Causal structures in the four dimensional Euclidean space

TL;DR

This paper explores two distinct causal structures in four-dimensional Euclidean space, including a novel structure linked to a scalar field, extending the understanding of spacetime and gravity beyond Minkowski space.

Contribution

It introduces and analyzes a second causal structure in 4D Euclidean space, deriving exact solutions with scalar field energy-momentum tensor, and compares it to the well-known Minkowski spacetime.

Findings

Identifies two causal structures in 4D Euclidean space.

Derives exact solutions for the scalar field energy-momentum tensor case.

Proposes the second causal structure as physically interesting and comparable to Minkowski space.

Abstract

It is shown that in the 4d Euclidean space there are two causal structures defined by the temporal field. One of them is well-known Minkowski spacetime. In this case the gravitational potential (the positive definite Riemann metric) and temporal field satisfy the Einstein equations with trivial energy-momentum tensor. However, in the case of the second causal structure the gravitational potential and temporal field should be connected with some nontrivial energy-momentum tensor. We consider the simplest case with energy-momentum tensor of the real scalar field and derive exact solution of the field equations. It can be viewed as the ground to consider the second causal structure on the equal footing with the Minkowski spacetime, i.e., as an object interesting from the physical point of view, especially in the framework of the field theory.