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- 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!

$$R_{\alpha\beta\mu\nu}= \frac{1}{2}(h_{\alpha \nu,\beta \mu}+h_{\beta\mu,\alpha\nu}-h_{\alpha\mu,\beta \nu}-h_{\beta \nu , \alpha \mu})$$

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!