Reformulation of Special Relativity and Electromagnetism in terms of Reference Frames defined as maps from spacetime onto affine spaces

Abstract

A reference frame on a set $M$ is given by a 3-dimensional euclidean space $E$, a function from $M$ to $E$, a 1-dimensional affine space $A$ and a function from $M$ to $A$. The definition allows an intuitive and coordinate-free formulation of Newtonian mechanics, special relativity and electromagnetism in terms of frames instead of coordinates. In particular, the postulate of a set of charts on $M$ (an atlas) is replaced by the postulate of a set of frames. Then the charts can be re-obtained through the choice of affine coordinates. In addition, units enter the theory as elements of positive spaces and we obtain a geometric and manifestly unit-independent reformulation. Each frame allows us to identify spacetime with a 4-dimensional product-space and we can generalize the definition of differentiable manifolds such that the frames turn out to form an atlas. Then the only difference between Newtonian mechanics and special relativity is the assumed type of transition functions - Galilean transformations or Poincar\'e transformations between affine spaces. This allows us to highlight the common features and differences in the second chapter of the paper. For example, we define velocity reciprocity in mathematical terms and prove the phenomenon for both kind of transformations. The chapter concludes with a discussion of inertial and accelerated frames. In the third chapter we restrict our attention to the reformulation of special relativity and electromagnetism with a strong emphasis on covariance. World lines are introduced as particular subsets of spacetime, the proper time of world line is defined as an affine structure on the world line and the reformulation of electromagnetism is based on the representation of differential forms w.r.t. a frame.