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Holographic complexity growth in a FLRW universe
High Energy Physics - Theory · Submitted 26 Sep 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that there are two kinds of contributions to the growth rate: one is from the interaction among the degrees of freedom, while the other one from the change of the spatial volume of the universe.
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The abstract, sentence by sentence
We investigate the holographic complexity growth rate of a conformal field theory in a FLRW universe. We consider two ways to realize a FLRW spacetime from an Anti-de Sitter Schwarzschild geometry. The first one is obtained by introducing a new foliation of the Schwarzschild geometry such that the conformal boundary takes the FLRW form. The other one is to consider a brane universe moving in the Schwarzschild background. For each case, we compute the complexity growth rate in a closed universe and a flat universe by using both the complexity-volume and complexity-action dualities. We find that there are two kinds of contributions to the growth rate: one is from the interaction among the degrees of freedom, while the other one from the change of the spatial volume of the universe. The behaviors of the growth rate depend on the details to realize the FLRW universe as well as the holographic conjecture for the complexity. For the realization of the FLRW universe on the asymptotic boundary, the leading divergent term for the complexity growth rate obeys a volume law which is natural from the field theory viewpoint. For the brane universe scenario, the complexity-volume and complexity-action conjectures give different results for the closed universe case. A possible explanation of the inconsistency when the brane crosses the black hole horizon is given based on the Lloyd bound.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the holographic complexity growth rate of a conformal field theory in a FLRW universe. | contribution 0.94 | 0.90 | 0.04 | asserted 0.72 |
| s1 | We consider two ways to realize a FLRW spacetime from an Anti-de Sitter Schwarzschild geometry. | method 0.69 | 0.79 | 0.02 | asserted 0.92 |
| s2 | The first one is obtained by introducing a new foliation of the Schwarzschild geometry such that the conformal boundary takes the FLRW form. | method 0.90 | 0.75 | 0.02 | asserted 1.00 |
| s3 | The other one is to consider a brane universe moving in the Schwarzschild background. | method 0.86 | 0.51 | 0.02 | asserted 0.98 |
| s4 | For each case, we compute the complexity growth rate in a closed universe and a flat universe by using both the complexity-volume and complexity-action dualities. | method 0.89 | 0.95 | 0.07 | asserted 0.98 |
| s5 | We find that there are two kinds of contributions to the growth rate: one is from the interaction among the degrees of freedom, while the other one from the change of the spatial volume of the universe. | result 0.87 | 0.95 | 0.46 | asserted 1.00 |
| s6 | The behaviors of the growth rate depend on the details to realize the FLRW universe as well as the holographic conjecture for the complexity. | interpretation 0.49 | 0.65 | 0.24 | asserted 0.99 |
| s7 | For the realization of the FLRW universe on the asymptotic boundary, the leading divergent term for the complexity growth rate obeys a volume law which is natural from the field theory viewpoint. | result 0.93 | 0.90 | 0.18 | asserted 1.00 |
| s8 | For the brane universe scenario, the complexity-volume and complexity-action conjectures give different results for the closed universe case. | result 1.00 | 0.91 | 0.13 | asserted 1.00 |
| s9 | A possible explanation of the inconsistency when the brane crosses the black hole horizon is given based on the Lloyd bound. | interpretation 0.95 | 0.78 | 0.10 | hedged possibility 0.98 |
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The paper
Original abstract
We investigate the holographic complexity growth rate of a conformal field theory in a FLRW universe. We consider two ways to realize a FLRW spacetime from an Anti-de Sitter Schwarzschild geometry. The first one is obtained by introducing a new foliation of the Schwarzschild geometry such that the conformal boundary takes the FLRW form. The other one is to consider a brane universe moving in the Schwarzschild background. For each case, we compute the complexity growth rate in a closed universe and a flat universe by using both the complexity-volume and complexity-action dualities. We find that there are two kinds of contributions to the growth rate: one is from the interaction among the degrees of freedom, while the other one from the change of the spatial volume of the universe. The behaviors of the growth rate depend on the details to realize the FLRW universe as well as the holographic conjecture for the complexity. For the realization of the FLRW universe on the asymptotic boundary, the leading divergent term for the complexity growth rate obeys a volume law which is natural from the field theory viewpoint. For the brane universe scenario, the complexity-volume and complexity-action conjectures give different results for the closed universe case. A possible explanation of the inconsistency when the brane crosses the black hole horizon is given based on the Lloyd bound.
Human review
References
How the literature uses it
11 arXiv papers cite this in 28 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
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Motivated by the work in Ref. [ref] , we would like to extend it to the cases of some modified gravities, such as Gauss-Bonnet gravity and massive gravity, respectively.
Holographic Complexity Growth Rate in a dual FLRW Universe · Introduction · 1.00
Uses its method or tool 1 paper
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For the convenience to implement our goals, according to the previous process [ref] , one needs to introduce the Eddington-Finkelstein coordinates, so that the line element ([ref]) becomes [equation] Here, we have used the effective AdS radius.
Holographic Complexity Growth Rate in a dual FLRW Universe · The case for the neutral Gauss-Bonnet black hole · 1.00
Compares with it 2 papers
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To summarize, taking the large $R_max$ condition and keeping the time $V_R$ fixed, for holographic complexity of a field on the FLRW universe being dual to a five dimensional AdS Gauss-Bonnet black hole, we come to a conclusion that its growth behavior includes the first term, as a finite term, mainly relating to the conserved charge $E$ and the second term (the leading divergent term) being proportional to the rate of the spatial volume in the dual FLRW universe as well as the third term (the sub-leading divergent term) coming from the codimension-two spatial constant curvature, whose behaviors are very like the one presented in literature [ref] .
Holographic Complexity Growth Rate in a dual FLRW Universe · The case for the neutral Gauss-Bonnet black hole · 0.98
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As studied by the [ref] , the evolution mode of the closed brane universe is in a contracting phase on late time, namely $ȧ<0$.
Holographic complexity growth for a charged AdS-dilaton black holes with fixed and dynamical boundary respectively · Growth rate of holographic complexity for brane-bulk system · 0.83
Cites it as background 8 papers
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Recently, through constructing a model of moving brane in AdS-Schwarzchild black hole, the holographic complexity growth in a FLRW universe is considered by [ref] .
Brane universe and holography in spacetime of charged AdS dilaton black hole · Introduction · 0.87
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Although there have been extensive studies on CV and CA conjectures from the bulk side (see, e.g. charged black holes [ref] , UV divergences [ref] , subregion complexity [ref] , time-evolution [ref] , higher derivative gravities [ref] , Einstein-Maxwell-dilaton gravities [ref] , Vaidya spacetimes [ref] , switchback effect and quenches [ref] , and dS/FLRW boundaries [ref] ), the difficulty to establish a convincing holographic complexity lies in the lack of unique and well-defined notion for the complexity in the field theory from the boundary side [ref] .
Revisit on holographic complexity in two-dimensional gravity · Introduction · 1.00
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Refer to [ref] for recent works on the CV and CA hypotheses.
Generalized Volume-Complexity for RN-AdS Black Hole · Introduction · 1.00
What it cites · 28 sentences without a confident label
Uses the method or tool of 5 papers
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The calculation method mainly follows the procedure provided in Ref. [ref] .
On the Time Dependence of Holographic Complexity · The complexity growth with CV conjecture · 0.94
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The method for the calculation of the action in the presence of null boundary has been developed by Ref. [ref] ,where the action reads [equation] Terms in the expression above are respectively bulk term, Gibbons-Hawking-York(GHY) boundary term for space-like or time-like boundary, Hayward joint term [ref] , null boundary term, null joint term and counter term needed to cancel the dependence of arbitrary normalization parameter.The joints $η$ and $a$ are constructed by the rules summarized in Ref. [ref] .$h$,$σ$,$γ$ is the determinant of the induced metric of the corresponding hyper-surface and $ξ$ is the parameter of null hyper-surface, For simplicity, we will choose affine parametrization and set $κ=0$ in the following, so the contribution of null boundary vanishes.
Variational Principle for Gravity with Null and Non-null boundaries: A Unified Boundary Counter-term · The complexity growth with CA conjecture · 1.00
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The method for the calculation of the action in the presence of null boundary has been developed by Ref. [ref] ,where the action reads [equation] Terms in the expression above are respectively bulk term, Gibbons-Hawking-York(GHY) boundary term for space-like or time-like boundary, Hayward joint term [ref] , null boundary term, null joint term and counter term needed to cancel the dependence of arbitrary normalization parameter.The joints $η$ and $a$ are constructed by the rules summarized in Ref. [ref] .$h$,$σ$,$γ$ is the determinant of the induced metric of the corresponding hyper-surface and $ξ$ is the parameter of null hyper-surface, For simplicity, we will choose affine parametrization and set $κ=0$ in the following, so the contribution of null boundary vanishes.
A Boundary Term for the Gravitational Action with Null Boundaries · The complexity growth with CA conjecture · 1.00
Uses data or values from 1 paper
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The physical energy of the brane universe was given by the author of Ref. [ref] :
On the Holographic Principle in a Radiation Dominated Universe · Summary of Results and Discussion · 0.86
Compares with 1 paper
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Eq.([ref]) and Eq.([ref]) reduce to$dC_V/d t_R =-2 Ω_k,d-1/G_N L E$ and $d C_A/dt_R$ respectively, so the evolution is the same as in Ref. [ref] .
On the Time Dependence of Holographic Complexity · The complexity growth with CA conjecture · 0.93
Cites as background 45 papers
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Note added: As this paper was in preparation, there has been an related paper studying the holographic complexity in FLRW spacetimes [ref] .
Holographic Complexity in FRW Spacetimes · Summary of Results and Discussion · 0.86
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While most works on the complexity growth rate considered the static case, the generalization to the time dependent case is also quite interesting and it is worthwhile to study how the complexity evolves in a dynamical process, for related studies on the Vaidya spacetime, see Refs. [ref] .The investigation of the complexity can also be generalized to states on other dynamical backgrounds which correspond to different slices from gravity side, such as de-Sitter boundary in Ref. [ref] .
Holographic Complexity in Vaidya Spacetimes II · Introduction · 0.83
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While most works on the complexity growth rate considered the static case, the generalization to the time dependent case is also quite interesting and it is worthwhile to study how the complexity evolves in a dynamical process, for related studies on the Vaidya spacetime, see Refs. [ref] .The investigation of the complexity can also be generalized to states on other dynamical backgrounds which correspond to different slices from gravity side, such as de-Sitter boundary in Ref. [ref] .
Complexity in de Sitter Space · Introduction · 0.83
Discussion
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