Flat-Space Holography and Anisotrpic Conformal Infinity Reza Fareghbal Department of Physics, Shahid Beheshti University, Tehran Recent Trends in String Theory and Related Topics, IPM, May 26 2016
Based on: A. Bagchi, R. F.,JHEP 1210 (2012) 092 [arxiv:1203.5795] A. Bagchi, S. Detournay, R. F., Joan Simon,Phys.Rev.Lett. 110 (2013) 141302[arXiv:1208.4372] R. F., A. Naseh,JHEP 1403 (2014) 005 [arxiv:1312.2109] R. F., A. Naseh, JHEP 06 (2014) 134 [arxiv:1404.3937] R. F., A. Naseh, S. Rouhani [arxiv:1511.01774]. R. F., Y. Izadi, [arxiv:1603.04137] R. F., P. Karimi, work in progress
Motivation It is of interest to explore whether holography exists beyond the known example of AdS/CFT. A step in this direction is study of holography for asymptotically flat spacetimes. An approach is taking limit from the AdS/CFT calculation. Flat-space limit of AAdS spacetimes asymptotically flat space times What is the corresponding operation in the CFT? Our Proposal: It is a field theory with contracted Conformal Symmetry. ( Galilean Conformal symmetry in 2d) This correspondence: BMS/GCA or Flat/CCFT
Gravity in asymptotically AdS spacetimes AdS/CFT CFT Flat limit Gravity in asymptotically Flat spacetimes flat/ccft Contraction CCFT
Why contraction? Asymptotic symmetry of asymptotically AdS spacetimes in d+1 dimensions= Conformal symmetry in d dimensions Non-trivial ASG for asymptotically Minkowski spacetimes at null infinity in three and four dimensions: BMS 3 : [L m, L n ] = (m n)l m+n, [L m, M n ] = (m n)m m+n, [M m, M n ] = 0. (1) [Ashtekar, Bicak, Schmidt 1996], [Barnich,Compere 2006] BMS 4 : [l m, l n ] = (m n)l m+n, [ l m, l n ] = (m n) l m+n, [l m, l n ] = 0, [l l, T m,n ] = ( l + 1 m)t m+l,n, [ l l, T m,n ] = ( l + 1 n)t m,n+l. 2 2 [ Bondi, van der Burg, Metzner; Sachs 1962], [Barnich,Troessaert 2010]
Why contraction? Generators of two dimensional CFT on the cylinder: L n = e nw w, Ln = e n w w (w = t + ix, w = t ix) (2) Define L n = L n L n, M n = ɛ(l n + L n ). Use a spacetime contraction: t ɛt, x x. Final generators in the ɛ 0 limit: L n = e inx (i x + nt t ), M n = e inx t (3) The resultant algebra in the ɛ 0 limit is BMS 3 with central charges: c LL = C 1 = c c, c LM = C 2 = ɛ(c + c) [A. Bagchi, R. F. (2012)]
Next Step: Flat limit of BTZ No asymptotically flat black hole solutions in the three dimensional Einstein gravity. [D. Ida, 2000] The flat limit of BTZ is well-defind! BTZ black holes: l : ds 2 = (r 2 r+)(r 2 2 r ) 2 r 2 l 2 dt 2 r 2 l 2 + (r 2 r+)(r 2 2 r ) 2 dr 2 + r 2 ( dφ r +r lr 2 dt ) 2 r r 0 = 2G M J, r + lˆr + = l 8GM (4) Flat BTZ: a cosmological solution of 3d Einstein gravity ds 2 FBTZ = ˆr 2 +dt 2 r 2 dr 2 ˆr 2 +(r 2 r 2 0 ) + r 2 dφ 2 2ˆr + r 0 dtdφ (5) FBTZ is a shifted- boost orbifold of R 1,2 [ Cornalba, Costa(2002)] r = r 0 is a cosmological horizon with T = ˆr 2 + 2πr 0, S = πr0 2G
Dual field theory analysis: a Cardy-like formula 3d asymptotically flat spacetimes states of field theory with BMS symmetry. The states are labelled by eigenvalues of L 0 and M 0 : where L 0 h L, h M = h L h L, h M, M 0 h L, h M = h M h L, h M, h L = lim ɛ 0 (h h) = J, h M = lim ɛ 0 ɛ(h + h) = GM + 1 8 (6) Degeneracy of states with (h L, h M ) Entropy of cosmological solution with (r 0, ˆr + )
Dual field theory analysis: a Cardy-like formula Is there any Cardy-like formula? YES Modular invariance of CCFT partition function results in ) C M h M S = log d(h L, h M ) = 2π (h L + C L 2h M 2C M (7) [A. Bagchi, S. Detournay, R. F., Joan Simon (2012)] In agreement with the entropy of cosmological solution!
Flat-space Stress Tensor Use this method for calculation of flat-space Stress Tensor. Starting point is AdS/CFT and Brown and York s quasi-local stress tensor: T µν = 2 δs, (8) γ δγ µν The gravitational action is S = 1 d 3 x g (R 2l ) 16πG M 2 1 d 2 x γ K + 1 8πG M 8πG S ct(γ µν ), (9) where S ct = 1 d 2 x γ. (10) l M
Flat-space Stress Tensor We need a proper coordinate with well-defined flat-space limit. A choice is BMS gauge: ds 2 = ( r 2 ) l 2 + M du 2 2dudr + 2Ndudφ + r 2 dφ 2, (11) where M(u, φ) = 2 ( χ(x + ) + χ(x ) ), N(u, φ) = l ( χ(x + ) χ(x ) ), (12) and χ, χ are arbitrary functions of x ± = u l ± φ. The flat-space limit is well-defined: where ds 2 = Mdu 2 2dudr + 2Ndudφ + r 2 dφ 2. (13) M = lim G l 0 M = θ(φ), N = lim G l 0 N = β(φ)+ u 2 θ (φ), (14)
Flat-space Stress Tensor Components of stress tensor for the asymptotically AdS solutions (written in the BMS gauge) are T rr = O( 1 r 2 ), T r+ = O( 1 r 2 ), T r = O( 1 r 2 ), l T ++ = 8πG χ(x +) + O( 1 r ), T = l 8πG χ(x ) + O( 1 r ). T + = O( 1 r 2 ) (15) Energy-momentum one-point function of dual CFT: T ++ = lχ 8πG, T = l χ 8πG, T + = 0. (16) In the coordinate {u, φ} we have T uu = χ + χ 8πGl = T φφ = l(χ + χ) 8πG M 16πGl, T uφ = χ χ 8πG = N 8πGl, (17) = lm 16πG. (18)
Flat-space Stress Tensor l is not well-defined! However, the combinations G T 1 = lim G/l 0 l (T ++ + T ), T 2 = lim (T ++ T ), G/l 0 (19) are finite in the flat limit. We define the flat-space stress tensor, Tij, by T 1 = ( T ++ + T ), T 2 = ( T ++ T ), T+ = 0, (20) where x ± = u G ± φ [ R.F., Ali Naseh,2013]
Flat-space Stress Tensor Contraction of i T ij = 0 results in t T 1 = 0, x T 1 t T 2 = 0, (21) where t is the contracted time. The above equations are consistent with the Einstein equations in the bulk side. A useful result: l T uu = lim G l 0 G T uu, l T uφ = lim G l 0 G T uφ, G T φφ = lim G l 0 l T φφ. (22) The flat space stress tensor T ij is given by appropriate scaling of AdS counterpart.
Flat-space Stress Tensor The geometry of spacetime which contracted theory lives on it, is the same as parent theory. Its time coordinate is given by contraction of original one. Conserved charges of symmetry generators are given by Q ξ = dφ σv µ ξ ν T µν, (23) The final result is consistent with the known results. Σ
Anisotropic conformal infinity Asymptotically flat spacetimes which was used for calculation of holographic stress tensor: where ds 2 = Mdu 2 2dudr + 2Ndudφ + r 2 dφ 2. (24) M = lim G l 0 M = θ(φ), N = lim G l 0 N = β(φ)+ u 2 θ (φ). (25) CCFT lives on d s 2 = du 2 + G 2 dφ 2 (26) It is clear that the geometry which CCFT lives on it, is not given by isotropic scaling of generic asymtotically flat spacetimes (24)!
Anisotropic conformal infinity However, an anisotropic scaling like below is possible: g φφ Ω 2z g φφ, g uφ Ω 2 g uφ, g ur Ω 2 g ur, g uu g uu. (27) where z = 2, Ω = M 1 4 G (28) r The final metric is conformal to the metric of spacetime which dual CCFT lives on it.
Anisotropic conformal infinity The anisotropic scaling defined in this way is well-defined: Form of the asymptotically flat spacetimes defined as (24) are preserved under the transformation generated by χ u = F, χ φ = Y 1 r φf, χ r = r φ Y + 2 φ F 1 r N φf (29) where F = T + u φ Y and T and Y are arbitrary functions of φ coordinate. It is not difficult to show that: if ω generates the infinitesimal anisotropic conformal scaling for metric g µν then ω + δ χ ω scales metric g µν + δ χ g µν to a geometry which differs from the scaling of g µν upto a conformal factor. [R. F, P. Karimi, work in progress]
Holographic Anomaly of CCFT Use Flat/CCFT correspondence and calculate the anomaly of CCFT. Starting point asymptotically AdS solutions: ds 2 = A(u, r, φ)du 2 2e 2β(u,φ) dudr +2B(u, r, φ)dudφ+r 2 dφ 2. (30) The equations of motion determine A and B as where A(u, r, φ) = e 4β(u,φ) r 2 + M(u, φ), l2 B(u, r, φ) = r φ e 2β(u,φ) + N(u, φ), (31) φ M 2 u N + 4N u β = 0 8 φ β u φ β 4 u 2 φ β + 4 l 2 N φβ + 2 l 2 φn e 4β u M +4Me 4β u β = 0 (32)
Holographic Anomaly of CCFT (30) has non-flat conformal boundary ds 2 C.B = G 2 l 2 e4β(u,φ) du 2 + G 2 dφ 2 (33) Components of stress tensor: ( ) e 4β φ 2β + ( φβ) 2 T uu = + M 4πlG 16πlG, T uφ = N 8πlG, T φφ = lme 4β 16πG l( φβ) 2 4πG. (34) Anomoly is computed holographically as T = C 24π R C.B, (35) where C = 3l/2G is the Brown and Henneaux s central charge.
Holographic Anomaly of CCFT Take flat-space limit: G T ++ + T = lim G l 0 l (T ++ + T ) T ++ T = lim G l 0 (T ++ T ) G T + = lim G l 0 l T + (36) CCFT is on a spacetime with line-element: d s 2 = e 4β(u,φ) du 2 + G 2 dφ 2, (37) This metric is given by an anisotropic scaling from (30)! The trace anomaly is [R. F., A. Naseh, S. Rouhani, 2015] T = g ij Tij = 1 4π c M R (38)
Flat-space Holography and Stress Tensor of Kerr Black Hole Flat-space limit can be used for proposing a stress tensor for the Kerr black hole. The method is similar to three dimensions: Take limit from the holographic calculations of Kerr-AdS black hole: ds 2 = ( r ρ 2 dt a ) 2 sin2 θ Ξ dφ + ρ2 dr 2 + ρ2 dθ 2 (39) r θ + θ sin 2 θ (adt ρ 2 r 2 + a 2 ) 2 Ξ dφ, (40) r = (r 2 + a 2 ) (1 + r 2 ) l 2 2MGr, θ = 1 a2 l 2 cos2 θ, ρ 2 = r 2 + a 2 cos 2 θ, Ξ = 1 a2 l 2. (41) M is the mass of the black hole and a = J/M where J is the angular momentum of the black hole.
Flat-space Holography and Stress Tensor of Kerr Black Hole According to AdS/CFT correspondence, the non-zero components of Kerr-AdS stress tensor are 8πT tt = 2M rl, 8πT tφ = 2aM rlξ sin2 θ, 8πT θθ = Ml r θ, 8πT φφ = Ml rξ 2 sin2 θ ( Ξ + 3a2 sin 2 ) θ l 2. (42) l is not well-defined! However applying proper powers of l to the components of the stress tensor makes the flat limit well defined.
Flat-space Holography and Stress Tensor of Kerr Black Hole Our proposal for the Kerr stress tensor: 8πτ tt = 2M r G, 8πτ tϕ = 3aM r G sin2 θ, 8πτ θθ = M G, r 8πτ ϕϕ = M G sin 2 θ. (43) r [R. F., Y. Izadi (2016)] The anisotropic conformal boundary is given by d s 2 = r 2 [ dt 2 + Gdθ 2 + G sin 2 θdϕ 2]. (44) G Using (43) and (43) in the Brown and York formula results in the correct charges of Kerr!
Rindler-space Holography Rindler spacetime is the flat limit of Rindler-AdS spacetime, ds 2 = α 2 r 2 dτ 2 + dr 2 1 + r + (1 + r 2 ) 2 l 2 dχ 2, (45) l 2 Observer at r = r 0 perceive a constant acceleration a(3) 2 = 1 + 1. r0 2 l 2 Proper time of the observer is given by αr 0 τ. τ in metric (45) is the proper time of an observer located at r = r 0 = 1 α. Temperature is T = a (4) 2π = 1 2πr 0 = a 2 (3) 1 l 2 2π (46)
Rindler-space Holography Rindler-AdS covers a portion of global AdS which consists of two wedges The physics inside the two wedges of Rindler-AdS has a holographic description as entangled states of a pair of CFTs which live on the boundary of Rindler-AdS wedges. [Van Raamsdonk,et. al. (2012)]
Rindler-space Holography Our proposal: Dual theory of Rindler is the contracted CFT resulted in from Rindler-AdS/CFT correspondence. [R. F., A. Naseh (2014)] The symmetries of two dimensional CCFT predict the same two-point functions which one may find by taking the flat-space limit of three dimensional Rindler-AdS holographic results. It is possible to find an energy-momentum tensor for Rindler Gravity by using Flat/CCFT correspondence: T ττ = α2 16π, T χχ = 1 16πG 2. (47)
Rindler-space Holography Application in the black hole physics: Near horizon geometry of non-extreme black holes has a Rindler part. The dual theory at the horizon of non-extreme black holes is CCFT. Non-rotating BTZ: ds 2 = f (ρ)dτ 2 + f (ρ) 1 dρ 2 + ρ 2 dφ 2 (48) where f (ρ) = ρ2 8GM and M is the mass of black hole. l 2 Defining a new coordinate y = ρ ρ h and considering the region given by y ρ h results in ds 2 NH = f (ρ h )ydτ 2 + (f (ρ h )y) 1 dy 2 + ρ 2 h dφ2 (49)
Rindler-space Holography Defining new coordinate r by dr = dy/ f (ρ h )y results in dsnh 2 2 = f (ρ h ) r 2 dτ 2 + dr 2 + ρ 2 h 4 dφ2 (50) The above metric is Rindler with α = f (ρ h )/2 = 8GM/l and a compact χ coordinate given by χ = ρ h φ. Line element (50) is the flat-space limit of a Rindler-AdS metric which has the same α and φ. Dual of Rindler-AdS is a CFT. Demanding that Cardy formula gives the same result as Bekenstein-Hawking entropy of Rindler-AdS results in h = h = ρ 2 h 16lG ρ 2 h 16lG 2 αl2 + 2 ρ h 2 αl2 ρ h 2 1 αl2 ρ h, 1 αl2 (51) ρ h
Rindler-space Holography h L and h M are well-defined in the G/l 0 limit: h L = ρ2 h α, 4G ρ h h M = αρ h 8. (52) The Cardy-like formula of CCFT precisely result in Bekenstein-Hawking entropy of non-rotating BTZ!
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