AdS Geometry, projective embedded coordinates and associated isometry groups

TL;DR

This paper explores the geometry of anti-de Sitter spacetime using the Laplacian Comparison Theorem, introduces projective coordinates, and analyzes isometry groups through embedding and geodesic representations.

Contribution

It provides a novel description of AdS geometry via LCT, introduces projective coordinates for de Sitter spacetime, and connects these to isometry groups and geodesic models.

Findings

Reproduces AdS geometrical properties using LCT.

Introduces projective coordinates via stereographic projection.

Analyzes isometry groups and geodesic representations.

Abstract

This work is intended to investigate the geometry of anti-de Sitter spacetime (AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to give another description of the hyperbolical embedding standard formalism of the de Sitter and anti-de Sitter spacetimes in a pseudo-Euclidean spacetime. It is shown how to reproduce some geometrical properties of AdS, from the LCT in AdS, choosing suitable functions that satisfy basic properties of Riemannian geometry. We also introduce and discuss the well-known embedding of a 4-sphere and a 4-hyperboloid in a 5-dimensional pseudo-Euclidean spacetime, reviewing the usual formalism of spherical embedding and the way how it can retrieve the Robertson-Walker metric. With the choice of the de Sitter metric static frame, we write the so-called reduced model in suitable coordinates. We assume the existence of projective coordinates, since de Sitter spacetime is orientable. From these coordinates, obtained when stereographic projection of the de Sitter 4-hemisphere is done, we consider the Beltrami geodesic representation, which gives a more general formulation of the seminal full model described by Schrodinger, concerning the geometry and the topology of de Sitter spacetime. Our formalism retrieves the classical one if we consider the metric terms over the de Sitter splitting on Minkowski spacetime. From the covariant derivatives we find the acceleration of moving particles, Killing vectors and the isometry group generators associated to de the Sitter spacetime.