## Saturday, April 22, 2006

Gravity as a gauge theory and Dark Energy

The only difference between the electro-weak-strong force standard model of leptons and quarks with weak parity violation and Einstein's theory of gravity is that in the former it is internal compact continuous unitary symmetry groups SU(N) that are locally gauged with subgroup symmetries hidden (Meissner-Higgs effect), and in the latter it is the non-compact continuous orthogonal space-time symmetry groups O(n,n'). The local Einstein equivalence principle is simply the minimal coupling assumption use of locally gauge covariant derivatives applied to space-time symmetries rather than internal symmetries. The Cartan exterior derivative d is nilpotent, i.e. d^2 = 0, only for the 10-parameter Poincare group not for the 10-parameter DeSitter group with anti-gravity dark zero point energy density ~ /\zpf > 0 for G = c = h = 1 unit convention. Therefore, Wheeler's "boundary of a boundary vanishes" is violated by dark energy in DeSitter space because of homology/cohomology duality. The exterior derivative is dual to the boundary operator.

The Cartan exterior derivative operator is

d = dx^a&a

Where dx^u is a basis of forms and &u is a dual basis of co-forms (tangent vectors)

In general

F = dA = (1/2)(&aAb - &bAa)dx^a/\dx^b

the "/\" here is the antisymmetric exterior product not to be confused with /\zpf.

In particular let A = d, therefore

d^2 = (1/2)(&a&b - &b&a)dx^a/\dx^b

But this depends on the group structure!

For the Poincare group T4*O(3,1) = 4DTranslations*4DSpace-Time Rotations

(&a&b - &b&a) = 0

However, for the 5D DeSitter group O(4,1)

&a ~ Q5a

a,b = 0,1,2,3

(&a&b - &b&a) = /\zpfO(3,1)ab

The SUPERSYMMETRY limit of ZERO DARK ENERGY /\zpf --> 0 (positive side) is the Wigner-Inonu space dimension contraction of constantly curved 5D DeSitter space-time to globally flat 4D Poincare space-time.

Let {Qab,Q5a} be the 10 Lie algebra "charges" Q@, where @ = 0,1,2,3,4,5,6,7,8,9 infinitesimal generators of the DeSitter group O(4,1) in an obvious reassignment of labels that I do explicitly in my archive paper. I locally gauge this entire group to get the curved-torsioned Yang-Mills gravity tetrad 1-form

A = Au^@Q@dx^u

Where now {Q@} is the Lie algebra of the 5D dark energy DeSitter group O(4,1)

The FOUR curved tetrad potentials are the SUBSET

where a = 0,1,2,3 = (5,0),(5,1),(5,2), (5,3) i.e. @ = 0,1,2,3

Einstein's 1915 GR metric tensor field is

guv = (Iu^a + Au^a)(Minkowski)ab(Iv^a + Au^b)

In the limit /\zpf --> 0

In addition, in the archive paper I relate A^a to the 8 Goldstone phases of the inflation vacuum ODLRO field - that is a separate conceptual module.

In general we need to locally gauge O(4,1) for /\zpf =/= 0 and hide 6 symmetries of the 10 to get the 10^500 Calabi-Yau spaces.

*In this more general case d^2 = 0 is not true!

That is main NEW INSIGHT here.