15117Re: [EMHL] Re: Locus
- Apr 1, 2007Dear Jeff
Sorry, Cte means "constant".
So you look at events (m,t(0)) in H where t(0) is some fixed time that I
have named "Cte" in my previous post.. As H is an hyperboloid, these events
are on a conic, intersection of H in W = P x T with the plane t = t(0).
So you see why the Newtonian spacetime W is useful, allowing some synthetic
proofs in this difficult theory, even if it has no physical existence.
In fact you have a group G(alilean) acting not only on the set W of events
but also on the set of world lines, that's just the old theory of Galilean
Of course, the (choo-choo) theory is the study of the subgroup of G leaving
hyperboloid H globally invariant. This group G' is known for a long time as
H is a ruled quadric.
G' is itself the union of 2 distinct subsets G" and G'" where G" is the
subgroup of G' leaving globally invariant each of the 2 set of generators of
H and G'" the subset (G'" is not a subgroup) of G' swapping these 2 distinct
sets of generators, that is to say G" is a subgroup of G' of index 2.
On 4/1/07, Jeff Brooks <trigeom@...> wrote:
> Dear Francois,
> what is Cte?
> > I want to give a name to theses conicsbut I don't dare choose!
> > Geometrically they are the projections along time axis on plane P of
> > the sections of the hyperboloid H by the planes t = Cte.
> Sincerely, Jeff
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