[Paper Review] A note on the string spectrum at the Hagedorn temperature

J.D. Madrigal Martínez, Pedro Talavera|ArXiv.org|May 21, 2009

Black Holes and Theoretical Physics28 references3 citations

TL;DR

This paper investigates semi-classical string configurations in a curved background near the Hagedorn temperature, finding that solitonic string solutions vanish or diverge as the Hagedorn limit is approached in the strong coupling regime. The results suggest that Hawking radiation is driven by highly excited string states, and the system exhibits dynamical instability before reaching the Hagedorn temperature, indicating a possible second-order phase transition where high-energy modes disappear in favor of low-energy ones.

ABSTRACT

We discuss semi-classical string configurations at finite temperature. We find that those soliton solution in the background describing type IIA strings disappear or become divergent when we approach the Hagedorn temperature in the strong coupling regime. These findings together with a semi-classical analysis for the Hawking radiation let us to think that Hawking radiation is mainly driven by the existence of highly excited states. As by side, we check that beside the thermodynamical instability the system is dynamical unstable before reaching the Hagedorn temperature.

Motivation & Objective

• To investigate the behavior of semi-classical string configurations near the Hagedorn temperature in a strong coupling regime.
• To assess whether soliton solutions persist or diverge as the Hagedorn temperature is approached.
• To explore the connection between Hawking radiation and the existence of highly excited string states.
• To examine the dynamical stability of the system before reaching the Hagedorn temperature.
• To determine whether the Hagedorn transition corresponds to a phase transition, particularly a second-order one, in the context of strongly interacting strings.

Proposed method

• Analysis of semi-classical string trajectories in a near-extremal NS5-brane background with a warped geometry.
• Use of the near-horizon limit of the NS5-brane solution in type IIA string theory, including dilaton and metric fields with non-trivial warping factors.
• Computation of the effective potential and analysis of circular orbits to study stability via the linearized stability matrix and Lyapunov exponents.
• Evaluation of the Lyapunov timescale relative to gravitational wave timescale to assess observational relevance of instabilities.
• Application of the AdS/CFT correspondence and comparison with previous results on weakly interacting strings and black hole phase transitions.
• Use of canonical coordinates and energy conservation to derive eigenvalues of the stability matrix, ensuring phase space volume preservation.

Experimental results

Research questions

• RQ1 Do solitonic string solutions in the NS5-brane background remain finite or become divergent as the Hagedorn temperature is approached?
• RQ2 Is there evidence of a second-order phase transition at the Hagedorn temperature, marked by the disappearance of highly excited string modes?
• RQ3 How does the dynamical stability of the system evolve as the temperature increases toward the Hagedorn limit?
• RQ4 Can Hawking radiation be attributed to the presence of highly excited string states in the semi-classical regime?
• RQ5 What is the role of strong string interactions in modifying the phase structure compared to weakly interacting string gas models?

Key findings

• Soliton solutions in the NS5-brane background become divergent or disappear as the Hagedorn temperature is approached in the strong coupling regime.
• The system exhibits dynamical instability before reaching the Hagedorn temperature, with Lyapunov timescales shorter than gravitational wave timescales for κ=1 and trajectories near the horizon.
• For κ=1, the eigenvalues of the stability matrix are real and non-zero, indicating unstable circular orbits with λ± = ±2ρ₀√(N+ρ₀²)/(ρ(N+ρ²)), confirming dynamical instability.
• The Lyapunov timescale is significantly shorter than the gravitational timescale only for κ→1 and ρ₀ ≤ ρ ≲ 4πρ₀, making the instability observationally relevant in this regime.
• The Hagedorn transition does not exhibit discontinuities in thermodynamic quantities, suggesting a second-order phase transition where high-energy string modes vanish and are replaced by low-energy modes.
• Hawking radiation is primarily driven by highly excited string states, with the emission becoming thermal only after the high-energy modes have disappeared.