Complexity of AdS_5 black holes with a rotating string
Koichi Nagasaki
hep-th








Complexity of AdS_5 black holes with a rotating string



Koichi Nagasaki[koichi.nagasaki24@gmail.com]



Department of Physics and Center for High Energy Physics,
Chung Yuan Christian University

Address: 200 Chung Pei Road, Chung Li District, Taoyuan City, Taiwan, 32023





We consider computational complexity of AdS_5 black holes. 
Our system contains a particle moving on the boundary of AdS.
This corresponds to the insertion of a fundamental string in AdS_5 bulk spacetime.
Our results give a constraint for complexity. 
This gives us a hint for defining complexity in quantum field theories.




§ INTRODUCTION



 §.§ Background


Computational complexity is originally introduced in physics of quantum information <cit.> and its application to black hole physics is considered in many works <cit.>. 
It behaves in a similar way to the entropy as it is expected to satisfy the second law of thermodynamics <cit.>.
The growth of complexity is related to problems of black holes, for example, information problem, the transparency of horizons or the existence of firewalls <cit.>.


Complexity-Action (CA) conjecture <cit.> predicts that complexity of the black hole is equal to the the bulk action integrated in the region called the Wheeler-de Witt patch (WdW).
This conjecture is checked by many works <cit.>.
The time dependence is studied in <cit.>.
This duality is studied also in a deformed case by adding probe branes <cit.>.


Nevertheless, the strict and valid definition of complexity is still unclear.
The most standard one so far is by the method of quantum operators which are called gates.
Let us define a reference state as the simplest quantum state.
Quantum gates operate a state and change its state.
Computational complexity of a test state is defined as the minimum number of gates to prepare it from the reference state.
But this definition has some unsatisfied points. 
One is that the definition of the reference state is unclear.
Another is that the number of the gates depends on the choices of the primitive gates.
 


 §.§ Our approach


To treat this problem, in this paper, we shed light on a new property of complexity and suggest a new constraint which complexity should possess.

The above works so far treat stationary systems. 
A generalization to this conjecture to the system other than stationary states is an important work.
As an example, we consider here a system moved by a drag force. 
Such a work is motivated by jet-quenching phenomena in heavy ion collisions.
Energy loss of a charged particle in quark-gluon plasma (QGP) is calculated in several approaches <cit.>.
Especially in <cit.> a particle moving in AdS_5-Schewarzchild spacetime was considered and its action was calculated.
The particle is moved on the boundary of the AdS_5 space.
Because of the effect of shear viscosity in QGP, this particle losses the energy. 
This system is a dissipative system.


While his work considered the action in the outside of the horizon, that is usual spacetime, we apply here this method also to the inside of the horizon. 
According to CA relation, we will obtain complexity of a time dependent system.
In this work we consider a Wilson line operator located in AdS_5 spacetime by inserting a fundamental string. 
This Wilson line moves on a great circle in S^3 part in AdS_5.
Such a non-local operator describes a test particle moving on the boundary gauge theory.
This shows how complexity is deformed when adding a time-dependent operator, especially also non-local operator.
Since inserting the Wilson loop is described by adding a Nambu-Goto (NG) term, the action is expected to consist of the  Einstein-Hilbert term, the Nambu-Goto term and the boundary term.
We study on the effect of the Wilson line operator focusing on the Nambu-Goto term and show the black-hole mass and particle's velocity dependences of the action.
This will show the growth of complexity for a dissipative system.


This paper is constructed as follows.
In section <ref>, we explain our setup which includes the AdS black hole with a kind of non-local operator, a Wilson line, and the action.
In section <ref>, in order to find the effect of the test particle, we focus on the NG action which is a part of the action we introduced the former section.
Section <ref> gives an interpretation of our calculation and some suggestions of the constraint for complexity.




§ SETUP

We consider a Wilson loop inserted on AdS_5 spacetime.
The black hole geometry in AdS_5 spacetime is described by the metric:

    ds_AdS^2    = -f(r)dt^2 + dr^2/f(r) + r^2dΩ_3^2,
    
    f(r)    = 1 - 16π GM/5Ω_3r^2
      + r^2/ℓ_AdS^2
    = 1 - r_m^2/r^2 + r^2/ℓ_AdS^2,    
    r_m^2 := 16π GM/5Ω_3,

where dΩ_3^2 is the square of the line element on three sphere, while dΩ_3 is the volume form on the three sphere and Ω_3 is the volume obtained by integrating it.

[
The volume of (n-1)-dim unit sphere S^n-1 is Ω_n-1 = 2π^n/2/Γ(n/2).
]

We see in appendix <ref> that ℓ_AdS and other parameters are related to the gauge theory parameters as

    r_m^2/ℓ_AdS^2 
    = 16π GM/5Ω_3ℓ_AdS^2
    = 8GM/5πℓ_AdS^2
    = 4M/5√(α'g_YM)/N^7/4.

This geometry has some characteristic length; one is the Schwarzschild radius defined in eq.(<ref>) and the other is r_h which is the radius of the black-hole horizon which is determined by the equation f(r) = 0:

    r_h = ℓ_AdS(-1+√(1+4r_m^2/ℓ_AdS^2)/2)^1/2.

We parametrize S^3 part of the AdS spacetime (<ref>) by 

    x^1    = cosθcosϕ,  
    x^2 = cosθsinϕ,
    
    x^3    = sinθcosφ,  
    x^4 = sinθsinφ;
       θ∈[0,π/2), ϕ,φ∈[0,2π).



Let us take parameters τ and σ on the world-sheet of the fundamental string:

    t = τ,  
    r = σ,  ϕ = ωτ + ξ(σ),

where ω is the constant angular velocity and ξ(σ) is a function which determines the shape of the string. 
Thus, the induced metric on the world-sheet is

    ds_ind^2    = -f(σ)dτ^2 + dσ^2/f(σ),
    
    f(σ)    = 1 - r_m^2/σ^2 + σ^2/ℓ_AdS^2.


The action consists of the Einstein-Hilbert term and the Nambu-Goto (NG) term:

    S_EH   = 1/16π G∫ d^6x√(|g|)(ℛ-2Λ),  Λ = -6/ℓ_AdS^2,
    
    S_NG   = -T_s∫ d^2σ√(- g_ind),

where T_s is the tension of the fundamental string.
We are afraid for the necessity of adding a boundary term like the York-Gibbos-Hawking (YGH) term.

The development of the Einstein-Hilbert action is given in <cit.> as

    dS_EH/dt   = -1/2π Gℓ_AdS^2∫_Σ r^3dr dΩ_3
    
    = -1/8π Gr_h^4Ω_3/ℓ_AdS^2.

This is the leading contribution to complexity. In the next section we consider the effect of the addition of the Wilson loop.




§ EVALUATION OF THE NG ACTION


  

The addition of the Wilson loop corresponds to inserting a fundamental string whose world-sheet has a boundary on the Wilson loop.
We calculate here the NG action of this fundamental string.
The time derivative of the NG action is obtained by integrating the square root of the determinant of the induced metric over the WdW (figure <ref>):

    dS_NG/dt   = T_s∫_0^r_h dσ√(1 - σ^2ω^2/f(σ) + σ^2ξ'(σ)^2f(σ)).


The Lagrangian is

    ℒ
    = T_s√(1 - σ^2ω^2/f(σ) + σ^2ξ'(σ)^2f(σ)).

Then the equation of motion for ξ(σ) is 

    0 = 1/T_sd/dσ∂ℒ/∂ξ'(σ)   = d/dσ(σ^2 ξ'(σ)f(σ)/√(1- σ^2ω^2/f(σ) + σ^2 ξ'(σ)^2 f(σ))),
    
    c_ξ   :=  σ^2 ξ'(σ)f(σ)/√(1- σ^2ω^2/f(σ) + σ^2 ξ'(σ)^2 f(σ))
    =  σ^2 ξ'(σ)f(σ)/ℒ/T_s,

where by the second line we defined a conserved constant c_ξ.
Solving it for ξ(σ), this function is given by integrating the following:

    ξ'(σ) 
    = c_ξ/σ f(σ)√(%s/%s)f(σ) - σ^2ω^2σ^2f(σ) - c_ξ^2.

In order for this expression to give real values, the denominator in the square root must be negative when the numerator factor f(σ) - σ^2ω^2 becomes negative. 
Thus they become zero coincidentally.
From this condition, the constant c_ξ is determined to be 

    c_ξ = ωσ_H^2
       = ω-1+√(1+4r_m^2(1/ℓ_AdS^2-ω^2))/2(1/ℓ_AdS^2-ω^2),
    σ_H^2    = -1+√(1+4r_m^2(1/ℓ_AdS^2-ω^2))/2(1/ℓ_AdS^2-ω^2),

where σ = σ_H is the solution for numerator of the square root in eq.(<ref>).
We assumed that c_ξ is positive.
Since the numerator and the denominator have the coincident solution, the cancelation gives

    ξ'(σ) 
    
    
       = ωσ_H^2/(σ^2 - r_h^2)(1/ℓ_AdS^2(σ^2 + r_h^2) 
    + 1)√(%s/%s)(1/ℓ_AdS^2-ω^2)(σ^2 + σ_H^2) 
    + 11/ℓ_AdS^2(σ^2 + σ_H^2) + 1,
    ℒ   = T_sσ^2ξ'(σ)f(σ)/c_ξ
    = T_s√(%s/%s)(1/ℓ_AdS^2-ω^2)(σ^2 + σ_H^2) 
    + 11/ℓ_AdS^2(σ^2 + σ_H^2) + 1.

The integral of (<ref>) over the WdW is 

    dS_NG/dt   = ∫_WdWℒ
    = T_s∫_0^r_h dσ√(%s/%s)(1/ℓ_AdS^2-ω^2)(σ^2 + σ_H^2) + 11/ℓ_AdS^2(σ^2 + σ_H^2) + 1.

Using the integral formula (<ref>) shown in appedix,

    dS_NG/dt 
    = -iT_s√(σ_H^2(1-ℓ_AdS^2ω^2)+ℓ_AdS^2)
      E[arcsin(ir_h/√(σ_H^2 + ℓ_AdS^2)),
        √(%s/%s)(σ_H^2 + ℓ_AdS^2)(1-ℓ_AdS^2ω^2)σ_H^2(1-ℓ_AdS^2ω^2) + ℓ_AdS^2 ].

Substituting eq.(<ref>) and the horizon radius r_h, (<ref>), we can express the growth of the action as

    dS_NG/dt   = -iT_sℓ_AdS(1+√(1+4s^2
        (1-v^2))/2)^1/2×
           
    E[arcsin(i(
      (-1+√(4s^2+1))(1-v^2)/(1 - 2v^2) +√(1+4s^2(1-v^2)))^1/2),
        ((1 - 2v^2) + √(1+4s^2(1-v^2))/1+√(1+4s^2(1-v^2)))^1/2 ],
 
where for simplicity we defined the following dimensionless constants:

    s := r_m/ℓ_AdS,   
    v := ℓ_AdSω.




  
Limit behavior 
The behavior in the limits s→ 0 and s→∞ are, respectively, as follows:

    dS_NG/dt|_s→ 0   = -i4T_sℓ_AdS E[0, √(1-v^2)] = 0,
    dS_NG/dt|_s→∞
    
       = 4T_sℓ_AdS√(s)(1-v^2)^1/2.

The behavior in the limits v→ 0 and v→ 1 are 

    dS_NG/dt|_v=0
    
    
       = √(2)T_sℓ_AdS(-1+√(4s^2+1)/2)^1/2∼√(s)  (s≫ 1),
    dS_NG/dt|_v→ 1
    
    
    
       = -iT_sℓ_AdSarcsin i(-1+√(4s^2+1)/2(1+s^2))^1/2
    = T_sℓ_AdSarcsinh(-1+√(4s^2+1)/2(1+s^2))^1/2.

The figures <ref> and <ref> show these behavior and also the curves of all parameter ranges.
The function (<ref>) has an extremum at s=√(2).


  

  




§ DISCUSSION


Figure <ref> shows the relation between the growth of the action and the angular velocity (v = ℓ_AdSω). 
We can see that the effect of the drag force to the growth of the action is larger as the mass increases when the string moves relatively slower. As the string moves faster the effect become smaller.
For any mass, the effect vanishes in the limit v→ 1.
This behavior is similar to the one that of a rotating black holes — the complexity growth bound changes from 2M to 2√(M^2 - J^2/ℓ_AdS^2), where J is the angular momentum of the black holes.


Figure <ref> shows mass dependence of the growth of the action. 
When the angular velocity is small, it is a monotonically increasing function of the mass.
As the angular velocity becomes larger, it ceases to increase rapidly and the extremum appears at about v≳ 0.97.
That is a notable phenomena — complexity has different velocity dependence in the relativistic region.

According to CA duality these represent the growth of the black-hole complexity.
For a large mass black hole, complexity changes rapidly when the particle moves slowly and this effect becomes small for a relativistic particle.
As the black hole becomes smaller, the complexity does not change whether the particle moves quickly or not.
Complexity is expected to increase as the entropy increases by the second law of thermodynamics <cit.>.
So the results shown in figure <ref> and figure <ref> are consistent with this expectation.

The velocity dependence of energy loss is changed by particle's mass.
It is due to collisions between quarks and gluons, and gluon bremsstrahlung and which is dominant depends on its velocity <cit.>.
The behavior of the action for large mass, (∼√(s)), is consistent with the expected behavior from CA relation <cit.>, that is, the growth of complexity is bounded by two times mass. 
These behavior characterize the growth of complexity and gives a hint to define complexity in black hole spacetime.


In our analysis the boundary term, which was detaily discussed in <cit.>, is not taken into consideration.
An example of the boundary term calculation is given in <cit.>.
They considered AdS boundaries.
It is different from our case since in our calculation we need boundary terms which come from the boundary of region 1 (see figure <ref>). 
There is a cancelation and the contributions only from r=0 (black-hole singularity) and r=r_h (horizon) survive.
The detailed analysis will be considered as a future work.
However, while the Einstein-Hilbert term includes the second derivative of the metric, the NG action only includes the first derivative of it.
The value at the boundary is completely determined by the metric and the derivative of it is unnecessary.

Then we can expect that the boundary term is not need in this situation.




§ ACKNOWLEDGMENTS

I would like to thank Satoshi Yamaguchi, Hiroaki Nakajima and Sung Soo Kim for useful discussions.
This work is supported in parts by the Chung Yuan Christian University, the Taiwan's Ministry of
Science and Technology (grant No. 105-2811-M-033-011)





§ GAUGE THEORY PARAMETER


The Newton's gravitational constant we used is related to the gauge theory parameter.
Since 5-dimensional gravitational constant and AdS radius are as follows <cit.>:

    G_5 = πℓ_AdS^3/2N^2,  ℓ_AdS^2 = α'√(g_YM^2N)  ⇒  
    G_5 
    = π/2N^2α'^3/2(g_YM^2N)^3/4
    = π/2(α' g_YM)^3/2/N^5/4,

we obtain

    G_5/ℓ_AdS^2 
    = π/2N^2√(α')(g_YM^2N)^1/4
    = π√(α'g_YM)/2N^7/4.

Note that some people may use the different convention where ℓ_AdS^2 is different by the factor √(2), for example <cit.>, but we follow the above definition in this paper.




§ BLACK HOLE TEMPERATURE

Here we derive the temperature of AdS_5 black hole.
This result can be seen in <cit.>.
We consider the Euclidean version of the metric (<ref>) and take the coordinates:

    r = r_h(1+ρ^2).

Near the horizon the metric is approximated as

    ds_AdS^2
       ≈ 
    2ρ^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2) dt_E^2
     + 2r_h^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^-1dρ^2
     + r_h^2dΩ_3^2
       = 2r_h^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^-1(dρ^2
     + ρ^2/r_h^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^2 dt_E^2)
     + r_h^2dΩ_3^2.

Seeing the period of Euclidean time t_E, the black hole temperature (1/β) is 

    β   = 2π r_h
      (r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^-1.


Expressed it by using mass, this leads 

    T
       = 1/2π r_h(r_m^2/ℓ_AdS^2/r_h^2/ℓ_AdS^2 
      + r_h^2/ℓ_AdS^2)
       = 1/2πℓ_AdS(-1+√(1+4x)/2)^-1/2(2x/-1+√(1+4x) 
      + -1+√(1+4x)/2),  
    x := 8GM/5πℓ_AdS^2,
    
    Tℓ_AdS   = 1/2π(x(-1+√(1+4x)/2)^-3/2
      + (-1+√(1+4x)/2)^1/2).

where we used eqs. (<ref>) and (<ref>) and in the last line we used dimensionless quantities Tℓ_AdS and x.
This function is plotted in figure <ref>.

  




§ ELLIPTIC INTEGRAL

For calculating the action in the Wheeer-de Witt patch (WdW) we use the elliptic integral of second kind.
The elliptic integral of the second kind is defined as 

    E[φ, k]
    := ∫_0^sinφ√(%s/%s)1-k^2t^21-t^2 dt.

We need the following integral formula:

    ∫_0^c√(%s/%s)x^2+ax^2+bdx
    = -i√(a) E[arcsin(ic/√(b)),√(%s/%s)ba  ].

It can be proved as

    ∫_0^c√(%s/%s)x^2+ax^2+bdx
       = √(%s/%s)ab∫_0^c√(%s/%s)1 + x^2/a1 + x^2/bdx
       =√(%s/%s)ab∫_0^ic/√(b)√(%s/%s)1 - (b/a)t^21 - t^2(-i√(b)dt),  
    	ix/√(b) = t,
       = -i√(a) E(arcsin(ic/√(b), √(b/a))).





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