The variation of the Einstein Hilbert action is usually done in coordinate basis where there is a crucial divergence term one can neglect which arise in the variation of the Ricci tensor, and is given by ##g^{ab}\delta R_{ab} = \nabla_c w^c## where(adsbygoogle = window.adsbygoogle || []).push({});

$$w^c = g^{ab}(g^{db} \delta \Gamma^{c}_{db} - g^{cb} \delta\Gamma^c_{db})$$.

However, when one varies in a noncoordinate basis, one supposedly (see link below) get's an extra term and arrive at

$$g^{ab}\delta R_{ab} = \nabla_c w^c - C^c_{cd} \delta \Gamma^d_{ab}g^{ab}.$$

How is this result derived?

My calculations so far is the following: From the Riemann tensor in a noncoordinate basis ##\{e_a \}## with structure constants ##[e_b, e_c] = C^a_{bc}e_a## given by

$$R^a_{bcd} = e_c \Gamma^a_{db} - e_d\Gamma^a_{cb} + \Gamma^f_{db} \Gamma^a_{cf} - \Gamma^f_{cb} \Gamma^a_{df} - C^f_{cd} \Gamma^a_{fb}$$

the variation yields

$$\delta R^a_{bcd} = e_c \delta \Gamma^a_{db} - e_d\delta \Gamma^a_{cb} + \delta\Gamma^f_{db} \Gamma^a_{cf} + \Gamma^f_{db} \delta\Gamma^a_{cf} - \delta\Gamma^f_{cb} \Gamma^a_{df} - \Gamma^f_{cb} \delta \Gamma^a_{df} - C^f_{cd} \delta \Gamma^a_{fb}- \delta C^f_{cd} \Gamma^a_{fb}$$

From here we can extract three terms of the respective covariant derivatives ##\nabla_d \delta \Gamma^a_{cb}## and ##\nabla_c \delta \Gamma^a_{db}##; however the terms ##\Gamma^f_{cd}\delta \Gamma^a_{fb}## and ##\Gamma^f_{dc} \delta \Gamma^a_{fb}## are not present. Taking these into account, I get

$$\delta R^a_{bcd} = \nabla_c \delta \Gamma^a_{db} - \nabla_d \delta \Gamma^a_{cb} + (\Gamma^f_{cd} - \Gamma^f_{dc}) \delta \Gamma^a_{fb} - C_{cd}^f\delta\Gamma^a_{fb} - \delta C_{cd}^f\Gamma^a_{fb}= \nabla_c \delta \Gamma^a_{db} - \nabla_d \delta \Gamma^a_{cb}- \delta C_{cd}^f\Gamma^a_{fb}.$$

If there are no error in my calculations, where do I go from here? Assuming I have done nothing wrong so far, it seems like one must achieve the equality ##\delta C_{cd}^f\Gamma^c_{fb} g^{db} = C^c_{cd} \delta \Gamma^d_{ab}g^{ab}##.. Any ideas on how this equality might be obtained?

Equation (2.9) in: http://scitation.aip.org/content/aip/journal/jmp/15/6/10.1063/1.1666735

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# Variation of the Einstein-Hilbert action in noncoordinate basis

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