Abstract
Euclidean embedding of the 11-dimensional M-theory turned out to require a very large space leaving lavish amounts of 242 dimensional pseudo truly empty “regions” devoid of space and time and consequently of anything resembling ordinary physical energy density. It is shown here using Nash embedding that the ratio of “solid” M-theory spacetime to its required embedding “non-spacetime” is 1/22 for a classical theory and 1/22.18033989 for an analogous fractal theory. This then leads to a maximal ordinary energy density equation equal to that of Einstein’s famous formula E=mc2 but multiplied with in full agreement with previous results obtained using relatively more conventional methods including running the electromagnetic fine structure constant in the exact solution of the hydrogen atom. Consequently, the new equation corresponds to a quantum relativity theory which unlike Einstein’s original equation gives quantitative predictions which agree perfectly with the cosmological measurements of WMAP and the analysis of certain supernova events. Never the less in our view dark energy also exists being the energy of the quantum wave amounting to 95.5 present of the total Einstein theoretical energy which is blind to any distinction between ordinary energy of the quantum particle and the dark energy of the quantum wave. However, since measurement leads to the collapse of the Hawking-Hartle quantum wave, dark energy being a quantum wave non-ordinary energy could not possibly be measured in the usual way unless highly refined quantum wave non-demolition technology is developed if possible. It is a further reason that dark energy having a different sign to ordinary energy is the cause behind the anti gravity force which is pushing the universe apart and accelerating cosmic expansion. Consequently it can be seen as the result of anticlastic Cartan-like curvature caused by extra compactified dimensions of spacetime. A simple toy model demonstration of the effect of curvature in a “material” space is briefly discussed.
Highlights
The new equation corresponds to a quantum relativity theory which unlike Einstein’s original equation gives quantitative predictions which agree perfectly with the cosmological measurements of WMAP and the analysis of certain supernova events
Never the less in our view dark energy exists being the energy of the quantum wave amounting to 95.5 present of the total Einstein theoretical energy which is blind to any distinction between ordinary energy of the quantum particle and the dark energy of the quantum wave
Since measurement leads to the collapse of the Hawking-Hartle quantum wave, dark energy being a quantum wave non-ordinary energy could not possibly be measured in the usual way unless highly refined quantum wave non-demolition technology is developed if possible
Summary
Nash embedding which is a powerful topological method in the theory of Riemannian manifolds [1,2,3,4] is brought to bear on a fundamental and major cutting edge research problem in relativistic quantum cosmology, namely the dark energy of the cosmos [4,5,6,7,8]. 6) Here is the discovery: In similarity to what happened to Witten’s eleven dimensional M-Theory which unifies all the five known superstring theories, space-time at large has truly empty regions which empty, are needed to embed M-Theory Euclideanly. We start here by giving first some relevant applications of Nash’s embedding formula [1,2] and proceed from there to derive the above mentioned quantum relativity formula as well as the quantum relativistic formula for dark energy. The example is even more interesting as it can be used to make the fundamentality of the four dimensionality of spacetime and its intimate connection to Cantor sets plausible [15] To do this we need to extend Nash’s formula by replacing the factor 11 by the self similar eleven [13,14]. P(F) may be seen as a probability which plays the same role as 2 of the electromagnet force where is the Sommerfeld fine structure constant
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