$k$-calculus in $3+1$ dimensions? So I've recently been intrigued by $k$-calculus. Is there a formulation of $k$-calculus in $3+1$ dimensions? More specifically the relation between the $k$-factor and and a $4$-vector?
 A: I haven't read Bondi's whole book, but he does derive the Lorentz transformation in 3+1 dimensions at the end.
The earlier arguments (the $k$-calculus proper) only work for collinear motion, because the Doppler shift between general inertially moving objects varies with time. But they work in any direction. Distance is defined by assuming a constant speed of light, which is justified by the fact that it doesn't lead to any contradictions. To get coordinates in 3+1 dimensions, you just need to assume a constant speed of light in all directions (although the fact that that doesn't lead to contradictions is in conflict with most people's common sense).
The connection to 4-vectors depends on which 4-vector. Position is covered in the previous paragraph, velocity and acceleration are derivatives of position with respect to proper time, and for anything else (like energy-momentum or force) you need dynamical laws of some sort. It doesn't seem obvious in Bondi's approach that there is a natural norm/inner product on four-vectors, but that wasn't obvious in Einstein's approach either.
