What does the frame-invariance of the spacetime interval really mean? Minkowski said: "From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right." Consider a three-dimensional object, say a stick. It casts a two-dimensional shadow against a wall. As we turn the stick about, the length of the shadow changes, even though the stick itself remains the same length. In an analogous fashion, we can imagine a four-dimensional spacetime "object." All inertial observers agree that this object has the same "length" (interval) in spacetime. However, different observers see different lengths for the three-dimensional "shadow" of the object in space.

With dual numbers the mapping is t ′ + x ′ ϵ = ( 1 + v ϵ ) ( t + x ϵ ) = t + ( x + v t ) ϵ . {\displaystyle t^{\prime }+x^{\prime }\epsilon =(1+v\epsilon )(t+x\epsilon )=t+(x+vt)\epsilon .} [30]