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

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.