Boundary Term in Metric f(R) Gravity: Field Equations in the Metric Formalism
aa r X i v : . [ g r- q c ] J un Boundary Term in Metric f ( R ) Gravity: Field Equations in theMetric Formalism
Alejandro Guarnizo , Leonardo Casta˜neda & Juan M. Tejeiro Grupo de Gravitaci´on y Cosmolog´ıa, Observatorio Astron´omico Nacional,Universidad Nacional de ColombiaBogot´a-Colombia
Abstract
The main goal of this paper is to get in a straightforward form the field equations in metric f ( R )gravity, using elementary variational principles and adding a boundary term in the action, insteadof the usual treatment in an equivalent scalar-tensor approach. We start with a brief review of theEinstein-Hilbert action, together with the Gibbons-York-Hawking boundary term, which is mentionedin some literature, but is generally missing. Next we present in detail the field equations in metric f ( R ) gravity, including the discussion about boundaries, and we compare with the Gibbons-York-Hawking term in General Relativity. We notice that this boundary term is necessary in order to havea well defined extremal action principle under metric variation. Keywords : Modified Theories of Gravity, f ( R ) gravity, Variational Principles. General Relativity (GR) is the most widely accepted gravity theory proposed by Einstein in 1916, andit has been tested in several field strength regimes being on of the most successful and accurate the-ories in physics [1]. The field equations can be obtained using a variational principle, from the wellknow Einstein-Hilbert action [2]-[6]. The methodology leads to a boundary contribution which is usuallydropped out [7],[8], setting null fluxes through Gauss-Stokes theorem. It can be done by imposing thatthe variation of the metric and its first derivative vanishes in the boundary [3]. These conditions canbe relaxed whether a boundary term is introduced, called the Gibbons-York-Hawking boundary term[9],[10]. With this boundary term is necessary only to fix the variation of the metric in the boundary.There are some references [3]-[5] where this boundary term is shown explicitly.However, GR is not the only relativistic theory of gravity. In the last decades several generalizationsof Einstein field equations have been proposed [11]-[14]. Within these extended theories of gravity nowa-days a subclass, known as f ( R ) theories, are an alternative for classical problems, as the acceleratedexpansion of the universe, instead of Dark Energy and Quintessence models [15]-[21]. f ( R ) theories ofgravity are basically extensions of the Einstein-Hilbert action with an arbitrary function of the Ricciscalar R [22]-[25]. The field equations were founded in [26], and including boundary terms in fourth ordergravity in [27]-[29]. The Gibbons-York-Hawking like term in f ( R ) gravity was explored in [30], with anaugmented variational principle in [31],[32], and using a scalar-tensor framework in [33]-[38]. Here we ob-tain the field equations from a metric f ( R ) action with boundary terms, using only variational principles.We get a well constrained mathematical problem setting δg αβ = 0 and δR = 0 in the boundary. [email protected] [email protected] [email protected] General Relativity: The Einstein-Hilbert action with the Gibbons-York-Hawking boundary term
We consider the space-time as a pair ( M , g ) with M a four-dimensional manifold and g a metric on M . GR is based on the Einstein’s Field equations (without cosmological constant and geometrical units c = 1), which gives the form of the metric g αβ on the manifold M : R αβ − Rg αβ = κT αβ , (2.1)where R αβ = R ηαηβ is the Ricci tensor, R = R αβ R αβ the Ricci scalar, and T αβ the stress-energy tensor,with κ = 8 πG , and sign convention ( − , + , + , +). The Riemann tensor is given by: R αβγδ = ∂ γ Γ αδβ − ∂ δ Γ αγβ + Γ αγσ Γ σδβ − Γ ασδ Γ σγβ , (2.2)in terms of the connections Γ αβγ . The Einstein field equations can be recovered by using the variationalprinciple δS = 0, with S expressing the total action. In terms of Einstein-Hilbert action S EH , Gibbons-York-Hawking boundary term S GY H and the action associated with all the matter fields S M , the totalaction can be written by [4]: S = 12 κ (cid:0) S EH + S GY H (cid:1) + S M , (2.3)where S EH = Z V d x √− gR, (2.4) S GY H = 2 I ∂ V d y ε p | h | K, (2.5)here V is a hypervolume on M , ∂ V its boundary, h the determinant of the induced metric, K is the traceof the extrinsic curvature of the boundary ∂ V , and ε is equal to +1 if ∂ V is timelike and − ∂ V isspacelike (it is assumed that ∂ V is nowhere null). Coordinates x α are used for the finite region V and y α for the boundary ∂ V . Now we will obtain the Einstein field equations varying the action with respect to g αβ . We fixed the variation with the condition [3],[4] δg αβ (cid:12)(cid:12)(cid:12)(cid:12) ∂ V = 0 , (2.6)i.e., the variation of the metric tensor vanishes in the boundary ∂ V . We use the results [4],[7] δg αβ = − g αµ g βν δg µν , δg αβ = − g αµ g βν δg µν , (2.7) δ √− g = − √− gg αβ δg αβ , (2.8) δR αβγδ = ∇ γ ( δ Γ αδβ ) − ∇ δ ( δ Γ αγβ ) , (2.9) δR αβ = ∇ γ ( δ Γ γβα ) − ∇ β ( δ Γ γγα ) . (2.10)We give a detailed review for the variation principles in GR following [3],[4] and [7],. The variation ofthe Einstein-Hilbert term gives δS EH = Z V d x (cid:0) Rδ √− g + √− g δR (cid:1) . (2.11)Now with R = g αβ R αβ , we have that the variation of the Ricci scalar is δR = δg αβ R αβ + g αβ δR αβ . (2.12)2sing the Palatini’s identity (2.10) we can write [7]: δR = δg αβ R αβ + g αβ (cid:0) ∇ γ ( δ Γ γβα ) − ∇ β ( δ Γ γαγ ) (cid:1) , = δg αβ R αβ + ∇ σ (cid:0) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) (cid:1) , (2.13)where we have used the metric compatibility ∇ γ g αβ ≡ δS EH = Z V d x (cid:0) Rδ √− g + √− g δR (cid:1) , = Z V d x (cid:18) − Rg αβ √− g δg αβ + R αβ √− gδg αβ + √− g ∇ σ (cid:0) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) (cid:1)(cid:19) , = Z V d x √− g (cid:18) R αβ − Rg αβ (cid:19) δg αβ + Z V d x √− g ∇ σ (cid:0) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) (cid:1) . (2.14)Denoting the divergence term with δS B , δS B = Z V d x √− g ∇ σ (cid:0) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) (cid:1) , (2.15)we define V σ = g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) , (2.16)then the boundary term can be written as δS B = Z V d x √− g ∇ σ V σ . (2.17)Using Gauss-Stokes theorem [4],[7]: Z V d n x p | g |∇ µ A µ = I ∂ V d n − y ε p | h | n µ A µ , (2.18)where n µ is the unit normal to ∂ V . Using this we can write (2.17) in the following boundary term δS B = I ∂ V d y ε p | h | n σ V σ , (2.19)with V σ given in (2.16). The variation δ Γ σβα is obtained by using that Γ σβα is the Christoffel symbol (cid:8) σβα (cid:9) :Γ αβγ ≡ n αβγ o = 12 g ασ (cid:2) ∂ β g σγ + ∂ γ g σβ − ∂ σ g βγ (cid:3) , (2.20)getting δ Γ σβα = δ (cid:18) g σγ (cid:2) ∂ β g γα + ∂ α g γβ − ∂ γ g βα (cid:3)(cid:19) , = 12 δg σγ (cid:2) ∂ β g γα + ∂ α g γβ − ∂ γ g βα (cid:3) + 12 g σγ (cid:2) ∂ β ( δg γα ) + ∂ α ( δg γβ ) − ∂ γ ( δg βα ) (cid:3) . (2.21)From the boundary conditions δg αβ = δg αβ = 0 the variation (2.21) gives: δ Γ σβα (cid:12)(cid:12)(cid:12) ∂ V = 12 g σγ (cid:2) ∂ β ( δg γα ) + ∂ α ( δg γβ ) − ∂ γ ( δg βα ) (cid:3) , (2.22)and V µ (cid:12)(cid:12)(cid:12) ∂ V = g αβ (cid:20) g µγ (cid:2) ∂ β ( δg γα ) + ∂ α ( δg γβ ) − ∂ γ ( δg βα ) (cid:3)(cid:21) − g αµ (cid:20) g νγ ∂ α ( δg νγ ) (cid:21) , (2.23)3e can write V σ (cid:12)(cid:12)(cid:12) ∂ V = g σµ V µ (cid:12)(cid:12)(cid:12) ∂ V = g σµ g αβ (cid:20) g µγ (cid:2) ∂ β ( δg γα ) + ∂ α ( δg γβ ) − ∂ γ ( δg βα ) (cid:3)(cid:21) − g σµ g αµ (cid:20) g νγ ∂ α ( δg νγ ) (cid:21) , = 12 δ γσ g αβ (cid:2) ∂ β ( δg γα ) + ∂ α ( δg γβ ) − ∂ γ ( δg βα ) (cid:3) − δ ασ g νγ (cid:2) ∂ α ( δg νγ ) (cid:3) , = g αβ (cid:2) ∂ β ( δg σα ) − ∂ σ ( δg βα ) (cid:3) . (2.24)We now evaluate the term n σ V σ (cid:12)(cid:12) ∂ V by using for this that g αβ = h αβ + εn α n β , (2.25)then n σ V σ (cid:12)(cid:12)(cid:12) ∂ V = n σ ( h αβ + εn α n β )[ ∂ β ( δg σα ) − ∂ σ ( δg βα )] , = n σ h αβ [ ∂ β ( δg σα ) − ∂ σ ( δg βα )] , (2.26)where we use the antisymmetric part of εn α n β with ε = n µ n µ = ±
1. To the fact δg αβ = 0 in theboundary we have h αβ ∂ β ( δg σα ) = 0 [4]. Finally we get n σ V σ (cid:12)(cid:12)(cid:12) ∂ V = − n σ h αβ ∂ σ ( δg βα ) . (2.27)Thus the variation of the Einstein-Hilbert term is: δS EH = Z V d x √− g (cid:18) R αβ − Rg αβ (cid:19) δg αβ − I ∂ V d y ε p | h | h αβ ∂ σ ( δg βα ) n σ . (2.28)Now we consider the variation of the Gibbons-York-Hawking boundary term: δS GY H = 2 I ∂ V d y ε p | h | δK. (2.29)Using the definition of the trace of extrinsic curvature [4]: K = ∇ α n α , = g αβ ∇ β n α , = ( h αβ + εn α n β ) ∇ β n α , = h αβ ∇ β n α , = h αβ ( ∂ β n α − Γ γβα n γ ) , (2.30)the variation is δK = − h αβ δ Γ γβα n γ , = − h αβ g σγ (cid:2) ∂ β ( δg σα ) + ∂ α ( δg σβ ) − ∂ σ ( δg βα ) (cid:3) n γ , = − h αβ (cid:2) ∂ β ( δg σα ) + ∂ α ( δg σβ ) − ∂ σ ( δg βα ) (cid:3) n σ , = 12 h αβ ∂ σ ( δg βα ) n σ . (2.31)This comes from the variation δ Γ γβα evaluated in the boundary, and the fact that h αβ ∂ β ( δg σα ) = 0, h αβ ∂ α ( δg σβ ) = 0. Then we have for the variation of the Gibbons-York-Hawking boundary term: δS GY H = I ∂ V d y ε p | h | h αβ ∂ σ ( δg βα ) n σ . (2.32)4e see that this term exactly cancel the boundary contribution of the Einstein-Hilbert term. Now, if wehave a matter action defined by: S M = Z V d x √− g L M [ g αβ , ψ ] , (2.33)where ψ denotes the matter fields. The variation of this action takes the form: δS M = Z V d x δ ( √− g L M ) , = Z V d x (cid:18) ∂ L M ∂g αβ δg αβ √− g + L M δ √− g (cid:19) , = Z V d x √− g (cid:18) ∂ L M ∂g αβ − L M g αβ (cid:19) δg αβ , (2.34)as usual, defining the stress-energy tensor by: T αβ ≡ − ∂ L M ∂g αβ + L M g αβ = − √− g δS M δg αβ , (2.35)then: δS M = − Z V d x √− gT αβ δg αβ , (2.36)imposing the total variations remains invariant with respect to δg αβ . Finally the equations are writingas: 1 √− g δSδg αβ = 0 , = ⇒ R αβ − Rg αβ = κT αβ , (2.37)which corresponds to Einstein field equations in geometric units c = 1. f ( R ) gravity As we mentioned above the modified theories of gravity have been studied in order to explain among theaccelerated expansion of the universe. One of these theories is the modified f ( R ) gravity which consistsin add additional higher order terms of the Ricci scalar in the Einstein-Hilbert action [15],[16],[25]. Thereare three versions of f ( R ) gravity: Metric formalism, Palatini formalism and metric-affine formalism [25].Here we focus only in the metric formalism; for a detailed deduction of field equations in the Palatiniand the metric-affine formalism see [39],[40]. Again, we consider the space-time as a pair ( M , g ) with M a four-dimensional manifold and g αβ a metric on M . Now the lagrangian is an arbitrary function ofthe Ricci scalar L [ g αβ ] = f ( R ), the relation of the Ricci scalar and the metric tensor is given assuming aLevi-Civita connection of the manifold. i.e. a Christoffel symbol. This lagrangian was presented in [31]using augmented variational principles. The general action can be written as [38]: S mod = 12 κ (cid:0) S met + S ′ GY H (cid:1) + S M , (3.1)with the bulk term S met = Z V d x √− gf ( R ) , (3.2)and the Gibbons-York-Hawking like boundary term [30],[38] S ′ GY H = 2 I ∂ V d y ε p | h | f ′ ( R ) K, (3.3)5ith f ′ ( R ) = df ( R ) /dR . Again, S M represents the action associated with all the matter fields (2.33).We fixed the variation to the condition δg αβ (cid:12)(cid:12)(cid:12)(cid:12) ∂ V = 0 . (3.4)First, the variation of the bulk term is: δS met = Z V d x (cid:0) f ( R ) δ √− g + √− g δf ( R ) (cid:1) , (3.5)and the functional derivative of the f ( R ) term can be written as δf ( R ) = f ′ ( R ) δR. (3.6)Using the expression for the variation of the Ricci scalar: δR = δg αβ R αβ + ∇ σ (cid:0) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) (cid:1) , (3.7)where the variation of the term g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) is given in A. With this result the variation ofthe Ricci scalar becomes δR = δg αβ R αβ + ∇ σ (cid:0) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) (cid:1) , = δg αβ R αβ + g µν ∇ σ ∇ σ ( δg µν ) − ∇ σ ∇ γ ( δg σγ ) , = δg αβ R αβ + g αβ (cid:3) ( δg αβ ) − ∇ α ∇ β ( δg αβ ) . (3.8)Here we define (cid:3) ≡ ∇ σ ∇ σ and relabeled some indices. Putting the previous results together in thevariation of the modified action (3.5): δS met = Z V d x (cid:0) f ( R ) δ √− g + √− g f ′ ( R ) δR (cid:1) , = Z V d x (cid:18) − f ( R ) 12 √− g g αβ δg αβ + f ′ ( R ) √− g (cid:16) δg αβ R αβ + g αβ (cid:3) ( δg αβ ) − ∇ α ∇ β ( δg αβ ) (cid:17)(cid:19) , = Z V d x √− g (cid:18) f ′ ( R ) (cid:16) δg αβ R αβ + g αβ (cid:3) ( δg αβ ) − ∇ α ∇ β ( δg αβ ) (cid:17) − f ( R ) 12 g αβ δg αβ (cid:19) . (3.9)Now we will consider the next integrals: Z V d x √− gf ′ ( R ) g αβ (cid:3) ( δg αβ ) , Z V d x √− gf ′ ( R ) ∇ α ∇ β ( δg αβ ) . (3.10)We shall see that these integrals can be expressed differently performing integration by parts. For thiswe define the next quantities: M τ = f ′ ( R ) g αβ ∇ τ ( δg αβ ) − δg αβ g αβ ∇ τ ( f ′ ( R )) , (3.11)and N σ = f ′ ( R ) ∇ γ ( δg σγ ) − δg σγ ∇ γ ( f ′ ( R )) . (3.12)The combination g στ M τ + N σ is g στ M τ + N σ = f ′ ( R ) g αβ ∇ σ ( δg αβ ) − δg αβ g αβ ∇ σ ( f ′ ( R )) + f ′ ( R ) ∇ γ ( δg σγ ) − δg σγ ∇ γ ( f ′ ( R )) , (3.13)in the particular case f ( R ) = R , the previous combination reduces to the expression (2.16) with equation(A.11). The quantities M τ and N σ allow us to write the variation of the bulk term (3.9) in the followingway (for details see B): δS met = Z V d x √− g (cid:18) f ′ ( R ) R αβ + g αβ (cid:3) f ′ ( R ) − ∇ α ∇ β f ′ ( R ) − f ( R ) 12 g αβ (cid:19) δg αβ + I ∂ V d y ε p | h | n τ M τ + I ∂ V d y ε p | h | n σ N σ . (3.14)6n the next section we will work out with the boundary contribution from (3.14), and show how thisterms cancel with the variations of the S ′ GY H action. f ( R ) gravity We express the quantities M τ and N σ calculated in the boundary ∂ V . Is convenient to express them infunction of the variations δg αβ . Using the equation (2.7) in (3.11) and (3.12) yields : M τ = − f ′ ( R ) g αβ ∇ τ ( δg αβ ) + g αβ δg αβ ∇ τ ( f ′ ( R )) , (3.15)and N σ = − f ′ ( R ) g σµ g γν ∇ γ ( δg µν ) + g σµ g γν δg µν ∇ γ ( f ′ ( R )) . (3.16)To evaluate this quantities in the boundary we use the fact that δg αβ | ∂ V = δg αβ | ∂ V = 0, then the onlyterms not vanishing are the derivatives of δg αβ in the covariant derivatives. Hence we have M τ (cid:12)(cid:12)(cid:12)(cid:12) ∂ V = − f ′ ( R ) g αβ ∂ τ ( δg αβ ) , (3.17)and N σ (cid:12)(cid:12)(cid:12)(cid:12) ∂ V = − f ′ ( R ) g σµ g γν ∂ γ ( δg µν ) , (3.18)We now compute n τ M τ (cid:12)(cid:12) ∂ V and n σ N σ (cid:12)(cid:12) ∂ V which are the terms in the boundary integrals (3.14) n τ M τ (cid:12)(cid:12)(cid:12)(cid:12) ∂ V = − f ′ ( R ) n τ ( εn α n β + h αβ ) ∂ τ ( δg αβ ) , = − f ′ ( R ) n σ h αβ ∂ σ ( δg αβ ) , (3.19)where we rename the dummy index τ . In the other hand n σ N σ (cid:12)(cid:12)(cid:12)(cid:12) ∂ V = − f ′ ( R ) n σ ( h σµ + εn σ n µ )( h γν + εn γ n ν ) ∂ γ ( δg µν ) , = − f ′ ( R ) n µ ( h γν + εn γ n ν ) ∂ γ ( δg µν ) , = − f ′ ( R ) n µ h γν ∂ γ ( δg µν )= 0 , (3.20)where we have used that n σ h σµ = 0, ε = 1 and the fact that de tangential derivative h γν ∂ γ ( δg µν )vanishes. With this results the variation of the action S met becomes: δS met = Z V d x √− g (cid:18) f ′ ( R ) R αβ + g αβ (cid:3) f ′ ( R ) − ∇ α ∇ β f ′ ( R ) − f ( R ) 12 g αβ (cid:19) δg αβ − I ∂ V d y ε p | h | f ′ ( R ) n σ h αβ ∂ σ ( δg αβ ) . (3.21)We proceed with the boundary term S ′ GY H in the total action. The variation of this term gives δS ′ GY H = 2 I ∂ V d y ε p | h | (cid:0) δf ′ ( R ) K + f ′ ( R ) δK (cid:1) , = 2 I ∂ V d y ε p | h | (cid:0) f ′′ ( R ) δR K + f ′ ( R ) δK (cid:1) . (3.22)7sing the expression for the variation of K , equation (2.31), we can write δS ′ GY H = 2 I ∂ V d y ε p | h | (cid:18) f ′′ ( R ) δR K + 12 f ′ ( R ) h αβ ∂ σ ( δg βα ) n σ (cid:19) , = 2 I ∂ V d y ε p | h | f ′′ ( R ) δR K + I ∂ V d y ε p | h | f ′ ( R ) h αβ ∂ σ ( δg βα ) n σ . (3.23)We see that the second term in (3.23) cancels the boundary term in the variation (3.21), and in additionwe need to impose δR = 0 in the boundary. Similar argument is given in [38].Finally, with the variation of the matter action, given in (2.36), the total variation of the action ofmodified f ( R ) gravity is: δS mod = 12 κ Z V d x √− g (cid:18) f ′ ( R ) R αβ + g αβ (cid:3) f ′ ( R ) − ∇ α ∇ β f ′ ( R ) − f ( R ) g αβ (cid:19) δg αβ − Z V d x √− gT αβ δg αβ . (3.24)Imposing that this variation becomes stationary we have:1 √− g δS mod δg αβ = 0 = ⇒ f ′ ( R ) R αβ + g αβ (cid:3) f ′ ( R ) − ∇ α ∇ β f ′ ( R ) − f ( R ) g αβ = κT αβ , (3.25)which are the field equations in the metric formalism of f ( R ) gravity. We have obtained the field equations in the metric formalism of f ( R ) gravity by using the direct resultsfrom variational principles. The modified action in the metric formalism of f ( R ) gravity plus a Gibbons-York-Hawking like boundary term must be written as: S mod = 12 κ (cid:20)Z V d x √− g (cid:16) f ( R ) + 2 κ L M [ g αβ , ψ ] (cid:17) +2 I ∂ V d y ε p | h | f ′ ( R ) K (cid:21) , (4.1)with f ′ ( R ) = df ( R ) /dR and L M the lagrangian associated with all the matter fields. From the quantities M σ and N σ , defined in (3.11) and (3.12) respectively, we recovered GR plus Gibbons-York-Hawkingboundary term in the particular case f ( R ) = R . We see that including the boundary term, we have awell behaved mathematical problem setting both, δg αβ = 0 and δR = 0 in ∂ V . Acknowledgements:
The authors are grateful with the Observatorio Astron´omico Nacional, Bo-got´a, Colombia, where this paper was carried out. A. Guarnizo acknowledges the financial support bythe Programa de Becas para Estudiantes Sobresalientes de Posgrado, Universidad Nacional de Colombia.8 eferences [1] C. M. Will.
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Evaluation of the term g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) We already have calculated the variation δ Γ σβα : δ Γ σβα = 12 δg σγ (cid:2) ∂ β g γα + ∂ α g γβ − ∂ γ g βα (cid:3) + 12 g σγ (cid:2) ∂ β ( δg γα ) + ∂ α ( δg γβ ) − ∂ γ ( δg βα ) (cid:3) , (A.1)writing the partial derivatives for the metric variations with the expression for the covariant derivative: ∇ γ δg αβ = ∂ γ δg αβ − Γ σγα δg σβ − Γ σγβ δg ασ , (A.2)and also using that we are working in a torsion-free manifold i.e., the symmetry in the Christoffel symbolΓ αβγ = Γ αγβ , we can write: δ Γ σβα = 12 δg σγ (cid:2) ∂ β g γα + ∂ α g γβ − ∂ γ g βα (cid:3) + 12 g σγ (cid:2) ∇ β ( δg γα ) + ∇ α ( δg γβ ) − ∇ γ ( δg βα ) + Γ λβα δg γλ + Γ λαβ δg λγ (cid:3) , = 12 δg σγ (cid:2) ∂ β g γα + ∂ α g γβ − ∂ γ g βα (cid:3) + g σγ Γ λβα δg γλ + 12 g σγ (cid:2) ∇ β ( δg γα ) + ∇ α ( δg γβ ) − ∇ γ ( δg βα ) (cid:3) , (A.3)using equation (2.7) in the second term: δ Γ σβα = 12 δg σγ (cid:2) ∂ β g γα + ∂ α g γβ − ∂ γ g βα (cid:3) − δg µν g σγ g γµ g λν Γ λβα + 12 g σγ (cid:2) ∇ β ( δg γα ) + ∇ α ( δg γβ ) − ∇ γ ( δg βα ) (cid:3) , = δg σν g λν Γ λβα − δg µν δ σµ g λν Γ λβα + 12 g σγ (cid:2) ∇ β ( δg γα ) + ∇ α ( δg γβ ) − ∇ γ ( δg βα ) (cid:3) , = δg σν g λν Γ λβα − δg σν g λν Γ λβα + 12 g σγ (cid:2) ∇ β ( δg γα ) + ∇ α ( δg γβ ) − ∇ γ ( δg βα ) (cid:3) . (A.4)Then we have δ Γ σβα = 12 g σγ (cid:2) ∇ β ( δg αγ ) + ∇ α ( δg βγ ) − ∇ γ ( δg βα ) (cid:3) , (A.5)and similarly δ Γ γαγ = 12 g σγ (cid:2) ∇ α ( δg σγ ) (cid:3) . (A.6)However it is convenient to express the previous result in function of the variations δg αβ , we again use(2.7): δ Γ σβα = 12 g σγ (cid:2) ∇ β ( − g αµ g γν δg µν ) + ∇ α ( − g βµ g γν δg µν ) − ∇ γ ( − g βµ g αν δg µν ) (cid:3) , = − g σγ (cid:2) g αµ g γν ∇ β ( δg µν ) + g βµ g γν ∇ α ( δg µν ) − g βµ g αν ∇ γ ( δg µν ) (cid:3) , = − (cid:2) δ σν g αµ ∇ β ( δg µν ) + δ σν g βµ ∇ α ( δg µν ) − g βµ g αν g γσ ∇ γ ( δg µν ) (cid:3) , = − (cid:2) g αγ ∇ β ( δg σγ ) + g βγ ∇ α ( δg σγ ) − g βµ g αν ∇ σ ( δg µν ) (cid:3) , (A.7)where we write ∇ σ = g σγ ∇ γ . In a similar way: δ Γ γαγ = − g µν ∇ α ( δg µν ) . (A.8)11ow we compute the term g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) = − (cid:16)(cid:2) g αβ g αγ ∇ β ( δg σγ ) + g αβ g βγ ∇ α ( δg σγ ) − g αβ g βµ g αν ∇ σ ( δg µν ) (cid:3) (A.9) − (cid:2) g ασ g µν ∇ α ( δg µν ) (cid:3)(cid:17) , = − (cid:16)(cid:2) δ βγ ∇ β ( δg σγ ) + δ αγ ∇ α ( δg σγ ) − δ αµ g αν ∇ σ ( δg µν ) (cid:3) − (cid:2) g µν g ασ ∇ α ( δg µν ) (cid:3)(cid:17) , = − (cid:16)(cid:2) ∇ γ ( δg σγ ) + ∇ γ ( δg σγ ) − g µν ∇ σ ( δg µν ) (cid:3) − (cid:2) g µν ∇ σ ( δg µν ) (cid:3)(cid:17) , = − (cid:16) ∇ γ ( δg σγ ) − g µν ∇ σ ( δg µν ) (cid:17) , (A.10)then we have, g αβ ( δ Γ σβα ) − g ασ ( δ Γ γαγ ) = g µν ∇ σ ( δg µν ) − ∇ γ ( δg σγ ) . (A.11) B Integrals with M τ and N σ Taking the covariant derivative in M σ : ∇ τ M τ = ∇ τ (cid:0) f ′ ( R ) g αβ ∇ τ ( δg αβ ) (cid:1) − ∇ τ (cid:0) δg αβ g αβ ∇ τ ( f ′ ( R )) (cid:1) , = ∇ τ ( f ′ ( R )) g αβ ∇ τ ( δg αβ ) + f ′ ( R ) g αβ (cid:3) ( δg αβ ) − ∇ τ ( δg αβ ) g αβ ∇ τ ( f ′ ( R )) − δg αβ g αβ (cid:3) ( f ′ ( R )) , = f ′ ( R ) g αβ (cid:3) ( δg αβ ) − δg αβ g αβ (cid:3) ( f ′ ( R )) . (B.1)Here we have used the metric compatibility ∇ τ g αβ = 0, integrating this expression Z V d x √− g ∇ τ M τ = Z V d x √− gf ′ ( R ) g αβ (cid:3) ( δg αβ ) − Z V d x √− gδg αβ g αβ (cid:3) ( f ′ ( R )) , (B.2)using again the Gauss-Stokes theorem (2.18), the first integral can be written as a boundary term: I ∂ V d y ε p | h | n τ M τ = Z V d x √− gf ′ ( R ) g αβ (cid:3) ( δg αβ ) (cid:1) − Z V d x √− gδg αβ g αβ (cid:3) ( f ′ ( R )) , (B.3)then we can write: Z V d x √− gf ′ ( R ) g αβ (cid:3) ( δg αβ ) = Z V d x √− gδg αβ g αβ (cid:3) ( f ′ ( R )) + I ∂ V d y ε p | h | n τ M τ . (B.4)In a similar way, taking the covariant derivative of N σ : ∇ σ N σ = ∇ σ (cid:0) f ′ ( R ) ∇ γ ( δg σγ ) (cid:1) − ∇ σ (cid:0) δg σγ ∇ γ ( f ′ ( R )) (cid:1) , = ∇ σ ( f ′ ( R )) ∇ γ ( δg σγ ) + f ′ ( R ) ∇ σ ∇ γ ( δg σγ ) − ∇ σ ( δg σγ ) ∇ γ ( f ′ ( R )) − δg σγ ∇ σ ∇ γ ( f ′ ( R )) , = f ′ ( R ) ∇ σ ∇ β ( δg σβ ) − δg σβ ∇ σ ∇ β ( f ′ ( R )) , (B.5)integrating: Z V d x √− g ∇ σ N σ = Z V d x √− gf ′ ( R ) ∇ σ ∇ β ( δg σβ ) − Z V d x √− gδg σβ ∇ σ ∇ β ( f ′ ( R )) , (B.6)using again the Gauss-Stokes theorem we can write: Z V d x √− gf ′ ( R ) ∇ σ ∇ β ( δg σβ ) = Z V d x √− gδg σβ ∇ σ ∇ β ( f ′ ( R )) + I ∂ V d y ε p | h | n σ N σ ..