_{1}

^{*}

Ordinary energy and dark energy density are determined using a Cosserat-Cartan and killing-Yano reinterpretation of Einstein’s special and general relativity. Thus starting from a maximally symmetric space with 528 killing vector fields corresponding to Witten’s five Branes model in eleven dimensional M-theory we reason that 504 of the 528 are essentially the components of the relevant killing-Yano tensor. In turn this tensor is related to hidden symmetries and torsional coupled stresses of the Cosserat micro-polar space as well as the Einstein-Cartan connection. Proceeding in this way the dark energy density is found to be that of Einstein’s maximal energy *mc*^{2} where *m* is the mass and *c* is the speed of light multiplied with a Lorentz factor equal to the ratio of the 504 killing-Yano tensor and the 528 states maximally symmetric space. Thus we have *E* (dark) = *mc*^{2} (504/528) = *mc*^{2} (21/22) which is about 95.5% of the total maximal energy density in astounding agreement with COBE, WMAP and Planck cosmological measurements as well as the type 1a supernova analysis. Finally theory and results are validated via a related theory based on the degrees of freedom of pure gravity, the theory of nonlocal elasticity as well as ‘t Hooft-Veltman renormalization method.

The aim of the present note is to demonstrate how easy it is to reformulate and rephrase Einstein’s general relativity [

We start from the premise that both Einstein’s spacetime and the maximally symmetric Witten’s five Branes model leads to the same Lorentzian factor γ = 1 for the maximal Einstein energy density, E = γmc^{2} where m is the mass and c is the speed of light and will look upon Cartan’s affine connection from a Lie symmetry groups view point [

where γ is the Lorentz factor,

The 528 killing vector fields on the other hand are interpreted by us here in two ways. First it is the number of components of the killing-Yano conformal tensor [

Adding |E4| = 24 where |E4| is simply another name for |SU(5)| of GUT unification [

In other words we have [

Consequently the number of the killing components [

It is vital at this point not to confuse dark energy with torsion energy due to the 24 Riemann-Cartan connection components in four dimensions because the concept torsion has various meanings in string theory compared to Einstein-Cartan theory [

Two further relevant observations regarding the vital number 24. First it is exactly equal to the number of the gauge bosons of SU(5) GUT unification which adds 12 bosons to the well known and experimentally found 12 bosons of the SU(3) SU(2) U(1) standard model of high energy physics [

Contemplating the situation a little it is not particularly difficult to convince oneself that the Lorentzian factor of Einstein’s energy density corresponding to 24 non-trivial Bianchi identities [

and therefore ordinary is given by [

Exactly as expected from previous analysis. Note that 22 could be interpreted as the number of two forms, i.e. harmonic forms of a K3 Kähler compactification representing nontrivial deformations of the anti-symmetric string tensor. Dark energy on the other hand is squarely connected to the negative energy of the non-vanishing torsional part in the Cartan connection, namely the 504 known also from the particle physics spectroscopy of Heterotic string theory [

leading to a dark energy density [

Einstein’s energy density

We note on passing that 528 is divided in Witten’s model into 1D strings, 2D membranes and 5D Branes [

The corresponding E8E8 expression includes the point-like particles as well as the 3D and 4D Branes [

The hidden Yano-killing 504 on the other hand are given in Heterotic super string theory by three groups of states, namely 480, 16 and 8 leading to [

and

as explained in great detail in [

exist [

Unlike the engineering theories of elasticity and elastoplastic structures as well as the theory of micropolar media, there is no meaning at all for the rotation of a point in the Riemannian geometry of Einstein’s field equation. Consequently the details of the energy household are glossed over by taking the continuum limit. That way the role of anticlastic curvature as well as shear forces and torsional deformation is not brought to the fore [

The preceding analysis is intended to rectify the situation and may be seen as a Cosserat-like material space based analysis in the spirit of the analogies discussed in connection with the pinched elastic cylindrical shell and the corresponding anti-curvature caused by local curvature [

which is essentially the Magueijo-Smolin ingenious extension of Einstein’s celebrated formula. Needless to say, this result agrees completely with our derivation as shown in detail in earlier publications in the present journal [

Deriving dark energy from pure gravity [

where D are the degrees of freedom and d is the dimension. Since Einstein’s formula

Consequently α could be regarded as the scaling exponent

The Equation (16) for D well deserves a closer look as there is more to it than meets the eyes. Setting

Whether we look upon the problem of dark energy exclusively from the point of view of the special theory of relativity or exclusively from the view point of pure energy of general relativity, the end result is the same. As many leading scientists such as Nobel Laureate G. ‘t Hooft [

Finally nothing could testify to the power of the fractal Cantorian picture of spacetime as the origin of all fundamental physics more than the agreement of the cosmological measurements of COBE, WMAP and type 1a supernova measurement with the results of our equation

sity is given by

The Author is indebted for the constructive suggestions and criticisms of an anonymous Referee. He is also deeply thankful to Prof. Dr. Dr. Otto Bruhns who communicated to me some exceptionally relevant papers by our common teacher, the eminent Prof. Dr. Dr. T. Lehmann. Prof. Lehmann was the first to formulate the theory of elasticity in Einstein’s four dimensional spacetime. Memories of our student years in the sixties of last century in Hannover, Germany are highly treasured.