content_image/415139.jpg content_image/415137.jpg content_image/415142.jpg content_image/415151.jpg content_image/415152.jpg content_image/415144.jpg content_image/415145.jpg content_image/415138.jpg content_image/415140.jpg content_image/415141.jpg
Furthermore, we can also write 

For a fixed r c and spacetime dimension D , the existence of a regular event horizon imposes that the charge Q must be smaller than a certain value Q ext . With our units this maximum charge is 

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Gravitational perturbations of this spacetime couple to the electromagnetic field, and were completely character- ized by Kodama and Ishibashi [8]. They can be reduced to a set of two second order ordinary differential equa- tions of the form, 

The variables x, y and parameters µ, m are defined through 

and the quantity δ is implicitly given by µ = (1+2 mδ ) M . Note that the following relations holds Q 2 = ( n + 1) $^{2}$M$^{2}$δ (1 + mδ ). 

Note also that for the spacetime considered in this pa- per K = 1, which means that the eigenvalues k 2 are given 

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where the tortoise coordinate r$_{∗}$ and the potentials V S ± are defined through 

We have 

and the quantities U$_{±}$ are given by 

by k 2 = l ( l + n − 1), where l is the angular quantum number, that gives the multipolarity of the field. The behavior of the potentials varies considerably over the range of parameters. In Fig. 1 we show V$_{−}$ for D = 8, r c = 1 / 0 . 95, l = 2 and three different values of the charge, Q = 0 . 2 , 0 . 35 , 0 . 44. 

## III. A CRITERIUM FOR INSTABILITY 

A sufficient (but not necessary) condition for the exis- tence of an unstable mode has been proven by Buell and 

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