb/Dzgye+vPUnO06PEHnCH3TGnmllsbRZbs37/+WGQb7In+1KdoWbm3/5eBWf+h0Ef8MF2Z0DClbjuPkCCeF5MH1ZKO4xVhZNmNyzI+31uAv+b3QgbKquM8pDbadNsfugwntGUDlC30kysK6+aq86LpXc5+Dc5+ZO2as4qF4y9qUDYmJWTAspwxwg9vVhA4PxX0gy5O6QMm2CZjAaQmoP/xzSeUV/04j/tX73RSGCQWr6HiZK0u/6303N8uLm578w4CBtF6F9IN7uC7tC6Vdw8+XJnAWAilvy0m5bJ3tQ/0NY8j6Vh1W8bIzAkO/1TWQLwKc5CdL2SYsG+6NbQ+SmBxi6vfOW5fz/ZrRMWRuOe4jR97TyQPsi3F2W9blPe+9bMbrHDc6Z+cKr3nhaD74JsjehjE/998u0ywj9UBx2FV/0tp8HrGr4k2VXXtiB2waEpO5p1zIcKP6SNccAS4wwzlhCiP/QiiyPakLUGybgAmckoD6wabi7EZuPOV6LJsnX1MNT9eetUQKk4nmDa5vmTdpNd8t0UyadE+9CqAJxNeHI9D8z9QrGCbksbagpVw2+jLGgbtt1akL5gDnScv2AHzbqsBOHO6/TcO9mBeYX81pmgFPcG1GxxCy5biPHBlHj91lyMIkxtTfhaKpz0Fphb6EXYVvg3tEm/F+mE9UxaBv1FFdc6JrVu0xRiqUiLi1GPw3n0Pca8l4MydVfNWJndJji+nFkkddIwQAY4eG0Teho76nndSJV7ixM1hve7cdfLI0MesWM+8shbWg0Ga9AN5wVJZqM2GdZCQT+l/iXTyVlztuKNC4gYf+WZfFjZHJRNtNtEhLfhdpFZEufwgXVqVc+mx4pXIdtg8di80MxpJaFk6cw4Cy7T8Z3DYfOLbVIuWyUV7KJ2vOAwTiFn2VbMaKmLFbdW5DZ/4LOx3UIrSxeo2CW5R2Vy/Ikc7Fm3sk7zTlo0LFZFk2rNnR8/mR6jq3LuJhRnPhJRTPctxeGGLM+Jijuf2yL3Pewc9/jHtnMPRBzbF4uAeEMNgosd7XHTY4577c9k5/sqLPLMql8mMzTv+8URc5XZjiocaFywYXuyreOso/d2LHAOhDM01BZRklzjSwMFhqndARSGEJx+q4ZkPCu24It/ukrl6Atc9VZ15Q/lJ2/ab+m66/0JH1ICpm3WLmvXYbST09s95cQjzdmnKjvpFMWI4eHohw4w6F5OZHP8pkouupG/0327jq/Q/xusKHtLEpa8i37r7Zucp52D50LDQzGEtqVLi/laGC3RaJsQr/o6ZhoIsZGsvchtrmN+WyrVHj0Vt11sgssTRylm31RbnEmA4Vh4ngt+pjmVzycBybyWMxt9G5za0y3Yzybwlu69vKkDED4+ux4wP0K7xC4bniyCrMff3yXyOdBzrn/W+sfrvRNasW2eFCPrjjzwMb3F/qwD2Mz5kXfKaD++kS80qRSb/NUBbKSr4sKhqj10CvFMqo023MvW2SnZTqnIkdKyZ46kMDqt6tIcBM0MLqAwSL0EMja5vQKXihEU1yUkfhtjLiwuQYhWNV95IX1/BFLlmamHWLmXeWwlpQaLNeAK8nasmVEPSnrB6mPyhM6ccACYVTMPQjF8vR5VH1yyHQSJv+mxtftRxcaTPJ6L1BKwxlvZFdV9jhtJlRXoftQ8dCM4OxpFYNl/K2nJTLtoYQGJhXY6Y1EswojaPLNjlRqH9l8rv5RDC5ik8okBlNgJVwUMtxW+GIL7qQ8F7SwczK8EF/chVe/l3j8j73Ve+dlEHHjQ6+NH7xMELXk/pOhUdBdyO7s85XEGY63J0Zb1E0VQwNKsfXSoiJXQWsdP++FiaEq/KUH1+S+ljHxWRM1yja8Luj46GOOsAwoavS4URhENxFOhcBbsPsPqlrlmGDa+pcca+l/6vOH4tJUeeae06nMesWM++cZLRGWc16DYq1NPS/R5n0qWyWl6NM+0R2XZn1Xm6s/C3CyKaPrfvrcr5RXqSHIq+4CU5IqbV/nxB/TlC3v1uF6FHvI3PaxNI4/y0TCPZVevpvoNR+L5uxU2F0zv2ah5E85Y1mUi7bUiiqG30Sx6QVwEvzTSV+5rL9dyocXQ4TMAETMIHVCKw19udhP2ltbu5unkMjA928o0zslO/cCR01WEuwDRpRL3lyyyS6acIkGv9cTcy6xcw7V3nNLbdZzyXXHY+n68UKN/WZLBV/Uw8qN55YsRr2Fe66ZiXx2k+xKAMPUEYp/xWOewpfSA19F0Xbw7j93b4f9aj3kT3a0Nw8WHXK1gm2UvAA8586UtmClnLZ5vImHkq3KVt1luSVatyjyjZV3i6XCZiACZhABwGNf3gQyetpuD/PMmVc0pi7U2dSvvcmhV4nMJMqVgrcqJKtEzu580Sx2GqkcyZTq0yolBaTOQaqL3V8oAxDRuFiTeqGijbbX3UaM6H9ZHYGESPGrFvMvCMij5K1WW+Gnb4hKFJQLL0W6+rlzmWuj+TWtspptUIpfd4DN6afIs8fFXZUf75WAUeWLcs+dCwjMxhLav1wZXtPcgtaymVbIgnVi4e3o7fqLMkr1bhHlW2qvF0uEzABEzCBQQLolHgVwtxXZLHYYLfx1O4r3lS51omd3Otms4mdBg7Vu4PqGfacM6ljwHUkEyaEfZPVsZPe1LjErFvMvFOTw9blMettCNM/8n7MsI2NiWZgfSN3+oVd+kPl1dc/VbUfG66KsM5JYNJXxlz70LGEzGAsKYczARMwARMwARMwARNYlUA5B2BhFav/J5kyDnFXWeA1JvMYK97CxI53iWGY2FWTF1WeycqmEzvlUeVHAfrMlLB96WTodz/DMo8tcsy6xcx7LJ+jhDPriZJUf8e20c6to2V/WKxYnpj0GYO7/d3cmMEZW77rbAImYAImYAImYAI7ENDcBMVZ9ZG3sVkq3uQ4Y9PuCofijcKyxC5sL+oK23QfFU+V+rMZse1a4aov5+E/Nl5bWnu4qXx39shnozz6ZB1WMfDp8hxNzLotzRuF827LXQeEO+r/PZDGlt5LWW9Ztl3SVh/EQ4p/6cAea75UvLchcOr9bCjnHHvjPvr07U8yOQKDlPrcOc3ccUzABPIjkPr4Kj+iLrEJmIAJbE9g7pixiofijRfSofTCcfRKsLHxNp78qBg2UwlIJnz9lWhtE/bgxsAgOxOzbivk/VjA+S/u8oLHAeHO7RcGkl3HewXW6xQkYiowUPYPlxRBaeT8AGFJ1RfFdfsrHo4d4T6SUp+7qE06sgmYQDYEkh5fZUPRBTUBE1idgMa3jIvQBdBPsSuRj5ehI7K5/aDYnHl6Nda8Z4qnJcCfiD9V04QVbzn/yWLWLWbeTVke/dqsjy7htOvn9nf7wO6o95G0W59LZwImYAImYAImYALrEkDR9jFJykYB9x8dfkgPkBVM9oq3slG8KFmECcAzubMaxKabAF8A+bbFmxU0bzPnF7NuMfNuEeehncw6cfEevH92+7v9ktRR7yOJ/7tcPBMwARMwARMwARNYlUB9Jw16FetTVsR7dxiQzgAAACAASURBVMW0YiX1rSZ335QHL/3mvTO/xSpMLvmKF9sZ38sOXy+80Tma7a90PMulHm3ljFm3mHm3sTiym1lnId3D9s9uf8XT0MPeR7L4d7mQJmACJmACJmACJrASAY1t66+a4sFq9A+qqUxPdITFVavUVOlN/grqGhlnv+JNEJ4LXn3/MY0Etwc6qpeIrwHrgGmg1WZi/JlsPqaA/flBuMWsW8y8JcJTGbNOW9xH75/d/m7fM3jU+0ja/y6XzgRMwARMwARMwARWJiBdAO8yY2HTLF2K4j0v4/OeuNmmTOd32ZVCUOcsFBrc7ahwKOu+0fGOAui6+oop5zq+17HrRw2PoHgD2BuA2kwjoMbG8tFdG9y0Es4PHbNuMfOeTyzPmGadvNwO3T+7/d1+ZEGt8JD3kQj/Ln/tMAJ0Z3l4AryP033U4cXsCpqACaxBQGNbFjGhdOMjWl/rqBRWE9JHN4PybrZRvijYvpDdXHXHw96qT9c5+bDbsVLylXFx/1Ln1OM3HT9wLrdgWLhV7MwJDlvb2W81bYGIINCMztLQbg3c6ZuACZjAWQi4fz6LpF3PlQjwdJaBYvjI0UrJOhkTODWBx6r9ogngqem58iZgAqchoHE7Ciy2Yf5H53/KrhRcUyCgh9ExR2FXzwYFYFvfzW4a+vVgCPd3uT0IDrJ5DzOKtaBo41Vk9Tg38uOhzGPZKPh2MUdY8VaBEjiA884ytv/YmIAJmIAJJELA/XMignAxDkFA/ycGmhheE8FTXp7aVlsx8LCZRsBMh3mZ0TAjhzABEzCBVAmoD0dP8oWO1zp4yIfSqRpDyH+0DkVhUdAx7iAd0ngnN1aVkSaKL5RaxRdRa248YGQFGzbHZ/JrrmiTc2GI36b4w61zt6PikC5xqWcwlLHNFMo3efzc5rm22921E4yVXgmZgehDnQftZqziOF8TMAETMIGSgPtnNwUTWI+A/k88kf5VNltBeKLMO0xel/+z9TI6UUpmOixsMxpm5BAmYAImkDgBFFAoxf5JOdWvM4Z4pOMB12ON4rF67L7sn3XwoS1WphUrx3SNHuZCYVZzQynG6jTikTfn1YcedV0YuRGuVZ8jv+aWUfKq73ZkTIRCrW6oX9vDyV/lXlfQ1eOsfn6IFW+lcBiAFuDK6xvZbYBXh+gETcAETMAE2gmU/bH753Y8djWBSQT0f+KFxR/Jrp7O6pz3l3DNwHe3AeSkgicc2EyHhWNGw4wcwgRMwARSJqB+HOVTWOGFogqFGeaZzqe+ooutm7z/DWUbq+d+0Xl9ayn+XaauFCMcSramwa0vjSK88qROzd2OKAWpX9gZUKSv67dFpMsflHtt+V+GWukqe8WbIAKLwSYTu6CtRfOJttPGBEzABEwgEgH3z5HAO9sjE2BLRtvgkae2xSBY/7vWp8RHhrKwbmY6DNCMhhmtHkL/5TBx9JbyDrpm1AEmM2fLcXuBiXExdpDNA7pXIcfgHq7H2MQp00Hfwgq6G6511JVqrUkpTH2MUj+vh/9IF11+RTilgw6obbcj7pSlWIAlGz1RUDIWcWs/g8q9WtjFp3eVwt/KVII9NtEQPthj460dji0XaDaxw/G8IdS183R6JrAFgf/ZItGZaYb/dbBnJuNoJyfg/vnkDSDx6qfU545FxXjnfUvgsMIff5tpBMx0mJcZDTMaGyKMq4LdGk/zGO6f3lLeSufW0Yx64GTkZTnuLiz687dLcpXMWOz0RjYKLt7jxqKn8KBgSdIhLmMalG+tRnmiXCt20+icVf9sWcWtMDoPYyKuv9LRVTbi1MMSvsvMHTNW8e4p5f+WqQe7K7Omewgf7Kb/LtcC+/EuGTkTE9iewL+3z2J0DuF/HezRER3QBAIB98+BhO1ECaTU5w4i0v+pcxBai/xJ7TypU5X/gQr0VMcDnTNYLwbOuuapdvFSZtm7mtyZ7gHLjFanHMZVwb7KQMy9pfyKyqWDGV3yyPXKctxXcuLNfZfXVQyuTBsoGemgbOO4UXq8d63rIwkE6TKt4xqlxYo68rgypXvfbkfiUkcUcoR7JrtLuUYe764yaXeYO2as4t1tT9euJmACJmACJmACJmACCREISjUUVV2mdRDbFXhn96ca/BavASkHw/+QXQzaVY6up9FbFzF3plvzIX0z2oPyZR59W3v5Wl/K//PLmmx3ZUbbsd0zZctxT9q3q8iqd8QuzVp9Ea+4YKch9/CghKN/Ch9vQPF1I3/civu8zoMbDxj4sANjA1bhNQ3vaWtTvg3tpmGc8UJxyY8PT/XVl223ff7NMi26ZsWbjQmYgAmYgAmYgAmYQP4E7qdYhXLwHJ4qo8hhW8inZVkZkHe9fyWF6iTJNAUwtTKYUQ3GCqdMQtv+E2HVBv67TRZXqM8WSZjRFlT3T9Ny3JG57rtsMZ2zMu2ilKUyq7UPkt+HZh4dbvRxbf1cyCso84oHdsFRafXudpQ/q/kGV/QpHGOPG9mhXw1ZbGbf2yxlJ2wCJmACJmACJmACJrAWgfc9CYVVSX/0hFnkVQ5S/6VEisHqyMS+VDwG+sT5sYzzQHZY6cagtzov/fe0ojLds6IL8jKjBfCmRi3/Z0PRwv99KNwh/c3oGGK1HI8hx61qofbBl1IZQ/CgbgvlWFDsbVWFq3QPoXiTMIqli6odA06eoH67kYCuAObucGR2MesWM+/c2+TU8pv1VGLrhjf/dXnmmJrbwD5SE2feV0JmbYqv4LbF4LSoIPnr5GFxMfFHcVG+3chmdQMmKOFuryL9Uicd5B741UsS3DZjWs8s1XMz2l0yQanG/63LhLbZ5X90dzM6hoQtx2PIcbNa6P7DKynQ67zkXrRWRkqLba4/yd71/p79O94EzF/9mdkKj8wuZt1i5j2zKWQbzazjis784/JPIXe3gd2lwPaJtneehAnM4PaK3Ut8mSFbXHhvy2oD6MvkZ13lznRWpSdGMqOJwDYO7q29w4DNaJhRDiEsxxyktGEZNV642Gq6UlY/Kt3dx0tZr3gTMH/1Z2brOzK7mHWLmffMppBtNLOOKzrzj8s/hdzdBqJI4SflGlb51wvASjS+5JWSQqtevnDOirfW98KEABHs3JnugcyM9qB8m4e39g6zNqNhRjmEsBxHSEn3dVa4zn3NQ5WD0vmzusjwROVftdTN9HR9Z9UMWhLLWvGm+vR9CYWvbBSfkm2pt52OzS5mu4iZ99natVnHlbj5x+WfQu5uAztLQeOaH3R8o+OJjkKBJZtB+Vc6Pt+5OJOyUzlZqcfxelLEjQOrXNky3RhNlbwZVSg2PxFrb38eoGxGA4Ay8bYcxwkKTgrJw7VFRulsrlhaVMATRM5d8caT07avYYT9uvin9mQ1lWZ1ZHYx6xYz71Ta1l7lMOu9SLfnY/7tXM7k6jYQR9oMwHnnyWeyebct9ue6Lt6jpvNUDUo3tpnuvr1jBJBcmY6o2mpBzGg1lIMJeWvvIKLiq4U5b7sfruE5Qritn0POrqUI3MuVggZuPOEdMuGdJ0PhTuV/ZHYx6xYz71M1YFXWrONK3PyLNvhAUniq44F4fKGDCQDvoeDJ5Dtdtz0UktcxjNtAPDmKPW0s5pdAZ1Ve5WaCxQewkjO5Mt0TpBntSfvGW3uHcZvRMKMcQliOOUhppTLqPoL+5kWZXFCcPyvvLyvlkm4yd9Mt2mDJglKNAWiXQbg21wSOzC5m3WLmfS3lY7uYdVz5mr+UbhooFC98lf29xMGXl4IypO0dXHEltn7ubgPrM3WKJmACJnCjewkPbt7LfhJw6Jw5DVvKnwW3M9tmdAzpW47HkOOEWrBan9dlcPC6Et7zx4cyT2GyXfE2Ujr3R4ZzsGsCR2YXs24x876W8rFdzDqufA/LX4MFntK9K/GigPq73MJKHiZHh17tVtZ7jHXYNjCm8g5jAiZgAgsIPFTcHLeUL6jy5KhmNBlZkhEsxyTFskmhnmu8/JOO8MoJHlTjxu6R1F+XsRhIzoo3fwllvviPzC5m3WLmPb815BnTrOPK7ez8Ua79WIrggeyw0u1GA4fqPK6INs/97G1gc8DOwARM4LwEdC/5oNqf5X4yS9BmNAtbcpEsx+REsmWB6NPebJlBymlnu9W0/JPClglQ0wS38JGFpv+pr4/MLmbdYuZ9tgZt1nElfnb+qv9bGOjg4wKYoIS7vTrB79nbwAlE7CqagAmYgAmYgAmYwGoENHbkK+I8VAgGRRwfXXobHI5sZ6t4K4XiL6HMb51HZhezbjHznt8a8oxp1nHlZv43N7yfggFDfRARVyr75u42sC9v52YCJmACJmACJmAC2RPQ2JkdI7zHkq3GpzC5K974EsqjFkkhwGJFQoufnW4JHJldzLrFzPtsbdus40rc/G9uWPH2c1wxRM3dbSAqfmduAiZgAiZgAiZgAnkRkNKNdyXzfreHZ3p4nbXiTYLyV39m/s+OzC5m3WLmPbMpZBvNrOOK7uz8VX8GDRyv40oiXu5nbwPxyDtnE7gmoP/jYx1PdHyt43uur0PZxQRMwARMwATiEdC9ibEzXzX9QgevbeEDZbgd3uT8cYUgHH8JJZCYbh+ZXcy6xcx7eivIO4ZZx5XfmfkzSGCbKdstz2zO3AbOLHfXPT0CfCnuY4olm3cd/0fHHa5tTMAETMAETCA2Ad2bGDt/rwPFG1tNMbzn7Zvi7OA/2SveJDTereOv/sxoqEdmF7NuMfOe0QyyjmLWccV3Zv6qOwq3T+NKIH7uZ24D8em7BCZwQQAleDBMbqK/e1L9A+/v4dUvq3zsTOmwmu+7UEnbJmACJmACWRH4TaXlwRB2ZdSvn0KXk73irZKYT0zABEzABEzABEzABEzghAQayi3encPHXyYbpfNckViNsOjBQpkOq4IrpZvOmXC9KAsVthY9k/uFklDX+LEC4h1hdV0o27B1sI32FJM06m5jAjkQ0H+Sre38v/nv0newAvfsOwKEwaZOQG2iWJVddzvT+d0zVdZ1NQETMAETMAETMAETMIGjEtDEJmzjmTvpfSM2pDHbqAxMwHl/T7MM38oNpR4HisH3OporH4gb6oDC7anC4hYME3oUizYmYALpEOB/+bMO/rMozU/7/tt0ROKSpEbAirfUJOLymIAJmIAJmIAJmIAJmMBEAqVCCqUW2zu/nhi9CF7GXbqdE8VYm/LuudJnZUwwhOPF2uFdP7jztWQm8R+4kEE5V8WRO8o8PiRRV8YRzsYETCAegSS3uqufYAXeKkZpzepTV8nciRyCgLeaHkKMroQJmIAJmIAJmIAJmMBRCWjSx/vSvtDBSpJPdKB4YksXq8jY0snKMRRYvAdN1g1bPFsVaPJnAok/6ZDGO7n9oIM0UXyh2Co+zFBzYwLLKjVsjs/k17Wdlfht20FxY0Vdq1Ec0iUu9QyGMjZNoXyT489ND1+bgAnsT0D/2WpLuXJPbqu7ykff9qIkQz+DGb3NncBKw1vdAWEzm4AVb7PROaIJmIAJmIAJmIAJmIAJ7EIABVSxGkw2q9qYBAZlG4q3+oqTzgIpHKvH7hOfQLKZhKLU4/yDDpRjxbvVWtxYnVYou2T/puNJuCYsRtek96G4aPzI74eGE3lR9relO1vUmttTUSbWJ/UE/VUHyjkr3qBhM4qA2tko5cuoxByolYAYh23i4T/dGq7HEcV822rZniiXXqWc2epefzDAA4rqYUBZTvrP6l2WZTzy/lLn9IX0cTyQqPdnxVZ3uZ3iK5yXZH21lMDdpQk4vgmYgAmYgAmYgAmYgAmYwDYENMlD+YRCCqUWiqqgwGLFxlTlE1s3WRXHRwpQuL2XXV8Zh3+XqSvFCEd5mga3vjSK8MqTOpF/XWGIUvBGfkySOVA0cv0Wu2aYCLflXQviUxO4IkCbQmnN0fqOwasYdhhNQExZ6ZbqVvfF29wBoToWq21lo8S1MYFJBLzibRIuBzYBEzABEzABE+giUA5GB7dzdMW3+zgC4swEB/OHjmq7YeES+SflsvWhSbncKluheJLNKq9XoR7BPVyPsYlTpsOKjaDYavsQwlVyildf+VE/r4dlQtrlV4RTOijNaMMPG2niTlmKFW6yWX0SlIw6rcygYq8K6RMT+IsAyhdWLAUFMm0Qtwc6msrdv2L57EZ8ct/qvtY2d1pDoXyTPfWhh1vSyQncPXn9XX0TMAETMAETMIH1CHhFwXosW1PSBIjtMb/KLrYb6rz4gpyuo68ASrlsrTBLx4zKzYqwRQoC1ZWVZm9ko+DiPW5MSIMiV6eLDUozlG+tRnnSTmm75M92LravVm1X5/VtpV8pbFvZCF8P15qXHU2gQaBX+dII23lZb6+dgRoec+I0koh9+YkKgKL+nxRE9WGV7CMd9Cdco0S/UzuqLZz4ByP/sNWdL6CiVEe5XvQXukZhX20HJU7Njf98sdVdbuTNOcrACyM3wl0p/uXe3DJKPnO2uZNf2OrOuY0JjCZwd3RIBzQBEzABEzABEzCBfgJjtnP0pyDfcvA8GK4eYE6cevwczlXH5yrnR7KrJ+06Z5LBNROYaCblsvVByaXcKicTSmQfVuv0VavPj3QqZZbSY/I7ZwVZMVluZqT0UAySx5Up60A7ZZsrq4yYtKM4DvmzGq9IVzbhnsluU7CRfvUeOp3bmMAgAbWlIeXLmDT4MEmhbBoMfBkARVGWX8VUuakv/Q7/u+y3upf1mbvNHalyz23t4/C0MYEuAne7POxuAiZgAiZgAiZgAhMJLF5RUE5OTjWxmcCY9xK1rXjiCfxjsWtVhkxIf0nQlMvWV69cyo1sK4VrX4XG+PE/04GiHCVcscqkbD9hRUuhyC3dCkWdzoMbCmBWuzyVW/FetkaeTM7bJqas1iQ8djgoAxNZDEq4F7omv9eyu+p7ig8rwKE8wjv52pjC7fBGHFBc/UcHbW+xUTrcY5rKl950FYfwfJikq13eyA9ZBblV9zG5FQpz2auUv7egK3uqzCjEUYCvstW9TIe2zAo6ZNrWh8jr0ihc6CfwqJ/XA9JPdvndKA3ypX9p2+b+D/mzGjc8DGjb5k5e4UEB5zYNAuIX2v/p+60Gmpt7TYccrxFwWe4/ZCf1rpPUeR6ZXcy6xcw79Ta3dvnMem2i09I7O/8j1V91YUDKZJiBZ9eAs7eBtMRrbucYih8mNvWXvV/EUR6sGrhfOr7SdaGIkv2LDlbRMJGfVf6LjNK8YILSVrewKgj/zknhxlVKuWx9Vc+i3GrTtHOUhIuM0qF9tLYR+TFhvcijw4022NYOQ9kYl/PfZwJbGaX1cXXRciJ/lBO9K/oUhon1jezQ5ltSyt9J9aMvfim7kJVs6s1XFqt34OVfy0k14P4Eg/rHOCYlEAKLYZfyJQS5skv+KIU785ffa0XkHXJ8cbiQl66rLZelOzL8UceHq0zSd6CvLJTvc4uqeqOMLLa6k4auUUQWijCuVzD0C7C/MsoLuRfb3PEsr7GLviTYZUS2uXfJmnQO3f+UDCZbYuh+q4fa3R6/LLxKASf5rpPUAR6ZXcy6xcw79Ta3dvnMem2i09I7O/8D1p/B5CoTG1qS+ExaUaDw5M3E5mKyTlrByI+JDe+FIsxLHTwxr4zcUdjx1Lp14F0FzPBkZJ0+iVG1lMvWxyPXcvfVKQU/cUV5xrZY+pS1TVDqrZ1uMumJG8qIJLeUx4JUtqmPZRerM+eWo2yThfJF59xLWEk3pp3S7jqVTkoD/xvZhUJa9gddkna16g1/GdIowhZXmfyoHsX4QHavYnxEdUinqr/Sg9ecFWSt93ilxwOKK3mW5Yf90m3uSqJI31vdIVEzYux+q8aj7fRum2MubhbwfEkdmV3MusXMe35ryDOmWceV29n5H7H+qhMD6sUTG1qm0gqD6+Z2jr6Ge+qJTR+Y0i8o1ZjQdZnWyUhX4BXdUy5bXzVzLXdfnZLwUx+AgmRVJbjSZGLHiqKjrzbJZfvzrm1Ncu/r+wbLovjcl/qUL31pfKX4ras85Y5yjZXYlUKpltCFEqhMg7Ri9dW1ok06pbytK2UnpVIGVv3ZhsjqdJgVytSSyVZb3VmJxYo97HCQf2hTPMwbs81dwYott6uxIMGDGPdbA4K8N+CfunefgPlD87Qo/KFSr8ve5Tsyu5h1i5n33m0odn5mHVcCZ+d/yPqvcc9UGkw0Ordz9DRbJiOtqxnkHiY2vGOmaciv2G6Kh8LyAm3eG0MZzjYGCFtwm4xSuE65bH18ci13X5128Sv/g0zY1zK5btGbWv8stj9PrVQC4VG40B6xK6N22nrfCQHkzysQ+pS9L/BXuGo1mM5Duw/K/ZAcNmmxlbFVkUeA1Izqk/VWd5V/8TZ3ZBLkKruvPaQmvr3K435rgHTuijcLeEDAPd5HZhezbjHz7hH3Ib3MOq5Yz87/7PVvbX0ajIYVBSi9UJZhmNR0bh8lgMKefmIDhwHTtx0nTO7+GEhjK++Uy9ZX5yzKrf/Hn32VSN1P5V+tiG1pye3OahlETkh1CQqbvpKE/3tfmCh+Kj/9/lMdvG+T99EVD2J0/UHHO123KpvKeNxXCRcYPJU7q6ZJg9cKYPMuTx583cgmL1ZI4f6jDlZPcS/BfKaDVxFV7wvVea/ypYjV/sMDn0qpVg+iNCkreTbvcY/KcG0KGtIizVYWZTxbMwlIJkUbkc1W3zb+M1MuolUr9JYkcrS45f9gqFrJ9ltDBV/L/95aCe2djgU8n/iR2cWsW8y857eGPGOadVy5nZ1/7Por/2QnNmqZs1YUKJ4nNgN/a8md9xERKkxK6zGC29qTjHoenecpl62z0PLIpdwq52EUS33ysF9BIExOP/TwCP/3niDRvFCW8eCFr9KyrZN+iy3HKCw4rpRN8qM+fAnxYkUz8eR+I5t+DQVcc6Uaq7Bw531bcEPZVyjaZHOf5EMGP+tY2i+iRKMubYbtz5imYo78MW15U16UjDYbEZDMizYn+6WOvv/S6BIonbNsdR/NpBYw936rVpXtTrNVvAmJBTy/XRyZXcy6xcx7fmvIM6ZZx5Xb2fnHrn+yExsNTOeuKPDEZtx/mskdqzuaJrTJ5uSvGW7L65TL1lfvXMvdVyf7HZtAktuf1f/TN4WXztMnseIofNUT5dqV0q0UE/3/I4VtviKoqezqWqGKcuux4hcr4UhT5291cIqCqytf/McYyt6VN6v7MLwfrDgpf57IRunYpngjrbZ+vB7f5wsJiD0KYGS3ljnLVve1eDXTSbLfahZyy+ucFW9juJxewGMgdYQ5MruYdYuZd4eoD+ts1nFFe3b+m9Rfg0hPbP5q12ec2LDdilUjTfNQDkw0V3my30x85HXKZeurQq7l7qvT6n7lBJZ3WWGC0uBZ5DZ3W5pj/XYpeKhlULBvtqW8lPO/lNcUhcWXive2jMOWT8wDHdW70+RfnRe+tR/5sTWQevNuThThfL2alWrVNtFa8K7TNx0eU+rRkUTBvatvpZ7V9lcSULnJk/vTS65bDMq4NcrVkvRfTmU55sqySkjpeKt7SUMsKi7hRG5ekdytmAbT5v1WkEXqds6Kt6g3ptQFO1C+I7OLWbeYeQ+I/HDeZh1XpGfnH7P+DNY9sUlkYrP331ADfD4cwVP8JzqKr6rJpk3wou7P9y5PPT+VI9my1cvZPM+13M167HDNVsBKeaJzViOx9S+saNqhCMfPQlyjbiknf1FGkT/ZKO5bIsl+XEYO96oxaZEnil0UVsW2VKVDn1K1uTGJ7BlGZQvKMxSFdUN/jOlaaYciAs6bGpVvtizrBVM6VizVgfj8igBtTQfu4T9RDxPc2lZ/1sMd/vxurjUsOxOKH4RZr0pwO72A61DC+ZHZxaxbzLyDbM9im3VcSZ+df8z6K+9iVZPsuRMbVhCwWoWJDS+6bm7lidu4GrmrfOF+nuTEplHcvS6ZoPLSchQhX+ucl4t/rvNi0rtXITrySblsHUUunHMtd1+d1vZ7rjYW+h3Spg9hKyErfmzWJZD79me2fPKVzw9jsChcsYJSNg8VPtWBogeFG20uhfbFw7ZwL2qrUrPv5UMLKA276k9apGljAkcikHu/tbks7m2ew7YZWMDz+R6ZXcy6xcx7fmvIM6ZZx5Xb2fnHrv+cic17TQSYEBRfX9M5Lwr+XjZHc+Kwd+vyxGYkccmKyVySq0BSLlsf3lzL3VenDfxoc13b+TbI7tRJ5r79GQVtsSJ3pBRRrqF8q7aW6j+J4oqPLTzSEfv+xEKOQjkouzL0Gzqqa050zYo9VrQV91ncWgz+XhzSAqbNSUxRVHqbexuctNxy77c2p3l38xy2zQAB0yE3DU8uY7/rpFmm1K6PzC5m3WLmnVob27o8Zr014f70z84/dv3nTGzC19cKyWowyzYYJkdt99F+6a/v2zmxaWbliU2TiK9NYHsC9Bc66it4UMSxqim2UmT7yu+cA6yVJQ9KUOIURucoH9jC+OzWJc1flRMFFUdzlfJQgfk4AXWsG655yFU3zTD4ocja0tDGP+vIgHtoUaay/KwEZQXyh47wOBfz1B5/e10SYHU3qyE5eOjIg7qLL9xeBvdVDAKSTbb91l68sla8WcDzm8mR2cWsW8y857eGPGOadVy5nZ1/zPorb09sPLGJ2wE491MTUB/ECiWUQigQbLYhkOv2Z+5PKGSbCrM+SiipmLSztfTr8gjveSOtBzrCwy6+fvqTjo9q7rTH4E5Y3oEZFH8o9Ii7xLxSZPJoMyhC2frPtn/KzPmQMpoHZ6F8OrUZIOBt7gOAEvLOtd/aBeG9XXLZNhMEjCacJxF85Qc7lXedqChJmyOzi1m3mHkn3eA2KJxZbwB1QpJn5x+r/ksnNkHE93XC/bOY2OicrRzF6je5MVFhQkFeuDPpYFsN7i914B62OzKx+UwHT6LnGiY2XZMjykE5yZMXuTOxGdqmw8QmlE+nWRnevdcsMF/4W8K3mZ6vTWAWgfJ/iILhoc77VvXMSn9JJJXnWtWo3wAAArdJREFUneLTT2RvSrbZ9WEqNwq3SR/cKON0KurkjyKrrf/rc5+y1bW3vZC/jhsdKPUulGq65j8wWk4KX7RP2Z317S3MOT3h++acVc+r1lP/D3nVbnlps1e8WcDzG8GR2cWsW8y857eGPGOadVy5nZ1/rPorX09sepq++OQ6sUGZ2DWBG1I09hCxlwmsQ6D8b7Hd6wtSrP3XUmmfKARtTGALArQt+ueuPnpsnrz7ze10LC2FUz/Dasi6QQbe5l4n4vMsCGSveMuCsgtpAiZgAiZgAiYwRODUExtNLsJ2qyFO9jeB3QmUSja+goziLWy7YwLc9xL5XcvZMkHfNX9ndlwCtC0dr3XwJd9ZimbiihDxlyrvjgt6oGZiR9/jbe4DnOydJoG7aRbLpTIBEzABEzABEzgTAQ2oearNpKRYsTan7mVc0mg+IZ+TnOOYgAn8RYCXmT/WgR0O3r2U1HbTv4rrMxNYnQDbXVE+zzXEtdJtJr3y/s4DuuS2uc+skqOdjIBXvJ1M4K6uCZiACZiACSRMgIkN73ortrLNKKcnNjOgOYoJDBHQpPfjoTD2N4EjE0DJrOMfOvgAxHdT6kochSfurNVyU/I6Ylhx44Fcytvcj4jddVqZgBVvKwN1ciZgAiZgAiZgAvMIaHDtic08dI5lAiZgAiawMYFScTZJ6UaRFG9ynI2rkk3yYofSLelt7tnAdEGjEqgr3uZ+RWtuvKgVd+YmEJOAbiK5fHnL/++YDcV5m8AJCah/ZEXA5ElK38Qmoz73hBJ3lU3glAQ8vjql2F3pGQTY2v6RDuzK6L7ubbsVDZ+sSWDumHEoHoo3BrhdDbdvOezceGtycVomkCuB1L9o5P93ri3L5TYBE2gjkHqf21Zmu5mACRyPgMdXx5Opa7QhASkzvM19Q75OupXA3DFjb7z/BxJkc7oB54A1AAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & \\frac{\\frac{d}{d r} A{\\left(r \\right)}}{2} & 0 & 0\\\\\\frac{\\frac{d}{d r} A{\\left(r \\right)}}{2} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}- \\frac{\\frac{d}{d r} A{\\left(r \\right)}}{2} & 0 & 0 & 0\\\\0 & - \\frac{\\frac{d}{d r} B{\\left(r \\right)}}{2} & 0 & 0\\\\0 & 0 & r & 0\\\\0 & 0 & 0 & r \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & - r & 0\\\\0 & - r & 0 & 0\\\\0 & 0 & 0 & \\frac{r^{2} \\sin{\\left(2 \\theta \\right)}}{2}\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & - r \\sin^{2}{\\left(\\theta \\right)}\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin{\\left(2 \\theta \\right)}}{2}\\\\0 & - r \\sin^{2}{\\left(\\theta \\right)} & - \\frac{r^{2} \\sin{\\left(2 \\theta \\right)}}{2} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ ⎡ d ⎤ ↪\n",
"⎢⎡ d ⎤ ⎢-──(A(r)) ⎥ ↪\n",
"⎢⎢ ──(A(r)) ⎥ ⎢ dr ⎥ ↪\n",
"⎢⎢ dr ⎥ ⎢────────── 0 0 0 ⎥ ⎡0 0 0 ↪\n",
"⎢⎢ 0 ──────── 0 0⎥ ⎢ 2 ⎥ ⎢ ↪\n",
"⎢⎢ 2 ⎥ ⎢ ⎥ ⎢0 0 - ↪\n",
"⎢⎢ ⎥ ⎢ d ⎥ ⎢ ↪\n",
"⎢⎢d ⎥ ⎢ -──(B(r)) ⎥ ⎢0 -r 0 ↪\n",
"⎢⎢──(A(r)) ⎥ ⎢ dr ⎥ ⎢ ↪\n",
"⎢⎢dr ⎥ ⎢ 0 ────────── 0 0 ⎥ ⎢ ↪\n",
"⎢⎢──────── 0 0 0⎥ ⎢ 2 ⎥ ⎢ ↪\n",
"⎢⎢ 2 ⎥ ⎢ ⎥ ⎢0 0 0 ↪\n",
"⎢⎢ ⎥ ⎢ 0 0 r 0 ⎥ ⎣ ↪\n",
"⎢⎢ 0 0 0 0⎥ ⎢ ⎥ ↪\n",
"⎢⎢ ⎥ ⎢ 2 ⎥ ↪\n",
"⎣⎣ 0 0 0 0⎦ ⎣ 0 0 0 r⋅sin (θ)⎦ ↪\n",
"\n",
"↪ ⎤\n",
"↪ ⎡0 0 0 0 ⎤⎥\n",
"↪ ⎢ ⎥⎥\n",
"↪ 0 ⎤ ⎢ 2 ⎥⎥\n",
"↪ ⎥ ⎢0 0 0 -r⋅sin (θ) ⎥⎥\n",
"↪ r 0 ⎥ ⎢ ⎥⎥\n",
"↪ ⎥ ⎢ 2 ⎥⎥\n",
"↪ 0 ⎥ ⎢ -r ⋅sin(2⋅θ) ⎥⎥\n",
"↪ ⎥ ⎢0 0 0 ─────────────⎥⎥\n",
"↪ 2 ⎥ ⎢ 2 ⎥⎥\n",
"↪ r ⋅sin(2⋅θ)⎥ ⎢ ⎥⎥\n",
"↪ ───────────⎥ ⎢ 2 ⎥⎥\n",
"↪ 2 ⎦ ⎢ 2 -r ⋅sin(2⋅θ) ⎥⎥\n",
"↪ ⎢0 -r⋅sin (θ) ───────────── 0 ⎥⎥\n",
"↪ ⎣ 2 ⎦⎥\n",
"↪ ⎦"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"chr_lll = chr.change_config('lll') \n",
"print(chr_lll.config)\n",
"chr_lll.tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Speziell die $ \\Gamma_{1 1 1} = \\Gamma_{r r r}$ Komponente: "
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle - \\frac{\\frac{d}{d r} B{\\left(r \\right)}}{2}$"
],
"text/plain": [
" d \n",
"-──(B(r)) \n",
" dr \n",
"──────────\n",
" 2 "
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"chr_lll.tensor()[1,1,1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Der Riemann und Ricci Tensor"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Der Riemann Tensor:\n",
"$$ R^{\\mu}{}\\!_{\\nu \\rho \\sigma} = \\frac{\\partial \\Gamma^{\\mu}_{\\nu \\sigma}}{\\partial x^{\\rho}} - \\frac{\\partial \\Gamma^{\\mu}_{\\nu \\rho}}{\\partial x^{\\sigma}} + \\Gamma^{\\alpha}_{\\nu \\sigma}\\Gamma^{\\mu}_{\\rho \\alpha} - \\Gamma^{\\alpha}_{\\nu \\rho}\\Gamma^{\\mu}_{\\sigma \\alpha} $$\n",
"\n",
"Hier speziell $$ R^{0}{}\\!_{2 0 2} = R^{t}{}\\!_{\\theta t \\theta} $$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ulll\n"
]
},
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle - \\frac{r \\frac{d}{d r} A{\\left(r \\right)}}{2 A{\\left(r \\right)} B{\\left(r \\right)}}$"
],
"text/plain": [
" d \n",
"-r⋅──(A(r)) \n",
" dr \n",
"────────────\n",
"2⋅A(r)⋅B(r) "
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rm = ep.RiemannCurvatureTensor.from_christoffels(chr)\n",
"print(rm.config)\n",
"rm[0,2,0,2]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Der Riemann Tensor (kovariante Form):\n",
"$$ R_{\\mu \\nu \\rho \\sigma} = \\frac{\\partial \\Gamma_{\\mu \\nu \\sigma}}{\\partial x^{\\rho}} - \\frac{\\partial \\Gamma_{\\mu \\nu \\rho}}{\\partial x^{\\sigma}} + \\Gamma^{\\alpha}_{\\ \\nu \\sigma}\\Gamma_{\\mu \\rho \\alpha} - \\Gamma^{\\alpha}_{\\ \\nu \\rho}\\Gamma_{\\mu \\sigma \\alpha} - \\frac{\\partial g_{\\mu \\alpha}}{\\partial x^{\\rho}}\\Gamma^{\\alpha}_{\\ \\nu \\sigma} + \\frac{\\partial g_{\\mu \\alpha}}{\\partial x^{\\sigma}}\\Gamma^{\\alpha}_{\\ \\nu \\rho} $$\n",
"\n",
"Hier speziell $$ R_{1 3 1 3} = R_{t \\theta t \\theta} $$"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle - \\frac{r \\frac{d}{d r} A{\\left(r \\right)}}{2 B{\\left(r \\right)}}$"
],
"text/plain": [
" d \n",
"-r⋅──(A(r)) \n",
" dr \n",
"────────────\n",
" 2⋅B(r) "
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rm_llll = rm.change_config('llll')\n",
"rm_llll[0,2,0,2]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Der Ricci Tensor:\n",
"\n",
"$$ R_{\\mu \\nu} = R^{\\sigma}{}\\!_{\\mu \\sigma \\nu} = \\frac{\\partial \\Gamma^{\\sigma}_{\\mu \\nu}}{\\partial x^{\\sigma}} - \\frac{\\partial \\Gamma^{\\sigma}_{\\mu \\sigma}}{\\partial x^{\\nu}} + \\Gamma^{\\rho}_{\\mu \\nu}\\Gamma^{\\sigma}_{\\rho \\sigma} - \\Gamma^{\\sigma}_{\\mu \\rho}\\Gamma^{\\rho}_{\\nu \\sigma} $$"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ll\n"
]
},
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{\\frac{d^{2}}{d r^{2}} A{\\left(r \\right)}}{2 B{\\left(r \\right)}} - \\frac{\\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{4 B^{2}{\\left(r \\right)}} - \\frac{\\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2}}{4 A{\\left(r \\right)} B{\\left(r \\right)}} + \\frac{\\frac{d}{d r} A{\\left(r \\right)}}{r B{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & - \\frac{\\frac{d^{2}}{d r^{2}} A{\\left(r \\right)}}{2 A{\\left(r \\right)}} + \\frac{\\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{4 A{\\left(r \\right)} B{\\left(r \\right)}} + \\frac{\\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2}}{4 A^{2}{\\left(r \\right)}} + \\frac{\\frac{d}{d r} B{\\left(r \\right)}}{r B{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & \\frac{r \\frac{d}{d r} B{\\left(r \\right)}}{2 B^{2}{\\left(r \\right)}} - \\frac{r \\frac{d}{d r} A{\\left(r \\right)}}{2 A{\\left(r \\right)} B{\\left(r \\right)}} + 1 - \\frac{1}{B{\\left(r \\right)}} & 0\\\\0 & 0 & 0 & \\frac{\\left(r A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)} - r B{\\left(r \\right)} \\frac{d}{d r} A{\\left(r \\right)} + 2 \\left(B{\\left(r \\right)} - 1\\right) A{\\left(r \\right)} B{\\left(r \\right)}\\right) \\sin^{2}{\\left(\\theta \\right)}}{2 A{\\left(r \\right)} B^{2}{\\left(r \\right)}}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ 2 ↪\n",
"⎢d 2 ↪\n",
"⎢───(A(r)) d d ⎛d ⎞ d ↪\n",
"⎢ 2 ──(A(r))⋅──(B(r)) ⎜──(A(r))⎟ ──(A(r)) ↪\n",
"⎢dr dr dr ⎝dr ⎠ dr ↪\n",
"⎢───────── - ───────────────── - ─────────── + ──────── ↪\n",
"⎢ 2⋅B(r) 2 4⋅A(r)⋅B(r) r⋅B(r) ↪\n",
"⎢ 4⋅B (r) ↪\n",
"⎢ ↪\n",
"⎢ 2 ↪\n",
"⎢ d ↪\n",
"⎢ ───(A(r)) d ↪\n",
"⎢ 2 ──(A(r) ↪\n",
"⎢ dr dr ↪\n",
"⎢ 0 - ───────── + ─────── ↪\n",
"⎢ 2⋅A(r) 4⋅A( ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ 0 ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ 0 ↪\n",
"⎢ ↪\n",
"⎣ ↪\n",
"\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 2 ↪\n",
"↪ d ⎛d ⎞ d ↪\n",
"↪ )⋅──(B(r)) ⎜──(A(r))⎟ ──(B(r)) ↪\n",
"↪ dr ⎝dr ⎠ dr ↪\n",
"↪ ────────── + ─────────── + ──────── 0 ↪\n",
"↪ r)⋅B(r) 2 r⋅B(r) ↪\n",
"↪ 4⋅A (r) ↪\n",
"↪ ↪\n",
"↪ d d ↪\n",
"↪ r⋅──(B(r)) r⋅──(A(r)) ↪\n",
"↪ dr dr 1 ↪\n",
"↪ 0 ────────── - ─────────── + 1 - ──── ↪\n",
"↪ 2 2⋅A(r)⋅B(r) B(r) ↪\n",
"↪ 2⋅B (r) ↪\n",
"↪ ↪\n",
"↪ ⎛ ↪\n",
"↪ ⎜r ↪\n",
"↪ ⎝ ↪\n",
"↪ 0 0 ── ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"\n",
"↪ ⎤\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 0 ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 0 ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 0 ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ d d ⎞ 2 ⎥\n",
"↪ ⋅A(r)⋅──(B(r)) - r⋅B(r)⋅──(A(r)) + 2⋅(B(r) - 1)⋅A(r)⋅B(r)⎟⋅sin (θ)⎥\n",
"↪ dr dr ⎠ ⎥\n",
"↪ ──────────────────────────────────────────────────────────────────⎥\n",
"↪ 2 ⎥\n",
"↪ 2⋅A(r)⋅B (r) ⎦"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ric = ep.RicciTensor.from_metric(g)\n",
"print(ric.config)\n",
"ric.tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Überprüfung durch Kontraktion des Riemann Tensors:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{\\frac{d^{2}}{d r^{2}} A{\\left(r \\right)}}{2 B{\\left(r \\right)}} - \\frac{\\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{4 B^{2}{\\left(r \\right)}} - \\frac{\\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2}}{4 A{\\left(r \\right)} B{\\left(r \\right)}} + \\frac{\\frac{d}{d r} A{\\left(r \\right)}}{r B{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & - \\frac{\\frac{d^{2}}{d r^{2}} A{\\left(r \\right)}}{2 A{\\left(r \\right)}} + \\frac{\\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{4 A{\\left(r \\right)} B{\\left(r \\right)}} + \\frac{\\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2}}{4 A^{2}{\\left(r \\right)}} + \\frac{\\frac{d}{d r} B{\\left(r \\right)}}{r B{\\left(r \\right)}} & 0 & 0\\\\0 & 0 & \\frac{r \\frac{d}{d r} B{\\left(r \\right)}}{2 B^{2}{\\left(r \\right)}} - \\frac{r \\frac{d}{d r} A{\\left(r \\right)}}{2 A{\\left(r \\right)} B{\\left(r \\right)}} + 1 - \\frac{1}{B{\\left(r \\right)}} & 0\\\\0 & 0 & 0 & \\frac{r \\sin^{2}{\\left(\\theta \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{2 B^{2}{\\left(r \\right)}} - \\frac{r \\sin^{2}{\\left(\\theta \\right)} \\frac{d}{d r} A{\\left(r \\right)}}{2 A{\\left(r \\right)} B{\\left(r \\right)}} + \\frac{\\left(B{\\left(r \\right)} - 1\\right) \\sin^{2}{\\left(\\theta \\right)}}{B{\\left(r \\right)}}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ 2 ↪\n",
"⎢d 2 ↪\n",
"⎢───(A(r)) d d ⎛d ⎞ d ↪\n",
"⎢ 2 ──(A(r))⋅──(B(r)) ⎜──(A(r))⎟ ──(A(r)) ↪\n",
"⎢dr dr dr ⎝dr ⎠ dr ↪\n",
"⎢───────── - ───────────────── - ─────────── + ──────── ↪\n",
"⎢ 2⋅B(r) 2 4⋅A(r)⋅B(r) r⋅B(r) ↪\n",
"⎢ 4⋅B (r) ↪\n",
"⎢ ↪\n",
"⎢ 2 ↪\n",
"⎢ d ↪\n",
"⎢ ───(A(r)) d ↪\n",
"⎢ 2 ──(A(r) ↪\n",
"⎢ dr dr ↪\n",
"⎢ 0 - ───────── + ─────── ↪\n",
"⎢ 2⋅A(r) 4⋅A( ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ 0 ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ 0 ↪\n",
"⎢ ↪\n",
"⎣ ↪\n",
"\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 2 ↪\n",
"↪ d ⎛d ⎞ d ↪\n",
"↪ )⋅──(B(r)) ⎜──(A(r))⎟ ──(B(r)) ↪\n",
"↪ dr ⎝dr ⎠ dr ↪\n",
"↪ ────────── + ─────────── + ──────── 0 ↪\n",
"↪ r)⋅B(r) 2 r⋅B(r) ↪\n",
"↪ 4⋅A (r) ↪\n",
"↪ ↪\n",
"↪ d d ↪\n",
"↪ r⋅──(B(r)) r⋅──(A(r)) ↪\n",
"↪ dr dr 1 ↪\n",
"↪ 0 ────────── - ─────────── + 1 - ──── ↪\n",
"↪ 2 2⋅A(r)⋅B(r) B(r) ↪\n",
"↪ 2⋅B (r) ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ r⋅ ↪\n",
"↪ ↪\n",
"↪ 0 0 ── ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"\n",
"↪ ⎤\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 0 ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 0 ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 0 ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ 2 d 2 d ⎥\n",
"↪ sin (θ)⋅──(B(r)) r⋅sin (θ)⋅──(A(r)) 2 ⎥\n",
"↪ dr dr (B(r) - 1)⋅sin (θ)⎥\n",
"↪ ──────────────── - ────────────────── + ──────────────────⎥\n",
"↪ 2 2⋅A(r)⋅B(r) B(r) ⎥\n",
"↪ 2⋅B (r) ⎦"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.tensorcontraction(rm.tensor(), (0, 2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Der Ricci Skalar ergibt sich aus der Kontraktion des Ricci Tensors: $R = R^{\\mu}{}_{\\!\\mu} = g^{\\mu \\nu}R_{\\nu \\mu}$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{\\frac{d^{2}}{d r^{2}} A{\\left(r \\right)}}{A{\\left(r \\right)} B{\\left(r \\right)}} - \\frac{\\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{2 A{\\left(r \\right)} B^{2}{\\left(r \\right)}} - \\frac{\\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2}}{2 A^{2}{\\left(r \\right)} B{\\left(r \\right)}} - \\frac{2 \\frac{d}{d r} B{\\left(r \\right)}}{r B^{2}{\\left(r \\right)}} + \\frac{2 \\frac{d}{d r} A{\\left(r \\right)}}{r A{\\left(r \\right)} B{\\left(r \\right)}} - \\frac{2}{r^{2}} + \\frac{2}{r^{2} B{\\left(r \\right)}}$"
],
"text/plain": [
" 2 ↪\n",
"d 2 ↪\n",
"───(A(r)) d d ⎛d ⎞ d d ↪\n",
" 2 ──(A(r))⋅──(B(r)) ⎜──(A(r))⎟ 2⋅──(B(r)) 2⋅──(A(r)) ↪\n",
"dr dr dr ⎝dr ⎠ dr dr 2 ↪\n",
"───────── - ───────────────── - ──────────── - ────────── + ─────────── - ── + ↪\n",
"A(r)⋅B(r) 2 2 2 r⋅A(r)⋅B(r) 2 ↪\n",
" 2⋅A(r)⋅B (r) 2⋅A (r)⋅B(r) r⋅B (r) r ↪\n",
"\n",
"↪ \n",
"↪ \n",
"↪ \n",
"↪ \n",
"↪ 2 \n",
"↪ ───────\n",
"↪ 2 \n",
"↪ r ⋅B(r)"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"R = ep.RicciScalar.from_riccitensor(ric)\n",
"R.tensor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Überprüfung durch Kontraktion des Ricci Tensors:"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{\\frac{d^{2}}{d r^{2}} A{\\left(r \\right)}}{A{\\left(r \\right)} B{\\left(r \\right)}} - \\frac{\\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)}}{2 A{\\left(r \\right)} B^{2}{\\left(r \\right)}} - \\frac{\\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2}}{2 A^{2}{\\left(r \\right)} B{\\left(r \\right)}} - \\frac{2 \\frac{d}{d r} B{\\left(r \\right)}}{r B^{2}{\\left(r \\right)}} + \\frac{2 \\frac{d}{d r} A{\\left(r \\right)}}{r A{\\left(r \\right)} B{\\left(r \\right)}} - \\frac{2}{r^{2}} + \\frac{2}{r^{2} B{\\left(r \\right)}}$"
],
"text/plain": [
" 2 ↪\n",
"d 2 ↪\n",
"───(A(r)) d d ⎛d ⎞ d d ↪\n",
" 2 ──(A(r))⋅──(B(r)) ⎜──(A(r))⎟ 2⋅──(B(r)) 2⋅──(A(r)) ↪\n",
"dr dr dr ⎝dr ⎠ dr dr 2 ↪\n",
"───────── - ───────────────── - ──────────── - ────────── + ─────────── - ── + ↪\n",
"A(r)⋅B(r) 2 2 2 r⋅A(r)⋅B(r) 2 ↪\n",
" 2⋅A(r)⋅B (r) 2⋅A (r)⋅B(r) r⋅B (r) r ↪\n",
"\n",
"↪ \n",
"↪ \n",
"↪ \n",
"↪ \n",
"↪ 2 \n",
"↪ ───────\n",
"↪ 2 \n",
"↪ r ⋅B(r)"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ric_ul = ric.change_config('ul')\n",
"sp.tensorcontraction(ric_ul.tensor(), (0, 1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Der Einstein Tensor"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Der Einstein Tensor:$$ G_{\\mu \\nu} = R_{\\mu \\nu} - \\frac{1}{2}g_{\\mu \\nu}R $$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ll\n"
]
},
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{1.0 \\left(r \\frac{d}{d r} B{\\left(r \\right)} + B^{2}{\\left(r \\right)} - B{\\left(r \\right)}\\right) A{\\left(r \\right)}}{r^{2} B^{2}{\\left(r \\right)}} & 0 & 0 & 0\\\\0 & \\frac{1.0 \\frac{d}{d r} A{\\left(r \\right)}}{r A{\\left(r \\right)}} - \\frac{1.0 B{\\left(r \\right)}}{r^{2}} + \\frac{1.0}{r^{2}} & 0 & 0\\\\0 & 0 & \\frac{0.5 r \\left(1.0 r A{\\left(r \\right)} B{\\left(r \\right)} \\frac{d^{2}}{d r^{2}} A{\\left(r \\right)} - 0.5 r A{\\left(r \\right)} \\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)} - 0.5 r B{\\left(r \\right)} \\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2} - 1.0 A^{2}{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)} + 1.0 A{\\left(r \\right)} B{\\left(r \\right)} \\frac{d}{d r} A{\\left(r \\right)}\\right)}{A^{2}{\\left(r \\right)} B^{2}{\\left(r \\right)}} & 0\\\\0 & 0 & 0 & \\frac{0.5 r \\left(1.0 r A{\\left(r \\right)} B{\\left(r \\right)} \\frac{d^{2}}{d r^{2}} A{\\left(r \\right)} - 0.5 r A{\\left(r \\right)} \\frac{d}{d r} A{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)} - 0.5 r B{\\left(r \\right)} \\left(\\frac{d}{d r} A{\\left(r \\right)}\\right)^{2} - 1.0 A^{2}{\\left(r \\right)} \\frac{d}{d r} B{\\left(r \\right)} + 1.0 A{\\left(r \\right)} B{\\left(r \\right)} \\frac{d}{d r} A{\\left(r \\right)}\\right) \\sin^{2}{\\left(\\theta \\right)}}{A^{2}{\\left(r \\right)} B^{2}{\\left(r \\right)}}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡ ⎛ d 2 ⎞ ↪\n",
"⎢1.0⋅⎜r⋅──(B(r)) + B (r) - B(r)⎟⋅A(r) ↪\n",
"⎢ ⎝ dr ⎠ ↪\n",
"⎢──────────────────────────────────── 0 ↪\n",
"⎢ 2 2 ↪\n",
"⎢ r ⋅B (r) ↪\n",
"⎢ ↪\n",
"⎢ d ↪\n",
"⎢ 1.0⋅──(A(r)) ↪\n",
"⎢ dr 1.0⋅B(r) 1.0 ↪\n",
"⎢ 0 ──────────── - ──────── + ─── ↪\n",
"⎢ r⋅A(r) 2 2 ↪\n",
"⎢ r r ↪\n",
"⎢ ↪\n",
"⎢ ⎛ ↪\n",
"⎢ ⎜ ↪\n",
"⎢ 0.5⋅r⋅⎜1 ↪\n",
"⎢ ⎜ ↪\n",
"⎢ ⎝ ↪\n",
"⎢ 0 0 ──────── ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ ↪\n",
"⎢ 0 0 ↪\n",
"⎢ ↪\n",
"⎣ ↪\n",
"\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 2 ↪\n",
"↪ d d d ⎛d ↪\n",
"↪ .0⋅r⋅A(r)⋅B(r)⋅───(A(r)) - 0.5⋅r⋅A(r)⋅──(A(r))⋅──(B(r)) - 0.5⋅r⋅B(r)⋅⎜──(A(r ↪\n",
"↪ 2 dr dr ⎝dr ↪\n",
"↪ dr ↪\n",
"↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
"↪ 2 2 ↪\n",
"↪ A (r)⋅B (r) ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 2 ⎞ ↪\n",
"↪ ⎞ 2 d d ⎟ ↪\n",
"↪ ))⎟ - 1.0⋅A (r)⋅──(B(r)) + 1.0⋅A(r)⋅B(r)⋅──(A(r))⎟ ↪\n",
"↪ ⎠ dr dr ⎟ ↪\n",
"↪ ⎠ ↪\n",
"↪ ─────────────────────────────────────────────────── ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ⎛ ↪\n",
"↪ ⎜ ↪\n",
"↪ 0.5⋅r⋅⎜1.0⋅r⋅A(r)⋅B(r)⋅ ↪\n",
"↪ ⎜ ↪\n",
"↪ ⎝ ↪\n",
"↪ ─────────────────────── ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 0 ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ ↪\n",
"↪ 2 2 ↪\n",
"↪ d d d ⎛d ⎞ 2 ↪\n",
"↪ ───(A(r)) - 0.5⋅r⋅A(r)⋅──(A(r))⋅──(B(r)) - 0.5⋅r⋅B(r)⋅⎜──(A(r))⎟ - 1.0⋅A (r ↪\n",
"↪ 2 dr dr ⎝dr ⎠ ↪\n",
"↪ dr ↪\n",
"↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
"↪ 2 2 ↪\n",
"↪ A (r)⋅B (r) ↪\n",
"\n",
"↪ ⎤\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎥\n",
"↪ ⎞ ⎥\n",
"↪ d d ⎟ 2 ⎥\n",
"↪ )⋅──(B(r)) + 1.0⋅A(r)⋅B(r)⋅──(A(r))⎟⋅sin (θ)⎥\n",
"↪ dr dr ⎟ ⎥\n",
"↪ ⎠ ⎥\n",
"↪ ────────────────────────────────────────────⎥\n",
"↪ ⎥\n",
"↪ ⎦"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"einst = ep.EinsteinTensor.from_metric(g)\n",
"print(einst.config)\n",
"einst.simplify()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Der Einsteintensor $G_{\\mu \\nu}$ ist der entscheidende Tensor der allgemeinen Relativitätstheorie. Die Einsteingleichung (in natürlichen Einheiten c=G=1) lautet\n",
"\n",
"$$\n",
"R_{\\mu\\nu} - \\frac{1}{2} g_{\\mu\\nu} R ~=~ -8 \\pi \\, T_{\\mu\\nu} \\quad,\n",
"$$\n",
"wobei $T_{\\mu\\nu}$ der Energie-Impuls-Tensor der Materie ist. A. Einstein stellte im Jahre 1915 seine revolutionäre allgemeine Relativitätstheorie der wissenschaftlichen Öffentlichkeit vor (siehe Einstein, A. 1915, Preuss. Akad.Wiss. Berlin, Sitzungsber., 778-786. und Einstein, A. 1915, Preuss. Akad.Wiss. Berlin, Sitzungsber., 844-847.). Seine Feldgleichungen leitete er mittels eines allgemeinen Kovarianzprinzips her und über seine neue Theorie schrieb er selbst \"Dem Zauber der Theorie wird sich kaum jemand entziehen können, der sie wirklich erfasst hat;...\".\n",
"\n",
"Die Schönheit der Einstein Gleichung liegt neben ihrem zugrunde liegenden Kovarianzprinzip in der Einfachheit ihrer fundamentalen Aussage. Nach Einstein verbiegt jede Energieansammlung (Energie-Impuls-Tensor, rechte Seite der Gleichung) die Struktur der Raumzeit (Einstein Tensor, linke Seite der Gleichung) und diese gekrümmte Raumzeit ist der ursächliche Grund der Gravitation. Nach Einstein fällt der Apfel vom Baum zu Boden, da der große Energiegehalt der Erde die raumzeitliche Struktur so stark verbiegt, dass der Apfel sich in dieser gekrümmten Raumzeit nach geodätischen Gesetzen zu Boden bewegen muss."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "venv",
"language": "python",
"name": "venv"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.12.3"
}
},
"nbformat": 4,
"nbformat_minor": 4
}