- #1
Destroxia
- 204
- 7
Homework Statement
Given some 2D line element, ## ds^2 = -dt^2 +x^2 dx^2 ##, find the Christoffel Symbols, ## \Gamma_{\beta \gamma}^{\alpha} ##.
Homework Equations
## \Gamma_{\beta \gamma}^{\alpha} = \frac {1}{2} g^{\delta \alpha} (\frac{\partial g_{\alpha \beta}}{\partial x^\gamma} + \frac{\partial g_{\alpha \gamma}}{\partial x^{\beta}} - \frac{\partial g_{\beta \gamma}}{\partial x^\alpha}) ##
## ds^2 = g_{\alpha \beta} dx^\alpha dx^\beta ##
The Attempt at a Solution
While I understand the theory behind what I'm doing, I'm lost when it comes to solving these Christoffel Symbols. The main issue for me is, although we know the equation ## ds^2 = g_{\alpha \beta} dx^\alpha dx^\beta ##, I don't know how to derive a metric from this for ## ds^2 = -dt^2 +x^2 dx^2 ##, such that I could then simply use the Christoffel Symbol Equation to read off the coordinates.
What I assume I should do is somehow create a metric for this line element, and then proceed to relate the coefficients of this metric to the Christoffel Symbol Equation, so for example, we have some metric (not the correct one):
\begin{equation} g_{\alpha \beta} =
\left({\begin{array}{cc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}}\right)
\end{equation}
and then in order to solve these symbols, we need to follow the form of ## \Gamma_{\beta \gamma}^{\alpha} = \frac {1}{2} g^{\delta \alpha} (\frac{\partial g_{\alpha \beta}}{\partial x^\gamma} + \frac{\partial g_{\alpha \gamma}}{\partial x^{\beta}} - \frac{\partial g_{\beta \gamma}}{\partial x^\alpha}) ##
Although, another hang up is index notation, I just have no idea how these ## \alpha, \beta, and \gamma ## relate to each other. Any explanation, or resource for reading this index notation would be very helpful.