Derivation of Riemann tensor's first order-equation

[MathematicalPhysicist]

TL;DR Summary

Trying to derive Eq. (8.25) for Riemann tensor in the book of Schutz's in GR.

According to Schutz's second edition book, the equation for Riemann tensor to first order in $h_{\mu\nu}$ is:

$$R_{\alpha\beta\mu\nu}= \frac{1}{2}(h_{\alpha \nu,\beta \mu}+h_{\beta\mu,\alpha\nu}-h_{\alpha\mu,\beta \nu}-h_{\beta \nu , \alpha \mu})$$
which (as stated in the book) can be derived easily by using Eq. (8.12) which is:

$$g_{\alpha \beta}=\eta_{\alpha \beta}+h_{\alpha\beta}$$
All this is covered in pages 189-192.

How to derive the above identity? and how to derive the n-th order in $h_{\mu\nu}$ equation for Riemann tensor?
How come this equation is of first order in $h_{\mu\nu}$ if the terms in the above equation are second order derivatives of $h$.

Thanks in advance!

 

[Orodruin]

It is first order in $h$ (and its derivatives), not in the order of the derivatives. The assumption is that $h$ and its derivatives are small and you ignore terms that are of higher order in $h$. The derivation is just inserting your metric into the expression for the Christoffel symbols (keeping only terms linear in $h$) and then inserting the expression for the Christoffel symbols into the expression for the curvature components in terms of the Christoffel symbols (again, keeping only terms linear in $h$).