2 Field Equations in the Metric Formalism

We start with the 4-dimensional action in f (R) gravity:

1 ∫ 4 √ --- ∫ 4 S = ---2 d x − gf(R ) + d xℒM (gμν,ΨM ), (2.1 ) 2κ
where κ2 = 8πG, g is the determinant of the metric gμν, and ℒM is a matter Lagrangian1 that depends on gμν and matter fields ΨM. The Ricci scalar R is defined by R = gμνR μν, where the Ricci tensor R μν is
α λ λ λ ρ λ ρ Rμν = R μαν = ∂λΓ μν − ∂ μΓλν + ΓμνΓ ρλ − Γ νρΓ μλ. (2.2 )
In the case of the torsion-less metric formalism, the connections Γ α βγ are the usual metric connections defined in terms of the metric tensor gμν, as
1 ( ∂gγλ ∂gλβ ∂gβγ) Γ αβγ = -gαλ --β--+ ---γ-− ---λ- . (2.3 ) 2 ∂x ∂x ∂x
This follows from the metricity relation, ∇ λgμν = ∂g μν∕∂xλ − gρνΓ ρ − gμρΓ ρ = 0 μλ νλ.

 2.1 Equations of motion
 2.2 Equivalence with Brans–Dicke theory
 2.3 Conformal transformation

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