Different representations of the Levi-Civita Bertotti Robinson solution

TL;DR

This paper explores various coordinate representations of the Levi-Civita Bertotti Robinson spacetime, analyzing their transformations, reference frame motions, and embeddings, including models with domain walls and Milne-LBR universe scenarios.

Contribution

It introduces a general formalism for coordinate transformations in conformally flat spacetimes and presents a new k-function calculus for analyzing LBR spacetime.

Findings

Derived coordinate transformations between different LBR representations

Characterized reference frame motions via normalized timelike Killing vectors

Presented embedding formulae in a 6D flat manifold

Abstract

The Levi-Civita Bertotti Robinson (LBR) spacetime is investigated in various coordinate systems. By means of a general formalism for constructing coordinates in conformally flat spacetimes, coordinate transformations between the different coordinate systems are deduced. We discuss the motion of the reference frames in which the different coordinate systems are comoving. Furthermore we characterize the motion of the different reference frames by their normalized timelike Killing vector fields, i.e. by the four velocity fields of the reference particles. We also deduce the formulae in the different coordinate systems for the embedding of the LBR spacetime in a flat 6-dimensional manifold. In particular we discuss a scenario with a spherical domain wall having LBR spacetime outside the wall and flat spacetime inside. We also discuss the internal flat spacetime using the same coordinate systems as in the external LBR spacetime with continuous metric at the wall. Among the different cases one represents a Milne-LBR universe model with a part of the Milne universe inside the wall and an infinitely extended LBR universe outside it. In an appendix we define combinations of trigonometric and hyperbolic functions that we call k-functions and present a new k-function calculus.