Compact Four-Dimensional Euclidean-Space Immersed in the Universe

TL;DR

This paper introduces Euclid balls, compact four-dimensional manifolds with positive metric determinant, which can be immersed in Minkowski space, behave like dark matter, and potentially seed supermassive black holes and gamma-ray bursts in the early universe.

Contribution

It proposes Euclid balls as stable, high-mass entities that can explain dark matter, black hole formation, and early universe signals, linking geometry with cosmological phenomena.

Findings

Euclid balls can mimic black holes on their surface.

They may account for dark matter in the universe.

Potential signals include early gamma-ray bursts.

Abstract

A compact four-dimensional manifold whose metric tensor has a positive determinant (named the "Euclid ball") is considered. The Euclid ball can be immersed in the Minkovskian space (which has the negative determinant) and can exist stably through the history of the universe. Since the Euclid ball has the same solution as the Schwarzschild black hole on its three-dimensional surface, an asymptotic observer can not distinguish them. If large fraction of whole energy of the pre-universe was encapsulated in Euclid balls, they behave as the dark matter in the current universe. Euclid balls already existed at the end of the cosmological inflation, and can have a heavy mass in this model, they can be a seed of supper-massive black-holes, which are necessary to initiate a forming of galaxies in the early universe. The $\gamma$-ray burst at early universe is also a possible signal of the Euclidean ball.