Holographic complexity of braneworld in Horndeski gravity Fabiano F. Santosa Oleksii Sokoliukbcand Alexander Baranskyb aInstituto de F sicaUniversidade Federal do Rio de Janeiro

2025-05-06 0 0 684.76KB 18 页 10玖币

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Holographic complexity of braneworld in Horndeski gravity

Fabiano F. Santosa, Oleksii Sokoliukb,c and Alexander Baranskyb∗

aInstituto de F´ısica,Universidade Federal do Rio de Janeiro,

Caixa Postal 68528, Rio de Janeiro-RJ, 21941-972 – Brazil†and

bMain Astronomical Observatory of the NAS of Ukraine (MAO NASU), Kyiv, 03143, Ukraine

cAstronomical Observatory, Taras Shevchenko National University of Kyiv,

3 Observatorna St., 04053 Kyiv, Ukraine

This work investigates the inﬂuence of a probe string on the complexity of braneworld

according to the CA (Complexity equals action) conjecture within the Horndeski gravity. In

the current study, it is considered that scalar ﬁelds that source Horndeski gravity has a spatial

dependence. In addition, our system contains a particle moving on the boundary, which

corresponds to the insertion of a fundamental string in the higher dimensional bulk. Such an

eﬀect is given by the Nambu-Goto term, which also incorporates the time-dependence and

evolution in our system. Both warp factor, scalar ﬁeld, and superpotential values are derived

numerically assuming appropriate initial conditions, and the growth rate of holographic

complexity is analyzed within the so-called Wheeler-De Witt (WDW) patch with null-like

hypersurfaces present.

I. INTRODUCTION

In recent years, Horndeski’s theory of gravitation has been used widely in the studies of black

hole information paradox [1,2] and holographic thermodynamics (BH entropy, heat capacity, etc.)

[1]. Also, beyond Horndeski theories are now being developed and incorporated [3,4]. The afore-

mentioned black hole entropy is the thermodynamic quantity, which is extracted through the

AdS/BCFT correspondence as presented by the authors [2,4]. This correspondence [5–8] is the

most important realization of the holographic principle [9,10], which relates a gravity theory in a

higher dimensional anti-de Sitter (AdS) bulk to a conformal ﬁeld theory (CFT), but without grav-

ity living on the bulk boundary. As we know, this theory suggests non-trivial connections between

diﬀerent areas of physics where an example is a particular connection between general relativity

and quantum information theory. One of the outstanding developments in this correspondence was

∗Electronic address: fabiano.ﬀs23-at-gmail.com

†Electronic address: oleksii.sokoliuk-at-mao.kiev.ua, abaransky-at-ukr.net

arXiv:2210.11596v2 [hep-th] 30 Dec 2022

proposed by the work of Ryu and Takayanagi [11,12], which gives a holographic dictionary for the

calculation of entanglement entropy from BCFT. Recently an extension of this proposal was in the

Horndeski gravity [13]. However, according to the proposal of Ryu and Takayanagi and beyond,

the entanglement entropy of the boundary theory is equivalent to the area of a certain minimal

surface in the bulk geometry. Thus, we have that the dynamics of the bulk spacetime emerge from

the quantum entanglement of the boundary theory [14]. Furthermore, the entanglement entropy

may not be enough to probe the suﬃcient number of degrees of freedom in the black hole interior,

since the volume of a black hole usually continues to grow even if spacetime reaches its thermal

equilibrium phase [15]. It is believed that quantum complexity is the quantity that can continue

to grow even after reaching thermal equilibrium, which is similar to the growth of a black hole

interior.

From the viewpoint of quantum information theory, quantum complexity is deﬁned by the

minimal number of quantum gates that are needed to build a target state from a reference state [16,

17]. Additionally, in the framework of AdS/CFT correspondence, one can compute the complexity

of states in the boundary quantum ﬁeld theory of the two-sided AdS black hole through the

complexity =action (CA) conjecture. In this conjecture, it is assumed that quantum complexity on

the boundary is associated with the gravitational action evaluated on a region of Wheeler-DeWitt

(WDW) patch in the bulk spacetime [18,19]. In the WDW patch, space/time-like boundaries

include null boundary surfaces [20] which can join with each other (for extensive information on

the derivation of total WDW action in CA conjecture, refer to the work of [21]). To analyze the

evolution of the complexity of braneworld according to the CA conjecture within the Horndeski

theory of gravity, we respectively sketch the Penrose diagram of the braneworld causal structure,

see Fig.∼1. In the previously mentioned ﬁgure, the WDW patch is denoted by the shaded region,

which intersects with cutoﬀs at times tLand tRrespectively. In the present study, we have chosen

the symmetrical conﬁguration for the time slices, i.e. tL=tR≡t/2. Now it will be a handful

to evaluate the gravitational action on this patch as the boundary time increases. In our case,

the WDW patch includes two UV cutoﬀ surfaces near the asymptotic boundary regions at r=

rmax which are denoted by black dashed lines on Fig.∼1and are used to omit IR divergencies.

Besides, there are two meeting points in the bulk due to the intersection with the future boundary

hypersurface at r=r1

mand with the past one at r=r2

m. As we know, the time evolution from the

WDW patch can be encoded in the time dependence of these points.

In this work, we investigate the holographic complexity of the braneworld. This model, being an

interesting approach to the hierarchy problem, assumes that the observable universe corresponds

r= 0

r=∞

III

rmax

r= 0

r=∞

III

rmax

FIG. 1: Bulk conformal diagram with WDW patch at early (tR=tL=τ/2 = 0) and late (tR=tL=τ/2>

0) times with the present singularity at the origin.

to a four-dimensional brane located at the boundary of the (warped) extra-dimensional space [22–

26]. Our boundary term (namely Gibbons-Hawking-Yang term, deformed because of the presence

of additional matter ﬁelds), does not contribute to the total Nambu-Goto action, since in our

conﬁguration we are only taking into account null-like hypersurfaces. Within the braneworld

formalism, gravity usually is conﬁned to the brane and therefore holographic conﬁnement could be

easily investigated through the Nambu-Goto term with null-like boundaries. The idea behind our

prescription is to investigate ﬁeld derivative coupling.

II. METHODOLOGICAL ROUTE AND ACHIEVEMENTS

Motivated by the recent applications of the Complexity equals action conjecture we present the

investigation of the conﬁnement and complexity of a Horndeski braneworld geometry. Here we

present a summary of the main results achieved in this work:

•We study the inﬂuence of the Horndeski parameters on the braneworld geometry and provide

a complete numerical solution for EoMs through ﬁrst-order formalism;

•We construct the braneworld spectrum through the linearization process executing the tensor

perturbations around the metric;

•We construct the null-like boundary term for the Horndeski action integral, which does not

contribute to the Nambu-Goto total action;

•Additionally, we compute the holographic conﬁnement and complexity of the braneworld

(both of the quantities depend on the Horndeski).

This work is organized as follows. In Sec.∼III, we present our gravitational setup. In Sec.∼V,

we found the numerical solutions through the ﬁrst-order formalism. In Sec.∼VI, we derive the

contribution of the null boundary intersections to the Gibbons-Hawking term. In Sec.∼VII, using

the well-known Nambu-Goto action integral, we analyze the holographic conﬁnement with the

varying values of modiﬁed gravity parameters and we show that the behavior of the energy where

E∼L. In Sec.∼VIII, we present the framework for the holographic complexity and analyze its

growth with the variation of free parameters. Finally, in Section IX we present our conclusions

and ﬁnal remarks on the key topics of our study.

III. THE HORNDESKI GRAVITY WITH A SCALAR POTENTIAL

In the current study, we are going to address the holographic properties of braneworld in the

framework of the Horndeski gravity and analyze the linear conﬁnement. The action with a scalar

potential within the Horndeski theory is therefore given as follows

I[gMN , φ] = Z√−gd5xκ(R−2Λ) −1

2(αgMN −γGMN )∇Mφ∇Nφ−V(φ)(1)

Note that we have a non-minimal scalar-tensor coupling where we can deﬁne a new ﬁeld φ0≡Ψ with

κ= (16πGN), GNbeing the Newton’s gravitational constant. This ﬁeld has dimension of (mass)2

and the parameters αand γcontrol the strength of the kinetic couplings, αis dimensionless and

γhas dimension of (mass)−2. The Einstein ﬁeld equations can be therefore derived with the use

of well-known least-action principle

GMN + ΛgMN =1

2kTMN (2)

where TMN =αT (1)

MN −gMN V(φ) + γT (2)

MN with

T(1)

MN =∇Mφ∇Nφ−1

2gMN ∇Pφ∇Pφ

T(2)

MN =1

2∇Mφ∇NφR −2∇Pφ∇(MφRP

N)− ∇Pφ∇KφRMP NK

−(∇M∇Pφ)(∇N∇Pφ)+(∇M∇Nφ)φ+1

2GMN (∇φ)2

−gMN −1

2(∇P∇Kφ)(∇P∇Kφ) + 1

2(φ)2−(∇Pφ∇Kφ)RP K 

(3)

and the scalar ﬁeld EoM is

∇M[(αgMN −γGMN )∇Nφ] = Vφ(4)

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HolographiccomplexityofbraneworldinHorndeskigravityFabianoF.Santosa,OleksiiSokoliukb;candAlexanderBaranskybaInstitutodeFsica,UniversidadeFederaldoRiodeJaneiro,CaixaPostal68528,RiodeJaneiro-RJ,21941-972{BrazilyandbMainAstronomicalObservatoryoftheNASofUkraine(MAONASU),Kyiv,03143,UkrainecAstronomica...

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Holographic complexity of braneworld in Horndeski gravity Fabiano F. Santosa Oleksii Sokoliukbcand Alexander Baranskyb aInstituto de F sicaUniversidade Federal do Rio de Janeiro

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