Re: Orthogonality in GR (kst+)

From: xxein (xxein_at_bellsouth.net)
Date: 10/06/04 

Date: 5 Oct 2004 19:46:38 -0700

dynamics@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.0410051408.6144e65a@posting.google.com>...
> Recently, I was asked by a poster (Bjoern F) to post a derivation
> of some equations. The equations to be derived intend to describe
> "orthogonal" distances in "nonorthogonal" fields, one may imagine
> the character of the field to be gravitational, but it's not
> restricted to g-fields.
>
> In accord with the International System of Units (ISU),
>
> r = ct where the length "r", speed"c" and time "t"
>
> are all measured in a vacuum, and are all *finite* quantities.
> The *finite* condition is what the ISU stipulates. The "r" and
> "t" may be small, but they do not tend to zero. That protects
> the Quantum Theory (QT), wherein only small quantities
> (not differentials) are measureable. That also poses problems
> where "positional vectors" and "positional operators" are concerned.
>
>
> We may begin with a CS wherein r, c and t are constant in the
> vacuum and find,
>
> r^2 = x_i x^i {i = 1,2,3}
>
> c^2 t^2 = x_0 x^0 ,
>
> and find,
>
> x^2 = x_i x^i = x_0 x^0 , described as,
>
> (invariant spatial length) = (invariant time interval),
>
> is true for all systems of reference, thus those invariants are
> true in all CS's. We should find the ISU definition to be true
> always and the "local" speed of light is always "c".
>
> The condition of invariance detailed above may be expressed by
> the more detailed formula,
>
> X^2 = d^uv x_u x_v
> a b
>
> wherein the "a" and "b" are fixed points in 3D space delineating
> the length "r".
>
> We further set
>
> x_i = - x_i
> a b
>
> to represent the relatively opposite directions "a" and "b" are in
> space and
>
> x_0 = x_0
> a b
>
> to represent "a" and "b" propagate in time positively.
>
> In this static circumstance, we'll employ the conventional
> Minkowski metric trace, [1,-1,-1,-1] in d^uv to get,
>
> X^2 = x_i x_i + x_0 x_0 .
>
> That is the circumstance when a small region of a curving
> field is held to be orthogonal and the Minkowski metric is
> applicable, even in a g-field.
>
> Now, we need to find a simulataneous solution for the invariants
> X^2 and x^2, in order to conform to ISU standards.
>
> In a non-vacuum medium the speed of light is different from "c"
> and is denoted "C", (albiet locally constant),
>
> R = CT
>
> where the R and T are measured in the "non-vacuum".
>
> Using the following transformation,
>
> R = r(1 - o) and T = t(1 + o)
>
> where "o" is a small perturbation, (o ~ m/r in a g-field)
>
> We obtain an equivalent set of equations to consolidate
> all of the above.
>
> X^2 = r^2 +c^2 t^2 = R^2 + c^2 T^2 = 2x^2 .
>
> The coordinate speed of light in a g-field, (as found by Shapiro's
> experiments) validates the vertical speed of light to be,
>
> C = R/T = 1 - 2*o = 1 - 2m/r
>
> ((see Weinberg's Grav & Cosmo Eq. (9.2.5), in order to retain
> a connection to real physicality herein)).
>
> In compliance with the above requirements, we can solve
> the *finite* orthogonal components to be invariant in a
> nonorthogonal space by using,
>
> x_i = cT , x_0 = CT
>
> x^i = CT , x^0 = cT .
>
> Summary
> The importance of obtaining an invariant, orthogonal spacetime X
> (or 2x) in a *nasty* nonorthogonal field allows one to readily
> define (piggy-back) an invariant of choice onto X like,
>
> S = X + INVARIANT
>
> or
>
> S^2 = X^2 + INVARIANT^2 .
>
> That permits complicated spacetime nonorthogonal finite distances
> "S" to be computed on the basis of the imposed INVARIANT's.
>
> Ken S. Tucker

xxein: Can you say that without math and include the non-mathematical
concepts of subjective observation and an objective reality at the
same time?