Variation of the Einstein-Hilbert action in noncoordinate basis

[center o bass]

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

 

[center o bass]

As the subject of the thread states, I wonder about noncoordinate bases. I.e. bases for which $e_a \neq \partial_a$.

 

[Mentz114]

Tangent space basis vectors like $e_a = f \partial_a$ commute, so the problem goes away for them.
If ${C^f}_{cd}$ is constant, then is $\delta {C^f}_{cd} = 0$ ?

In which case you need ${C^c}_{cd} = 0$, which could be the case.

[Edit]
I guess you saw my deleted post.

 

[center o bass]

Tangent space basis vectors like $e_a = f \partial_a$ commute, so the problem goes away for them.
If ${C^f}_{cd}$ is constant, then is $\delta {C^f}_{cd} = 0$ ?

In which case you need ${C^c}_{cd} = 0$, which could be the case.

[Edit]
I guess you saw my deleted post.

When one varies the components of the
metric (expressed in a basis), one also varies the associated basis, and since the structure constants are spacetime functions associated to a particular basis, these varies too. Thus, it seems like the variation of the structure constant does not disappear.

 

[center o bass]

Tangent space basis vectors like $e_a = f \partial_a$ commute, so the problem goes away for them.
If ${C^f}_{cd}$ is constant, then is $\delta {C^f}_{cd} = 0$ ?

In which case you need ${C^c}_{cd} = 0$, which could be the case.

[Edit]
I guess you saw my deleted post.

To correct my previous answer to you, I do agree that the structure constants are constant, since the variation is performed for a fixed basis.

Thus, the mystery where the additional term involving $C_{cd}^{\ \ c}$ comes from is even bigger.