Relativity on rotated graph paper

Alternate conventions for signs and factors of $1/2$ would complicate the counting calculations.

Indeed, this is one property that makes Minkowski spacetime diagrams difficult to interpret. Note, however, that Galilean spacetime diagrams (position-time graphs) also have this property.

A parallelogram with edges $au→$ and $bv→$ has diagonals $(au→+bv→)$ and $(au→−bv→)$⁠, with dot-product $(a2u→·u→−b2v→·v→)$⁠. Using the dot-product of Minkowski spacetime, the diagonals of a causal diamond are perpendicular to each other since the edges are lightlike (⁠$u→·u→=0$ and $v→·v→=0$⁠).

This is equivalent to Minkowski's definition: the radius vector drawn to a point on the hyperbola is “normal” (perpendicular) to the tangent vector at that point (see Ref. 2, p. 85).

When $(Δx)2+(Δs)2=(Δt)2$ for nonnegative integers $Δx, Δs$⁠, and Δt, with $Δx<Δt$ and $Δs≤Δt$ for future-timelike displacements, then $(Δx,Δs,Δt)$ form a Pythagorean triple. For triples generated by $Δx=λ(m2−n2)$⁠, $Δs=λ(2mn)$⁠, and $Δt=λ(m2+n2)$⁠, with positive integers λ, m, and n (with $m≥n$⁠), we find $k=m/n$⁠. Alternatively, when $Δs=λ(m2−n2)$ and $Δx=λ(2mn)$⁠, we find $k=(m+n)/(m−n)$⁠.

The inverse transformations express the coordinates of the “lab frame” (here, Alice) in terms of those of the “moving frame” (Bob). The standard transformations (with the minus signs) are obtained by solving these equations for $tEB$ and $xEB$⁠. See p. 107 in Ref. 17.

The relations $Δu=ΔtrA$ and $Δv=ΔteA$ hold only when the timelike diagonal of the causal diamond is on the forward (⁠$x≥0$⁠) side of the observer. However, if, for example, $xB3>0$ but $xB2<0$⁠, then we have the less-elegant relations $Δu=tr3A−te2A$ and $Δv=te3A−tr2A$⁠. Hence, we restrict to the case $x≥0$ for simplicity.