Fuzziness at the horizon Davide Batic Departamento de Matematica, Universidad de los Andes, Cra 1E, No. 18A-10, Bogot´ a, Colombia Department of Mathematics, University of West Indies, Kingston, Jamaica Piero Nicolini † Frankfurt Institute for Advanced Studies (FIAS), Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨ at, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany (Dated: November 13, 2018) We study the stability of the noncommutative Schwarzschild black hole interior by analysing the propagation of a massless scalar field between the two horizons. We show that the spacetime fuzzi- ness triggered by the field higher momenta can cure the classical exponential blue-shift divergence, suppressing the emergence of infinite energy density in a region nearby the Cauchy horizon. PACS numbers: Valid PACS appear here Keywords: Suggested keywords Black holes can exhibit not only an event horizon, namely the outermost surface that physically separates two noncommunicating regions of spacetime, but also inner Cauchy horizons. These internal horizons, null surfaces beyond which spacetime predictability breaks down, have the intriguing properties of showing up a “dual effect” of the conventional red shift, i.e. the blue shift. To understand the physics of this blue shift, it is common procedure to study a radiation represented by a scalar field, propagating in the region between the two horizons. For the sake of clarity we consider a spherically symmetric spacetime region, whose metric can be cast in the form where U µ is the 4-velocity of the observer (the dot de- notes differentiation with respect to proper time). Ac- tually the FFO measures a flux of energy given by the square of the above quantity, that is even more diver- gent. This mechanism of instability due to the infinite blue shift at the Cauchy horizon can be explained in these terms. An external observer would require an infinite time to reach the future null infinity ( x$_{+}$ = ∞ ) since at the best its velocity is ˙ x$_{+}$ ≃ 1. On the other hand, a FFO can reach the Cauchy horizon in a finite proper time, which implies that ˙ x$_{+}$ will diverge as x → ∞ . From Eq. (4) we see that this divergence overcomes the field decay. As a result the Cauchy horizon is unstable. An extensive study on this subject has basically led to the general conclusion that this infinite amount of energy density at the Cauchy horizon can develop unbounded curvature, disrupting the spacetime geometry [1–18]. Up to now we have mentioned purely classical solu- tions, since the above analyses have concerned quantum effect at the most in the matter fields propagating on the spacetime manifold. On the other hand, there is a recent class of black hole solutions (QGBHs) obtained by means of quantum gravity arguments, like loop quan- tum black holes [19–21], asymptotically safe gravity black holes [22, 23], generalized uncertainty principle [24, 25] and noncommutative geometry inspired black holes [26– 32] (for a review see [33] and the references therein). In- dependently on their starting point the above solutions, converge on a unique qualitative behavior, namely the absence of any curvature singularity and the presence of more than a horizon. In other words, as far as some sort of smearing effect is concerned due to the fuzziness of spacetime in its quantum gravity regime, the physics of QGBHs has a universal character. This fact has its equivalent on the thermodynamics side: the Hawking temperature admits a maximum, followed by a “SCRAM phase”, a thermodynamic stable shut down, character- The solution of the above equation, let us conclude that the field, in the vicinity of the Cauchy horizon where x$_{+}$ → ∞ , decays as φ ∼ x − 2 ℓ − 2 + , with ℓ the multipole order of the field. However this is no longer true for the energy of the field. Indeed, if we consider, the field’s rate variation as measured by a free falling observer (FFO) crossing the Cauchy horizon we obtain the infinite result where κ$_{±}$ ≡ ( r$_{+}$ − r$_{−}$ ) /r 2 $_{±}$, we can define null coordinates x$_{−}$ = − r ∗ − t and x$_{+}$ = − r $^{∗}$+ t to study the propagation of a scalar field on this background in terms of the scalar wave equation where r$_{−}$ is the Cauchy horizon, r$_{+}$ is the event hori- zon, i.e. r$_{−}$ < r < r$_{+}$ , r plays the role of a temporal coordinate and t is a spatial one. Introducing tortoise coordinates arXiv:1001.1158v2 [gr-qc] 3 Aug 2010 ds 2 = r $^{2}$dr2 ( r$_{+}$ − r )( r − r$_{−}$ ) − ( r$_{+}$ − r )( r − r$_{−}$ ) r 2 dt 2 − r $^{2}$d Ω 2 (1) r ⋆ = − r − 1 κ$_{+}$ ln( r$_{+}$ − r ) + 1 κ$_{−}$ ln( r − r$_{−}$ ) (2) g $^{µν}$∇ µ ∇ $_{ν}$φ = 0 . (3) φ$_{,µ}$ U µ ≃ φ$_{,x}$$_{+}$ ˙ x$_{+}$ ∼ x − 2 ℓ − 3 + e κ − x$_{+}$ (4)