Kent davidge: yeah you’re right, I’ve got a typo in the definition of ##\nabla_\alpha Z^\mu##.Ok, it seems that there is a conflict of notations here. In the text I am using, the points along a curve ##\gamma## are denoted as ##\gamma^\mu \partial_\mu## with ##\partial_\mu## being the basis...
Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
I’ve already done that. The geodesic is of the form ##(x(\tau))=(u^0,0,0,u^3)\tau## with ##u^0=\sqrt{1+(u^z)^2}##. Here ##u^0## and ##u^3## are constant.
However that does not answer the questions formulated in my OP.
Thank you for your answers!
I know that ##\eta_{\mu \nu}=g_{\mu \nu}## only holds point for point, which is why I was asking if this was the justification for setting ##\eta_{\mu \nu}=g_{\mu \nu}##: I couldn’t see how this would hold for all of ##\gamma## in those coordinates.
And just to...
Hey, (I didn't know my PW by heart so I had to create a new account)
See the proof that every Lorentzian Metric is locally Minkowski on p.31 of this script: https://gravity.univie.ac.at/fileadmin/user_upload/i_gravity_physics/material/teaching/rt2/SS_2013/Vienna2013.pdf