It has been around 8 years since I first realized that space, not spacetime, should be a four-sphere of fixed radius. I was not trained in physics with a mathematics background. Since then I have gradually learned some of what the physicists had accomplished and looking at the issues that have animated their work it is now fairly clear that in terms of empirical successes, the quantum field theory model of particles and gauge forces of Glashow, Weinberg and Salam (including the Higgs boson, which they predicted and which was found in 2012) is correct and quite likely supersymmetry is not a feature of nature. This is a 2009 talk of Weinberg who is without questions a great physicist Weinberg talking about HiggsWeinberg talking about Higgs. I am still quite sure that S4 theory is correct with the Standard Model, and Big Bang cosmology is wrong. In fact, it is pretty clear to me now that there is nothing beyond the Standard Model in particle physics and the project of quantum gravity is the wrong direction. I don’t think there is a graviton. The Universe is a constructed in my view as a hypersurface of a scaled four-sphere where the quantum field theory is somehow attaining a complicated (29 adjustable parameters) theory despite the fact that there is a totally wrong cosmology, which leads to the ‘cosmological constant’ problem. The cosmological constant can be calculated in the ‘Einstein static model’ — and Einstein’s first instinct was in the right direction of an eternal three-sphere universe. The measured cosmological constant is positive, . People have calculated the vacuum energy density (see energydensityeinsteinuniversein the Einstein model and it is

where is the radius of the universe, and the standard explanation that the Einstein-static universe is unstable is not quite true, see stabilityeinsteinuniverse) If we take the estimated radius of the universe, then we actually get an energy density . We get much closer to the measured energy density if instead we use a radius , then .

So what makes the most sense to me is that the correct model of the universe is approximately the Einstein static universe, where the cosmological constant problem is not a big deal — and the claims of a flat universe by the establishment is incorrect. They are simply calling the curvature ‘dark energy’. There is no dark energy; the positive cosmological constant is the curvature of the static eternal universe of fixed radius. The key new point that I bring is that on a spherical universe, whether three dimensional or a hypersurface of a four-sphere, the wavelength-frequency relation of light is distorted sufficiently that we will see a fake redshift, that the redshift observed is actually just an artifact of a curved universe. The dark energy is a concept invented to protect the Big Bang model which is not real and has not been observed in any way. The really interesting new direction is not quantizing gravity but to understand the quantum theory and particle theory that arises geometrically in a static curved universe whose curvature is obviously small — quantization of energy is automatic in a four-sphere universe which suggests that there can be a much cleaner standard model. I also think that there is a macroscopic fourth space dimension and that quantum fields can be described purely geometrically which is probably already done by theorists of the brane-worlds etc.). Gravity is a feature of the universe that is not so much like the gauge forces in my opinion..

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