Field equations

9.1 Field equations

Let us derive field equations by treating $g μν$ and $Γ α βγ$ as independent variables. Varying the action (2.1

) with respect to $gμν$ , we obtain

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. (2.5

). 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. (9.1

) gives

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 [556

]

where a prime represents a derivative in terms of $R(T )$ . The variation of the action (2.1

) 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 (9.2 ) and (9.4 ) are identical to the equations (2.7 ) and (2.4 ), 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. (2.7 ), 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 [100 ] 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 (2.21 ) from (2.18 ), the action in Palatini f (R) gravity is equivalent to

where

Since the derivative of $U$ in terms of $φ$ is $U = R∕(2κ2 ) ,φ$ , we obtain the following relation from Eq. (9.2

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

Comparing this with Eq. (2.23

) in the unit $κ2 = 1$ , we find that Palatini f (R) gravity is equivalent to BD theory with the parameter $ω = − 3∕2 BD$ [262

, 470

, 551

]. As we will see in Section 10.1, this equivalence can be also seen by comparing Eqs. (9.1

) and (9.4

) with those obtained by varying the action (2.23

) 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 [299 , 558 , 557 , 121 ]. 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 [557 , 556 ].