Some subtleties of Riemannian geometry

TL;DR

This paper discusses issues with formulating Maxwell's equations in Riemannian space-time, highlighting that solutions in Minkowski space do not straightforwardly translate to Riemannian geometry when replacing the world function.

Contribution

It identifies a fundamental problem in applying standard Maxwell solutions from Minkowski space to Riemannian space-time via world function substitution.

Findings

Standard Minkowski solutions do not directly extend to Riemannian space-time.

Replacing Minkowskian world function with Riemannian one is insufficient for solutions.

Highlights the need for revised approaches in Riemannian electromagnetic theory.

Abstract

It is shown, that the conventional presentation of the Maxwell equations for the electromagnetic field in the Riemannian space-time appears to be problematic. The reason of hesitations is the fact, that a solution of the Maxwell equations in the space-time of Minkowski do not turn into solution of the Maxwell equations in the Riemannian space-time after replacement of Minkowskian world function $\sigma_{M}$ by the world function $\sigma_R$ of the Riemannian space-time in the solution.