content_image/413234.jpg
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. 

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- 

<img src='content_image/413234.jpg'>

## Acknowledgments 

fensive zur EntwicklungWissenschaftlich- ¨ Okonomischer Exzellenz) launched by the State of Hesse. 

∗ Electronic address: dbatic@uniandes.edu.co † 

Electronic address: nicolini@th.physik.uni-frankfurt.de 

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