[OTO] Added solution to exercise 3

2 files changed, 96 insertions, 0 deletions

diff --git a/Year 3/General theory of relativity/Project.pdf b/Year 3/General theory of relativity/Project.pdf
index 42ef5ba..4401d98 100644
--- a/Year 3/General theory of relativity/Project.pdf
+++ b/Year 3/General theory of relativity/Project.pdf
diff --git a/Year 3/General theory of relativity/Project.tex b/Year 3/General theory of relativity/Project.tex
index 4c5c7ef..d27d789 100644
--- a/Year 3/General theory of relativity/Project.tex
+++ b/Year 3/General theory of relativity/Project.tex
@@ -21,6 +21,8 @@
 \usepackage{enumitem}
 \setlist[enumerate]{label=\arabic*.}
 
+\newcommand{\der}{\mathrm{d}}
+
 \usepackage[pdfauthor={Камен Димитров Младенов, },
 		    pdftitle={Решения на задачи за изпит по ОТО},
 		    pdfsubject={},
@@ -49,4 +51,98 @@
 
 Разглеждайки два перпендикулярни снопа, изминаващи еднакво разстояние от еднакъв източник, стигайки еднаква дестинация, Майкелсон и Морли експериментално намират обратното: скоростта на светлината никога не се променя (и етер не съществува).
 
+\paragraph{3. Решение}
+
+$00800028 \; mod \; 3 = 0$, тоест ще докажем, че ковариантната производна на тензор от тип $(0, 2)$ е тензор от тип $(0, 3)$.
+
+\begin{align*}
+	\nabla g = \nabla_k \, g_{ab} \, \der x^a \der x^b \der x^k = \left( \frac{\partial g_{ab}}{\partial x^k} - {\Gamma^m}_{ak} \, g_{mb} - {\Gamma^m}_{kb} \, g_{ma} \right) \der x^a \der x^b \der x^k
+\end{align*}
+
+Правим смяна на променлива, от $x$ към $y$, което означава че:
+
+\begin{align*}
+	\tilde{\Gamma}^i {}_{jk} = {\Gamma^a}_{bc} \frac{\partial y^i}{\partial x^a} \frac{\partial x^b}{\partial y^j} \frac{\partial x^c}{\partial y^k} + \frac{\partial y^i}{\partial x^p} \frac{\partial^2 x^p}{\partial y^j \partial y^k}
+	\implies {\Gamma^a}_{bc} = \left( \tilde{\Gamma}^i {}_{jk} - \frac{\partial y^i}{\partial x^p} \frac{\partial^2 x^p}{\partial y^j \partial y^k} \right) \frac{\partial x^a}{\partial y^i} \frac{\partial y^j}{\partial x^b} \frac{\partial y^k}{\partial x^c}
+	\qquad \der x^a = \frac{\partial x^a}{\partial y^i} \der y^i
+\end{align*}
+
+\begin{align*}
+	& g = g_{ab} \der x^a \der x^b 
+	    = g_{ab} \frac{\partial x^a}{\partial y^i} \der y^i \frac{\partial x^b}{\partial y^j} \der y^j 
+	    = g_{ab} \frac{\partial x^a}{\partial y^i} \frac{\partial x^b}{\partial y^j} \der y^i \der y^j 
+	    = \tilde{g}_{ij} \der y^i \der y^j 
+	    \implies g_{ab} 
+	    = \tilde{g}_{ij} \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \\
+	& \text{Заедно с}
+	  \qquad
+	  \frac{\partial}{\partial x^i} 
+	  = \frac{\partial}{\partial y^a} \frac{\partial y^a}{\partial x^i}
+	  \qquad
+	  \text{следва че:} \\
+    & \frac{\partial g_{ab}}{\partial x^k} 
+	  = \frac{\partial y^l}{\partial x^k} \frac{\partial}{\partial y^l} \left( \tilde{g}_{ij} \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \right)
+	  = \frac{\partial y^l}{\partial x^k} \left( \tilde{g}_{ij} \frac{\partial}{\partial y^l} \left( \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \right) + \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \frac{\partial}{\partial y^l} \left( \tilde{g}_{ij} \right) \right) \\
+	& = \frac{\partial y^l}{\partial x^k} \left( 
+	\tilde{g}_{ij} \left( \frac{\partial y^i}{\partial x^a} \frac{\partial}{\partial y^l} \left( \frac{\partial y^j}{\partial x^b} \right) + \frac{\partial y^j}{\partial x^b} \frac{\partial}{\partial y^l} \left( \frac{\partial y^i}{\partial x^a} \right) \right) + 
+	\frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \frac{\partial}{\partial y^l} \left( \tilde{g}_{ij} \right) \right) \\
+	& = \frac{\partial y^l}{\partial x^k} \left( 
+	\tilde{g}_{ij} \left(
+		\frac{\partial y^i}{\partial x^a} \frac{\partial^2 y^j}{\partial y^l \partial x^b} + 
+		\frac{\partial y^j}{\partial x^b} \frac{\partial^2 y^i}{\partial y^l \partial x^a} \right) + 
+	\frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \frac{\partial \tilde{g}_{ij}}{\partial y^l} \right) \\
+	& = \frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial^2 y^j}{\partial y^l \partial x^b} \tilde{g}_{ij} + 
+		\frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \frac{\partial^2 y^i}{\partial y^l \partial x^a} \tilde{g}_{ij} + 
+	    \frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \frac{\partial \tilde{g}_{ij}}{\partial y^l}
+\end{align*}
+
+Тоест:
+
+\begin{align*}
+	\nabla g = & \left(
+		\frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial^2 y^j}{\partial y^l \partial x^b} \tilde{g}_{ij} + 
+		\frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \frac{\partial^2 y^i}{\partial y^l \partial x^a} \tilde{g}_{ij} + 
+	    \frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \frac{\partial \tilde{g}_{ij}}{\partial y^l} \right. \\
+	& \quad - \left(
+			\tilde{\Gamma}^n {}_{il} -
+			\frac{\partial y^n}{\partial x^p} \frac{\partial^2 x^p}{\partial y^i \partial y^l} 
+		\right) \cancel{\frac{\partial x^m}{\partial y^n}} \frac{\partial y^i}{\partial x^a} \frac{\partial y^l}{\partial x^k}
+			\tilde{g}_{nj} \cancel{\frac{\partial y^n}{\partial x^m}} \frac{\partial y^j}{\partial x^b} \\
+	& \quad - \left(
+			\tilde{\Gamma}^n {}_{lj} - 
+			\frac{\partial y^n}{\partial x^p} \frac{\partial^2 x^p}{\partial y^l \partial y^j} 
+		\right) \cancel{\frac{\partial x^m}{\partial y^n}} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b}
+			\tilde{g}_{ni} \cancel{\frac{\partial y^n}{\partial x^m}} \frac{\partial y^i}{\partial x^a} \\
+	& \; \bigg) \frac{\partial x^a}{\partial y^i} \der y^i \frac{\partial x^b}{\partial y^j} \der y^j \frac{\partial x^k}{\partial y^l} \der y^l
+\end{align*}
+
+\begin{align*}
+	\tilde{g}_{nj} = \tilde{g}_{ij} \frac{\partial y^i}{\partial y^n} \implies
+	& \frac{\partial y^n}{\partial x^p} \frac{\partial^2 x^p}{\cancel{\partial y^i} \partial y^l} \frac{\cancel{\partial y^i}}{\partial x^a} \frac{\partial y^l}{\partial x^k} \tilde{g}_{nj} \frac{\partial y^j}{\partial x^b}
+	= \frac{\cancel{\partial y^n}}{\partial x^p} \frac{\partial^2 x^p}{\partial x^a \partial y^l} \tilde{g}_{ij} \frac{\partial y^i}{\cancel{\partial y^n}} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \\
+	& = \frac{\partial y^i}{\partial x^p} \frac{\partial}{\partial x^a} \left( \frac{\partial x^p}{\partial y^l} \right) \tilde{g}_{ij} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b}
+	= \left( \frac{\partial}{\partial x^a} \left( \frac{\partial y^i}{\cancel{\partial x^p}} \frac{\cancel{\partial x^p}}{\partial y^l} \right) - \frac{\partial x^p}{\partial y^l} \frac{\partial}{\partial x^a} \left( \frac{\partial y^i}{\partial x^p} \right) \right) \tilde{g}_{ij} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \\
+	& = 0 - \frac{\cancel{\partial x^p}}{\partial y^l} \frac{\partial^2 y^i}{\partial x^a \cancel{\partial x^p}} \tilde{g}_{ij} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} = - \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \frac{\partial^2 y^i}{\partial y^l \partial x^a} \tilde{g}_{ij} \\
+	& \text{Аналогично } \frac{\partial y^n}{\partial x^p} \frac{\partial^2 x^p}{\partial y^l \partial y^j} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \tilde{g}_{ni} \frac{\partial y^i}{\partial x^a} = - \frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial^2 y^j}{\partial y^l \partial x^b} \tilde{g}_{ij}
+\end{align*}
+
+Следователно:
+
+\begin{align*}
+	\nabla g = & \left(
+		\cancel{\frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial^2 y^j}{\partial y^l \partial x^b} \tilde{g}_{ij}} + 
+		\cancel{\frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \frac{\partial^2 y^i}{\partial y^l \partial x^a} \tilde{g}_{ij}} + 
+	    \frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b} \frac{\partial \tilde{g}_{ij}}{\partial y^l} \right. \\
+	& \quad - \tilde{\Gamma}^n {}_{il} \frac{\partial y^i}{\partial x^a} \frac{\partial y^l}{\partial x^k}
+			\tilde{g}_{nj} \frac{\partial y^j}{\partial x^b}
+	  - \cancel{\frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b} \frac{\partial^2 y^i}{\partial y^l \partial x^a} \tilde{g}_{ij}} \\
+	& \quad - \tilde{\Gamma}^n {}_{lj} \frac{\partial y^l}{\partial x^k} \frac{\partial y^j}{\partial x^b}
+			\tilde{g}_{ni} \frac{\partial y^i}{\partial x^a}
+		- \cancel{\frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial^2 y^j}{\partial y^l \partial x^b} \tilde{g}_{ij}} \\
+	& \; \bigg) \frac{\partial x^a}{\partial y^i} \frac{\partial x^b}{\partial y^j} \frac{\partial x^k}{\partial y^l} \der y^i \der y^j \der y^l \\
+	= & \left( \frac{\partial \tilde{g}_{ij}}{\partial y^l} - \tilde{\Gamma}^n {}_{il} \tilde{g}_{nj} - \tilde{\Gamma}^n {}_{lj} \tilde{g}_{ni} \right)
+	\cancel{\frac{\partial y^l}{\partial x^k} \frac{\partial y^i}{\partial x^a} \frac{\partial y^j}{\partial x^b}} \; \cancel{\frac{\partial x^a}{\partial y^i} \frac{\partial x^b}{\partial y^j} \frac{\partial x^k}{\partial y^l}} \der y^i \der y^j \der y^l \\
+	= & \tilde{\nabla}_l \, \tilde{g}_{ij} \, \der y^i \der y^j \der y^l = \tilde{\nabla} \tilde{g}
+\end{align*}
+
 \end{document}
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