2 be cast in the form where we introduced the new variables τ and ρ in place of r and t , since in this region they become a temporal and a spatial coordinate, respectively. It is convenient to introduce a temporal tortoise coordinate τ ∗ defined as dτ ∗ = dτ/ (2 m ( τ ) /τ − 1) such that τ ∗ → ±∞ as τ → r$_{±}$ . In the sequel we shall introduce null coordinates x$_{−}$ ≡ − τ ∗ − ρ and x$_{+}$ ≡ − τ ∗ + ρ as in [5, 6]. Then the metric reads ds 2 = ( 2 m ( τ ) τ − 1 ) dx$_{−}$dx$_{+}$ − τ $^{2}$d Ω $^{2}$. (8) ds 2 = dτ 2 ( 2 m ( τ ) τ − 1 ) − ( 2 m ( τ ) τ − 1 ) dρ 2 − τ $^{2}$d Ω 2 (7) The event horizon becomes the null hypersurface x$_{−}$ = −∞ and the left and right branches of the Cauchy hori- zon r$_{−}$ are null hypersurfaces x$_{−}$ = ∞ and x$_{+}$ = ∞ , respectively. In the region between the two horizons (see Fig. 1) we consider the propagation of a massless scalar test field φ = φ ( τ, ρ, ϑ, ϕ ) governed by the equation (3) where the metric is associated to the background geome- try (7). Now it is the time to invoke the noncommutative nature of the field. Indeed, up to now, noncommutative effects have been considered to smear the matter gener- ating the background geometry only. We need to extend this procedure to matter propagating over the manifold too. To this purpose, we follow the formulation proposed in [34–40] to get a modified integral measure in the mo- mentum representation of the field, which between the two horizons can be written as We shall start recalling some properties of the non- commutative geometry inspired black hole, whose line element is given in [27] ized by a positive black hole heat capacity. As a con- sequence, also for the neutral solution, in place of the runaway behavior of the temperature, one finds that the evaporation ends up with a zero temperature extremal black hole, a final configuration entirely governed by a quantum gravity induced minimal length. This new scenario of the evaporation implies a further virtue of QGBHs: a finite temperature prevents any relevant back reaction, namely a self interaction of the radiated energy with its source. Thus we can conclude that these solu- tions are stable versus back reaction and can describe the entire black hole life until the final configuration. How- ever, having an inner Cauchy horizon is again a source of concern, since we might have the suspect that the in- terior of these black holes is unstable. As a result the solutions could be no longer singularity free, since the singularity might occur on the inner horizon rather than at the origin, frustrating the efforts that vivified the for- mulations at the basis of their derivation. In some sense, the instability of QGBHs is even worse with respect to the conventional classical analogs, since it affects the neu- tral, static case too. To address this problem we need to change our perspective. If we do believe in the tenets of quantum gravity, we have to accept the possibility for a quantum manifold to provide a natural ultraviolet cut-off for any field propagating over it in order to prevent any growth of energy beyond Planckian magnitude. Without loss of generality we will consider the neutral noncommu- tative spacetime only, even if our analysis holds for the charged solution too and other QGBHs. Indeed noncom- mutative geometry is just one of the possible effective ways to implement a natural cut-off. The presence of an exponential damping factor encodes the effect of the noncommutative UV regularization. Here τ has to be thought as an implicit function of the variable τ $^{∗}$, while Y$_{ℓm}$ = Y$_{ℓm}$ ( ϑ, ϕ ) denotes the spherical harmonics. Analogous modifications have been already efficiently employed in a variety of contexts, namely to describe a traversable wormhole sustained by quantum geometry fluctuations [41], to remove the initial cosmo- logical singularity and drive the inflation without an in- flaton field [42] and to get corrections to the Unruh ther- mal bath by means of a nonlocal deformation of conven- tional field theories [43–46]. Further contributions con- cern the modification of the Newton potential in the pres- ence of noncommutative spacetime coordinates [47], the evaporation of the noncommutative black holes in terms of gravitational amplitudes for boson and fermion fields [48–50] and for up to ten spatial dimensions in particle detectors at the LHC [51] and the calculation of the spec- tral dimension of a quantum spacetime [52]. As a result is the incomplete lower gamma function. The above line element is clearly regular at the origin, where a de Sitter core accounts for the mean value of the quantum fluctu- ations of the manifold. The metric admits one, two or no horizon depending on the total mass M , respectively equal, larger or smaller than extremal black hole total mass M$_{0}$ ≈ 1 . $_{9}$√$_{θ}$$_{. In the following we shall restrict our}$ attention to the case of two horizons. In this scenario the line element (5) in the interior region r$_{−}$ < r < r$_{+}$ can with the mass function m ( r ) = 4 M γ ( 3 / 2 , r $^{2}$/ 4 θ ) $_{/}$√$_{π}$$_{,}$ where M is the total mass in the spacetime manifold, θ is a parameter encoding noncommutativity and having the dimension of a length squared, while γ ( 3 / 2 , r $^{2}$/ 4 θ ) = ∫ r $^{2}$/ 4 θ 0 dt t 1 / 2 e − t (6) ds 2 = (1 − 2 m ( r ) /r ) dt 2 − dr 2 (1 − 2 m ( r ) /r ) − r $^{2}$d Ω 2 (5) φ ( τ $^{∗}$, ρ, ϑ, ϕ ) = (9) = ∞ ∑ ℓ =0 ℓ ∑ m = − ℓ ∫ ∞ −∞ dke − k $^{2}$θ/ $^{4}$e − ikρ 1 τ ψ$_{ℓmk}$ ( τ $^{∗}$) Y$_{ℓm}$.