- #1
ozone
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Homework Statement
Hello, I posted a similar question in the physics section but no one was able to help, I am first going to include a link to the older problem where I was attempting to find the ,(Finding the local flat space of the Poincare half disk metric), and explain what is different this time~~
I still haven't figured out how to answer the previous problem and that is probably why I am struggling with the new one, but here is the statement for my new problem:
In an earlier problem you were instructed to find locally flat coordinates, and after hte first step, with the metric already in the form [itex] g_{\mu\nu} = \delta_{\mu\nu} + A_{\mu\nu,\lambda} x^{\lambda} + ... [/itex] we claimed that by using the transformation [itex] x^{\mu} = x'^{\mu} + L_{\nu\lambda}^{\mu}x'^{\nu}x'^{\lambda} + ... [/itex] we could get rid of the linear terms in the metric. Using the transformation property of Christoffel symbols, determine [itex] L^{\mu}_{\nu\lambda}. [/itex] ("Just from the index structure you should probably be able to guess the answer"~~This makes me think the answer is just a Christoffel symbol, but I get nothing close to that!).
Homework Equations
The transformation property of Christoffel symbols is given as
[itex] \Gamma'^{\lambda}_{\mu\nu} = \frac{\partial x'^{\lambda}}{\partial x^\eta} \frac{\partial x^{\omega}}{\partial x'^{\mu}} \frac{\partial x^{\sigma}}{\partial x'^{\nu}} \Gamma^{\eta}_{\omega\sigma} + \frac{\partial x'^{\lambda}}{\partial x^\eta} \frac{\partial^2 x^\eta}{\partial x'^{\mu} \partial x'^{\nu}} [/itex]
Now we defined before [itex] A_{\mu\nu,\lambda} x^{\lambda} = \partial_\lambda g_{\mu\nu}(0) x^{\lambda} [/itex].. Now I began thinking on my own here and my intuition told me to write out this as a sum of Christoffel symbols, namely [itex] \partial_\lambda g_{\mu\nu}(0) x^{\lambda} = (\Gamma_{\lambda \nu . \mu} + \Gamma_{\lambda\mu . \nu})x^{\lambda} [/itex]
From here I played around a lot and didn't really get anywhere.. I guess I'd like to know that I'm even going in the right direction and what might be an appropriate next step to take to surmise what L must be.