[OTO] Added solution to exercise 4

2 files changed, 58 insertions, 0 deletions

diff --git a/Year 3/General theory of relativity/Project.pdf b/Year 3/General theory of relativity/Project.pdf
index 4401d98..53f4b8d 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 d27d789..2ad12e6 100644
--- a/Year 3/General theory of relativity/Project.tex
+++ b/Year 3/General theory of relativity/Project.tex
@@ -145,4 +145,62 @@ $00800028 \; mod \; 3 = 0$, тоест ще докажем, че ковариа�
 	= & \tilde{\nabla}_l \, \tilde{g}_{ij} \, \der y^i \der y^j \der y^l = \tilde{\nabla} \tilde{g}
 \end{align*}
 
+\paragraph{4. Решение}
+
+Правим смяна на променлива, от $x$ към $y$, което означава че:
+
+\begin{align*}
+	{\Gamma^i}_{jk} = \tilde{\Gamma}^a {}_{bc} \frac{\partial x^i}{\partial y^a} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k} + \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k}
+	\qquad \frac{\partial}{\partial x^i} = \frac{\partial}{\partial y^a} \frac{\partial y^a}{\partial x^i}
+	\qquad \der x^a = \frac{\partial x^a}{\partial y^i} \der y^i
+\end{align*}
+
+Тоест:
+
+\begin{align*}
+	& {R^i}_{jkl} \frac{\partial}{\partial x^i} \der x^j \der x^k \der x^l = 
+	\left( \frac{\partial}{\partial x^l} {\Gamma^i}_{jk} - \frac{\partial}{\partial x^k} {\Gamma^i}_{jl} + {\Gamma^i}_{ml} {\Gamma^m}_{jk} - {\Gamma^i}_{mk} {\Gamma^m}_{jl} \right) \frac{\partial}{\partial x^i} \der x^j \der x^k \der x^l \\
+	& = \left( \frac{\partial}{\partial y^d} \frac{\partial y^d}{\partial x^l} \left( \tilde{\Gamma}^a {}_{bc} \frac{\partial x^i}{\partial y^a} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k} + \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k} \right)
+	  - \frac{\partial}{\partial y^c} \frac{\partial y^c}{\partial x^k} \left( \tilde{\Gamma}^a {}_{bd} \frac{\partial x^i}{\partial y^a} \frac{\partial y^b}{\partial x^j} \frac{\partial y^d}{\partial x^l} + \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} \right) \right. \\
+	& \qquad + \left( \tilde{\Gamma}^a {}_{nd} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^d}{\partial x^l} + \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^l} \right)
+	\left( \tilde{\Gamma}^n {}_{bc} \frac{\partial x^m}{\partial y^n} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k} + \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k} \right) \\
+	& \qquad - \left( \tilde{\Gamma}^a {}_{nc} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^c}{\partial x^k} + \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^k} \right)
+	\left( \tilde{\Gamma}^n {}_{bd} \frac{\partial x^m}{\partial y^n} \frac{\partial y^b}{\partial x^j} \frac{\partial y^d}{\partial x^l} + \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} \right) \\
+	& \qquad \bigg) \frac{\partial}{\partial y^a} \frac{\partial y^a}{\partial x^i} \frac{\partial x^j}{\partial y^b} \der y^b \frac{\partial x^k}{\partial y^c} \der y^c \frac{\partial x^l}{\partial y^d} \der y^d \\
+	& = \left( \frac{\partial}{\partial y^d} \tilde{\Gamma}^a {}_{bc} \left( \frac{\partial y^d}{\partial x^l} \frac{\partial x^i}{\partial y^a} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k} \right) + \cancel{\frac{\cancel{\partial y^d}}{\partial x^l} \frac{\partial}{\cancel{\partial y^d}} \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k}} \right. \\
+	& \qquad - \frac{\partial}{\partial y^c} \tilde{\Gamma}^a {}_{bd} \left( \frac{\partial y^c}{\partial x^k} \frac{\partial x^i}{\partial y^a} \frac{\partial y^b}{\partial x^j} \frac{\partial y^d}{\partial x^l} \right) - \cancel{\frac{\cancel{\partial y^c}}{\partial x^k} \frac{\partial}{\cancel{\partial y^c}} \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l}} \\
+	& \qquad + \tilde{\Gamma}^a {}_{nd} \frac{\partial x^i}{\partial y^a} \cancel{\frac{\partial y^n}{\partial x^m}} \frac{\partial y^d}{\partial x^l} \tilde{\Gamma}^n {}_{bc} \cancel{\frac{\partial x^m}{\partial y^n}} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k}
+	+ \tilde{\Gamma}^a {}_{nd} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^d}{\partial x^l} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k} \\
+	& \qquad + \tilde{\Gamma}^n {}_{bc} \frac{\partial x^m}{\partial y^n} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k} \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^l}
+	+ \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^l} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k}
+\end{align*}
+
+\begin{align*}
+	& \qquad - \tilde{\Gamma}^a {}_{nc} \frac{\partial x^i}{\partial y^a} \cancel{\frac{\partial y^n}{\partial x^m}} \frac{\partial y^c}{\partial x^k} \tilde{\Gamma}^n {}_{bd} \cancel{\frac{\partial x^m}{\partial y^n}} \frac{\partial y^b}{\partial x^j} \frac{\partial y^d}{\partial x^l}
+	- \tilde{\Gamma}^a {}_{nc} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^c}{\partial x^k} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} \\
+	& \qquad - \tilde{\Gamma}^n {}_{bd} \frac{\partial x^m}{\partial y^n} \frac{\partial y^b}{\partial x^j} \frac{\partial y^d}{\partial x^l} \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^k}
+	- \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^k} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} \\
+	& \qquad \bigg) \frac{\partial y^a}{\partial x^i} \frac{\partial x^j}{\partial y^b} \frac{\partial x^k}{\partial y^c} \frac{\partial x^l}{\partial y^d} \frac{\partial}{\partial y^a} \der y^b \der y^c \der y^d	
+\end{align*}
+
+\begin{align*}
+	\frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^l} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k}
+	= \frac{\partial x^i}{\partial x^m} \frac{\partial^2 y^p}{\partial y^p \partial y^p} \frac{\partial x^m}{\partial x^l} \frac{\partial^2 x^l}{\partial x^j \partial x^k} = 0 \quad \text{Аналогично } \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^k} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} = 0
+\end{align*}
+
+\begin{align*}
+	& \text{По формулата виждаме, че } \tilde{\Gamma}^a {}_{nd} = \tilde{\Gamma}^a {}_{nc} \cancel{\frac{\partial y^a}{\partial y^a}} \cancel{\frac{\partial y^n}{\partial y^n}} \frac{\partial y^c}{\partial y^d} + \frac{\partial y^a}{\partial y^p} \frac{\partial^2 y^p}{\partial y^n \partial y^d} = \tilde{\Gamma}^a {}_{nc} \frac{\partial y^c}{\partial y^d} \text{ , следователно: } \\
+	& \tilde{\Gamma}^a {}_{nd} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^d}{\partial x^l} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k} - \tilde{\Gamma}^a {}_{nc} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^c}{\partial x^k} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} = \\
+	& \tilde{\Gamma}^a {}_{nc} \frac{\partial y^c}{\cancel{\partial y^d}} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\cancel{\partial y^d}}{\partial x^l} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^k} - \tilde{\Gamma}^a {}_{nc} \frac{\partial x^i}{\partial y^a} \frac{\partial y^n}{\partial x^m} \frac{\partial y^c}{\partial x^k} \frac{\partial x^m}{\partial y^p} \frac{\partial^2 y^p}{\partial x^j \partial x^l} = 0 \\
+	& \text{Аналогично } \tilde{\Gamma}^n {}_{bc} \frac{\partial x^m}{\partial y^n} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k} \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^l} - \tilde{\Gamma}^n {}_{bd} \frac{\partial x^m}{\partial y^n} \frac{\partial y^b}{\partial x^j} \frac{\partial y^d}{\partial x^l} \frac{\partial x^i}{\partial y^p} \frac{\partial^2 y^p}{\partial x^m \partial x^k} = 0
+\end{align*}
+
+Следователно:
+
+\begin{align*}
+	& {R^i}_{jkl} \frac{\partial}{\partial x^i} \der x^j \der x^k \der x^l \\
+	& = \left( \frac{\partial}{\partial y^d} \tilde{\Gamma}^a {}_{bc} - \frac{\partial}{\partial y^c} \tilde{\Gamma}^a {}_{bd} + \tilde{\Gamma}^a {}_{nd} \tilde{\Gamma}^n {}_{bc} - \tilde{\Gamma}^a {}_{nc} \tilde{\Gamma}^n {}_{bd} \right) \cancel{\frac{\partial y^d}{\partial x^l} \frac{\partial x^i}{\partial y^a} \frac{\partial y^b}{\partial x^j} \frac{\partial y^c}{\partial x^k}} \cancel{\frac{\partial y^a}{\partial x^i} \frac{\partial x^j}{\partial y^b} \frac{\partial x^k}{\partial y^c} \frac{\partial x^l}{\partial y^d}} \frac{\partial}{\partial y^a} \der y^b \der y^c \der y^d \\
+	& = \tilde{R}^a {}_{bcd} \frac{\partial}{\partial y^a} \der y^b \der y^c \der y^d
+\end{align*}
+
 \end{document}
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