Can material time derivative be objective?

TL;DR

This paper explores the concept of objectivity in classical field theories within a four-dimensional framework, deriving covariant and material derivatives to assess their invariance under spacetime transformations.

Contribution

It introduces a covariant approach to defining objective time derivatives in a four-dimensional setting, extending traditional Euclidean-based concepts.

Findings

Derived Christoffel symbols for spacetime transformations

Formulated covariant and material time derivatives

Analyzed the invariance of objective derivatives

Abstract

The concept of objectivity in classical field theories is traditionally based on time dependent Euclidean transformations. In this paper we treat objectivity in a four-dimensional setting, calculate Christoffel symbols of the spacetime transformations, and give covariant and material time derivatives. The usual objective time derivatives are investigated.