## Monday, April 17, 2006

Dark Energy, DeSitter Space & Catan's Forms

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http://stardrive.org/title.shtml

for more current up-dates

This is starting something very new! Uncharted territory. Before Waldyr jumps on this:

"Anyone who has never made a mistake has never tried anything new." Einstein

I. Relation of DeSitter Group to Poincare Group

First, Klein 1872 groups define geometries.

Second, orthogonal groups of nxn matrices have n(n - 1)/2 parameters.

The invariant is x^2 = xi^2 (summation convention), xi real, i = 1 ... n

The anti-gravtating w = -1 (no boundaries) dark energy density ~ /\ > 0 corresponds to a global DeSitter group O(4,1) with invariant of Kaluza-Klein type 4 + 1 space-time with one extra space dimension.

x1^2 + x2^2 + x3^2 + x4^2 - x0^2

n(n - 1)/2 = 5x4/2 = 10

The "charges" of the Lie algebra are the 4x3/2 = 6 space-space rotations Mij = - Mji, i,j = 0,1,2,3 and the 4 space-time rotations Pi = Mi4, i = 0,1,2,3

The commutator of interest is

[Pi,Pj] = /\zpfMij

The other commutators

[Pi,Mjk] and [Mij,Mkl] are exactly the same as the Lie algebra of the 10 parameter Poincare group T4*O(3,1) in 3 + 1 space-time.

When the CONSTANT dark energy density vanishes /\zpf ---> 0 in the fermion-boson supersymmetry limit of 3 + 1 space-time then this "Wigner-Inonu contraction is from constantly curved 4 + 1 space-time to globally flat 3 + 1 Minkowski space-time, so that [Pi,Pj] = 0 where Pi is the total Energy-Momentum of 3+1 flat space-time.

Relationship to Cartan's forms.

d = dx^iPi = exterior derivative 1-form, h = c = 1 for now

d^2 = 0

means

d/\d = 0

That is only true when

[Pi,Pj] = 0

Therefore, when /\zpf =/= 0 we can no longer assume that d/\d = 0 in a 3 + 1 space-time with a dark energy density.

Note "/\" = Cartan exterior multiplication of forms

"/\zpf" is scalar constant curvature of 4 + 1 DeSitter space-time.

d/\d = dx^iPi/\dx^jPj = dx^i/\dx^j[Pi,Pj] = dx^i/\dx^j/\zpfMij

So for example suppose

B = dTheta

Theta is a 0-form

B is a 1-form

Therefore

dB = d^2Theta = d/\d = dx^i/\dx^j/\zpfMijTheta =/= 0

This will create ANOMALIES.

For example, Maxwell's EM equations

F = dA

dF = d^2A = dx^i/\dx^jMij/\zpfA =/= 0

This begins to look like the Meissner effect in superconductors ? and also magnetic monopoles!

dF = 0 when /\zpf = 0 corresponds to

curlE + Bt = 0 i.e. induction

divB = 0 no magnetic monopoles

Then also second half of Maxwell's equations

d*F = *J

d^2*F = d*J = dx^i/\dx^jMij/\zpf*F =/= 0 current anomaly

When /\zpf = 0 this is Ampere's law with displacement current

curlH + D,t = j

divD = rho

and Gauss's law.

This is before we get /\zpf as a local dark energy/matter field including /\zpf < 0 by locally gauging all of O(4,1).