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In this work, we consider a noncommutative (NC) massless scalar field coupledto the classical nonrotational BTZ geometry. In a manner of the theories wherethe gravity emerges from the underlying scalar field theory, we study theeffective action and the entropy derived from this noncommutative model. Inparticular, the entropy is calculated by making use of the two differentapproaches, the brick wall method and the heat kernel method designed forspaces with conical singularity. We show that the UV divergent structures ofthe entropy, obtained through these two different methods, agree with eachother. It is also shown that the same renormalization condition that removesthe infinities from the effective action can also be used to renormalize theentanglement entropy for the same system. Besides, the interesting feature ofthe NC model considered here is that it allows an interpretation in terms of anequivalent system comprising of a commutative massive scalar field, but in amodified geometry; that of the rotational BTZ black hole, the result that hintsat a duality between the commutative and noncommutative systems in thebackground of a BTZ black hole.

Entanglement Entropy Renormalization for the NC scalar field coupled to classical BTZ geometry