The Structure of the Big Bang from Higher-Dimensional Embeddings

TL;DR

This paper explores how 4D cosmological models of the Big Bang can be embedded into higher-dimensional flat manifolds, revealing geometric structures and singularities related to the universe's origin.

Contribution

It provides explicit relations and diagrams for embedding flat Friedmann-Robertson-Walker models into higher-dimensional spaces, linking hypersurface geometry to matter equations of state.

Findings

Hypersurfaces depend on matter's equation of state.

A line-like curvature singularity exists near the Big Bang.

The big bang corresponds to a caustic in 5D geodesics.

Abstract

We give relations for the embedding of spatially-flat Friedmann-Robertson-Walker cosmological models of Einstein's theory in flat manifolds of the type used in Kaluza-Klein theory. We present embedding diagrams that depict different 4D universes as hypersurfaces in a higher dimensional flat manifold. The morphology of the hypersurfaces is found to depend on the equation of state of the matter. The hypersurfaces possess a line-like curvature singularity infinitesimally close to the $t = 0^+$ 3-surface, where $t$ is the time expired since the big bang. The family of timelike comoving geodesics on any given hypersurface is found to have a caustic on the singular line, which we conclude is the 5D position of the point-like big bang.