ブラックホール地平面における粒子と弦の運動のカオス
棚橋典大 [阪大理]based onUniversality in Chaos of Particle Motion near Black Hole Horizon
橋本幸士、棚橋典大 [arXiv:1610.06070]
Work in progress (橋本幸士、村田佳樹、棚橋典大)Classical particle moving near black hole (BH) horizonGravity

External force: electric, scalar, ...

String moving near BH horizon in AdS spacetime

 Particle & string motion become chaotic due to BH gravity
 Lyapunov exponent 𝜆 of the chaos is bounded by surface gravity 𝜅 Motivation:
A bound on chaos in QFT at temperature T :[Maldacena-Shenker-Stanford '15]

 Probing the effect of temperature T to chaos in QFT.

 We study effect of temperature to chaos in classical gravity.
Use BH surface gravity

instead.

 To probe effect of 𝜅, we look at trajectories very close to BH.

4CONTENTS
1. Chaos bound for particle near BH
2. Chaos bound for AdS string
3. Summary

5Classical Chaos
Classic chaos in deterministic dynamical systems= Non-periodic bounded orbits sensitive to initial conditions

Diagnostics of chaos
Poincaré plot = Section of orbits in phase space

Lyapunov exponent 𝜆 = Separation growth rate of nearby orbits

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Chaos Bound for Particle near BH
 To realize a particle moving very close to BH horizon,
1. put a particle in a trapping harmonic potentialr
θ
2. take it close to a BH horizonr( no chaos)

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Chaos Bound for Particle near BH
Top view:: Lyapunov exponent

r0r

2. take it close to a BH horizon & look at the separatrix
Gravity

Potential force

r0r
Total potentialfor particle

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Chaos Bound for Particle near BH
Gravity

Potential V(r)rr0• Slow radial velocity near r = r0

• Linear approximation for V(r) ~ (slope) ⨉ (r − rhorizon)• Near-horizon limit r0 → rhorizon

9Chaos Bound for Particle near BH
Lyapunov exponent

Head-on view:

r0r• Slow radial velocity near r = r0

• Linear approximation for V(r) ~ (slope) ⨉ (r − rhorizon)• Near-horizon limit r0 → rhorizon

 A generic trajectory obeys

Examples
Charged particle near charged black hole:withParticle with scalar force:

These two examples give 𝜆 = 𝜅 for any m and c.

10

Numerical check

Particle near Potential Minimum

Particle near BH Horizon

Numerical check

Particle near Potential Minimum

Particle near BH Horizon

Periodic motion, no chaos

Chaotic motion

Poincaré plot at
Particle near Potential Minimum

Particle near BH Horizon

Regular KAM tori, no chaos

Lyapunov exponent 𝜆 ~ 0.2 𝜅Satisfies the bound 𝜆 ≤ 𝜅14CONTENTS
1. Chaos bound for particle near BH
2. Chaos bound for AdS string
3. Summary

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Chaos Bound for AdS String
Gravity

Potential V(r)rr0GravityString tension

r0
Fundamental string in AdS = “quark-anti quark pair”

Maldacena ‘98
Rey & Yee ‘98

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Chaos Bound for AdS String“Square-shaped string” approximation

r

L

17

Chaos Bound for AdS String“Square-shaped string” approximation
Gravitational forcer
String tension
 Linear potentialL

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Chaos Bound for AdS String“Square-shaped string” approximationr

For a string near horizon,

⇒L

19

Chaos Bound for AdS String
Nonlinear dynamics of string
Shake the string end points
by amplitude Nonlinear string motion⊥//

Shake the quark－quark pair
 Nonlinear flux tube dynamics

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Chaos Bound for AdS String
Nonlinear dynamics of string for slightly different

⊥
//

21

Chaos Bound for AdS String
Nonlinear dynamics of string for slightly different

⊥
//

22

Chaos Bound for AdS String
Nonlinear dynamics of string for slightly different
String tip position difference|Δr (t)|⊥//

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SUMMARYWe got a bound on chaos from classical BH-particle system
which coincides with the bound by Maldacena-Shenker-Stanford.

Independent of particle mass, external force & metric form.
Extension to string in AdS
 Unstable mode similar to the BH-particle system
 Instability growth rate:?: Does this govern chaotic motion of string in AdS?
?: Interpretation as chaos in the gauge theory?