Riemannian Structure imposed on Friedmann and more general spacetimes

TL;DR

This paper introduces a Riemannian metric into general relativity, compares its geometric structure with the Lorentzian metric in Friedmann universes, and suggests it as a simpler alternative to Wick rotation.

Contribution

It presents a novel approach of embedding a nondegenerate Riemannian metric into spacetime, providing new insights into geometric structures in cosmological models.

Findings

Comparison of Riemannian and Lorentzian geometries in Friedmann universes

Introduction of a simpler, more general Riemannian metric for spacetime

Potential advantages over Wick rotation in geometric analysis

Abstract

In the paper we consider two Finsler-like Riemannian metrics, which can be in a natural way introduced into general relativity. One of those metrics $\gamma_{ab}$ is degenerate and the second $h_{ab}$ is nondegenerate. We are mainly interested with the metric $h_{ab}$ and comparing the geometric structure determined by this metric $h_{ab}$ with the geometric structure determined by the Lorentzian metric $g_{ab}$ of the underlying spacetime. Full comparison we have given for Friedmann Universis. The preliminary version of the paper was presented by one of us (J.G.) on conference POTOR 6 in Szczecin 2019. We think that the our introduction of the Riemannian metric $h_{ab}$ into spacetime is simpler and more general than to so-called Wick rotation.