On the General Covariance of Natural Laws

Abstract

This paper explores the concept of general covariance in natural laws through the lens of projective geometry and tensor algebra. By introducing the notions of covariance and contravariance using intuitive examples from projections and the scalar product, we illustrate how the covariance of natural laws ensures their universality and objectivity. We also discuss the role of symmetries and conservation principles in relation to the covariant nature of physical equations, highlighting the deep interplay between the mathematical structure of physical theories and the fundamental principles of nature.