9.1 Field equations

Let us derive field equations by treating g μν and Γ α βγ as independent variables. Varying the action (2.1View Equation) with respect to gμν, we obtain
1 F (R )Rμν(Γ ) − -f(R )gμν = κ2Tμ(Mν ), (9.1 ) 2
where F(R ) = ∂f ∕∂R, R μν(Γ ) is the Ricci tensor corresponding to the connections Γ αβγ, and (M ) Tμν is defined in Eq. (2.5View Equation). Note that R μν(Γ ) is in general different from the Ricci tensor calculated in terms of metric connections R (g ) μν. The trace of Eq. (9.1View Equation) gives
2 F (R )R − 2f(R ) = κ T , (9.2 )
where T = gμνT (μMν). Here the Ricci scalar R(T ) is directly related to T and it is different from the Ricci scalar R (g) = gμνR μν(g) in the metric formalism. More explicitly we have the following relation [556Jump To The Next Citation Point]
R(T ) = R (g ) +------3------(∇ f′(R(T )))(∇ μf ′(R (T))) + ----3----□f ′(R (T )), (9.3 ) 2 (f ′(R (T)))2 μ f′(R(T ))
where a prime represents a derivative in terms of R(T ). The variation of the action (2.1View Equation) with respect to the connection leads to the following equation
2 R μν(g) − 1gμνR (g) = κ-Tμν-− FR-(T-) −-f-gμν + 1-(∇ μ∇ νF − gμν□F ) 2 F [ 2F F ] -3-- 1- 2 − 2F 2 ∂μF ∂νF − 2 gμν(∇F ) . (9.4 )

In Einstein gravity (f(R) = R − 2Λ and F (R ) = 1) the field equations (9.2View Equation) and (9.4View Equation) are identical to the equations (2.7View Equation) and (2.4View Equation), respectively. However, the difference appears for the f (R) models which include non-linear terms in R. While the kinetic term □F is present in Eq. (2.7View Equation), such a term is absent in Palatini f (R) gravity. This has the important consequence that the oscillatory mode, which appears in the metric formalism, does not exist in the Palatini formalism. As we will see later on, Palatini f (R) theory corresponds to Brans–Dicke (BD) theory [100Jump To The Next Citation Point] with a parameter ωBD = − 3∕2 in the presence of a field potential. Such a theory should be treated separately, compared to BD theory with ωBD ⁄= − 3∕2 in which the field kinetic term is present.

As we have derived the action (2.21View Equation) from (2.18View Equation), the action in Palatini f (R) gravity is equivalent to

∫ √ ---[ 1 ] ∫ S = d4x − g --2φR (T ) − U(φ ) + d4x ℒM (gμν,ΨM ), (9.5 ) 2κ
′ φ = f′(R(T )), U = R-(T)f-(R-(T-)) −-f(R-(T-)). (9.6 ) 2κ2
Since the derivative of U in terms of φ is U = R∕(2κ2 ) ,φ, we obtain the following relation from Eq. (9.2View Equation):
4U − 2φU, φ = T . (9.7 )

Using the relation (9.3View Equation), the action (9.5View Equation) can be written as

∫ √ --- [ 1 3 1 ] ∫ S = d4x − g --2-φR (g) + --2--(∇ φ)2 − U (φ) + d4xℒM (gμν,ΨM ). (9.8 ) 2κ 4κ φ
Comparing this with Eq. (2.23View Equation) in the unit κ2 = 1, we find that Palatini f (R) gravity is equivalent to BD theory with the parameter ω = − 3∕2 BD [262Jump To The Next Citation Point470Jump To The Next Citation Point551Jump To The Next Citation Point]. As we will see in Section 10.1, this equivalence can be also seen by comparing Eqs. (9.1View Equation) and (9.4View Equation) with those obtained by varying the action (2.23View Equation) in BD theory. In the above discussion we have implicitly assumed that ℒM does not explicitly depend on the Christoffel connections Γ λμν. This is true for a scalar field or a perfect fluid, but it is not necessarily so for other matter Lagrangians such as those describing vector fields.

There is another way for taking the variation of the action, known as the metric-affine formalism [299Jump To The Next Citation Point558Jump To The Next Citation Point557Jump To The Next Citation Point121]. In this formalism the matter action SM depends not only on the metric gμν but also on the connection Γ λμν. Since the connection is independent of the metric in this approach, one can define the quantity called hypermomentum [299], as Δ μν ≡ (− 2∕√ −-g)δℒM ∕δ Γ λ λ μν. The usual assumption that the connection is symmetric is also dropped, so that the antisymmetric quantity called the Cartan torsion tensor, λ λ Sμν ≡ Γ [μν], is defined. The non-vanishing property of λ S μν allows the presence of torsion in this theory. If the condition Δ [μν]= 0 λ holds, it follows that the Cartan torsion tensor vanishes (S λ = 0 μν[558]. Hence the torsion is induced by matter fields with the anti-symmetric hypermomentum. The f (R) Palatini gravity belongs to f (R) theories in the metric-affine formalism with μν Δ λ = 0. In the following we do not discuss further f (R) theory in the metric-affine formalism. Readers who are interested in those theories may refer to the papers [557556].

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