2 where the tortoise coordinate r$_{∗}$ and the potentials V S ± are defined through 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 We have 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, λ = r − 4 − n c ( r n +2 c − r 3 $_{c}$)( r n +2 c − Q $^{2}$r3 $_{c}$) r n +2 c − r c . (4) r$_{∗}$ ≡ ∫ f − $^{1}$dr , V S ± = fU$_{±}$ 64 r $^{2}$H 2 ± . (7) Q 2 ext = r n c ( − 2 r c + ( n + 1) r n c − ( n − 1) r n +2 c ) − r c r c ( n + 1) − 2 nr n c + ( n − 1) r 2 n +1 c . (5) H + = 1 − n ( n + 1) 2 δx , (8) H$_{−}$ = m + n ( n + 1) 2 (1 + mδ ) x , (9) d 2 dr 2 ∗ Φ$_{±}$ + ( ω 2 − V S ± ) Φ$_{±}$ = 0 , (6) and the quantities U$_{±}$ are given by U + = [ − 4 n $^{3}$( n + 2)( n + 1) $^{2}$δ$^{2}$x2 − 48 n $^{2}$( n + 1)( n − 2) δx − 16( n − 2)( n − 4)] y − δ $^{3}$n$^{3}$(3 n − 2)( n + 1) $^{4}$(1 + mδ ) x 4 +4 δ $^{2}$n$^{2}$( n + 1) $^{2}${ ( n + 1)(3 n − 2) mδ + 4 n 2 + n − 2 } x 3 +4 δ ( n + 1) { ( n − 2)( n − 4)( n + 1)( m + n $^{2}$K ) δ − 7 n 3 + 7 n 2 − 14 n + 8 } x 2 + { 16( n + 1) ( − 4 m + 3 n $^{2}$( n − 2) K ) δ − 16(3 n − 2)( n − 2) } x +64 m + 16 n n + 2) K, (10) U$_{−}$ = [ − 4 n $^{3}$( n + 2)( n + 1) $^{2}$(1 + mδ ) $^{2}$x2 + 48 n $^{2}$( n + 1)( n − 2) m (1 + mδ ) x − 16( n − 2)( n − 4) m $^{2}$] y − n $^{3}$(3 n − 2)( n + 1) $^{4}$δ (1 + mδ ) $^{3}$x4 − 4 n $^{2}$( n + 1) $^{2}$(1 + mδ ) $^{2}${ ( n + 1)(3 n − 2) mδ − n $^{2}$} x 3 +4( n + 1)(1 + mδ ) { m ( n − 2)( n − 4)( n + 1)( m + n $^{2}$K ) δ +4 n (2 n 2 − 3 n + 4) m + n $^{2}$( n − 2)( n − 4)( n + 1) K } x 2 − 16 m { ( n + 1) m ( − 4 m + 3 n $^{2}$( n − 2) K ) δ +3 n ( n − 4) m + 3 n $^{2}$( n + 1)( n − 2) K } x +64 m 3 + 16 n ( n + 2) m $^{2}$K. (11) The variables x, y and parameters µ, m are defined through 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. 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 ). A sufficient (but not necessary) condition for the exis- tence of an unstable mode has been proven by Buell and Note also that for the spacetime considered in this pa- per K = 1, which means that the eigenvalues k 2 are given x ≡ 2 M r n − 1 , y ≡ λ r $^{2}$, (12) µ 2 ≡ M 2 + 4 mQ 2 ( n + 1) 2 , m ≡ k 2 − nK , (13) III. A CRITERIUM FOR INSTABILITY