Time dependent electromagnetic fields and 4-dimensional Stokes' theorem

TL;DR

This paper explores the application of Stokes' theorem in four-dimensional spacetime with time-dependent electromagnetic fields, highlighting differences from the traditional static case and discussing implications for gauge transformations and topology.

Contribution

It provides explicit examples and analysis of Stokes' theorem in 4D spacetime with time-dependent fields, filling a gap in pedagogical and research literature.

Findings

Demonstrates how Stokes' theorem extends to 4D with time-dependent fields

Identifies unusual features related to gauge transformations in dynamic scenarios

Discusses topological implications for non-simply connected spaces

Abstract

Stokes' theorem is central to many aspects of physics -- electromagnetism, the Aharonov-Bohm effect, and Wilson loops to name a few. However, the pedagogical examples and research work almost exclusively focus on situations where the fields are time-independent so that one need only deal with purely spatial line integrals ({\it e.g.} $\oint {\bf A} \cdot d{\bf x}$) and purely spatial area integrals ({\it e.g.} $\int (\nabla \times {\bf A}) \cdot d{\bf a} = \int {\bf B} \cdot d{\bf a}$). Here we address this gap by giving some explicit examples of how Stokes' theorem plays out with time-dependent fields in a full 4-dimensional spacetime context. We also discuss some unusual features of Stokes' theorem with time-dependent fields related to gauge transformations and non-simply connected topology.