Characterizations Of Umbilically Synchronized Space-times

TL;DR

This paper characterizes umbilically synchronized space-times with vanishing electric Weyl tensor components, establishing conditions for conformal flatness and analyzing their properties in various dimensions.

Contribution

It provides new characterizations of umbilically synchronized space-times, especially relating conformal flatness to the structure of spatial slices and electric Weyl tensor conditions.

Findings

In 4D, such space-times are conformally flat.

In higher dimensions, conformal flatness depends on spatial slices.

Vacuum conditions with zero electric Weyl tensor are characterized.

Abstract

We study umbilically synchronized space-times $M$ that have vanishing electric part of the Weyl tensor and show that (i) If $dim(M) = 4$, then $M$ is conformally flat, (ii) If $dim(M) > 4$, $M$ is conformally flat if and only if the spatial slices are conformally flat. Next, we characterize umbilically synchronized spacetimes that are vacuum and have zero electric components of its Weyl tensor. Finally, we characterize conformal vector fields of an umbilically synchronized space-time.