[Paper Review] Chaos in the black hole S-matrix

Joseph Polchinski|arXiv (Cornell University)|May 29, 2015

Black Holes and Theoretical PhysicsPhysics and Astronomy33 references72 citations

TL;DR

This paper extends the connection between black hole chaos and the S-matrix by deriving an identity for the change in the black hole S-matrix due to an infalling particle, showing exponential growth in the effect over time—evidence of Lyapunov chaos—until the scrambling time. The result links horizon physics to quantum chaos and implies that outgoing Hawking radiation carries information about infalling particles, challenging the idea of a smooth horizon and reinforcing firewall-like implications.

ABSTRACT

Recent work by Shenker, Stanford, and Kitaev has related the black hole horizon geometry to chaotic behavior. We extend this from eternal black holes to black holes that form and then evaporate. This leads to an identity for the change in the black hole S-matrix (over times shorter than the scrambling time) due an addition infalling particle, elaborating an idea of 't Hooft.

Motivation & Objective

• To extend the connection between black hole chaos and the S-matrix beyond eternal black holes to dynamical black holes that form and evaporate.
• To derive a quantitative identity for how an additional infalling particle alters the black hole S-matrix, reflecting chaotic dynamics.
• To clarify the role of the horizon in encoding information and the implications for the black hole information paradox.
• To connect Lyapunov growth in the commutator-squared to the redshift near the horizon, identifying it as a signature of quantum chaos.
• To examine the consistency of the S-matrix framework with effective field theory and the firewall paradox.

Proposed method

• Derives an S-matrix identity under plausible assumptions, using the commutator-squared of operators as a probe of chaos.
• Applies the commutator [W(t), V(0)] to analyze sensitivity to initial conditions, with the squared commutator revealing Lyapunov growth.
• Uses the redshift factor dt/dτ ∝ e^{2πt/β} as a signature of exponential growth in the commutator-squared, linking it to Lyapunov exponents.
• Assumes low-energy effective field theory holds outside the horizon for few-particle scattering, while allowing breakdown near the horizon to permit an S-matrix.
• Compares the black hole S-matrix effect to a laboratory analog involving vacuum fluctuations and detector measurements, illustrating information transfer.
• Analyzes the implications of the identity for the nature of the horizon, concluding it cannot be information-free if an S-matrix exists.

Experimental results

Research questions

• RQ1 How does the black hole S-matrix change when an additional particle falls in, and what is the time evolution of this change?
• RQ2 What is the quantum mechanical signature of chaos in a dynamical black hole, and how does it relate to the horizon geometry?
• RQ3 Can the exponential growth in the commutator-squared be linked to the redshift near the black hole horizon, and what does this imply for Lyapunov exponents?
• RQ4 How does the existence of an S-matrix conflict with or reinforce the firewall paradox, given that the horizon must carry information?
• RQ5 What is the role of effective field theory outside the horizon in enabling the S-matrix identity, and what assumptions are required?

Key findings

• The paper derives an identity for the change in the black hole S-matrix due to an infalling particle, showing that the effect grows exponentially with time.
• The exponential growth is linked to the Lyapunov exponent, with the rate matching the redshift factor dt/dτ ∝ e^{2πt/β}, confirming chaotic behavior.
• The commutator-squared exhibits early exponential growth (Lyapunov) and later decay (Ruelle), distinguishing quantum chaos from thermalization.
• The S-matrix identity implies that the horizon cannot be information-free; outgoing Hawking modes carry information about infalling particles.
• The result supports the idea that the black hole horizon is a chaotic boundary, and that the S-matrix framework implies a firewall-like energy flux due to measurement-like effects.
• The laboratory analog shows that measuring vacuum fluctuations correlates modes with detectors, and the black hole performs a similar measurement, leading to a non-trivial energy flux—consistent with a firewall.