This paper is devoted to the study of the Reissner-Nordstrøm-de Sitter black holes and their maximal analytic extensions. We study some of their properties that lays the groundwork for obtaining (in separate papers) decay results [17] and constructing conformal scattering theories for test fields on such spacetimes [16]. Here, we find the necessary and sufficient conditions on the parameters of the Reissner-Nordstrøm-de Sitter metric -namely, the mass , the charge, and the cosmological constant-to have three horizons. Under this conditions, we prove that there is only one photon sphere and we locate it. We then give a detailed construction of the maximal analytic extension of the Reissner-Nordstrøm-de Sitter manifold in the case of three horizons.

In this paper we use Morawetz and geometric energy estimates -the so-called vector field method-to prove decay results for the Maxwell field in the static exterior region of the Reissner-Nordstrøm-de Sitter black hole. We prove two types of decay: The first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of t −1 which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field.The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper. * 1 2 , and h is any twice differentiable function on R. Let u be such a solution and say h is a smooth function which is compactly supported in ]0, +∞[. This solution radiates away form the origin at speed 1 as t increases, and for some R > 0, it identically vanishes 2 for t > |x| + R (figure 1). Such a solution models a disturbance starting in a bounded region which then spreads outward and reaches every point in space, but for each point and after a finite amount of time, there is no disturbance left at all. Perhaps not seen as directly as in the previous simple case, but in fact, this is true for all the solutions of the above wave equation on R 1+3 which start in confined regions. This infinite fall off rate follows from Kirchhoff's formula (19th century) which can be proved by the method of spherical means. An equation having this property is said to satisfy the strong Huygens principle:Theorem (Huygens Principle). If the initial data, (u(0, x), ∂ t u(0, x)) with x ∈ R 3 , for the above wave equation are supported in the ball B(0, R), then the associated solution u satisfies, u(t, x) = 0 f or all |t| > |x| + R 1 A Killing horizon is a null hypersurface defined by the vanishing of the norm of a Killing vector field. 2 Actually these particular solutions, i.e. h having such support, vanish for t ≥ |x|.