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I am studying on my own from Weinberg's Gravitation and Cosmology and I cannot understand how he derives a solution (pg. 72). I did not know where else to post this thread since it is not homework exercise.

He takes a coordinate system ## \xi^a## "in which the equation of motion of a particle moving freely under the influence of purely gravitational forces is that of a straight line in space-time, that is:

[tex] \frac{d^2\xi^a}{d\tau^2}=0[/tex]

Next he takes another coordinate system ##x^\mu## which could be "what we will". He does some elementary differentiations, defines the affine connection

[tex]\Gamma^\lambda_{\mu\nu}\equiv\frac{\partial x^\lambda}{\partial \xi^a}\frac{\partial^2\xi^a}{\partial x^\mu\partial x^\nu}[/tex]

and finally gets the equation

[tex] \frac{\partial^2\xi^a}{\partial x^\mu\partial x^\nu}=\Gamma^\lambda_{\mu\nu}\frac{\partial \xi^a}{\partial x^\lambda}[/tex]

He states (without giving any other argument) that "The values of the metric tensor and the affine connection at a point ##X## in an arbitrary coordinate system ##x^\mu## provide enough information to determine the locally inertial coordinates ##\xi^a(x)## in a neighbourhood of ##X##. The solution (to the above equation) is:

[tex]\xi^a(x)=\alpha^a+b^a_\mu(x^\mu-X^\mu)+\frac{1}{2}b^a_\lambda\Gamma^\lambda_{\mu\nu}(x^\mu-X^\mu)(x^\nu-X^\nu)+...[/tex]

where ##\alpha^a=\xi^a(X)## and ##b^a_\lambda=\frac{\partial\xi^a(X)}{\partial X^\lambda}## [...] Thus given the affine connection and the metric tensor at ##X##, the locally inertial coordinates are determined to order ##(x-X)^2##"

Where does this solution come from?

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# I Determination of locally inertial coordinates

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