K-calculus in 4-dimensional optics
  4-dimensional optics is based on the use 4-dimensional movement space, resulting from the consideration of the usual 3-dimensional coordinates complemented by proper time. The paper uses the established K-calculus to make a parallel derivation of special relativity and 4-dimensional optics, allowing a real possibility of comparison between the two theories. The significance of proper time coordinate is given special attention and its definition is made very clear in terms of just send and receive instants of radar pulses. The 4-dimensional optics equivalent to relativistic Lorentz transformations is reviewed.   Special relativity and 4-dimensional optics are also compared in terms of Lagrangian definition of worldlines and movement Hamiltonian. The final section of the paper discusses simultaneity through the solution of a two particle head-on collision problem. It is shown that a very simple graphical construction automatically solves energy and momentum conservation when the observer is located at the collision position. A further discussion of the representation for a distant observer further clarifies how simultaneity is accommodated by 4DO. 
Jose B. Almeida