Thermo Field Dynamics, Boulware and Hartle-Hawking States (2017)
Abstract. In this thesis one studied a scalar field over a Schwarzschild background with an integrated interpretation of entanglement, thermo field dynamics formalism and correct ground states, in order to give an explanation of Bekenstein-Hawking entropy and where is located. Using statistical mechanics on a scalar field over a Schwarzschild background one obtain the number of modes of the field and with this, one can find the Helmholtz free energy, which has two terms: one dependent of the volume that is the contribution of the vacuum energy at large distances, and the second one is the contribution of the event horizon to the energy and is proportional to the horizon area. This term diverges when the altitude h tends to zero, ie, very close to the event horizon. This is a reproduction of Gerardus ’tHooft paper [1]. Taking the correct form of h, the total energy U and the entropy S could be obtained. The last one results to be exactly the Bekenstein-Hawking entropy. If one give to a scalar field an operator behavior, and taking into account that above the brick wall there are no horizons, it is possible to see that the correct state of the system in study is the Boulware state. One can conclude that an outside observer can see what it looks like as a black hole but actually does not contain a horizon. Using thermo field dynamic formalism (ThFD) in a quantum scalar field over a eternal Sch- warzschild background one shows that the system meets exactly the characteristics of ThFD, then is a thermal system.