Holographic entanglement entropy of deSitter braneworld with Lovelock
aa r X i v : . [ h e p - t h ] F e b Prog. Theor. Exp. Phys. , 00000 (10 pages)DOI: 10.1093 / ptep/0000000000 Holographic entanglement entropy of deSitterbraneworld with Lovelock
Kouki Kushihara , Keisuke Izumi , , and Tetsuya Shiromizu , Department of Mathematics, Nagoya University, Nagoya 464-8602, Japan Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We examine the deSitter entropy in the braneworld model with the Gauss-Bonnet/Lovelock terms. Then, we can see that the deSitter entropy computed throughthe Euclidean action exactly coincides with the holographic entanglement entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index E0
1. Introduction
In the development of adS/CFT correspondence [1–3], a remarkable one is the Ryu-Takayanagi proposal for the holographic entanglement entropy [4]. This is regarded as anatural extension of the Bekenstein-Hawking entropy for the black hole to general casesbased on the holographic aspect.On the other hand, the braneworld model inspired by string theory also has the holographicfeature [5, 6]. Therefore, it is natural to consider the holographic entanglement entropy inthe braneworld context too. Recent hybrid formulation of adS/CFT(or adS/BCFT [7]) andbraneworld, say Island formula, may be able to offer the solution to the information lossparadox in black hole evapolation [8] (See also Refs [9–11]). In this sense, the braneworldsetup contributes to understanding the quantum gravity.In this paper, we revisit the holographic entanglement entropy of the deSitter braneworld(See Ref. [12] for black hole). In Ref. [13], the authors showed the exact agreement betweenthe deSitter entropy computed from the Eucliedan path integral [14] and the Ryu-Takayanagiformula for holographic entanglement entropy (See also [15]). However, they also founds adisagreement between them in the braneworld model with the Gauss-Bonnet term. Thisis not surprising result because the Ryu-Tayakanagi formula should be improved for thehigher derivative theories. Indeed, motivated by the formula for the black hole entropy inthe Lovelock gravity theory [16], the authors in Ref. [17] proposed the new formula whichhas the correction terms to the Ryu-Takayanagi formula. Then, our purpose is to confirmthat the formula given in Ref. [17] coincides with the deSitter entropy in braneworld withthe Gauss-Bonnet/Lovelock terms.The remaining part of this paper is organized as follows. In Sect. 2, we describe thebraneworld model with the Gauss-Bonnet term and deSitter brane in the n -dimensional anti-deSitter spacetime. We give the detail of the calculation for the deSitter entropy through theEuclidean path integral and the holographic entanglement entropy. In Sect. 3, the analysis © The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. nd results in the braneworld model with the Lovelock gravity are shown. Finally, in Sect. 4,we give short summary.
2. Braneworld with Gauss-Bonnet
In this paper, we consider the Z -symmetric braneworld model in the anti-deSitter bulk. Thebulk Gauss-Bonnet term [18] is analysed first. The system is composed of the n -dimensionalbulk ( M ± n , g MN ) and the brane ( M n − , q µν ) (Ref. [13] focused on the n = 5 case. Here, weconsider any dimensions with n ≥ S = 116 πG n Z M + n ∪ M − n d n x √− g (cid:16) R −
2Λ + βℓ L GB (cid:17) + Z M n − d n − x √− q (cid:16) − σ + 116 πG n [ Q ] − (cid:17) , (1)where G n is the n -dimensional Newton constant, R is the n -dimensional Ricci scalar, Λ is anegative cosmlogical constant, ℓ is supposed to be the anti-deSitter curvature length and β is a dimensionless constant. Here, the Gauss-Bonnet Lagrangian L GB is given as L GB = R − R AB R AB + R ABCD R ABCD (2)and the gravitational surface term Q is written in Q = 2 K + βℓ ( J − ( n − G µν K µν ) , (3)where ( n − G µν is the ( n − J is the trace of J µν defined by J µν = −
13 (2 K µα K νβ K αβ − KK µα K αν − K µν K αβ K αβ + K K µν ) (4)and K µν is the extrinsic curvature of M n − (whose the normal direction is taken to outwardfor M + n ). Supposing that the locus of the brane is y = 0 in the Gaussian normal coordinate( y, x µ ) around the brane, [ F ] − is defined by[ F ] − := lim y → +0 F − lim y →− F. (5)Then, the bulk field equation is G MN + Λ g MN + βℓ H MN = 0 , (6)where H MN = RR MN − R MK R KN − R KL R MKNL + R MKLP R KLPN − g MN L GB . (7)The junction condition is[ K µν − δ µν K ] − + βℓ J µν − δ µν J − P µανβ K αβ ] − = 8 πG n τ µν , (8)where P µανβ = ( n − R µανβ − ( n − R µ [ ν q β ] α + 2 ( n − R α [ ν q β ] µ + ( n − Rq µ [ ν q β ] α (9)and τ µν is the energy-momentum tensor on the brane. Since we focus on the vacuum brane, τ µν = − σq µν , where σ is the brane tension. ereafter we consider the deSitter brane in the n -dimensional anti-deSitter spacetime(adS n ) which is the solution to the current model. The bulk metric is given by [19] ds = dr + ( ℓH ) sinh ( r/ℓ )[ − dt + H − cosh ( Ht ) d Ω n − ]= dr + ( ℓH ) sinh ( r/ℓ )[ − (1 − H ρ ) dT + (1 − H ρ ) − dρ + ρ d Ω n − ] , (10)where ℓ is the curvature length of adS n and H is the Hubble constant on the brane. Then,supposing that the brane is located at r = r , we see H − = ℓ sinh( r /ℓ ) . (11)The bulk field equation and junction condition imply usΛ = − ( n − n − ℓ (cid:18) − β ( n − n − (cid:19) (12)and ( n − ℓ cosh( r /ℓ ) (cid:20) − β ( n − n − (cid:18) − ( r /ℓ ) (cid:19)(cid:21) = 4 πG n σ, (13)respectively. In this subsection, we will give the deSitter entropy in the braneworld with the Gauss-Bonnetterm. One can compute the deSitter entropy through the Euclidean action I E = 116 πG n Z M + n ∪ M − n d n x √ g (cid:16) − R − βℓ L GB (cid:17) + Z M n − d n − x √ q (cid:16) σ − [ Q ] − πG n (cid:17) = ( n − − β ( n − n − πG n ℓ Z M + n ∪ M − n d n x √ g − coth( r /ℓ )8 πG n ℓ (cid:20) − β ( n − n − (cid:18) − ( r /ℓ ) (cid:19)(cid:21) Z M n − d n − x √ q = − ( n − ℓ n − πG n Ω n − (cid:20) − β ( n − n − (cid:21) Z r /ℓ dx sinh n − x + β ( n −
3) sinh n − ( r /ℓ ) cosh( r /ℓ ) ! , (14)where we used[ Q ] − = 4( n − ℓ coth( r /ℓ ) (cid:20) − β ( n − n − (cid:18) − ( r /ℓ ) (cid:19)(cid:21) . (15)and Ω n − is the surface area of the n dimensional unit sphere,Ω n − = 2 π n/ Γ( n/ . (16) hen, we see that the deSitter entropy in the braneworld with the Gauss-Bonnet term isgiven by S dS = − I E = ( n − ℓ n − πG n Ω n − (cid:20) − β ( n − n − (cid:21) Z r /ℓ dx sinh n − x + β ( n −
3) sinh n − ( r /ℓ ) cosh( r /ℓ ) ! . (17) For the current setup, following Ref. [16], we consider S JM = 14 G n Z Γ + ∪ Γ − d n − x √ h (cid:16) βℓ ( n − R (cid:17) + 12 G n Z ∂ Γ d n − x √ p βℓ ( n − k ] − , (18)where h ij is the induced metric of ( n − ± with ∂ Γ + = ∂ Γ − =: ∂ Γ, ( n − R is the Ricci scalar of h ij , p AB is the induced metric of ∂ Γ and ( n − k is the extrinsiccurvature of ∂ Γ. We suppose Γ + ⊂ M + n , Γ − ⊂ M − n and ∂ Γ ⊂ M n − in the braneworld setup.When β = 0, Eq. (18) becomes to be proportional to the volume of Γ + ∪ Γ − .Then, one takes the variation of Γ for S JM and the minimum value gives us the holographicentanglement entropy. From the variation, one obtains ( n − k − βℓ n − G ij ( n − k ij = 0 (19)in the bulk and β h ( n − k AB − ( n − kp AB i − ( n − k AB = 0 (20)on the brane, where ( n − k ij and ( n − k AB are the extrinsic curvatures of Γ and of ∂ Γ inΓ. These equations determine the geometry of Γ. When β = 0, Eq. (19) becomes ( n − k = 0and Eq. (20) becomes trivial.For the deSitter braneworld, it is easy to see that the 3-surface Γ ∗ with T =const. and ρ = H − in Eq. (10) satisfies Eq. (19). This is because Γ ∗ is the hyperboloid with thecurvature length ℓ , that is, the induced metric is h = dr + ℓ sinh ( r/ℓ ) d Ω n − , and thenEq. (10) implies [1 − β ( n − n − / ( n − k =0. In Ref. [13], it was shown that Γ ∗ satisfiesEq. (19). Moreover, since ( n − k ij = 0 is satisfied on Γ ∗ , we also see that Eq. (20) is triviallysatisfied.Since ( n − R = − ( n − n − /ℓ , ( n − k = [( n − /ℓ ] coth( r /ℓ ), the holographic entan-glement entropy is computed as S JM = 14 G n (cid:18) − β ( n − n − (cid:19) Z Γ ∗ d n − x √ h + βℓ ( n − G n coth( r /ℓ ) Z ∂ Γ ∗ d n − x √ p = ℓ n − G n Ω n − (cid:20) − β ( n − n − (cid:21) Z r /ℓ dr sinh n − x + β ( n −
3) sinh n − ( r /ℓ ) cosh( r /ℓ ) ! , (21) here Ω n − is the surface area of the ( n −
2) dimensional unit sphere. Since Ω n − is expressedwith Ω n − , Ω n − = Ω n − ( n − π (22)we can find that S dS = S JM (23)holds exactly!
3. Braneworld with Lovelock
We will consider more generic setting, the braneworld model with the Lovelock terms. Theanalysis can be proceeded in the same way as in the case with the Gauss-Bonnet term. Wefollow the definitions of geometrical quantities and objects given in the previous section. Theaction with boundary is given in Ref. [18], S = 116 πG n Z M + n ∪ M − n d n x √− g (cid:16) −
2Λ + X m c m L m (cid:17) + Z M n − d n − x √− q (cid:16) − σ + X m c m [ Q m ] − πG n (cid:17) , (24)where c m ’s are coefficients of Lovelock terms, L m = 12 m g K L ...K m L m M N ...M m N m R K L M N . . . R K m L m M m N m , (25) Q m = 4 m m Z ds q α β ...α m − β m − α m µ ν ...µ m − ν m − µ m (cid:16) ( n − R α β µ ν − s K µ α K ν β (cid:17) . . . (cid:16) ( n − R α m − β m − µ m − ν m − − s K µ m − α m − K ν m − β m − (cid:17) K µ m α m = 4 m m q α β ...α m − β m − α m µ ν ...µ m − ν m − µ m m − X k =0 (cid:18) m − k (cid:19) ( − k k + 1 K µ α K ν β . . . K µ k α k K ν k β k K µ m α m ( n − R α k +1 β k +1 µ k +1 ν k +1 . . . ( n − R α m − β m − µ m − ν m − , (26)and (cid:0) m − k (cid:1) is the binomial coefficients. The tensors g K L ...K m L m M N ...M m N m and q α β ...α m − β m − α m µ ν ...µ m − ν m − µ m aredefined as g K L ...K m L m M N ...M m N m := (2 m )! δ K [ M δ L N . . . δ K m M m δ L m N m ] , (27) q α β ...α m − β m − α m µ ν ...µ m − ν m − µ m := (2 m − q α [ µ q β ν . . . q α m − µ m − q β m − ν m − q α m µ m ] . (28)The brackets [ M . . . N m ] in index indicate antisymmetrization, for instance, T [ MN ] =(1 / T MN − T NM ). The sum in Eq. (24) is taken from m = 0 to [ n/ ... ] is thefloor function. he equation of motion and the junction condition are obtained by taking the variationof Eq. (24). The equation of motion becomesΛ g MN + X m c m E ( m ) MN = 0 , (29)where E ( m ) MN = m m g K L ...K m L m M N ...M m N m g K M R NL M N R K L M N . . . R K m L m M m N m − L m g MN . (30)This gives us Λ = 12 n X m c m ( n − m ) L m . (31)The junction condition is 16 πG n σq µν + X m c m [ J ( m ) µν ] − = 0 , (32)where J ( m ) µν = 4 m m q α β ...α m − β m − α m µ ν ...µ m − ν m − µ m m − X k =0 (cid:18) m − k (cid:19) ( − k k + 1 × (2 k + 1) K µ α K ν β . . . K µ k α k K ν k β k K µ m µ q α m ν × ( n − R α k +1 β k +1 µ k +1 ν k +1 . . . ( n − R α m − β m − µ m − ν m − +2( m − k − K µ α K ν β . . . K µ k α k K ν k β k K µ m α m × ( n − R α k +1 β k +1 µ k +1 ν k +1 . . . ( n − R α m − β m − µ m − ν m − ( n − R α m − β m − µ m − µ q ν m − ν ! − Q m q µν . (33)Then, the brane tension is expressed as16 πG n σ = X m c m n − mn − Q m ] − . (34) Let us calculate the deSitter entropy through the Euclidean action. Substituting Eqs. (31)and (34) to the Euclidean action, we have I E = 116 πG n Z M + n ∪ M − n d n x √ g (cid:16) − X m c m L m (cid:17) + Z M n − d n − x √ q (cid:16) σ − X m c m [ Q m ] − πG n (cid:17) = − πG n Z M + n ∪ M − n d n x √ g X m c m mn L m − πG n Z M n − d n − x √ q X m c m m − n − Q m ] − . (35)Since the Riemann curvature of bulk is R KLMN = − ℓ − ( g KM g LN − g KN g LM ) , (36) m is calculated as L m = ( − m n !( n − m )! ℓ − m . (37)On the other hand, the Riemann curvature and the extrinsic curvature of the brane are ( n − R αβµν = ℓ − sinh − ( r /ℓ )( q αµ q βν − q αν q βµ ) , (38) K µν = ℓ − coth( r /ℓ ) q µν (39)Then, we have Q m = 2 m ( n − n − m )! ℓ − m +1 cosh( r /ℓ )sinh m − ( r /ℓ ) Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − . (40)With Eqs. (37) and (40), the deSitter entropy S dS = − I E is calculated as S dS = Ω n − πG n X m mc m ℓ n − m ( n − n − m )! ( − m ( n − Z r /ℓ dx sinh n − x +(2 m −
1) cosh( r /ℓ ) sinh n − m ( r /ℓ ) Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − ! . (41) The holographic entanglement entropy is also given in Ref. [16]. Then, as subSect. 2.2, wefirst consider S JM = 14 G n Z Γ + ∪ Γ − d n − x √ h X m c m ˜ L m + 14 G n Z ∂ Γ d n − x √ p X m c m [ ˜ Q m ] − , (42)where˜ L m = m m − h i j ...i m − j m − k l ...k m − l m − ( n − R i j k l . . . ( n − R i m − j m − k m − l m − , (43)˜ Q m = 4 m ( m − m − Z ds p A B ...A m − B m − A m − C D ...C m − D m − C m − (cid:16) ( n − R A B C D − s n − k C A ( n − k D B (cid:17) . . . (cid:16) ( n − R A m − B m − C m − D m − − s n − k C m − A m − ( n − k D m − B m − (cid:17) ( n − k C m − A m − . (44)Here, h i j ...i m − j m − k l ...k m − l m − and p A B ...A m − B m − A m − C D ...C m − D m − C m − are defined by h i j ...i m − j m − k l ...k m − l m − := (2 m − h i [ k h j l . . . h i m − k m − h j m − l m − ] , (45) p A B ...A m − B m − A m − C D ...C m − D m − C m − := (2 m − p A [ C p B D . . . p A m − C m − q B m − D m − q A m − C m − ] . (46)Then the minimum value of S JM for the variation of Γ gives us the holographicentanglement entropy which is evaluated on a surface satisfying X m c m ˜ E ( m ) ij ( n − k ij = 0 (47)in the bulk and X m c m [ ˜ J ( m ) AB ] − ( n − k AB = 0 (48) n the brane, where˜ E ( m ) ij = m m − h i j ...i m − j m − k l ...k m − l m − g i i R jj k l R i j k l . . . R i m − j m − k m − l m − −
12 ˜ L m h ij , (49)˜ J ( m ) AB = 4 m ( m − m − p A B ...A m − B m − A m − C D ...C m − D m − C m − m − X l =0 (cid:18) m − l (cid:19) ( − l l + 1 × (2 l + 1) ( n − k C A ( n − k D B . . . ( n − k C l A l ( n − k D l B l ( n − k C m − A p A m − B × ( n − R A l +1 B l +1 C l +1 D l +1 . . . ( n − R A m − B m − C m − D m − +2( m − l − k C A k D B . . . k C l A l k D l B l k C m − A m − × ( n − R A l +1 B l +1 C l +1 D l +1 . . . ( n − R A m − B m − C m − D m − ( n − R A m − B m − C m − A p D m − B ! − ˜ Q m p AB . (50)As is the case in Sect. 2, on T =const. hypersurface, ρ = H − , which is a minimal surface ( n − k = 0, satisfies Eqs. (47) and (48). On this surface, we have˜ L m = m ( n − n − m )! ( − m − ℓ − m +2 , (51)˜ Q m = 2 m ( m −
1) ( n − n − m )! ℓ − m +3 sinh − m +3 ( r /ℓ ) cosh( r /ℓ ) × Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − . (52)Then, the holographic entanglement entropy is calculated in S JM = Ω n − G n X m mc m ( n − n − m )! ℓ n − m ( − m − ( n − Z r /ℓ dx sinh n − x +2( m −
1) sinh n − m ( r /ℓ ) cosh( r /ℓ ) Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − ! = Ω n − πG n X m mc m ℓ n − m ( n − n − m )! ( − m ( n − Z r /ℓ dx sinh n − x +(2 m −
1) cosh( r /ℓ ) sinh n − m ( r /ℓ ) Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − ! . (53) here, in the second equality, we used Eq. (22) and( n − Z r /ℓ dx sinh n − x = − ( n − Z r /ℓ dx sinh n − x + sinh n − ( r /ℓ ) cosh( r /ℓ ) , (54)2( m − Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − = ( − m sinh m − ( r /ℓ ) + (2 m − Z ds (cid:0) − s cosh ( r /ℓ ) (cid:1) m − . (55)As a summary, we can see that S JM = S dS (56)holds exactly in the braneworld with Lovelock terms.
4. Summary
In this paper, we revisited the comparison between the holographic entanglement entropyand deSitter entropy in the braneworld model with higher-curvature corrections, that is, theGauss-Bonnet/Lovelock terms. Employing the Jacobson-Myers formula for the holographicentanglement entropy, we could show the exact agreement of both. These results may encour-age to have the general formulation for holographic entanglement entropy in the braneworldcontext.
Acknowledgements
T. S. and K. I. are supported by Grant-Aid for Scientific Research from Ministry of Education,Science, Sports and Culture of Japan (No. 17H01091). K. I. is also supported by JSPS Grants-in-Aidfor Scientific Research (B) (20H01902). We thank Tadashi Takayanagi for useful discussion.
References [1] J. M. Maldacena, Int. J. Theor. Phys. , 1113-1133 (1999).[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B , 105-114 (1998).[3] E. Witten, Adv. Theor. Math. Phys. , 253-291 (1998).[4] S. Ryu and T. Takayanagi, Phys. Rev. Lett. , 181602 (2006).[5] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370-3373 (1999); Phys. Rev. Lett. , 3370-3373(1999).[6] S. S. Gubser, Phys. Rev. D , 084017 (2001); S. B. Giddings, E. Katz and L. Randall, JHEP ,023 (2000); T. Shiromizu and D. Ida, Phys. Rev. D , 044015 (2001); S. Nojiri, S. D. Odintsov andS. Zerbini, Phys. Rev. D , 064006 (2000); S. Nojiri and S. D. Odintsov, Phys. Lett. B , 119-123(2000); S. W. Hawking, T. Hertog and H. S. Reall, Phys. Rev. D , 043501 (2000); K. Koyama andJ. Soda, JHEP , 027 (2001); S. Kanno and J. Soda, Phys. Rev. D , 043526 (2002).[7] A. Karch and L. Randall, JHEP , 063 (2001); T. Takayanagi, Phys. Rev. Lett. , 101602 (2011);M. Fujita, T. Takayanagi and E. Tonni, JHEP , 043 (2011).[8] G. Penington, JHEP , 002 (2020); A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, JHEP , 063 (2019); A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, JHEP , 149 (2020); M. Rozali,J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, JHEP , 004 (2020); H. Z. Chen, Z. Fisher,J. Hernandez, R. C. Myers and S. M. Ruan, JHEP , 152 (2020).[9] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes and J. Sandor, JHEP , 166 (2020).[10] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes and J. Sandor, [arXiv:2010.00018 [hep-th]].[11] J. Hernandez, R. C. Myers and S. M. Ruan, [arXiv:2010.16398 [hep-th]].[12] R. Emparan, JHEP , 012 (2006).[13] Y. Iwashita, T. Kobayashi, T. Shiromizu and H. Yoshino, Phys. Rev. D , 064027 (2006).
14] G. W. Gibbons and S. W. Hawking, Phys. Rev. D , 2752-2756 (1977).[15] S. Hawking, J. M. Maldacena and A. Strominger, JHEP , 001 (2001).[16] T. Jacobson and R. C. Myers, Phys. Rev. Lett. , 3684-3687 (1993).[17] L. Y. Hung, R. C. Myers and M. Smolkin, JHEP , 025 (2011).[18] R. C. Myers, Phys. Rev. D , 392 (1987): K. i. Maeda and T. Torii, Phys. Rev. D , 024002 (2004).P. A. Cano, Phys. Rev. D , no.10, 104048 (2018).[19] T. Nihei, Phys. Lett. B , 81-85 (1999); J. Garriga and M. Sasaki, Phys. Rev. D , 043523 (2000)., 043523 (2000).