Equations of Motion in Nonlocal Modified Gravity
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- Βαρβάρα Ταμτάκος
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1 Equations of Motion in Nonlocal Modified Gravity Jelena Grujić Teachers Training Faculty, University of Belgrade Kraljice Natalije 43, Belgrade, SERBIA Abstract We consider nonlocal modified gravity without matter, where nonlocality is of the form H(RF( G(R. Equations of motion are usually very complex and their derivation is not available in the research papers. In this paper we give the corresponding derivation in detail. Key words: Modified gravity, nonlocal gravity, variational principle, equations of motion. 1. Introduction Einstein s theory of gravity, which is General Relativity, is given by Einstein- Hilbert action which Lagrangian contains Ricci scalar in the linear form. The first attempts to modify Einstein s theory of gravity started soon after its appearance and they were mainly inspired by investigation of possible mathematical generalizations. The discovery of accelerating expansion of the Universe in 1998 has not so far generally accepted theoretical explanation and it has produced an intensive activity in general relativity modification during the last decade. In this paper, we consider nonlocal modification of gravity (for a review, see [1,, 3, 4]. Under nonlocal modification of gravity we understand replacement of the scalar curvature R in the Einstein-Hilbert action by a suitable function F (R,, where = µ µ is d Alembert operator and µ denotes the covariant derivative. Here, nonlocality means that Lagrangian contains an infinite number of space-time derivatives, i.e. derivatives up to an infinite order in the form of d Alembert operator which is argument of an analytic function. We consider a class of nonlocal gravity models without matter, given by the next action S = d 4 x ( R Λ g 16πG + CH(RF( G(R, (1 address: jelenagg@gmail.com 181
2 18 J. Grujic where F( = f n n, H and G are differentiable functions of the scalar n=0 curvature R, Λ is cosmological constant and C is a constant. The corresponding Einstein equations of motion are rather complex and their derivation is not available in the research papers. In this paper we will present their derivation. In order to obtain equations of motion for g µν we must find the variation of the action (1 with respect to metric g µν.. Some preliminaries At the beginning we would like to prove the following identities: δg = gg µν δg µν = gg µν δg µν, ( δ g = 1 g µν gδg µν, (3 δr = R µν δg µν + g µν δg µν µ ν δg µν, (4 g µν x σ = gµα Γ ν σα g να Γ µ σα, (5 Γ µ µν = x ν ln g, (6 = 1 µ gg µν ν, (7 g where g is the determinant of the metric tensor g µν. The determinant g can be expressed in the following way: g = g µ0 G (µ,0 + g µ1 G (µ, g µn 1 G (µ,n 1, (8 where G (µ,ν is the corresponding algebraic cofactor and n is space-time dimension. If we replace elements of the µ row with the elements of the ν row then the determinant g becomes equal zero, i.e. 0 = g ν0 G (µ,0 + g ν1 G (µ, g νn 1 G (µ,n 1. (9 From this expression we have or G (α,ν g µν g where δµ α is Kronecker s delta symbol. Metric tensor g αν is defined by g µν G (α,ν = gδ α µ, (10 = δ α µ, (11 g αν = G(α,ν. (1 g
3 Equations of motion in nonlocal modified gravity 183 From (10 (here we set α = µ we get From the last equation and from (1 we obtain δg = G (µ,ν δg µν. (13 δg = gg µν δg µν. (14 From g µν g µν = n we have g µν δg µν = g µν δg µν which completes the proof for (. We have Finally, using equation ( we obtain (3. We know that by definition where R µν = R η µην is the Ricci tensor. The Riemann tensor R µ νγη is given by δ g = 1 δg. (15 g R = g µν R µν, (16 R µ νγη = Γµ νη x γ Γµ γν x η + Γµ γσγ σ νη Γ µ σηγ σ γν, (17 where Γ µ νγ is the Christoffel symbol It can be shown that Γ µ νγ = 1 gµσ ( g σγ x ν + g σν x γ g νγ. (18 xσ δg µν = g µα g βν δg αβ, (19 δg µν = g µα g βν δg αβ, (0 δr νγη µ = γ δγ µ ην η δγ µ γν, (1 δr µν = γ δγ γ µν ν δγ γ γµ, ( δγ σ νµ = 1 δgσγ ( g γµ x ν + g νγ x µ g νµ x γ + 1 gσγ ( δg γµ x ν + δg νγ x µ δg νµ x γ. (3 From equation (16 we have that the variation of the Ricci scalar is δr = δg µν R µν + g µν δr µν. (4
4 184 J. Grujic Using equation ( we get δr = δg µν R µν + g µν ( γ δγ γ νµ ν δγ γ γµ = δg µν R µν + σ (g µν δγ σ νµ g µσ δγ γ µγ. (5 We have used γ g µν = 0 (the metric compatibility and relabeled some dummy indices. Now we need to compute the term g µν δγ σ νµ g µσ δγ γ µγ. We have γ δg µν = δg µν x γ Γ σ γµδg σν Γ σ γνδg µσ. (6 Using the last equation, equation (3 and the symmetry in the Christoffel symbol Γ µ νγ = Γ µ γν we obtain δγ σ νµ = 1 δgσγ( g γµ x ν + g γν x µ g νµ x γ + 1 gσγ( ν δg γµ + µ δg γν γ δg νµ + Γ λ νµδg γλ + Γ λ µνδg λγ = 1 δgσγ( g γµ x ν + g γν x µ g νµ x γ + g σγ Γ λ νµδg γλ + 1 gσγ( ν δg γµ + µ δg γν γ δg νµ. (7 Now using equation (19 in the second term we get δγ σ νµ = 1 δgσγ( g γµ x ν + g γν x µ g νµ x γ δg αβ g σγ g γα g λβ Γ λ νµ + 1 gσγ( ν δg γµ + µ δg γν γ δg νµ = δg σβ g λβ Γ λ νµ δg αβ δ σ αg λβ Γ λ νµ + 1 gσγ( ν δg γµ + µ δg γν γ δg νµ = δg σβ g λβ Γ λ νµ δg σβ g λβ Γ λ νµ + 1 gσγ( ν δg γµ + µ δg γν γ δg νµ. (8 Then we have δγ σ νµ = 1 gσγ ( ν δg µγ + µ δg νγ γ δg νµ. (9 Similarly, δγ γ µγ = 1 gσγ µ δg σγ. (30
5 Equations of motion in nonlocal modified gravity 185 In order to express the previous result as function of the variations δg µν we again use (19 and obtain δγ σ νµ = 1 gσγ( ν ( g µα g γβ δg αβ + µ ( g να g γβ δg αβ γ ( g να g µβ δg αβ = 1 gσγ (g µα g γβ ν δg αβ + g να g γβ µ δg αβ g να g µβ γ δg αβ = 1 (δσ β g µα ν δg αβ + δ σ β g να µ δg αβ g να g µβ g γσ γ δg αβ = 1 (g µγ ν δg σγ + g νγ µ δg σγ g να g µβ σ δg αβ, (31 where σ = g σγ γ. Similarly, δγ γ µγ = 1 g αβ µ δg αβ. (3 Finally, we obtain g µν δγ σ νµ g µσ δγ γ µγ = 1 ( g µν g µγ ν δg σγ + g µν g νγ µ δg σγ g µν g να g µβ σ δg αβ g µσ g αβ µ δg αβ = 1 (δ ν γ ν δg σγ + δ γ µ µ δg σγ δ αg µ µβ σ δg αβ g µσ g αβ µ δg αβ = 1 ( γ δg σγ + γ δg σγ g αβ σ δg αβ g αβ σ δg αβ = 1 ( γ δg σγ g αβ σ δg αβ. (33 Then we have g µν δγ σ νµ g µσ δγ γ µγ = g αβ σ δg αβ γ δg σγ. (34 Substituting this in (5 we obtain the variation of the scalar curvature δr = δg µν R µν + g αβ σ σ δg αβ σ γ δg σγ = δg µν R µν + g µν δg µν µ ν δg µν. (35 Now we want to prove the equation (5. Using the identity g µα g να = δ µ ν, we obtain g µα x σ g να = g µα g να x σ. (36
6 186 J. Grujic Using the definition of the Christoffel symbols it is easy to show Substituting this in (36 we obtain g να x σ = g βαγ β νσ + g βν Γ β ασ. (37 g µα x σ g να = g µα (g βα Γ β νσ + g βν Γ β ασ. (38 If we multiply the last equation with g νγ and using g νγ g να = δ γ α we have g µγ x σ = gµα g βα g νγ Γ β νσ g µα g βν g νγ Γ β ασ = g νγ Γ µ νσ g µα Γ γ ασ. (39 Finally, if we replace ν with α and γ with ν in the last equation we obtain the equation (5. In order to prove the equation (6 we use the equation (10 From this and using G(µ,α g µν = 0 and g µα g µν Using g αβ = G(α,β g Substituting g αβ x µ g = g µα G (µ,α. (40 = δ ν α we have g G (µ,α = g µα + G (µ,α g µα = G (µ,ν, (41 g µν g µν g µν g x µ = g g αβ g αβ x µ the last equation becomes = G(α,β g αβ x µ. (4 g x µ = ggαβ g αβ x µ. (43 from (37 we obtain From this we obtain equation (6 In order to prove (7 we write g x µ = gγα αµ + gγ β βµ = gγα αµ. (44 Γ α αµ = x µ ln g. (45 φ = µ µ φ = µ (g µρ µρ φ ρ φ = µ (g x ρ = φ (gµρ x µ x ρ + Γµ νρ φ µνg x ρ, (46
7 Equations of motion in nonlocal modified gravity 187 where φ is any scalar function. Using (6 we obtain φ = φ (gµν x µ x ν + = φ (gµν x µ x ν + x ν (ln gg x µ (ln gg νρ φ x ρ µν φ x ν. (47 On the other hand, we have 1 g x µ ( µν φ gg x ν = 1 ( g x µ ( µν φ gg x ν + g gµν x µ = x µ (ln µν φ gg x ν + gµν x µ = x µ (ln gg µν φ x ν + x φ x ν + φ gµν x µ x ν µ (gµν φ φ x ν + gg µν φ x µ x ν. (48 xν From (47 and (48 we can conclude φ = 1 g µ gg µν ν φ for any scalar function φ. 3. Derivation of equations of motion Let s introduce the following actions S 0 = S 1 = (R Λ g d 4 x, (49 H(RF( G(R g d 4 x. (50 then the variation of the action (1 can be expressed as δs = 1 16πG δs 0 + CδS 1. (51
8 188 J. Grujic 3.1. The variation of S 0 The variation of S 0 can be found as follows δs 0 = δ((r Λ g d 4 x = δ(r g d 4 x Λ δ g d 4 x = ( gδr + Rδ g d 4 x + Λ g µν gδg µν d 4 x = ( gδ(g µν R µν 1 R gg µν δg µν d 4 x + Λ g µν gδg µν d 4 x = R µν gδg µν d 4 x + g µν gδr µν d 4 x 1 Rg µν gδg µν d 4 x + Λ g µν gδg µν d 4 x = (R µν 1 Rg µν gδg µν d 4 x + Λ g µν gδg µν d 4 x + g µν δr µν g d 4 x = 0. (5 Here we have used equation (3. Now we want to show that g µν δr µν g d 4 x = 0. We first introduce W ν = g µα δγ ν µα + g µν δγ α µα. (53 We have 1 g x ν ( gw ν = W ν x ν + W ν 1 g g x ν. (54 Using equation (6 we get 1 g x ν ( gw ν = x ν (gµα δγ ν µα + + ( g µα δγ ν µα + g µν δγ α µαγ β νβ x ν (gµν δγ α µα + gµν = gµα x ν δγν µα g µα δ Γν µα x ν x ν δγα µα + g µν δ Γα µα x ν + ( g µα δγ ν µα + g µν δγ α µαγ β νβ. (55
9 Equations of motion in nonlocal modified gravity 189 Now using equation (5 we obtain 1 g x ν ( gw ν = g αβ Γ µ νβ δγν µα + g µβ Γ α νβ δγν µα g βν Γ µ νβ δγα µα g µβ Γ ν νβ δγα µα g µα δ Γν µα x ν + gµν δ Γα µα x ν g µα Γ β νβ δγν µα + g µν Γ β νβ δγα µα = g µν( δ Γα µν x α + δ Γα µα x ν + Γβ αµδγ α βν + Γα νβ δγβ µα ( = g µν δ Γα µν x α + Γα µα x ν + Γβ αµγ α βν Γα βα Γβ µν Here we have relabeled some dummy indices. Finally, we have Γ β νµδγ α βα Γα βα δγβ µν = g µν δr µν. (56 g µν δr µν = 1 g x ν ( gw ν, (57 g µν δr µν gd 4 x = x ν ( gw ν d 4 x. (58 Using the Gauss-Stokes theorem we have x ν ( gw ν d 4 x = W ν dσ ν, (59 Σ where Σ is boundary of a space-time region of integration. Since δg µν = 0 and δ( g µν x = 0 at the boundary of region of integration we α have W ν Σ = 0. Then we have Σ W ν dσ ν = 0 and finally g µν δr µν gd 4 x = 0. (60 Substituting this in (5 we obtain the variation of S 0 δs 0 = G µν gδg µν d 4 x + Λ g µν gδg µν d 4 x, (61 where G µν = R µν 1 Rg µν is the Einstein tensor.
10 190 J. Grujic 3.. Preliminaries for the variation of S 1 1. Using the equation (4, for any scalar function h we have hδr g d 4 x = (hr µν δg µν + hg µν δg µν h µ ν δg µν g d 4 x. (6 The second and third term in this formula can be transformed in the following way: hg µν δg µν g d 4 x = g µν hδg µν g d 4 x, (63 h µ ν δg µν g d 4 x = µ ν h δg µν g d 4 x. (64 To prove the first of these equations we use the Stokes s theorem and obtain hg µν δg µν g d 4 x = hg µν α α δg µν g d 4 x = α (hg µν α δg µν g d 4 x = α α (hg µν δg µν g d 4 x = g µν α α h δg µν g d 4 x = g µν h δg µν g d 4 x. (65 Here we have used γ g µν = 0 and α α = α α = to obtain the last integral. To obtain the second equation we first introduce vector From the above expression we have N µ = h ν δg µν ν hδg µν. (66 µ N µ = µ (h ν δg µν ν hδg µν = µ h ν δg µν + h µ ν δg µν µ ν h δg µν ν h µ δg µν = h µ ν δg µν µ ν h δg µν. (67 Integrating µ N µ yields
11 Equations of motion in nonlocal modified gravity 191 µ N µ g d 4 x = Σ N µ n µ dσ = 0, (68 where Σ is boundary of region of integration and n µ is the unit normal vector. Since N µ Σ = 0 we have that the last integral is zero, which completes the proof. Finally, we obtain: hδr g d 4 x = (hr µν + g µν h µ ν h δg µν g d 4 x. (69. Let θ and ψ be scalar functions such that δψ Σ = 0. Then we have θδ ψ g d 4 x = θ α δ( gg αβ β ψ d 4 x ( 1 + θδ α ( gg αβ β ψ g d 4 x g = α (θδ( gg αβ β ψ d 4 x α θ δ( gg αβ β ψ d 4 x + 1 θg µν ψδg µν g d 4 x. (70 It is easy to see that α (θδ( gg αβ β ψ d 4 x = 0. From this result it follows θδ ψ g d 4 x = g αβ α θ β ψδ( g d 4 x α θ β ψδg αβ g d 4 x g αβ g α θ β δψ d 4 x + 1 θg µν ψδg µν g d 4 x = 1 g αβ α θ β ψg µν δg µν g d 4 x µ θ ν ψδg µν g d 4 x β (g αβ g α θ δψ d 4 x + β (g αβ g α θ δψ d 4 x + 1 g µν θ ψδg µν g d 4 x = 1 g αβ α θ β ψg µν δg µν g d 4 x µ θ ν ψδg µν g d 4 x + θ δψ g d 4 x + 1 g µν θ ψδg µν g d 4 x. (71
12 19 J. Grujic At the end we have that θδ ψ g d 4 x = 1 g αβ α θ β ψg µν δg µν g d 4 x µ θ ν ψδg µν g d 4 x + θ δψ g d 4 x + 1 g µν θ ψδg µν g d 4 x. ( The variation of S 1 Now, after this preliminary work we can get the variation of S 1. The variation of S 1 can be expressed as δs 1 = H(RF( G(Rδ( g d 4 x + δ(h(rf( G(R g d 4 x + H(Rδ(F( G(R g d 4 x. (73 Let us introduce notation K µν = µ ν g µν, Φ = H (RF( G(R + G (RF( H(R, (74 where denotes derivative with respect to R. For the first two integrals in the last equation we have I 1 = H(RF( G(Rδ( g d 4 x = 1 g µν H(RF( G(Rδg µν g d 4 x, (75 I = δ(h(rf( G(R g d 4 x = H (RδR F( G(R g d 4 x. Substituting h = H (R F( G(R in equation (69 we obtain I = ( R µν H (RF( G(R K µν ( H (RF( G(R δg µν g d 4 x. (76 The third integral can be divided into linear combination of the following integrals J n = H(Rδ( n G(R g d 4 x. (77
13 Equations of motion in nonlocal modified gravity 193 J 0 is the integral of the same form as I so ( J 0 = R µνg ( (RH(R K µν G (RH(R δg µν g d 4 x. (78 For n > 0, we can find J n using (7. In the first step we take θ = H(R and ψ = n 1 G(R and obtain J n = 1 g αβ α H(R β n 1 G(Rg µν δg µν g d 4 x µ H(R ν n 1 G(Rδg µν g d 4 x + H(R δ n 1 G(R g d 4 x + 1 g µν H(R n G(Rδg µν g d 4 x. (79 In the second step we take θ = H(R and ψ = n G(R and get the third integral in this formula, etc. Using (7 n times we obtain J n = 1 n 1 n n 1 g µν g αβ α l H(R β n 1 l G(Rδg µν g d 4 x µ l H(R ν n 1 l G(Rδg µν g d 4 x g µν l H(R n l G(Rδg µν g d 4 x + ( R µν G (R n H(R K µν ( G (R n H(R δg µν g d 4 x. (80 Using the equation (69 we obtain the last integral in the above formula. Finally, we can put everything together and obtain δs 1 = I 1 + I + f n J n n=0 = 1 g µν H(RF( G(Rδg µν g d 4 x + (R µν Φ K µν Φ δg µν g d 4 x + 1 n=1 g µν g αβ α l H(R β n 1 l G(Rδg µν g d 4 x f n n 1
14 194 J. Grujic + 1 n=1 f n n 1 n=1 f n n 1 µ l H(R ν n 1 l G(Rδg µν g d 4 x g µν l H(R n l G(Rδg µν g d 4 x. ( The variation of S and the equations of motion We have δs = Gµν + Λg µν 16πG δgµν g d 4 x + CδS 1. (8 From δs = 0 we obtain the equations of motion (EOM. We can write the EOM in the form G µν + Λg ( µν + C 1 16πG g µνh(rf( G(R + (R µν Φ K µν Φ + 1 n 1 ( f n gµν g αβ α l H(R β n 1 l G(R n=1 µ l H(R ν n 1 l G(R + g µν l H(R n l G(R = 0, (83 where K µν and Φ are given by expressions (74. In the case when the metric is homogeneous and isotropic, i.e. Friedmann- Lemaître-Robertson-Walker (FLRW metric, the last equation is equivalent to the following system (trace and 00 component: 4Λ R ( 16πG + C H(RF( G(R + (RΦ + 3 Φ (84 n 1 ( + f n µ l H(R µ n 1 l G(R + l H(R n l G(R = 0, n=1 G 00 + Λg ( 00 16πG + C 1 g 00H(RF( G(R + (R 00 Φ K 00 Φ + 1 n 1 ( f n g00 g αβ α l H(R β n 1 l G(R n=1 0 l H(R 0 n 1 l G(R + g 00 l H(R n l G(R = 0. (85
15 Equations of motion in nonlocal modified gravity Conclusion In this paper we have considered a nonlocal gravity model without matter given by the action in the form S = d 4 x ( R Λ g 16πG + CH(RF( G(R. (86 We have derived the equations of motion for this action. In many research papers there are equations of motion which are a special case of our equations. In the case H(R = G(R = R we obtain S = d 4 x ( R Λ g 16πG + CRF( R. (87 Studies of this model can be found in [, 3, 4, 8]. The case H(R = R 1 and G(R = R we analyze in [10]. Investigation of equations of motion and finding its solutions is a very difficult task. Using ansätze we can simplify the problem and get some solutions. For some examples of ansätze see [7, 9]. Acknowledgements This investigation is supported by Ministry of Education and Science of the Republic of Serbia, grant No I would like to thank Branko Dragovich, Zoran Rakić and Ivan Dimitrijević for useful discussions. References [1] S. Nojiri, S. D. Odintsov, Unified cosmic history in modified gravity: from F (R theory to Lorentz non-invariant models, Phys. Rept. 505 ( [arxiv: v4 [gr-qc]]. [] T. Biswas, T. Koivisto, A. Mazumdar, Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity, JCAP 1011 ( [arxiv: v [hep-th]]. [3] A. S. Koshelev, S. Yu. Vernov, On bouncing solutions in non-local gravity, Phys. Part. Nuclei 43, (01 [arxiv:10.189v1 [hep-th]]. [4] T. Biswas, A. S. Koshelev, A. Mazumdar, S. Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology, JCAP 08 (01 04, [arxiv: v [astro-ph.co]]. [5] M. Pantic, Introduction to Einstein s theory of gravity (Novi Sad, 005, in Serbian. [6] A. Guarnizo, L. Castaneda, J. M. Tejeiro, Boundary Term in Metric f(r Gravity: Field Equations in the Metric Formalism, (010 [arxiv: v4 [gr-qc]]. [7] I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, On modified gravity, Springer Proceedings in Mathematics and Statistics 36 (013 [arxiv:10.35v [hepth]].
16 196 J. Grujic [8] I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, New cosmological solutions in nonlocal modified gravity, to appear in Romanian Journal of Physics [arxiv: [gr-qc]]. [9] I. Dimitrijevic, Some ansätze in nonlocal modified gravity, in these proceedings. [10] I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic, to be published.
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