From 015ca916ccc9f7c4501d858f20ad4843a5a8a156 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 9 Feb 2021 18:29:07 +0100 Subject: [PATCH] add lecture notes --- ws2020/theo/uebungen/vl16.xopp | Bin 495265 -> 526037 bytes ws2020/theo/uebungen/vl18.xopp | Bin 0 -> 211297 bytes ws2020/theo/uebungen/vl19.pdf | Bin 0 -> 105269 bytes ws2020/theo/uebungen/vl19.tex | 85 +++++++++++++++++++++++++++++++++ ws2020/theo/uebungen/vl19.xopp | Bin 0 -> 456632 bytes ws2020/theo/uebungen/vl20.xopp | Bin 0 -> 351124 bytes ws2020/theo/uebungen/vl21.xopp | Bin 0 -> 304794 bytes ws2020/tut-ana/praesenz10.xopp | Bin 0 -> 1085869 bytes ws2020/tut-ana/praesenz11.xopp | Bin 0 -> 1439925 bytes ws2020/tut-ana/praesenz12.xopp | Bin 0 -> 709910 bytes ws2020/wtheo/uebungen | 2 +- 11 files changed, 86 insertions(+), 1 deletion(-) create mode 100644 ws2020/theo/uebungen/vl18.xopp create mode 100644 ws2020/theo/uebungen/vl19.pdf create mode 100644 ws2020/theo/uebungen/vl19.tex create mode 100644 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Der Bereich der Raumzeit, wo $s^2 > 0$ gilt, +wird auch \textbf{Lichtkegel} genannt. Innerhalb des Lichtkegels +liegt eine absolute zeitliche Ordnung der Ereignisse vor: Ein Ereignis $B$, das in $S$ nach $A$ stattfindet, +findet auch in $S'$ nach $A$ statt. Außerhalb des Lichtkegels, also für raumartig getrennte Ereignisse +gilt das nicht: Es kann $v$ immer so gewählt werden, dass $B$ unterhalb der $x'$-Achse liegt. Dann +findet $B$ in $S'$ vor $A$ statt. + +Außerdem kann $A$ nur zeitartigen Ereignissen $B$ Lichtsignale senden, da dort das Lichtsignal ankommt, +bevor $B$ stattfindet. +Bei raumartigen Ereignissen $C$ kann jedoch ein Lichtsignal von $A$ nicht ankommen, +da wenn das Lichtsignal die räumlichen Koordinaten von $C$ erreicht hat, $B$ schon passiert ist. + +\section{Relativistische Effekte} + +\begin{itemize} + \item Relativität der Gleichzeitig: Wie bereits im vorangegangenen Abschnitt erwähnt, + ist keine zeitliche Ordnung der Ereignisse außerhalb des Lichtkegels gegeben. + \item Zeitdilatation: Eine in $S'$ ruhende Uhr, scheint von $S$ aus betrachtet langsamer + zu gehen. Genauer: $\Delta t = \gamma \Delta t'$. Dieser Effekt ist symmetrisch und wird + von der Transformation erzeugt. + \item Längenkontraktion: Längenmaßstäbe in einem bewegten System erscheinen verkürzt. Genauer: + $\Delta x = \frac{1}{\gamma} \Delta x'$. Auch dieser Effekt ist symmetrisch und wird nur durch + die Transformation erzeugt. +\end{itemize} + +\section{Eigenzeit} +\thispagestyle{plain} + +Sei eine Trajektorie $x^{\mu}(\tau)$ durch ein Koordinatensystem gegeben. Die Uhr eines Insassen eines +Raumschiffes, das sich auf dieser Trajektorie bewegt, zeigt die sogenannte \textbf{Eigenzeit} $\tau$ des +Systems an. Die Tangente $u^{\mu}(\tau) = \frac{\d{x}^{\mu}}{\d{\tau}}$ gibt die Geschwindigkeit +des Teilchens auf der Trajektorie an. + +Durch Berechnung der Lorentz-Invariante $s^2$ erhalten wir $s^2 = (c\tau)^2$ im Ruhesystem +des Teilchens. Da dieser Zusammenhang auch infinitesimal gilt, folgt insgesamt für +die vom Ereignis $A$ zum Ereignis $B$ vergangene Eigenzeit +\begin{salign*} + \tau = \int_{A}^{B} \d{\tau} = \int_{A}^{B} \frac{\d{t}}{\gamma} +\end{salign*} +für Start- und Endpunkte $A$ und $B$. + +\end{document} diff --git a/ws2020/theo/uebungen/vl19.xopp b/ws2020/theo/uebungen/vl19.xopp new file mode 100644 index 0000000000000000000000000000000000000000..e1bd0f533eb105a27ff646bf01d7fd72593c0b53 GIT binary patch literal 456632 zcmV(#K;*w4iwFP!000001GK!^vg10EE%;tv5f=R@>)-?sU|VKg(>%`;rI$8|gXUSJ zD1Ch|2lt6BF|xAj+O4yXWd`?gi=-`w}Xe|~@1KY7t1INpE!-#>0a@MC?ypKt$Mw^zYoPb@ z+M}f9d%0D-2uAqxDqr50`;!4Xc1?fk7Cgf`tYX8?54NG1Tt%EOmxpe+{xBqa=FCM{&O<8jBj?SJKlG=SDly*cQ)D9qpjJ#deh+hCrI5Bav$d5!vdC!g;Z^TX5C){e~U)>vFK zo#WD5ZiClNvn`WHzcMaVu%}%f%9(k(n^qvCR?-&-e<%N^sT>9eS{ zxz(dwE@ulCPdBGa+#+y~gP?IdvjFBWI?Ok-N7c5x zoVDmoppf-$(LPthqt11|G%VJ8C0y6p-X@#b;e6V(p1EFfn^%{m*)nkVyiyoXPJZuv z>+ElCYyVs?7Kg@)g}qZUZrWBjI7qaw3zPBPw&`xR{Kx5#Pzax-gxI? 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