# Reissner-Nordstr{\o}m-de Sitter Manifold : Photon Sphere and Maximal Analytic Extension

## TL;DR

This paper analyzes the properties of Reissner-Nordström-de Sitter black holes, focusing on conditions for three horizons, the photon sphere, and their maximal analytic extension, laying groundwork for future decay and scattering studies.

## Contribution

It provides necessary and sufficient conditions for three horizons, locates the photon sphere, and constructs the maximal analytic extension of these black holes.

## Key findings

- Conditions for three horizons in Reissner-Nordström-de Sitter spacetimes.
- Identification and location of the photon sphere.
- Construction of the maximal analytic extension for three-horizon cases.

## Abstract

This paper is devoted to the study of the Reissner-Nordstr{\o}m-de Sitter black holes and their maximal analytic extensions. In particular, we study some of their properties that lays the groundwork for separate papers where we obtain decay results and construct conformal scattering theories for test fields on such spacetimes. Here, we find the necessary and sufficient conditions on the parameters of the Reissner-Nordstr{\o}m-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-Nordstr{\o}m-de Sitter manifold in the case of three horizons.