Problems with the Riemann tensor in general relativity

Ineedhelpimbadatphys · Feb 20, 2024

Textbook problem 2 titled Expansion of the metric in local inertial coordinates, asking for g_ab(x^γ|_p + δx^γ) to second order in δx^γ in terms of the Riemann tensor, with a hint referring to equation (3.71).

Handwritten derivation on lined paper of Riemann tensor identities in general relativity: equation (3.71) for the Riemann tensor in terms of metric derivatives, then numbered equations (1) to (4) involving metric expansion, the metric and its derivatives at a point, and a sum of second-order metric derivative terms ending with a question mark.

After Taylor expansion and using equations (2), I have no problem getting to equation (1). Now obviously I have to somehow use (3.71) ,which I do know how, to derive to express the second order derivative.
On the internet I found equation (3), and I have tried to understand where this comes from (4).
Is equation (3) correct, if yes, how am i supposed to contineu from what I have in (4). It should equal 3 times the secobd order derivative?

If equation (3) is not correct, any tips how to continue from what I have?

Ineedhelpimbadatphys · Feb 20, 2024

The sentence after (3.71) was a little unclear. I meant that I do know how to derive (3.71) using equations (2).

Problems with the Riemann tensor in general relativity

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