[Paper Review] Colliding waves on a string in AdS$_3$

David Vegh|arXiv (Cornell University)|Sep 16, 2015

Black Holes and Theoretical Physics3 citations

TL;DR

This paper studies classical string dynamics in global AdS₃ with periodically kicked endpoints, revealing three distinct phases: transparent (standing waves, no horizon), gray (linearly growing string length without horizon), and black (worldsheet horizon forms, preventing cusp propagation). The key discovery is the existence of a novel 'gray phase' where the string length grows linearly despite no horizon, explained via a simple cusp-counting model that reproduces all three phases and captures the transition behavior.

ABSTRACT

This paper is concerned with the classical motion of a string in global AdS$_3$. The initially static string stretches between two antipodal points on the boundary circle. Both endpoints are perturbed which creates cusps at a steady rate. The cusps propagate towards the interior where they collide. The behavior of the string depends on the strength of forcing. Three qualitatively different phases can be distinguished: transparent, gray, and black. The transparent region is analogous to a standing wave. In the black phase, there is a horizon on the worldsheet and cusps never reach the other endpoint. The string keeps folding and its length grows linearly over time. In the gray phase, the string still grows linearly. However, cusps do cross to the other side. The transparent and gray regions are separated by a transition point where a logarithmic accumulation of cusps is numerically observed. A simple model reproduces the qualitative behavior of the string in the three phases.

Motivation & Objective

• To investigate the formation of worldsheet horizons in a classical string in global AdS₃ under periodic boundary kicks.
• To understand the transition from horizon-free to horizon-containing string dynamics as kick strength increases.
• To identify and characterize qualitatively distinct dynamical phases in the string's behavior under steady-state forcing.
• To develop and validate a minimal model that captures the essential features of cusp propagation and string length evolution across phases.

Proposed method

• Uses an exact discretization technique that adds elementary shockwaves (cusps) to model the string's evolution in global AdS₃.
• Applies a numerical method with high stability to simulate the steady-state behavior after initial transients decay.
• Models cusp propagation using a simplified framework where each cusp increases string length by a fixed amount ν and tracks cusp exit times via a delay equation.
• Derives the cusp response time using a delay equation: t_out = (t_in - ν·n(t_in)) / (1 - ν), where n(t_in) is the number of right-moving cusps present at entry time.
• Numerically integrates the cusp number function n(t) using a sum of delta functions at response times to compute N(t) = 2(t - n(t))
• Validates the model against full numerical simulations of the string in AdS₃, showing qualitative agreement in all three phases.

Experimental results

Research questions

• RQ1 What happens to a string in global AdS₃ when both endpoints are periodically kicked, and how does the string's length evolve over time?
• RQ2 Can a worldsheet horizon form in the absence of a bulk black hole, and under what conditions does it appear?
• RQ3 Is there a dynamical phase where the string length grows linearly without forming a horizon, and what distinguishes it from the transparent and black phases?
• RQ4 How does the cusp number function N(t) behave across the three phases, and what is the nature of the transition between them?

Key findings

• Three distinct dynamical phases emerge: transparent (constant cusp number, standing wave), gray (linearly increasing cusp number, no horizon), and black (cusp number saturates, horizon prevents cusp transmission).
• In the transparent phase (ν < 1/2), the cusp number stabilizes in the steady state, indicating no net growth.
• In the gray phase (1/2 < ν < 1), the cusp number grows linearly with time, corresponding to a continuously lengthening string without a worldsheet horizon.
• In the black phase (ν > 1), cusps never reach the opposite endpoint due to the formation of a worldsheet horizon, and the cusp number plateaus.
• The transition between transparent and gray phases occurs at ν = 1/2, where the cusp number grows as √t, suggesting a non-analytic behavior that may be logarithmic in the full system.
• A simple cusp-counting model successfully reproduces the three phases and captures the qualitative behavior of the full string dynamics, though it does not reproduce the quasi-logarithmic cusp accumulation observed numerically at the transition point.