[Paper Review] Black Hole Horizons and Bose-Einstein Condensation

Frank D. Ferrari|arXiv (Cornell University)|Jan 29, 2016

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper proposes a large N matrix quantum mechanics model to study the thermodynamics of a probe D-particle near a black hole horizon. It demonstrates a non-perturbative Bose-Einstein condensation of massless open strings above a critical temperature, signaling horizon crossing, with the transition being infinitely sharp yet undetectable in finite time due to its non-local, infinite-time correlation order parameter.

ABSTRACT

Consider a particle sitting at a fixed position outside of a stable black hole. If the system is heated up, the black hole horizon grows and there should exist a critical temperature above which the particle enters the black hole interior. We solve a simple model describing exactly this situation: a large N matrix quantum mechanics modeling a fixed D-particle in a black hole background. We show that indeed a striking phenomenon occurs: above some critical temperature, there is a non-perturbative Bose-Einstein condensation of massless strings. The transition, even though precisely defined by the presence of the condensate, cannot be sharply detected by measurements made in a finite amount of time. The order parameter is fundamentally non-local in time and corresponds to infinite-time correlations.

Motivation & Objective

• To model the thermodynamic behavior of a probe particle near a black hole horizon using a large N matrix quantum mechanics framework.
• To investigate whether a phase transition occurs as temperature increases, corresponding to the probe being captured by the black hole.
• To determine if the transition associated with horizon crossing can be detected via finite-time measurements or correlation functions.
• To analyze the role of massless string modes and their condensation in high-temperature black hole physics.
• To explore the nature of the phase transition—particularly its order and detectability—within a holographically inspired quantum model.

Proposed method

• Formulates a Hamiltonian (1.1) describing a D-particle probe coupled to a large N matrix model representing a black hole background via D-branes.
• Uses canonical commutation relations for matrix operators $X$, $Π$ and creation/annihilation operators $a^{†}$, $a$ for open strings.
• Applies finite-temperature field theory techniques, including Matsubara formalism, to compute thermal Green's functions $G_k$.
• Analyzes the zero-mode expectation value $n_0 = \langle a^i \rangle$ as a candidate order parameter, but shows it vanishes identically.
• Identifies the infinite-time limit of the two-point function as the true non-local order parameter for the phase transition.
• Performs a high-temperature analysis to derive the critical temperature $T_c$ and the critical line in the $M$-$T$ plane, where $M$ is the string mass parameter.

Experimental results

Research questions

• RQ1 Does a phase transition occur in a probe particle system near a black hole horizon as temperature increases, signaling horizon capture?
• RQ2 Can the transition associated with horizon crossing be detected via finite-time correlation functions?
• RQ3 What is the nature of the order parameter for the transition, given the infinite redshift at the horizon?
• RQ4 How does the Bose-Einstein condensation of massless strings emerge in this large N matrix model?
• RQ5 Is the transition of infinite order, and what does this imply for its detectability in physical measurements?

Key findings

• Above a critical temperature $T_c$, a non-perturbative Bose-Einstein condensation of massless open strings occurs, signaling the probe particle has entered the black hole interior.
• The transition is of infinite order: all Matsubara coefficients, thermodynamic potentials, and finite-time correlation functions are analytic at $T_c$, indicating no divergence in standard observables.
• The order parameter is fundamentally non-local in time, defined by the infinite-time limit of the two-point function, which remains non-zero in the condensed phase.
• The condensate $n_0$ is not detectable via $\langle a^i \rangle$, which vanishes even with explicit U(N)-symmetry breaking, confirming its non-local character.
• For $M < M_c(T)$, the system is in the condensed phase, with $M_c(T)$ interpreted as the horizon radius; reentrant behavior is observed for certain $\mu$ and $M$ values.
• The thermal spectrum of strings matches between $M$ and a larger $M > M_c$ when $M < M_c$, suggesting a duality-like behavior where the particle appears to bounce off the horizon.