Yes, but then a vector space without a norm. No inner product is
defined, so from a pure math point of view, you are free to define any
arbitrary inner product that satisfies the axioms for it. Alan the
physicist will prefer that inner product that is dictated by the
relevant physics, so this has to take the form of the indefinite Lorentz
inner product in a locally free-falling coordinate system. This means
that you must also consider the metric, so the distance between
infinitesimally separated points. So, labels are arbitrary and hey are
then meaningless, unless you also specify what the distance is between a
point at [ct, x, y, z] and at [c(t+dt), x+dx, y+dy, z+dz].
Saibal
On 06-09-2024 10:03, Alan Grayson wrote:
Do the 4-tuple labels on spacetime, (ct, x, y, z), form a vector
space? I was told on good authority that the answer is negative, but
now I have grave doubts of that conclusion. AG
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