On one connection between Lorentzian and Euclidean metrics

TL;DR

This paper explores a mathematical connection between Lorentzian and Euclidean metrics, proposing a bijective mapping and discussing implications for reformulating physical theories in different metric signatures.

Contribution

It introduces a bijective mapping between Euclidean and Lorentzian metrics and discusses the potential to reformulate physical theories using Euclidean metrics.

Findings

Constructed a bijective mapping between Euclidean and Lorentzian metrics.

Discussed the existence of such maps on differentiable manifolds.

Pointed out the possibility of reformulating physical theories in Euclidean metrics.

Abstract

We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. Similar relations for metrics of arbitrary signature on a manifold are considered. We point the possibility that any physical theory based on real Lorentzian metric(s) can be (re)formulated equivalently in terms of real Euclidean metric(s).