Undergrad Derivation of Riemann tensor's first order-equation

[MathematicalPhysicist]

TL;DR

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 $h$).