Abstract
The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.
Highlights
The present work is a quite comprehensive review of E-Infinity theory [1]-[60] in high energy physics and cosmology which addresses a fundamental question in the very nature of the geometry and topology of spacetime [60]-[402]
On the other hand at very small scales our inability to give a watertight definition for what a point is becoming critical while the very small scale is where quantum mechanics becomes very important indispensable for high energy physics let alone quantum gravity [1] [13] [411]
The present work shows that this particular phenomenon is intimately linked to the completion of Einstein’s space time via extending it to a gauge invariant manifold with D= 4 + φ3 rather than D = 4 where φ is the said golden mean
Summary
The present work is a quite comprehensive review of E-Infinity theory [1]-[60] in high energy physics and cosmology which addresses a fundamental question in the very nature of the geometry and topology of spacetime [60]-[402]. We insist that Einstein’s spacetime remain self-affine to four-dimensional manifold on all scales down to a point like “infinitesimally” small spacetime “point” which naturally cannot be a classical point but rather a “Cantorian” point This means a point which upon “magnification” i.e. upon increasing the resolution of our “observation” turns ours to be not a point but an entire Cantor set with uncountably infinitely many points and so on ad infinitum [1]-[50]. Our analytical tool to do that is the marvelous classical mathematical “technique” known in the literature as continued fraction This means that our space should be 4D inside 4D and so on like an infinite arrangement of Russian Dolls [3] each with D = 4 so that at the end we find that our dimension is given by. It turns out that this hunch is correct and we will attempt to explain it in what follows
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