On Complexity Growth for F (R) and Critical Gravity

Transcription

1 On Complexity Growth for F (R) and Critical Gravity Mohammad Hassan Vahidinia Institute for Research in Fundamental Sciences (IPM), Iran Based on: Mohsen Alishahiha, Amin Faraji Astaneh, Ali Naseh, Mohammad H. Vahidinia arxiv: (to appear in JHEP) Recent Trends in String Theory and Related Topics Tehran, Iran May 2017

2 Outline 1 Computational Complexity 2 Complexity growth in higher curvatures gravities 3 Sum up and Outlook Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 2 / 24

3 Outline 1 Computational Complexity 2 Complexity growth in higher curvatures gravities 3 Sum up and Outlook Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 3 / 24

4 Black holes are the best in information! Black holes are nature s extreme memory! Holographic bound implies amount of information can be stored in a region is limited by area of the region S A 4G. Black holes saturate this bound S = A 4G Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 4 / 24

5 Black holes are the best in information! Black holes are nature s extreme memory! Holographic bound implies amount of information can be stored in a region is limited by area of the region S A 4G. Black holes saturate this bound S = A 4G Black holes are the best scrambler! For a system with N dof., the scrambling time t is a measure of how long it takes for information about a small O(1) perturbation to spread over O(N) dof. Typical system: t N 1 d Black holes: t = 1 2πT log S log N Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 4 / 24

6 Black holes are the best in information! Black holes are nature s extreme memory! Holographic bound implies amount of information can be stored in a region is limited by area of the region S A 4G. Black holes saturate this bound S = A 4G Black holes are the best scrambler! For a system with N dof., the scrambling time t is a measure of how long it takes for information about a small O(1) perturbation to spread over O(N) dof. Typical system: t N 1 d Black holes: t = 1 2πT log S log N Are black holes also the fastest computer? Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 4 / 24

7 Limits of Computation Computation is the processing of information. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

8 Limits of Computation Computation is the processing of information. Information needs to be encoded in a physical system. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

9 Limits of Computation Computation is the processing of information. Information needs to be encoded in a physical system. Time evolution of a physical system presents a computation. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

10 Limits of Computation Computation is the processing of information. Information needs to be encoded in a physical system. Time evolution of a physical system presents a computation. There are some bounds on time evolution between two states: Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

11 Limits of Computation Computation is the processing of information. Information needs to be encoded in a physical system. Time evolution of a physical system presents a computation. There are some bounds on time evolution between two states: Aharonov-Anandan-Bohm bound: orthogonality time π 2 E Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

12 Limits of Computation Computation is the processing of information. Information needs to be encoded in a physical system. Time evolution of a physical system presents a computation. There are some bounds on time evolution between two states: Aharonov-Anandan-Bohm bound: orthogonality time π Margolus-Levitin bound: orthogonality time π 2<E> 2 E Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

13 Limits of Computation Computation is the processing of information. Information needs to be encoded in a physical system. Time evolution of a physical system presents a computation. There are some bounds on time evolution between two states: Aharonov-Anandan-Bohm bound: orthogonality time π Margolus-Levitin bound: orthogonality time Is there any bound on the computational speed? Lloyd s Conjecture max. rate of computation E π 2<E> 2 E Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 5 / 24

14 Computation and Complexity How difficult is it to do a task? How difficult is it to prepare a particular state? Computational Complexity Minimum number of quantum gates (operations) required to prepare the desired state. It grows linearly with time even after thermalization! It is bounded by Lloyd,s bound: dc( ψ(t)>) dt [Brown, Roberts,Susskind, Swingle,Zhao 15] 2Energy π Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 6 / 24

15 Holographic Complexity There are (at least) two proposals for holographic complexity: Complexity=Volume (CV)[Standford, Susskind 14] Complexity Volume Gl Complexity=Action (CA)[Brown, Roberts,Susskind, Swingle,Zhao 15] Complexity Action π See also complexity of reduced state [Alishahiha 15 ] and complexity=volume version 2.0 [Couch,Fischler,Nguyen 17] Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 7 / 24

16 Complexity=Volume (CV) [Susskind, Stanford 14] CV- duality The complexity of the boundary state is proportional to the spatial volume V of a maximal slice behind the horizon C Volume Gl. l is unfixed length scale that has to be chosen appropriately. It is typically AdS radius of horizon radius. It is a criterion of wormhole (Einstein-Rosen bridge) size. It grows linearly with time after thermalization: C V TS(t L + t R ) It passes several tests. It has two unpleasant features: Arbitrary scale l Why should the maximal slice play a preferred role? Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 8 / 24

17 [I ve borrowed this pic from Brown s slides!] Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 9 / 24

18 [I ve borrowed this pic from Brown s slides!] Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 9 / 24

19 Complexity=Action (CA) [Brown, Roberts,Susskind, Swingle,Zhao 15] CA- duality The complexity of the boundary state is equal to the classical action of Wheeler-DeWitt patch. Wheeler-DeWitt (WDW) patch is defined as the bulk domain of dependence of a Cauchy slice anchored at the boundary state. C( ψ(t R, t, L) ) = Action π d dt C( ψ(t R, t L ) ) 2E π Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 10 / 24

20 Outline 1 Computational Complexity 2 Complexity growth in higher curvatures gravities 3 Sum up and Outlook Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 11 / 24

21 F (R) gravity I = 1 d d+2 x g F (R) + 1 d d+1 x γ F (R)K, 16πG M 8πG M where F (R) = F (R) R. F (R)R µν 1 2 F (R)g µν ( µ ν g µν )F (R) = 0. AdS-Schwarzschild is a solution to this equation. ds 2 = fdt 2 + f 1 dr 2 + r 2 dω 2 d, f (r) = 1 + r 2 ADM mass of this solution is given as M = Ω dd 16πGl 2 (l2 + rh 2 )F (R)r d 1 h l 2 + M 0 r d 1 Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 12 / 24

22 Complexity/Action Growth in F (R) To evaluate the on-shell action on the WDW patch, besides the generalized Gibbons-Hawking terms, one also needs boundary terms for null boundaries and intersections. In general it is very hard to find them. However, they are known for Einstein gravity [Parattu,Chakraborty, Majhi Padmanabhan ], [Lehner, Myers,Poisson,Sorkin 16]. As far as the late time behavior of the complexity growth is concerned one my circumvent this challenge by following the approach considered in [Complexity=Action (CA)[Brown, Roberts,Susskind, Swingle,Zhao 15]]. It matches with more rigorous calculation based on all boundary terms (it has been shown for Einstein theory [Lehner, Myers,Poisson,Sorkin 16] ) I will calculate the rate of increase of action of a WDW patch of the two-sided black hole (which could be dual to the rate of growth of complexity of the boundary state). Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 13 / 24

23 Complexity/Action Growth in F (R) cont. WDW t+δt t L + δt t L t R r 0 WDW t r = r h Bulk term δi M = I M [WDW t+δt ] I M [WDW t ] 1 rh t+δt = L 16πG F (R) dω d dt dr (at late time) r 0 t S d Boundary term 1 t+δt δi M = dt dω d KF (R) r h 8πG t Ω d r 0 Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 14 / 24

24 Complexity/Action Growth in F (R) cont. Bulk Term Boundary Term ADM mass Bulk + Boundary δi M = Ω df (R) 16πG(d + 1) r d+1 h δt δi M = Ω df (R) 16πG(d + 1) ((d + 1)r h 2 + d l2 )r d 1 h δt. M = Ω dd 16πGl 2 (l2 + rh 2 )F (R)r d 1 h δ(i M + I M ) = 2Mδt Complexity growth It saturates complexity growth bound C = δi δt = 2M. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 15 / 24

25 Critical Gravity: Bulk Action I M = 1 d d+2 x g [R 2Λ 1m ( 16πG M 2 R µν R µν d + 2 )] 4(d + 1) R2, m is a dimensionful parameter. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 16 / 24

26 Critical Gravity: Bulk Action I M = 1 d d+2 x g [R 2Λ 1m ( 16πG M 2 R µν R µν d + 2 )] 4(d + 1) R2, m is a dimensionful parameter. It admits AdS and AdS black holes with radius l. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 16 / 24

27 Critical Gravity: Bulk Action I M = 1 d d+2 x g [R 2Λ 1m ( 16πG M 2 R µν R µν d + 2 )] 4(d + 1) R2, m is a dimensionful parameter. It admits AdS and AdS black holes with radius l. It is known that at the critical point m 2 = d2 the model degenerates 2l 2 yielding to a log-gravity [ Alishahiha,Fareghbal 11] Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 16 / 24

28 Critical Gravity: Boundary Term bulk term I M = 1 16πG M d d+2 x g [R 2Λ 1m ( 2 R µν R µν d + 2 )] 4(d + 1) R2, boundary term [Hohm,Tonni 10] I M = 1 d d+1 x ( ) γ 2K ˆf ij K ij + ˆf K. 16πG where M ˆf ij = f ij + 2h (i N j) + sn i N j, ( ) f µν s h i =. f µν = 2 (R m 2 µν h i f ij and N i is shift function in ADM decomposition. 1 2(d + 1) Rg µν Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 17 / 24 ).

29 Critical Gravity: Static Solution AdS-Schwarzschild is a solution to this theory. ds 2 = fdt 2 + f 1 dr 2 + r 2 dω 2 d, f (r) = 1 + r 2 2Λ d(d + 1) l4 + l 2 d(d 2) 4m 2 = 0. The ADM mass/energy of this solution is M = M 0 (1 d 2 2l 2 m 2 ) l πG Ω d d M 0 r d 1 Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 18 / 24

30 Complexity growth in critical gravity: Static bulk term boundary term term ( δi M = Ω d 8πG δi M = Ω dr d+1 h 8πGl 2 d r d 1 h + bulk + boundary δ(i M + I M ) = 2Mδt Complexity growth (1 d 2 2l 2 m 2 ) δt ) d+1 ( (d + 1)rh l 2 1 d 2 ) 2l 2 m 2 δt It saturates complexity growth bound C = 2M = 2M 0 (1 d2 ). 2l 2 m 2 at the critical point m 2 = d2 where the model develops a log gravity, the 2l 2 rate of growth vanishes! Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 19 / 24

31 3D Critical Gravity (New Massive Gravity): Stationary 3d critical gravity is known as New Massive Gravity (NMG). BTZ balck hole is a solution to this theory ds 2 = dr 2 ( f (r) f (r)dt2 + r 2 dφ 8GJ ) 2 0 2r 2 dt, f (r) = r 2 l 2 8GM 0 + (8GJ 0) 2 4r 2, The ADM mass and angular momentum of this solution is ( M = M ) ( 2l 2 m 2, J = J ) 2l 2 m 2. Note that at the critical point, m 2 = 1 log gravity, the rate of growth vanishes. 2l 2 where the model develops a Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 20 / 24

32 Complexity growth in NMG: Stationary (BTZ) bulk term δi M = I M [WDW t+δt ] I M [WDW t] = r+ t+δt 2π r t 0 L NMG (at late time) r r boundary term δi M = t+δt 2π t 0 (Boundary Term) r h r 0 t L + δt t L WDW t+δt WDW t r + r + t R Complexity growth C = 2 M 2 J 2 < 2M. It respects to Lloyd bound. l 2 Intuitively, the conservation of angular momentum set a barrier to rapid complexification. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 21 / 24

33 Outline 1 Computational Complexity 2 Complexity growth in higher curvatures gravities 3 Sum up and Outlook Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 22 / 24

34 Sum up Using complexity=action proposal we study complexity growth of f (R) and critical gravity. We show action that the action growth for neutral black hole saturate the complexity growth bound. We also study effect of shock wave on black hole in critical gravity. The presence of massive spin-2 slows down the rate of growth. Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 23 / 24

35 Outlook Higher derivative terms in critical gravity may slow down complexity growth. What does it mean in dual CFT? Finding the boundary terms for null boundary and intersections. Using them one may study complexity Not complexity growth. What about Gauss-Bonnet and Lovelock or any theory with Riemann? They suffers from singularity at r = 0 where Riem 2 blows up!!! What is the role of quantum gravity near the singularity! Promotion CV duality for higher curvature theory. We suggest L C V = 1 l B Lots to explore! [ L R µναβ C V = 1 l B ɛ µν ɛ αβ (1) R µναβ ) ] + c (2) ( a n µ n β h να + d + 5 d(d + 1) h µβh να Thank you! Mohammad Hassan Vahidinia On Complexity Growth for F (R) and Critical Gravity 24 / 24