English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 afunction from $M$ to $A$. The definition allows an intuitive andcoordinate-free formulation of Newtonian mechanics, special relativity andelectromagnetism in terms of frames instead of coordinates. In particular, thepostulate of a set of charts on $M$ (an atlas) is replaced by the postulate ofa set of frames. Then the charts can be re-obtained through the choice ofaffine coordinates. In addition, units enter the theory as elements of positivespaces and we obtain a geometric and manifestly unit-independent reformulation.Each frame allows us to identify spacetime with a 4-dimensional product-spaceand we can generalize the definition of differentiable manifolds such that theframes turn out to form an atlas. Then the only difference between Newtonianmechanics 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 secondchapter of the paper. For example, we define velocity reciprocity inmathematical terms and prove the phenomenon for both kind of transformations.The chapter concludes with a discussion of inertial and accelerated frames. Inthe third chapter we restrict our attention to the reformulation of specialrelativity and electromagnetism with a strong emphasis on covariance. Worldlines are introduced as particular subsets of spacetime, the proper time ofworld line is defined as an affine structure on the world line and thereformulation of electromagnetism is based on the representation ofdifferential forms w.r.t. a frame.

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