content_image/396060.jpg content_image/396074.jpg content_image/396063.jpg content_image/396075.jpg content_image/396062.jpg content_image/396064.jpg content_image/396070.jpg content_image/396071.jpg content_image/396076.jpg content_image/396081.jpg content_image/396082.jpg
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Shadwick [9] and is the following, 

The instability region is depicted in figure 2 for several 

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spacetime-dimension D , which can be compared with the numerical results by KZ, their figure 4. It is apparent that condition (14) very accurately describes the numer- ical results for r b /r c ∼ 1, a regime we explore below in Section IV. As one moves away from extremality cri- terium (14) is just too restrictive. An improved analysis and refined criterium would be necessary to describe the whole range of the numerical results. Nevertheless, figure 

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2 is very clear: higher-dimensional ( D > 6) RNdS black holes are unstable for a wide range of parameters. 

## IV. AN EXACT SOLUTION IN THE NEAR EXTREMAL RNDS BLACK HOLE 

Let us now specialize to the near extremal RNdS black hole, which we define as the spacetime for which the cos- mological horizon r c is very close (in the r coordinate) to the black hole horizon r b , i.e. r$_{c}$ − r$_{b}$ r$_{b}$ ≪ 1. The wave equation in this spacetime can be solved exactly, in terms of hypergeometric functions [10]. The key point is that the physical region of interest (where the boundary con- ditions are imposed), lies between r b and r c . Thus, 

where we have introduced the surface gravity κ b associ- ated with the event horizon at r = r b , as defined by the relation κ b = 1 $_{2}$df/dr r = r$_{b}$ . For near-extremal black holes, it is approximately 

In this limit, one can invert the relation r$_{∗}$ ( r ) of (7) to get 

Substituting this on the expression (15) for f we find 

As such, and taking into account the functional form of the potentials for wave propagation, we see that for the near extremal RNdS black hole the wave equation (6) is of the form 

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with 

The potential in (19) is the well known P¨ oshl-Teller po- tential [11]. The solutions to (19) were studied and they are of the hypergeometric type, (for details see Refs. [12, 13]). It should be solved under appropriate boundary conditions: 

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