Thermo Field Dynamics on BTZ spacetime

Abstract

Abstract A thin dust shell contracting from infinity to near its gravitational radius $r_{+}$, in a spacetime $AdS_{3}$ is analyzed; its equation of motion is determined and the solution $R(t)$ as seen by a FIDO observer is estimated. It is concluded that this Shell’s exterior looks like a BTZ black hole with similar properties.

Based on the Thermo Field Dynamics technique, a scalar field $\Phi$ in the proximity of a non-rotating BTZ $(2 + 1)$ black hole is studied. From the corresponding Killing-Boulware $\Ket{0}_{KB*}$ and Hartle-Hawking $\Ket{0}_{HH*}$ vacuum states, the associated Wightman function $W(x,x')_{HH^{*}}- W(x,x')_{KB^{*}}$ is determined and based on it, the time component of the momentum-energy tensor of the system $\partial_{0}\partial_{0'} \left( W_{HH^{*}}- W_{KB^{*}}\right)(x,x')\approx \left\langle T_{00} (x,x')\right\rangle=\sigma(r)$ is calculated. Which allows establishing the origin and location of the degrees of freedom responsible for the entropy that describes a source for the Bekenstein-Hawking $S_{BH}$ entropy.

The thermal environment described by this model manifests itself with a well-defined and concentrated energy density near the event horizon, according to a FIDO observer.