[Paper Review] Aspects of black hole entropy

Shinji Mukohyama|ArXiv.org|Dec 24, 1999

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper demonstrates that the entanglement entropy approach to black hole entropy reduces to the brick wall model under maximal entanglement, and shows the brick wall model remains stable in rotating Kerr backgrounds when the inner boundary is near the horizon (at Planck scale), implying the entanglement approach is also viable in rotating spacetimes. The key result is the suppression of complex-frequency modes to negligible levels relative to Hawking temperature under these conditions.

ABSTRACT

There have been many attempts to understand the statistical origin of black-hole entropy. Among them, entanglement entropy and the brick wall model are strong candidates. In this paper, first, we show that the entanglement approach reduces to the brick wall model when we seek the maximal entanglement entropy. After that, the stability of the brick wall model is analyzed in a rotating background. It is shown that in the Kerr background without horizon but with an inner boundary a scalar field has complex-frequency modes and that, however, the imaginary part of the complex frequency can be small enough compared with the Hawking temperature if the inner boundary is sufficiently close to the horizon, say at a proper altitude of Planck scale. Hence, the brick wall model is well defined even in a rotating background if the inner boundary is sufficiently close to the horizon. These results strongly suggest that the entanglement approach is also well defined in a rotating background.

Motivation & Objective

• To investigate the relationship between entanglement entropy and the brick wall model in black hole thermodynamics.
• To assess the stability of the brick wall model in rotating black hole backgrounds, particularly Kerr spacetime.
• To determine whether the brick wall model remains physically viable in rotating systems with an inner boundary.
• To evaluate whether the entanglement entropy approach is consistent in rotating spacetimes by analyzing the behavior of scalar fields.

Proposed method

• Analyzes the entanglement entropy of a scalar field across a horizon-like boundary in a non-rotating case.
• Compares the entanglement entropy derivation to the brick wall model's entropy calculation.
• Examines the scalar field propagation in a Kerr background with an inner boundary but no horizon.
• Uses linear perturbation theory to study the existence of complex-frequency modes in the scalar field.
• Evaluates the ratio of the imaginary part of the complex frequency to the Hawking temperature.
• Considers the limit where the inner boundary is at a proper distance of Planck scale from the horizon.

Experimental results

Research questions

• RQ1 Does the entanglement entropy approach reduce to the brick wall model under maximal entanglement conditions?
• RQ2 Can the brick wall model be consistently applied in a rotating spacetime without a horizon?
• RQ3 Do complex-frequency modes appear in scalar fields near a rotating inner boundary, and are they physically significant?
• RQ4 Is the imaginary part of the complex frequency small enough relative to the Hawking temperature when the inner boundary is close to the horizon?
• RQ5 Does the stability of the brick wall model in rotating spacetimes imply the viability of the entanglement entropy approach?

Key findings

• The entanglement entropy approach reduces to the brick wall model when maximal entanglement entropy is considered.
• In the Kerr background without a horizon but with an inner boundary, scalar fields exhibit complex-frequency modes.
• The imaginary part of the complex frequency becomes negligible compared to the Hawking temperature when the inner boundary is at a proper distance of the Planck scale from the horizon.
• The brick wall model remains well-defined in rotating backgrounds under these conditions due to the suppression of unstable modes.
• The stability of the brick wall model in rotating spacetimes strongly suggests the entanglement approach is also well-defined in such geometries.