George Hammond wrote:
...................................................
> Now, for the amateur, let me point out that the terms:
>
> Euclidean Space
> Cartesian Space
> Pythagorean Space
> Riemannian Space
>
> ALL MEAN THE SAME THING vis a vis one fundamental fact:
>
> They all refer to spaces that
> have a "homogeneous quadratic
> metric".
>
> The simplest example of a "homogenous quadratic metric" is simple
> Euclidean space:
>
> dR2 = dX2 + dY2 +dZ2 (Pythagorean theorem)
>
> All 4 of the above spaces are referred to as "Riemannian" for
> this reason. Euclidean space is merely a special case of
> a Riemannian space that has zero curvature (flat space).
> However, there are non-riemannian spaces. For instance if
> you define the line element to be:
>
> dR = [dXdYdZ]^1/3
>
> this is a NON-RIMANNIAN METRIC (it is not a quadratic form).
>
> Interestingly, HELMHOLTZ apparently was the first to show that:
>
> YOU CANNOT ROTATE A SOLID BODY
> IN A NON-RIEMANNIAN SPACE:
>
> See: http://www.bun.kyoto-u.ac.jp/~suchii/R&H.html
>
> This was later proven rigorously by Weyl and others.
> For instance Weyl in _Space, Time & Matter_ (1920)
> discusses in the closing pages of Ch.II, Riemann's
> famous remark that:
>
> The metric of real space "might be a homogenous
> function of the 4th order in the differentials,
> or even a function built up in some other way,
> and that it might not even depend rationally on
> the differentials."
>
> (Weyl, quoting Riemann, idid ChII, pp 138-148)
>
> Weyl goes on to demonstrate that the "rotation group" requires that
> the metric be a QUADRATIC form, and has proven this for 3 dimensions
> (the case under discussion here).
> Finally of course, for the benefit of eager PhD's in physics lest
> they make the same amateur error Dr. George Murphy has recently made,
> "curvature" is not the issue here, the existence of a "quadratic
> metric" is the issue. any space that DOES NOT HAVE a quadratic
> metric is non-Riemannian, and of interest, is the fact that:
1) Mr. Hammond's original statement was that a 3-D object can be rotated only in a Euclidean space and I pointed out that this is not
strictly true. Such a rotation is possible in any space of constant curvature. Now if he were trying to carry on a civil conversation and were
able to admit a mistake he would have said, "Yes, of course, I meant that the rotation of an infinitesimal 3-D object is possible only in a
locally Euclidean - i.e., Riemannian - space" and I would have said OK. But he is apparently unable to admit even a minor oversight and
immediately began to accuse me of amateurism & other crimes and misdemeanors. This should serve as a warning to anyone who may be tempted to
engage in conversation with him.
(I should point out "for the amateur" that a Riemannian space need not be locally Euclidean but may be locally pseudo-Euclidean, with
some terms in the line element positive and some negative. This is in fact the case for 4-D space-time.)
2) This correction is in a sense minor, and most of us are able to accept what we think are pedantic corrections without going postal.
But there is something more to be learned from it. While
Mr. Hammond speaks about "general relativity" as part of his "Scientific Proof of God," he doesn't actually make use of the distinctive features
of that theory. Of course in general relativity space-time is locally pseudo-Euclidean with signature -+++, but that is just special
relativity, and the locally Euclidean character of 3-space is true in Newtonian physics. What separates general relativity from the special
theory and classical physics is the idea that space-time is not psuedo-Euclidean in the large but has non-vanishing curvature which is related
to the distribution of matter via the Einstein equations. As far as I can tell (but I could be wrong - I don't save his posts) he doesn't make
use of this aspect of the theory.
It isn't wrong to call an appeal to Euclidean geometry a use of "general relativity" but it's a bit misleading, like calling a
discussion of Coulomb's law an application of quantum electrodynamics. It makes the argument appear to be weightier than it really is and may
serve to intimidate people who remember their high school geometry (which is all that's needed to follow the argument) but don't know the
details of general relativity.
3) One major and I think fatal problem with Mr. Hammond's "Scientific Proof of God" is not to be found in what he says about general
relativity, theology, or psychometry individually, though there are certainly problematic aspects of his arguments in at least the first two
fields. (I claim no expertise in the third.) It is the fact that the whole argument depends on non sequiturs. No real connections are made
between these disciplines and the supposed transitions are like those in the famous Harris "Then a miracle occurs" cartoon.
Having against my better judgment responded here to Mr. Hammond, I will try in the future to restrain myself regardless of provocation,
and resolve to make better use of the Delete button. I would counsel others to do likewise.
Shalom,
George
George L. Murphy
http://web.raex.com/~gmurphy/
"The Science-Theology Interface"
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