4 fensive zur EntwicklungWissenschaftlich- ¨ Okonomischer Exzellenz) launched by the State of Hesse. T $^{ab}$T$_{ab}$ will not take place. The above analysis concerns the leading contribution to the energy density of the field in the vicinity of the Cauchy horizon along the lines of [5, 6]. Since higher order terms fall off faster, we argue that the stability of QGBH interiors can be shown in gen- eral. However, according to the theory of the stability in [3, 4], an exponential decay is a mere necessary condition only: in other words even if each term of the expansion is vanishing, their global contribution could yet destabilize the solution. Furthermore, we have addressed here the most simple case of classical perturbation of the mani- fold, while in general the field could be quantized. In such a case, according to previous contributions [15], the stress tensor 〈 T$_{ab}$ 〉 is expected to have an even worse UV behavior. For the above reasons, we think that, after the present analysis, further investigations will be necessary, also to include the other spacetimes within the class of QGBHs. FIG. 1: The Carter-Penrose diagram of the manifold. The conformal diagram of the maximally extended noncommuta- tive inspired Schwarzschild spacetime. The radii $^{r}$± represent the outer and inner horizons, respectively. The central sin- gularity appearing in the Reissner-Nordstr¨ om metric is now replaced by a regular de Sitter core, dotted line. The upper and bottom part of the box indicated by the dashed line can be identified to make the manifold cyclic in the time coordi- nate. −1 +1 r − r − r − r − r + r + r + r + I II III IV r − r − r − r − r + r + r + r + II I III IV r − r − r − r − Acknowledgments P.N. would like to thank the Universidad de los An- des, Bogot´ a, Colombia for the kind hospitality dur- ing the period of work on this project. P.N. is sup- ported by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesof- [31] I. Arraut, D. Batic and M. Nowakowski, Class. Quant. Grav. 26 , 245006 (2009). ∗ Electronic address: dbatic@uniandes.edu.co † Electronic address: nicolini@th.physik.uni-frankfurt.de [1] M. Simpson and R. Penrose, Int. J. Theor. Phys. 7 , 183 (1973). [2] D. M. Eardley, Phys. Rev. Lett. 33 , 442 (1974). [3] J.M. McNamara, Proc. Roy. Soc. Lond. A 358 , 499 (1978). [4] J.M. McNamara, Proc. Roy. Soc. Lond. A 364 , 121 (1978). [5] Y. Gursel, V. D. Sandberg, I. D. Novikov and A. A. Starobinsky, Phys. Rev. D 19 , 413 (1979). [6] Y. Gursel, I. D. Novikov, V. D. Sandberg and A. A. Starobinsky, Phys. Rev. D 20 , 1260 (1979). [7] R. A. Matzner, N. Zamorano and V. D. Sandberg, Phys. Rev. D 19 , 2821 (1979). [8] E. Poisson and W. Israel, Phys. Rev. Lett. 63 , 1663 (1989). [9] E. Poisson and W. Israel, Phys. Rev. D 41 , 1796 (1990). [10] P. Brady and E. Poisson, Class. Quant. Grav. 9 , 121 (1992). [11] L. M. Burko and A. Ori, Phys. Rev. Lett. 74 , 1064 (1995). [12] D. Markovic and E. Poisson, Phys. Rev. Lett. 74 , 1280 (1995). [13] R. Balbinot, Phys. Lett. B 227 , 30 (1989). [14] R. Balbinot, P. R. Brady, W. Israel and E. Poisson, Phys. Lett. A 161 , 223 (1991). [15] R. Balbinot and E. Poisson, Phys. Rev. Lett. 70 , 13 (1993). [17] V. Cardoso, M. Lemos and M. Marques, arXiv:1001.0019 [gr-qc]. [16] R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett. 103 , 161101 (2009). [18] G. Dotti and R. J. Gleiser, arXiv:1001.0152 [gr-qc]. [19] L. Modesto, Adv. High Energy Phys. 2008 , 459290 (2008). [20] L. Modesto, arXiv:0811.2196 [gr-qc]. [21] L. Modesto and I. Premont-Schwarz, Phys. Rev. D 80 , 064041 (2009). [22] A. Bonanno and M. Reuter, Phys. Rev. D 62 , 043008 (2000). [23] A. Bonanno and M. Reuter, Phys. Rev. D 73 , 083005 (2006). [24] G. Amelino-Camelia, M. Arzano, Y. Ling and G. Man- danici, Class. Quant. Grav. 23 , 2585 (2006). [25] V. Husain and R. B. Mann, Class. Quant. Grav. 26 , 075010 (2009). [26] P. Nicolini, J. Phys. A 38 , L631 (2005). [27] P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 632 , 547 (2006). [28] T. G. Rizzo, JHEP 0609 , 021 (2006). [29] S. Ansoldi, P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 645 , 261 (2007). [30] E. Spallucci, A. Smailagic and P. Nicolini, Phys. Lett. B 670 , 449 (2009).