The structure of the extreme Schwarzschild-de Sitter space-time

TL;DR

This paper provides a detailed analysis of the global structure, geodesics, and observational properties of the extreme Schwarzschild-de Sitter space-time, revealing how certain observers can escape the singularity and approach asymptotic regions.

Contribution

The paper offers a comprehensive analysis of the global structure and geodesic behavior of the extreme Schwarzschild-de Sitter space-time, including conformal diagrams and observer dynamics.

Findings

Certain observers escape the singularity and reach asymptotic regions.

Geodesic deviations indicate stable observer families.

Redshift calculations show signals from falling particles.

Abstract

The extreme Schwarzschild-de Sitter space-time is a spherically symmetric solution of Einstein's equations with a cosmological constant Lambda and mass parameter m>0 which is characterized by the condition that 9 Lambda m^2=1. The global structure of this space-time is here analyzed in detail. Conformal and embedding diagrams are constructed, and synchronous coordinates which are suitable for a discussion of the cosmic no-hair conjecture are presented. The permitted geodesic motions are also analyzed. By a careful investigation of the geodesics and the equations of geodesic deviation, it is shown that specific families of observers escape from falling into the singularity and approach nonsingular asymptotic regions which are represented by special "points" in the complete conformal diagram. The redshift of signals emitted by particles which fall into the singularity, as detected by those observers which escape, is also calculated.