Particle motion and chaos

Zhenhua Zhou 1∗ and Jian-Pin Wu 2,3† 1 School of Physics and Electronic Information, Yunnan Normal University, Kunming, 650500, China 2 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China and 3 Institute of Gravitation and Cosmology, Department of Physics, School of Mathematics and Physics, Bohai University, Jinzhou 121013, China Abstract In this note, we explicitly illuminate that the velocity bound in the gravity corresponds to the chaos bound in the quantum system by studying the particle free falling in hyperscaling violating (HV) black brane. We also study the particle falling in a repulsive potential in HV black brane and in the AdS soliton geometry, respectively. We find that the momentum doesn’t grow exponentially and chaos is suppressed in the two cases. It is attributed to that when the repulsive potential is introduced or the black hole horizon is absent, the particle is slow down and its trajectory seen by a comoving observer is timelike, which corresponds a weak chaos system.


I. INTRODUCTION
In [1], Susskind proposes that there is a correspondence between the operator growth in the chaotic quantum systems and the momentum of the particle falling toward the black hole. In particular, they grow exponentially with the same Lyapunov exponent. For the AdS black hole, the particle's momentum grows at a maximal rate [1] and the exponent saturates the chaos bound proposed in [2]. It is a universal property because all the horizons are locally the Rindler-like. The same characteristic is also found in the the strongly coupling chaotic quantum system, SYK model. References [3,4] further study the particle falling toward charged black holes and confirm the Susskin proposal.
In this note, we first study the growth of the momentum of the particle with mass m and an external potential in a general hyperscaling violating (HV) black brane geometry.
By this simple example, we explicitly illuminate that the velocity bound in the gravity set the chaos bound in the quantum system. When the free falling particle travels with the light speed near the horizon 1 , the Lyapunov exponent saturates the chaos bound. If we add some repulsive potential, which results in the velocity of the particle being less than the light velocity, the exponent of the growth of the momentum will not saturate the bound, or even the momentum doesn't grow exponentially. In this case, chaos is suppressed.
We also study the growth of the momentum of the particle in AdS soliton geometry [5,6].
The holographic superconducting models have been built based on AdS soliton geometry [7,8]. Since the horizon is absent, the growth of the particle's momentum in AdS soliton geometry is different from that in the black holes background.

II. PARTICLE MOTION AND THE RADIAL MOMENTUM
Let us consider a particle with mass m and an external potential V moving in d + 1 spacetime dimensions. The metric is given by The action of the particle is where τ is arbitrary parameter of the particle world line and X µ is the spacetime coordination. V (X) is the external potential. From the above action, we can derive the equation of motion (EOM) asẌ where dot denotes the derivative with τ . Also, we can obtain the canonical momentum We choose the static gauge τ = t and take the ansatz r = r(t) and x, y = const. in what follows. In addition, we assume that the potential only depends on r, i.e., V = V (r). And then, Eqs.(3) and (4) reduce tȯ where A > 0 is the integral constant. To have a solution, the increase of the potential V should be slow enough such that According to Suskind proposal [1], we shall consider the growth rate of the Rindler momentum p ρ for the black hole, which is ρ ∼ √ r + − r and r + is the black hole horizon.

A. Hyperscaling violating black brane
In this section, we study the particle falling in 4 dimensional neutral HV black brane geometry, which takes the form [9, 10] The horizon of the black brane locates at r + and the boundary is at r → 0. z and θ are the Lifshitz dynamical exponent and hyperscaling violating exponent, respectively. The constraints from the null energy condition give B. Particle free falling We study the particle free falling in the HV background in this subsection. So we set V = 0 in this subsection and Eq. (5a) reduces to And the momentum p ρ is We first analyze the growth rate of the radial momentum near the horizon of the black brane. Near the horizon, f (r) can be expanded as And then, substituting (11) into the EOM of particle (9), we find that Immediately, the radial momentum p ρ can be obtained as This behavior coincides with the proposal in [1], which is independent of θ and z.
We can also study the growth of the radial momentum by solving the equation (9) in the full AdS spacetime. For θ = 2 and z < 2, we can obtain the analytical solution, which Note that we have take r + = A = 1 in the above equation. It is easy to find that at the late times the momentum exhibits exponential growth.
For general θ and z, it is hard to obtain the analytic solutions and we resort to the numerical method instead. FIG.1 shows the momentum time dependence p ρ (t) for different hyperscaling and Lifshitz parameters θ and z. Again, it confirms that the exponential growth of the momentum at late time.
In summary, for the free falling particle, the momentum grows as p ρ ∼ e 2πT t , which is independent of the hyperscaling and Lifshitz parameters θ and z. It corresponds to a complexity growth bound (the scramble time) in the dual quantum system. Following Suskind proposal [1], it is because the velocity bound in the gravity is dual to the chaos bound in the quantum system. In the present case, the free falling particle travel with the light speed near the horizon, which set the chaos bound in the quantum system. If the particle cannot reach the light speed, the momentum growth will not saturate the bound.
We can add a repulsive potential to achieve this goal.

C. Particle falling with repulsive potential
In this subsection, we turn on the potential V and Eq.(5) in HV background becomė We take the form of the potential as which is large enough to slow down the falling particle.
As a simple example, we first study the asymptotic flat background with θ = 2, z = 1.
Then Eq.(15) gives Since p ρ ∼ √ r + − rp r , the Rindler momentum p ρ tends to a constant but doesn't grow exponentially at late time. It is because the particle can not reach the light speed when the repulsive potential is added. As a result, the scramble bound is not saturated in the dual quantum system.
For general θ and z, we havė The numerical results for different θ and z are shown in FIG2. It confirm the result that the momentum p ρ tends to a constant but doesn't grow exponentially at late time.

IV. PARTICLE FALLING IN ADS SOLITON BACKGROUND
5 dimensional AdS soliton metric is given by [5][6][7][8]  where r + is the tip of this geometry. To avoid a conical singularity at the tip we must impose a period of χ ∼ χ + π r + . We can make a double Wick rotation of the AdS Schwarzschild black brane to obtain the geometry (19).
Taking the ansatz r = r(τ ), x = y = χ = const. with the static gauge τ = t and using EOM (3) with zero potential, we can obtain the EOM for the free falling particle in AdS The momentum p r (t) reduces to We numerically solve Eq.(20) and obtain the momentum time dependence p r (t) (FIG.3).
Since the horizon is absent in the AdS soliton background, the chaos is suppressed. Therefore, the growth of the particle's momentum in AdS soliton geometry is different from that in the black holes background. More detailed explorations deserve further studying.

V. CONCLUSIONS AND DISCUSSIONS
In this note, we study the momentum growth of a particle when it falls in HV black brane and the soliton background. For HV case, when the particle free falls toward the black brane, the momentum always grows exponentially, independent of the Lifshitz and HV exponents.
The exponent growth rate 2πT can be explained by the correspondence between the light speed in gravity and the chaos bound in quantum complexity. However, after a repulsive potential is introduced, the exponent of the growth of the momentum may not saturate the bound, or even the momentum doesn't grow exponentially, since the the falling particle is slow down and can not achieve the light speed. The similar situation happens when the free particle falls in the soliton background. Without a horizon, the light speed is also hard to achieve and then no exponent growth appears. In a dual manner, we can say that the chaos of the quantum system is suppressed.
Our results explicitly illuminate that the velocity bound in the gravity corresponds to the chaos bound in the quantum system. To have an well understanding on this point, it is help to specify the line element of the particle trajectory seen by a comoving observer.
When l 2 (λ) → 0, we say that the particle travels with a light speed to the comoving observer.
Then, for a free particle falling near the black hole horizon, its speed approaches the light speed as g tt → 0. When a repulsive potential V ∼ 1/ √ −g tt (as in (16) ) is introduced, or the black hole horizon is absent, which results in g tt = 0, l 2 (λ) is finite and so the growth rate can not reach the chaos bound any more.