I'm currently looking at the field equations and trying to understand where the second order terms in the metric come from?

I have an idea but can't really see how it can work.

Here goes...
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The christophel symbols are functions of the first derivative of the metric.

The Riemann tensor can be expressed as having two terms that involve the first derivative of the christophel symbols plus two quadratic terms in the christophel symbols.

The Einstein tensor is composed of the Ricci tensor and the Ricci scaler. Both the Ricci tensor and scaler are contractions of the Riemann tensor.

My initial thought had been that the second order terms of the metric in the einstein tensor came from the two quadratic terms of the christophel symbols that originate in the Riemann tensor.

However, as the christophel symbols are functions of the first derivative of the metric, they have to vanish, even if we are not in a local inertial frame.
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Is this idea even remotely along the right lines? Can anyone suggest where I can find those second order terms?

Thanks, sox