Skip to content

Commit

Permalink
Updated
Browse files Browse the repository at this point in the history
  • Loading branch information
@phoenfrisur
phoenfrisur committed Oct 4, 2024
1 parent 0f50f43 commit e4c16ab
Show file tree
Hide file tree
Showing 26 changed files with 368 additions and 5,166 deletions.
30 changes: 2 additions & 28 deletions README.md
View file
Original file line number Diff line number Diff line change
@@ -1,29 +1,3 @@
# JupyterLite Demo
# Zentrifugalkräfte durch Einsteins Linse

[![lite-badge](https://jupyterlite.rtfd.io/en/latest/_static/badge.svg)](https://jupyterlite.github.io/demo)

JupyterLite deployed as a static site to GitHub Pages, for demo purposes.

## ✨ Try it in your browser ✨

➡️ **https://jupyterlite.github.io/demo**

![github-pages](https://user-images.githubusercontent.com/591645/120649478-18258400-c47d-11eb-80e5-185e52ff2702.gif)

## Requirements

JupyterLite is being tested against modern web browsers:

- Firefox 90+
- Chromium 89+

## Deploy your JupyterLite website on GitHub Pages

Check out the guide on the JupyterLite documentation: https://jupyterlite.readthedocs.io/en/latest/quickstart/deploy.html

## Further Information and Updates

For more info, keep an eye on the JupyterLite documentation:

- How-to Guides: https://jupyterlite.readthedocs.io/en/latest/howto/index.html
- Reference: https://jupyterlite.readthedocs.io/en/latest/reference/index.html
In diesem Jupyter Notebook werden die Rechnungen aus dem Phoebe-Artikel "Zentrifugalkräfte durch Einsteins Linse" mithilfe der Bibliothek `EinsteinPy` (https://einsteinpy.org) nachgestellt. Die Rechnung gliedert sich in zwei Teile: Zu Beginn betrachten wir eine flache Raumzeit, die wir in Polarkoordinaten $(r,\\theta)$ ausdrücken $$\\mathrm{d}s^2 = -\\mathrm{d}t^2 + \\mathrm{d}r^2 + r^2\\mathrm{d}\\theta^2.$$ Im zweiten Teil schauen wir uns genauer an welche Form die Christoffel-Symbole in einem rotierenden Bezugssystem annehmen werden. Wie sich zeigen wird, lassen sich daraus Zentrifugal-, Coriolis- und Eulerkraft ableiten.
366 changes: 366 additions & 0 deletions content/centrifugal.ipynb
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,366 @@
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "78f323e5-01a6-451f-a0da-76844f34c463",
"metadata": {},
"outputs": [],
"source": [
"import sympy\n",
"from einsteinpy.symbolic import MetricTensor, ChristoffelSymbols"
]
},
{
"cell_type": "markdown",
"id": "2481b42d-c582-4bde-a058-8d5199812a78",
"metadata": {},
"source": [
"# Zentrifugalkräfte aus Sicht der Relativitätstheorie"
]
},
{
"cell_type": "markdown",
"id": "a68ba0f0-e68c-43a1-b939-ec136d8acb57",
"metadata": {},
"source": [
"In diesem Jupyter Notebook werden die Rechnungen aus dem Artikel mithilfe der Bibliothek `EinsteinPy` (https://einsteinpy.org) nachgestellt. Die Rechnung gliedert sich in zwei Teile: Zu Beginn betrachten wir eine flache Raumzeit, die wir in Polarkoordinaten $(r,\\theta)$ ausdrücken $$\\mathrm{d}s^2 = -\\mathrm{d}t^2 + \\mathrm{d}r^2 + r^2\\mathrm{d}\\theta^2.$$ Im zweiten Teil schauen wir uns genauer an welche Form die Christoffel-Symbole in einem rotierenden Bezugssystem annehmen werden. Wie sich zeigen wird, lassen sich daraus Zentrifugal-, Coriolis- und Eulerkraft ableiten."
]
},
{
"cell_type": "markdown",
"id": "8a8a1001-3706-4a90-878b-a1d356789be2",
"metadata": {},
"source": [
"## Flache Raumzeitmetrik in Polarkoordinaten"
]
},
{
"cell_type": "markdown",
"id": "78ff30e0-9b27-4933-b7a4-63c0765cbcee",
"metadata": {},
"source": [
"Wir beginnen zunächst damit die Koordinaten $(t,r,\\theta)$ zu definieren"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "174d89f1-d27d-4b0c-911b-5ffdf12a1f74",
"metadata": {},
"outputs": [],
"source": [
"syms = sympy.symbols('t r theta')\n",
"t, r, theta = syms"
]
},
{
"cell_type": "markdown",
"id": "1b454087-9f62-4511-81f3-0ce589c2cded",
"metadata": {},
"source": [
"und bauen daraus die Metrik zusammen"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "97afbb27-10b8-497e-b403-2fc3780a3267",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & r^{2}\\end{matrix}\\right]$"
],
"text/plain": [
"[[-1, 0, 0], [0, 1, 0], [0, 0, r**2]]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m = [[0 for i in range(3)] for i in range(3)]\n",
"m[0][0] = - 1\n",
"m[1][1] = 1\n",
"m[2][2] = r**2\n",
"metric = MetricTensor(m, syms)\n",
"metric.tensor()"
]
},
{
"cell_type": "markdown",
"id": "88b49809-c7a1-4d7f-897d-d143e515ce68",
"metadata": {},
"source": [
"Die Christoffel-Symbole lassen sich schnell berechnen:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "fc8ac94e-542f-41b4-8c40-7dbd1c60b7ca",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & - r\\end{matrix}\\right] & \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & \\frac{1}{r}\\\\0 & \\frac{1}{r} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
],
"text/plain": [
"[[[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, -r]], [[0, 0, 0], [0, 0, 1/r], [0, 1/r, 0]]]"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ch = ChristoffelSymbols.from_metric(metric)\n",
"sympy.simplify(ch.tensor())"
]
},
{
"cell_type": "markdown",
"id": "83a91eb2-fd9f-4b7b-9e65-3297815ac966",
"metadata": {},
"source": [
"Wie man sieht, sind die Christoffel-Symbole in den unteren beiden Indizes symmetrisch:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "3b919d1d-ed7b-4ca4-a2f7-ec8687562bfd",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{1}{r}$"
],
"text/plain": [
"1/r"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ch[2][1][2]"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "35ca5834-2f08-45ec-beaf-efe5853a2ec8",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{1}{r}$"
],
"text/plain": [
"1/r"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ch[2][2][1]"
]
},
{
"cell_type": "markdown",
"id": "0e34cc44-e7da-41f5-a36f-25bbd8e1c70e",
"metadata": {},
"source": [
"Es ergeben sich also nur zwei nicht-verschwindende Christoffel-Symbole: $$\\Gamma^r_{\\theta\\theta} = -r,\\quad \\Gamma^\\theta_{r\\theta} = \\Gamma^\\theta_{\\theta r} = \\frac{1}{r}$$"
]
},
{
"cell_type": "markdown",
"id": "968ef45f-6464-4f03-9c65-05fc9f4c8c5c",
"metadata": {},
"source": [
"## Rotierendes Bezugssystem"
]
},
{
"cell_type": "markdown",
"id": "7a68578b-5199-4151-b5b9-448c4bc832f9",
"metadata": {},
"source": [
"Wie oben beschrieben, schauen wir uns nun genauer an, was passiert, wenn das Koordinatensystem gegenüber dem ersten zu rotieren anfängt. Diese Koordinatentransformation $\\theta \\longrightarrow \\theta + \\omega(t)\\cdot t$ beeinflusst Form die Metrik, es taucht eine zeitabhängige Winkelgeschwindigkeit $\\omega(t)$ auf:\n",
"$$\\mathrm{d}s^2 = - (1 -\\omega^2r^2)\\mathrm{d}t^2 + \\mathrm{d}r^2 + r^2\\mathrm{d}\\theta^2 + 2\\omega r^2 \\mathrm{d}t\\mathrm{d}\\theta$$"
]
},
{
"cell_type": "markdown",
"id": "00d45632-9ae0-4bef-8ebd-ee675a8f8f3a",
"metadata": {},
"source": [
"Wir definieren zunächst wieder die Koordinaten:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "990dab4d-5ac5-4c1f-9a0a-356717cfa234",
"metadata": {},
"outputs": [],
"source": [
"syms_ = sympy.symbols('t r theta')\n",
"t, r, theta = syms_"
]
},
{
"cell_type": "markdown",
"id": "ba8d4795-f417-4b52-a9fb-3dddbf43da0a",
"metadata": {},
"source": [
"Die Winkelgeschwindigkeit wird als Funktion erklärt:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "bdcf53f7-4b9c-4141-955c-4845aed1eaca",
"metadata": {},
"outputs": [],
"source": [
"omega_t = sympy.Function('omega')(t)"
]
},
{
"cell_type": "markdown",
"id": "46c5e30b-b2cb-4d36-98f4-36a0d73e48fe",
"metadata": {},
"source": [
"Definiere nun die Metrik, die diesmal zeitabhängig sein wird:"
]
},
{
"cell_type": "markdown",
"id": "1567a3d2-1f9b-4679-bf52-8f4ae9a03bee",
"metadata": {},
"source": [
"Daraus bauen wir die Metrik zusammen:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "8e3649f5-ded8-4f20-ab13-ddcd9e8aae92",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}r^{2} \\omega^{2}{\\left(t \\right)} - 1 & 0 & r^{2} \\omega{\\left(t \\right)}\\\\0 & 1 & 0\\\\r^{2} \\omega{\\left(t \\right)} & 0 & r^{2}\\end{matrix}\\right]$"
],
"text/plain": [
"[[r**2*omega(t)**2 - 1, 0, r**2*omega(t)], [0, 1, 0], [r**2*omega(t), 0, r**2]]"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m_t = [[0 for i in range(3)] for i in range(3)]\n",
"m_t[0][0] = - (1 - omega_t**2 * r**2)\n",
"m_t[1][1] = 1\n",
"m_t[0][2] = omega_t * r**2\n",
"m_t[2][0] = omega_t * r**2\n",
"m_t[2][2] = r**2\n",
"metric_t = MetricTensor(m_t, syms_)\n",
"metric_t.tensor()"
]
},
{
"cell_type": "markdown",
"id": "bbc56c63-ec62-4198-98a5-011a75838fcb",
"metadata": {},
"source": [
"Wie oben lassen sich die Christoffel-Symbole bestimmen:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "1b1f317b-5976-4419-a305-38197ae48d2a",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right] & \\left[\\begin{matrix}- r \\omega^{2}{\\left(t \\right)} & 0 & - r \\omega{\\left(t \\right)}\\\\0 & 0 & 0\\\\- r \\omega{\\left(t \\right)} & 0 & - r\\end{matrix}\\right] & \\left[\\begin{matrix}\\frac{d}{d t} \\omega{\\left(t \\right)} & \\frac{\\omega{\\left(t \\right)}}{r} & 0\\\\\\frac{\\omega{\\left(t \\right)}}{r} & 0 & \\frac{1}{r}\\\\0 & \\frac{1}{r} & 0\\end{matrix}\\right]\\end{matrix}\\right]$"
],
"text/plain": [
"[[[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[-r*omega(t)**2, 0, -r*omega(t)], [0, 0, 0], [-r*omega(t), 0, -r]], [[Derivative(omega(t), t), omega(t)/r, 0], [omega(t)/r, 0, 1/r], [0, 1/r, 0]]]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ch_t = ChristoffelSymbols.from_metric(metric_t)\n",
"sympy.simplify(ch_t.tensor())"
]
},
{
"cell_type": "markdown",
"id": "316b9de5-6e62-4dcf-b740-98de405684cf",
"metadata": {},
"source": [
"Wie man sieht, kommen nun vier weitere Christoffel-Symbole hinzu:"
]
},
{
"cell_type": "markdown",
"id": "971c5c63-58b3-47e2-a3b3-c6f4c3145bce",
"metadata": {},
"source": [
"$$\\begin{align*}\n",
"\\Gamma^r_{tt} &= -\\omega^2r, & \\Gamma^r_{t\\theta} &= -\\omega r, & \\Gamma^r_{\\theta\\theta} &= -r \\\\\n",
"\\Gamma^\\theta_{tt} &= \\dot{\\omega}, & \\Gamma^\\theta_{tr} &=\\frac{\\omega}{r}, & \\Gamma^\\theta_{r\\theta} &= \\frac{1}{r}.\n",
"\\end{align*}$$"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.12.0"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
1 change: 0 additions & 1 deletion content/data/Museums_in_DC.geojson
View file

This file was deleted.

Loading

0 comments on commit e4c16ab

Please sign in to comment.