3 we obtain the “radial” function ψ$_{ℓmk}$ obeys the equation whereas for h < 0 Hence, for h > 0 and asymptotically for x$_{−}$ → ∞ we have d $^{2}$ψ$_{ℓmk}$ dτ ∗ 2 + [ k 2 − V ( τ $^{∗}$) ] ψ = 0 (10) U x$_{+}$ ≈ e α − ( x − + x$_{+}$ ) / $^{2}$, U x − ≈ 1 2 . (18) φ$_{, α}$U α ≈ e α − ( x − + x$_{+}$ ) / 2 ∂φ ∂x$_{−}$ + 1 2 ∂φ ∂x$_{+}$ (19) φ$_{, α}$U α ≈ 1 2 ∂φ ∂x$_{−}$ + e α − ( x − + x$_{+}$ ) / 2 ∂φ ∂x$_{+}$ . (20) whereas for h < 0 and x$_{+}$ → ∞ where the potential is given by For our purposes we are interested in the asymptotic so- lutions of (10) as τ ∗ → −∞ . The potential V ( τ $^{∗}$) decays exponentially in time as V ( τ $^{∗}$) = g$_{ρρ}$ τ [ ℓ ( ℓ + 1) τ + 2 g$_{ρρ}$ τ + ∂$_{τ}$ g$_{ρρ}$ ] . (11) V ( τ $^{∗}$) ≈ e ∓ α ± τ $^{∗}$, τ ∗ → ±∞ . (12) In order that the FFO can measure a nondivergent amount of field energy density near the r$_{−}$ horizon, the appropriate derivative of the field times the exponential blue-shift factor must be finite. In the classical picture [5, 7] from the last two relations above one concludes that the e − ikx$_{+}$ waves are singular along the left branch of r − and the e ikx − waves become singular along the right branch of r$_{−}$ . We shall see that due to the noncommu- tativity of the field, (19) and (20) stay bounded at the Cauchy horizon. Indeed, we find that for the left-going component where 2 α$_{±}$ ≡ ± ( dg$_{00}$/dr )$_{r}$$_{=}$$_{r}$$_{±}$. Finally, near the Cauchy horizon the asymptotic solutions of (10) are Up to exponentially vanishing corrections, the solution are plane waves approaching the left and right branch of the horizon r$_{−}$ , respectively. As in [7] the energy den- sity in the scalar field φ as measured by a freely falling observer near a horizon with four velocity U µ will be proportional to e − $^{ikρ}$ψ$_{ℓmk}$ ( τ $^{∗}$) ≈ e ± ikx ± [ 1 + O ( e α − τ $^{∗}$) ] . (13) E = ( φ$_{, α}$U $^{α}$)( φ$_{, β}$U $^{β}$) + 1 2 φ$_{, α}$φ ∗ $^{, α}$. (14) φ$_{, α}$U α ∼ x$_{−}$ θ 3 / 2 e α − ( x − + x$_{+}$ ) / $^{2}$e − x 2 $_{−}$/θ (21) which vanishes as x$_{−}$ → ∞ , keeping x$_{+}$ constant. Anal- ogously, we find for the right-going component φ$_{, α}$U α ∼ x$_{+}$ θ 3 / 2 e α − ( x − + x$_{+}$ ) / 2 e − x 2 $_{+}$/θ (22) which vanishes as x$_{+}$ → ∞ , for x$_{−}$ fixed. The above result confirms that, probing higher momenta the field basically triggers the noncommutative nature of the man- ifold, which shows graininess and prevents any spacetime resolution beyond the value √$_{θ}$$_{. This let us also conclude}$ that in this framework no mass inflation can occur. To this purpose, we define Since − τ $^{∗}$± ρ = const are null surfaces and taking into ac- count for the form of φ nearby the horizon, we have that the energy density is dominated by the term | φ$_{, α}$U $^{α}$| $^{2}$. Therefore we can restrict our analysis to the term φ$_{, α}$U α only. To this purpose, we need the form of the velocity vector field associated to the FFO which can be written as where as the two-dimensional section of the stress tensor, which describes the cross flowing stream of infalling and outgo- ing of light like particles following null geodesics. Here l$_{a}$ is the radial null vector pointing inwards, n$_{a}$ is the radial null vector pointing outwards, while E in and E out represent the energy density of the fluxes. The mass in- flation is a huge boom of the black hole internal mass parameter, which becomes classically unbounded at the Cauchy horizon. Contrary to the intuition, the inflation is due to both the outflux and the blueshifted influx of a collapsing star as shown in [9]. On the other hand, in the present framework, energy densities cannot diverge even at the Cauchy horizon. Therefore, the mass in- flation which is in general proportional to the product with h a dimensionless parameter. If h > 0 the FFO worldline enters region III from region I and exits region III through the left-hand branch ( x$_{−}$ = ∞ ) of the inner horizon. If h < 0 the worldline enters region III from region II and exits through the right-hand ( x$_{+}$ = ∞ ) branch of r$_{−}$ . If h = 0 the worldline will move through the region III passing through the bifurcation points of the horizon r$_{−}$ . In the vicinity of the Cauchy horizon, we have for h > 0 T$_{ab}$ = E $_{in}$l$_{a}$ l$_{b}$ + E $_{out}$n$_{a}$ n$_{b}$ (23) U = U x − ∂ ∂x$_{−}$ + U x$_{+}$ ∂ ∂x$_{+}$ (15) U x ± = − τ 2 m ( τ ) − τ ( h ∓ √ h 2 + 2 m ( τ ) − τ τ ) , (16) U x$_{+}$ ≈ 1 2 , U x − ≈ e − α − τ ∗ = e α − ( x − + x$_{+}$ ) / 2 (17)