I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an action in terms of tensors.i mean converting it into component notation. How can I do this calculation? Or can anyone introduce me a book or paper in this way?
for example the action of a theory is
S=∫gij(x)∂μXi∂μXj
but it is possible to write this action as
ˉδXS[X,γ]=−∫ΣημνˉδXμ∧d⋆γdXν=0
Now I want to know that how the first form can be achieved from the second one.
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