Quasilocal vs Holographic Stress Tensor in AdS Gravity

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1 Quasilocal vs Holographic Stress Tensor in AdS Gravity Rodrigo Olea Universidad Andrés Bello Quantum Gravity in the Southern Cone VI Maresias, Sept 11-14, 2013 Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 1 / 19

2 Outline 1 Quasilocal (Brown-York) stress tensor Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 2 / 19

3 Outline 1 Quasilocal (Brown-York) stress tensor 2 Holographic Renormalization and Local Counterterms Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 2 / 19

4 Outline 1 Quasilocal (Brown-York) stress tensor 2 Holographic Renormalization and Local Counterterms 3 Holographic stress tensor Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 2 / 19

5 Outline 1 Quasilocal (Brown-York) stress tensor 2 Holographic Renormalization and Local Counterterms 3 Holographic stress tensor 4 Alternative regularization scheme in AdS gravity: Kounterterms Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 2 / 19

6 Outline 1 Quasilocal (Brown-York) stress tensor 2 Holographic Renormalization and Local Counterterms 3 Holographic stress tensor 4 Alternative regularization scheme in AdS gravity: Kounterterms 5 No quasilocal stress tensor: two examples Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 2 / 19

7 Outline 1 Quasilocal (Brown-York) stress tensor 2 Holographic Renormalization and Local Counterterms 3 Holographic stress tensor 4 Alternative regularization scheme in AdS gravity: Kounterterms 5 No quasilocal stress tensor: two examples 6 Prospects Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 2 / 19

8 Variational problem in EH gravity EH gravity action in D = d + 1 dimensions 1 I EH = d d +1 x G (R 2Λ) 16πG M Variation of the action: 1 δi EH = d d +1 x G Eµ ν 16πG M Surface term: Θ = 1 16πG h nµ δ [µν] [αβ] G βε δγ α νε ( G 1 δg ) µ ν + d d x Θ Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 3 / 19

9 Variational problem in EH gravity In Gauss-normal coordinates ds 2 = N 2 (ρ)dρ 2 + h ij (ρ, x)dx i dx j Extrinsic curvature K ij = 1 2N ρh ij Γ ρ ij = 1 N K ij, Γ i ρj = NK i j On-shell variation of the action (n µ = (N, 0)): δi EH = 1 16πG d d x ( ) h 2δK + Kj i (h 1 δh) j i Gibbons-Hawking term: I Dirichlet = I EH 8πG 1 d d x hk Dirichlet problem for the metric: δi Dirichlet = 1 16πG d d x h ( K i j K δ i j ) (h 1 δh) j i Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 4 / 19

10 Quasilocal stress tensor Variation of the Dirichlet action respect to h ij δi Dir = d d x 1 2 h T ij [h] δh ij Brown-York tensor is conserved, i.e., i T ij = 0. Conserved current: J i = T ij ξ j = Q[ξ] = σ ui T ij ξ j Σ In AAdS spacetimes Q[ξ] is divergent, even in D = 3. We can always add local counterterms such that T ij [h] is regular I ren = I EH 8πG 1 d d x hk + d d x L ct (h, R, R) Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 5 / 19

11 Counterterm method Regularized AdS gravity action (holographic renormalization) [Henningson, Skenderis JHEP 9807:023(1998)] I ren = 1 16πG M d (d 1) Λ = 2l 2 d d +1 x G (R 2Λ) 1 8πG + d d x L ct (h, R, R) d d x h K + Renormalized quasi-local stress tensor: T ij ren[h] = 2 h δi ren δh ij. Background-independent charge definition Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 6 / 19

12 Holographic Renormalization For Asymptotically AdS (AAdS) spacetimes, Fefferman-Graham (FG) form of the metric ds 2 = l2 4ρ 2 dρ2 + 1 ρ g ij (x, ρ) dx i dx j (1) the boundary of the spacetime is at ρ = 0 g ij (x, ρ) accepts a regular expansion in powers of ρ g ij (x, ρ) = g (0)ij (x) + ρg (1)ij (x) + ρ 2 g (2)ij (x) +. g (0)ij is the boundary data for the holographic reconstruction of the spacetime, i.e., solving g (k ) as a covariant funtional of g (0) ( ) Holographic stress tensor T ij [g (0) ] = lim 1 T ij [h]. ρ 0 ρ d /2 1 Contains the holographic information of the theory (e.g., Weyl anomaly) Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 7 / 19

13 Counterterm method in AdS gravity L ct = d 1 l h + l h 2(d 2) R + l 3 h l + 5 ( h 3d 2 (d 2) 3 (d 4)(d 6) 2R ij R kl R ijkl 2(d 2) 2 (d 4) ( R ij R ij d 4(d 1) R2) 4(d 1) RRij d (d +2) R ij R 16(d 3 1) 2 ) 4(d d 1) i R i R + k R ij k R ij +... Full series for an arbitrary dimension is unknown. Terms do not seem to follow any pattern. Far more complicated for higher-curvature theories (e.g., Einstein-Gauss-Bonnet). Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 8 / 19

14 Kounterterms Extrinsic regularization Ĩ ren = I EH + c d d d x B d (h, K, R) Inspired by a simple observation (EH+GB in D = 4) I = 1 16πG M d 4 x G [ ] (R 2Λ) + α(r µναβ R µναβ 4R µν R µν + R 2 ) Euclidean action for Sch-AdS black hole: (1 + 4l ) 2 α TS + πr 3 (1 4l 2 α ) G = β 1 I E = M 2 4G l 2 GB coupling has to be fixed as α = l2 4 Finite Noether current also implies α = l2 4 [Aros, Contreras, Olea, Troncoso, Zanelli, PRL 84, 1647 (2000)] Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 9 / 19

15 MacDowell-Mansouri form of the action I ren = l2 256πG M d 4 x ( G δ [σλµν] R γδ [γδαβ] σλ + 1 ) ( l 2 δ[γδ] R αβ [σλ] µν + 1 ) l 2 δ[αβ]. [µν] The action has a minimum for global AdS spacetime Weyl tensor (on-shell) I ren = 0 W αβ µν = R αβ µν + 1 l 2 δ[αβ] [µν] (On-shell) regularized action is equal to Conformal Gravity action I ren = l2 64πG M d 4 x G W µναβ W µναβ O.Mišković and R.O., [arxiv: ]; J.Maldacena, [arxiv: ] Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 10 / 19

16 Kounterterms Euler Theorem in D = 4 dimensions d 4 x GB = 32π 2 χ(m) + M d 3 x B 3 Kounterterms = given polynomial in the extrinsic and intrinsic curvatures (K ij and R kl ij (h)) B 3 = 4 h [i ( ) 1i 2 i 3 ] [j 1 j 2 j 3 ] K j 1 12 i 1 R j 2j 3 i 2 i 3 (h) 1 3 K j 2 i 2 K j 3 i 3 = 4 [ ] h 2(R i j 2 1 δi j R)K j i 2 3 K j i K j k K i k + K (Kj i K j i 1 3 K 2 ) c 3 = l 2 /64πG Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 11 / 19

17 Kounterterms D = 2n dimensions [R.O., JHEP 0506: 023 (2005)] B 2n 1 = 2n 1 h dt δ [i 1 i 2n 1 ] [j 1 j 2n 1 ] K j 1 i 1 ( 1 2 Rj 2j 3 i 2 i 3 t 2 K j 2 i 2 K j 3 i 3 ) 0 ( ) 1 2 Rj 2n 2j 2n 1 i 2n 2 i 2n 1 t 2 K j 2n 2 i 2n 2 K j 2n 1 i 2n 1 c 2n 1 = ( l 2 ) n 1 /(16πGn(2n 2)!) Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 12 / 19

18 Kounterterms Kounterterms in D = 2n + 1 [R.O., JHEP 0704: 073 (2007)] 1 B 2n = 2n 0 t dt ds δ [j 1 j 2n ] [i 1 i 2n ] K i 1 j 1 δ i 2 j2 ( 1 2 Ri 3i 4 j 3 j 4 t 2 K i 3 j 3 K i 4 j 4 + s2 l 2 δi 3 j3 δ i 4 j4 ) 0 ( ) 1 2 Ri 2n 1i 2n j 2n 1 j 2n t 2 K i 2n 1 j 2n 1 K i 2n j 2n + s2 l 2 δi 2n 1 j 2n 1 δ i 2n j 2n. Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 13 / 19

19 Kounterterms Kounterterms in D = 2n + 1 [R.O., JHEP 0704: 073 (2007)] 1 B 2n = 2n 0 t dt ds δ [j 1 j 2n ] [i 1 i 2n ] K i 1 j 1 δ i 2 j2 ( 1 2 Ri 3i 4 j 3 j 4 t 2 K i 3 j 3 K i 4 j 4 + s2 l 2 δi 3 j3 δ i 4 j4 ) 0 ( ) 1 2 Ri 2n 1i 2n j 2n 1 j 2n t 2 K i 2n 1 j 2n 1 K i 2n j 2n + s2 l 2 δi 2n 1 j 2n 1 δ i 2n j 2n. Coupling constant c 2n = 1 ( 1) n l 2n 2 16πG 2 2n 2 n (n 1)! 2. Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 13 / 19

20 Quasilocal vs Holographic Stress Tensor Kounterterms: No clear identification of the boundary quasilocal stress tensor δĩ ren = d D 1 x ( 1 ( ) ) i h 2 τj i h 1 δh + j j i δk j i But what about holographic stress tensor? Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 14 / 19

21 Holographic stress tensor not from a quasilocal one Example 1: AAdS sector in 3D Topologically Massive Gravity 1 I TMG = I EH + d 3 x (ΓdΓ + 23 ) 32πG µ Γ3 M No Gibbons-Hawking term for L CS (Γ) No quasilocal stress tensor δi TMG = d 2 x ( 1 ( ) ) h 2 T ij i h 1 δh (EH ) + 1 j 16πG µ K i k δk kj ɛ ij However K ij = 1 l Holographic stress tensor g (0) ij ρ + g (1) ij +... ˆT ij [g (0) ] = T ij (EH ) [g (0)] 1 8πG µ ( g il (1) g (0) lk ) ɛ kj + i j Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 15 / 19

22 Holographic stress tensor not from a quasilocal one Example 2: EH AdS+ GB term in 4D Ĩ ren = 1 16πG M d 4 x G [ ] (R 2Λ) + l2 4 (R µναβr µναβ 4R µν R µν + R 2 ) For any D > 4 (arbitrary GB coupling α) δi GB = α 4πG d D 1 x [ h δ [jj 1j 2 ] 1 ( ) i h 1 δh [ii 1 i 2 ] 2 K j k k Gibbons-Hawking term for GB β GB = α 4πG ] ( ) + δkj i 1 2 Ri 1i 2 j 1 j 2 (h) K i 1 j 1 K i 2 j 2 ( [j hδ 1 j 2 j 3 ] [i 1 i 2 i 3 ] K i 1 1 j 1 2 Ri 2i 3 j 2 j 3 1 ) 3 K i 2 j 2 K i 3 j 3, Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 16 / 19

23 Holographic stress tensor not from a quasilocal one Dirichlet variation δi GB = α 8πG d D 1 x ( h δ [j j 1j 2 j 3 ] [i i 1 i 2 i 3 ] (h 1 δh) i j K i 1 1 j1 2 Ri 2i 3 j 2 j 3 1 ) 3 K i 2 j 2 K i 3 j 3 No Gibbons-Hawking term for 4D GB No quasilocal stress tensor δĩ ren = lim d 3 x ( 1 ( ) ) i h ρ 0 2 τj i h 1 δh + j j i δk j i where τ j i = 32πG 1 δ[jkl] [mnp] K i m ( R np kl + 1 δ [np] ) (, j l 2 [kl] i = 32πG 1 δ[jkl] R np [inp] kl + 1 δ [np] ) l 2 [kl] Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 17 / 19

24 Holographic stress tensor not from a quasilocal one Performing FG expansion of the fields h = g ρ 3/2 = g(0) ρ 3/2 + O(ρ 1/2 ) ( h 1 δh ) i j = ( g 1 δg ) ( ) i i j = g 1 (0) δg (0) + O(ρ), j τ j i = ρ 3/2 T j i [g (0)] + O(ρ 2 ) = j i + O(ρ2 ) δk i j = O(ρ), Holographic stress tensor δĩ ren = d 3 x 1 ( g(0) 2 T j i [g (0)] g 1 (0) ) i 1 δg (0) j Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 18 / 19

25 Conclusions and prospects The result of D = 4 is always true in D = 2n (requires the asymptotic behavior of the Weyl tensor) In D = 2n + 1 it is more subtle because counterterms ambiguity (finite counterterms that do not modify the Weyl anomaly). Reading off the holographic (quasilocal) stress tensor from the variation of the action in other gravity theories (EGB, Lovelock, etc.) where Holographic Renormalization would drive you insane. Rodrigo Olea (UNAB) () Quasilocal vs Holographic Stress Tensor in AdS 19 / 19