A simple header construction example based on Einstein's Special Theory of Relativity () will serve to illustrate the formalism introduced in this paper. We will construct dual coordinate representations, the first for the rest frame and the second for an observer in uniform motion.
Suppose we have a data cube that, in the rest frame, has the following simple header containing two spatial axes and one temporal axis:
This describes three linear coordinate axes with the reference point in the middle of the data cube.
The spatial and temporal coordinates measured by an observer moving with uniform velocity v in the + x direction are related to the rest coordinates by the Lorentz transformation:
and c is the velocity of light. Time in each system is measured from the instant when the origins coincide. From the above header we have
where, rj and si are given by CRPIXj and CDELTi. Thus
This set of equations may be rewritten to make the scales, CDELTi, the same as the rest frame header:
Using character "V" as the alternate representation descriptor, a, for the relatively moving frame, we have
Note that the elements of the PC i_j matrix are all dimensionless, = s1 / s3 = 3 × 108 m s-1 having the dimensions of a velocity. However, in this instance we have seen fit not to apply the strictures of Eq. (4) in normalizing the matrix. In fact, in Minkowski space-time the concept of "distance", on which Eq. (4) relies, differs from the Euclidean norm, the invariant being
so one may query the fundamental validity of Eq. (4) in this case. However, the intent of that equation is well served since PCi_j and CDELTi are divided in a physically meaningful way, especially considering that is often close to unity so that PCi_j is approximately the unit matrix. Nevertheless, the appearance of the factor in each of the elements of PCi_j suggests the factorization
and indeed this does also have a physically meaningful interpretation in that the scales are dilated by the Lorentz-Fitzgerald contraction factor, .