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:
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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:
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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
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where, rj and si are given by CRPIXj and CDELTi. Thus
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This set of equations may be rewritten to make the scales, CDELTi, the same as the rest frame header:
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Using character "V" as the alternate representation descriptor, a, for the relatively moving frame, we have
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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
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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
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and indeed this does also have a physically meaningful interpretation in
that the scales are dilated by the Lorentz-Fitzgerald contraction
factor, .