Differential Geometry - Finding Flat Coordinates

[ozone]

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.

 

[Dick]

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.

I think you are talking about Riemann normal coordinates. And I'm pretty rusty at this stuff so I'm not really sure how the direction you are outlining is going. But that should give you something to research. 'Locally flat' to me means the Christoffel symbols vanish at a point. This is exercise 11.9 in Meissner Thorne and Wheeler. Using the geodesic equation is useful for that. Hope this helps.

 

[ozone]

Thanks for that, I found the notation in that book to be a little bit to dense, but I think I managed to find the solution today. I went through slowly step by step throwing away second order terms and found that
[itex] 2( L_{\sigma,\lambda,\rho} + L_{\rho \lambda , \alpha}) = -A_{\rho\sigma, \lambda} [/itex] using the substitution from then it solves for [itex] L_{\lambda \sigma,\rho} = -\frac{1}{2} \Gamma_{\lambda \sigma,\rho} [/itex] while also redundantly solving [itex] L_{\rho \lambda , \alpha} = -\frac{1}{2} \Gamma_{\rho \lambda , \alpha} [/itex]