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Microscopic Origin of Bekenstein-Hawking Entropy in $(2+1)$ Gravity: A Thermo Field Dynamics Approach

W. A. Rojas C., J. R. Arenas S

Abstract

We compute the entanglement entropy of a real massive scalar field near a non-rotating BTZ black hole using Thermo Field Dynamics. Modeling the black hole as a collapsing dust shell in AdS3, we derive the shell trajectory R(t) as seen by a Fiducial Observer (FIDO). From the Hartle-Hawking and Killing-Boulware vacua, we obtain the Wightman function difference and compute energy density, revealing a sharply localized energy density just outside the horizon, consistent with the brick wall picture. A full thermodynamic analysis yields an entanglement entropy proportional to the horizon area, numerically matching the Bekenstein-Hawking entropy. All intermediate steps, including junction conditions, Kruskal extension, WKB modes, and UV regularization, are explicitly detailed.

Microscopic Origin of Bekenstein-Hawking Entropy in $(2+1)$ Gravity: A Thermo Field Dynamics Approach

Abstract

We compute the entanglement entropy of a real massive scalar field near a non-rotating BTZ black hole using Thermo Field Dynamics. Modeling the black hole as a collapsing dust shell in AdS3, we derive the shell trajectory R(t) as seen by a Fiducial Observer (FIDO). From the Hartle-Hawking and Killing-Boulware vacua, we obtain the Wightman function difference and compute energy density, revealing a sharply localized energy density just outside the horizon, consistent with the brick wall picture. A full thermodynamic analysis yields an entanglement entropy proportional to the horizon area, numerically matching the Bekenstein-Hawking entropy. All intermediate steps, including junction conditions, Kruskal extension, WKB modes, and UV regularization, are explicitly detailed.
Paper Structure (23 sections, 485 equations, 10 figures)

This paper contains 23 sections, 485 equations, 10 figures.

Table of Contents

  1. Introduction
  2. BTZ black hole structure
  3. Kruskal diagram for the BTZ black hole
  4. KINEMATICS OF HYPERSURFACES
  5. First Fundamental Form: Induced Metric
  6. Second Fundamental Form: Extrinsic Curvature
  7. Darmois-Israel Formalism
  8. First Junction Condition
  9. Second Junction Condition
  10. Junction Conditions for BTZ
  11. Inner Solution
  12. Outer Solution
  13. External estimation of Extrinsic Curvature
  14. Internal estimation of the extrinsic curvature seen
  15. Shell motion equation
  16. ...and 8 more sections

Figures (10)

  • Figure 1: Kruskal diagram for the BTZ black hole
  • Figure 2: Two spacetime regions meeting at a common boundary. Taken from poisson2004relativist.
  • Figure 3: Representation of $R(t)$ given by \ref{['eqn156']}.
  • Figure 4: Carter-Penrose diagram for a BTZ black hole.
  • Figure 5: Outgoing modes of the scalar field in the Carter-Penrose diagram for a BTZ black hole.
  • ...and 5 more figures