### 6. HEADER CONSTRUCTION EXAMPLE

A simple header construction example based on Einstein's Special Theory of
Relativity
([1905])
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, *r*_{j} and *s*_{i} are given by
`CRPIX`*j* and `CDELT`*i*. Thus

This set of equations may be rewritten to make the scales,
`CDELT`*i*, 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,
= *s*_{1}
/ *s*_{3} = 3 × 10^{8} 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 `PC`*i*_*j* and `CDELT`*i* are
divided in a physically meaningful
way, especially considering that is often close to unity so
that `PC`*i*_*j* is approximately the unit
matrix. Nevertheless, the
appearance of the factor
in each of
the elements of `PC`*i*_*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, .