Periodic signature change in spacetimes of embedding class one

TL;DR

This paper proposes a mathematical model of an oscillating universe based on periodic signature changes in the embedding space of curved spacetimes, using embedding class transformations in general relativity.

Contribution

It introduces a novel approach by making the embedding transformation parameter a periodic function of time, modeling universe oscillations through signature changes.

Findings

Model exhibits periodic signature change in embedding space.

Supports oscillating universe hypothesis within general relativity.

Complements existing cosmological models with a new mathematical framework.

Abstract

The idea of an oscillating Universe has remained a topic of interest even after the discovery of dark energy. This paper confirms this idea by means of another well-established theory in general relativity, the embedding of curved spacetimes in higher-dimensional flat spacetimes: an $n$-dimensional Riemannian space is said to be of embedding class $m$ if $m+n$ is the lowest dimension $d$ of the flat space in which the given space can be embedded; here $d=\frac{1}{2}n(n-1)$. So a four-dimensional Riemannian space is of class two since it can be embedded in a six-dimensional flat space. A line element of class two can be reduced to a line element of class one by a suitable coordinate transformation. The extra dimension can be either spacelike or timelike, leading to accelerating and decelerating expansions, respectively. Accordingly, it is proposed in this paper that the free parameter occurring in the transformation be a periodic function of time. The result is a mathematical model that can be interpreted as a periodic change in the signature of the embedding space. This signature change may be the best model for an oscillating Universe and complements various models proposed in the literature.