Hypersphere cosmological model: Why the Fourth Dimension of Space

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A hypersphere cosmology: Why the fourth dimension of space

A complete understanding and acceptance of cosmology, as described by the Standard Model, ultimately depends on individual perspectives and ways of thinking. Not entirely convinced by the FLRW solution, I directed my attention toward the concept of superluminal motion, which becomes conceivable within the framework of Galactic Recession [*]. With both Galactic Recession and relative motion in mind, I sought a model in which the principle of Relativity and the Recession mechanism could emerge together—yet ideally formulated in a way that distinguishes them as much as possible [**].

The development of a physical model is independent of the specific form we attribute to it; however, for my approach, a geometric structure serves as a crucial link to our perception of reality. Moreover, geometry not only shapes our understanding but also influences abstract reasoning [***]; in this context, we arrive at the conclusion that the rejection of Absolute Space inherently leads to the formulation of Relativity, and vice versa [****]. The connection between geometry and purely mathematical formulation should only be severed when absolutely necessary.

But no geometry known to us was able to explain isotropy and homogeneity except by resorting to another spatial dimension. With this idea as a new starting point, I looked then for a model in which the metric of the Universe is not what appears to us, but it is only the result of our perception of a four-dimensional space.

In this geometry the Universe extends on the surface of an expanding 4d-hypersphere and the geodesic is reduced to an expanding arc of 2d-circle.

[*] - In the FLRW (Friedmann-Lemaître-Robertson-Walker) model, which under current assumptions has a curvature parameter k close to zero, the possibility of space-like intervals is predicted, where distant regions of the universe are receding at speeds greater than the speed of light.
[**] - The substitution of the FLRW metric for another is not straightforward, given that the FLRW metric represents an exact solution to Einstein's field equations in general relativity. FLRW follows a deductive approach, beginning with Einstein's field equations and deriving cosmological laws from them, by introducing the scale factor.
According to the FLRW, the light emitted by distant galaxies is redshifted, meaning it is stretched towards longer wavelengths due to the expansion of space. This cosmological redshift allows us to measure the distances to galaxies. Special relativity can be seen as a limiting case of the FLRW metric in regions of spacetime where curvature is small and gravitational effects are negligible.
According to the 4-Sphere instead, the expansion of space has no influence on the wavelength of light, furthermore the gravitational effect of the CMB (cosmic microwave background) on the propagation of light is negligible. This allows us to approximate the observed redshift of the most distant galaxies with the Doppler redshift, even for objects on the boundary of the observable universe.
[***] – Geometric shapes enhance the philosophical understanding of physical reality, allowing us to abstract and generalize concepts, thus aiding in the construction of scientific models. Furthermore, Karl Popper's concept of falsifiability demonstrates that scientific thought and philosophical thought cannot do without each other.
[****] – See page 59 in 4-SPHERE FEATURE AND SPECULATION. If we follow this purely philosophical consideration with our hypothesis for the metric tensor to be applied, we obtain Einstein's Relativity. The inductive approach through this combination of abstract thought and mathematical formulation is not, in itself, scientific proof, but it certainly enhances the model.