Science > Physics > Sphere Control Across Physics and Engineering 3: General Relativity
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Science > Physics |
| User: |
"OsherD" |
| Date: |
17 Sep 2005 12:57:54 AM |
| Object: |
Sphere Control Across Physics and Engineering 3: General Relativity |
From Osher Doctorow
Francesco Catoni, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti
of ENEA Centro Recherche Casaccia Roma, Italy in "Lorentz surfaces with
constant curvature and their physical interpretation," math-ph/0508012
v1 3 Aug 2005, appear to derive General Relativity (GR) without
assuming the Principle of Equivalence or its direct analogs by a
"generalized sphere/circle" generalization of the complex numbers into
"hyperbolic numbers" which are a subset of Clifford algebras.
Hyperbolic numbers are defined as numbers of form z = x + hy where x
and y are real and h^2 = 1 with h not real but also not necessarily
complex.
Hyperbolic numbers do for Lorentz surfaces what complex numbers do for
Euclidean space surfaces in regard to constant curvature surfaces. The
Lorentz transformation group space-time symmetry physically yields the
link between fields and curvature corresponding to the Einstein Field
Equation(s). The hyperbolic plane is introduced analogously with the
complex Gauss-Argand plane as points of form P = (x, y) where z = x +
hy, and represented on a Cartesian plane the distance squared of P from
the origin of coodinate axes is zz-bar = x^2 - y^2 with z-bar = x - hy.
Lorentz (also called hyperbolic) surfaces are represented by a
non-definite quadratic form, unlike Riemann surfaces with definite line
element form.
The hyperbolic plane has the same characteristics as the
pseudo-Euclidean flat plane described by line element ds^2 = e(dx^2 -
dy^2) with e = +/- 1 such that ds^2 > = 0.
The equilateral hyperbolas represented in the pseudo-Euclidean plane
are analogs of and satisfy the same theorem as circles in the Euclidean
plane and in fact circles and equilateral hyperbolas have the same
equations expressed respectively in terms of complex and hyperbolic
variables.
It turns out that constant curvature surfaces with definite and
non-definite line elements have geodesics represented by circles
limited by "limiting circles" which latter are respectively x^2 + y^2 =
+/-R^2 vs x^2 - y^2 = +/-R^2 with - for positive constant curvature
surfaces and + for negative constant curvature surfaces.
With the x variable physically interpreted as a normalized (speed of
light = 1) time variable and y as a space variable, geodesics in a
plane with constant curvature metric are the same as those resulting
from constant acceleration motion, which relates the (gravitational)
fields and space curvature as with the Einstein Field equation (see p.
11 of their paper).
Note that in Probable Influence, the space variables are also taken
with opposite sign to the time variable, there with the interpretation
that time is the causal/driving variable.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Sphere Control Across Physics and Engineering 3: General Relativity |
17 Sep 2005 01:20:39 AM |
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From Osher Doctorow
K. Hallowell and A. Waldron of U. C. Davis Math. Dept. in "Constant
curvature algebras and higher spin action generating functions",
hep-th/0505255 v1 28 May 2005, develop remarkable algebras indicated in
the title of their paper which can formulate theories of arbitrary spin
in d dimensions in terms of a single scalar field in dimension 2d with
the d additional dimensions corresponding to coordinate differentials,
and develop an analogous machinery for spinor-tensor fields with
respect to the corresponding superalgebra.
I consider Catoni et al's paper from last time to be both simple and
(generalized) circular/spherical, while Hallowell and Waldron's paper
is simple and has a Riccati equation-like correspondence or
identification between the constant curvature algebra and the harmonic
oscilator via:
1) tr <--> -(1/h)p^2
2) div <--> -ip
3) N <--> (i/2h)(xp + px)
4) "square" operator <--> h
5) grad <--> - x
6) g <--> (1/h)x^2
and with the quantum commutator [p, x] = -i the two algebras coincide
where the right hand sides are the spectrum generating sl(2,R) algebra
of the harmonic oscillator. The symbols N, g, tr, etc. on the left
hand sides are elements of the constant curvature algebra (the first 3
generate the Lie algebra sl(2, R), with grad and div being the gradient
and divergence operators. N is defined by:
7) N PHI = s PHI with PHI = phi_(u1...us)dx^(u1)...dx^(us)
and operators g, tr act on eigenvectors of N by multiplication by the
metric and symmetrizing versus contracting a pair of indices
respectively. On a constant curvature manifold, we have:
8) {div, grad} = "square operator" - 2c
where c is the sl(2, R) Casimir:
9) c = g tr - N(N + n - 1), n - dim(M) - 1
with M the constant curvature manifold. The "square operator" is the
Lichnerowicz wave operator although it can be defined by:
10) square = DELTA + c = {div, grad} + 2c
Osher Doctorow
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