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A new definition of entropy is introduced using a model that simulates an expanding space-time compatible with the fundamental principle of cosmology. The entropy is obtained by mean of a state function that measures the variation of the space-time normal curvature, from a highly compressed space to a lower compressed space. The defined entropy leads to work out a new understanding of the earliest conditions that last for a period estimated to 380 000 years after the Big Bang. It leads to understand via a short period of inflation the process that generates the uniform distribution of matter and energy at the surface of the last scattering. It involves gravitational singularities in a process of gradual decompression propitious to the incubation of matter recombination, and it allows to trace back gravity origin., Comment: 26 Pages, 9 Figures

The use of an infinity of fluctuating paths of least time that are compatible with the quantum mechanics indeterminacy provides a new interpretation in geometrical optic of the interference pattern of Young's double slit experiment, which suggests that the wave behavior of matter and radiation is dictated by the space-time geodesics. Moreover, the association of a wave function to each path of least time as a probability amplitude together with an uncertainty for momentum and position allows to derive the Schrodinger's equation starting from the geodesic's characteristics. A new insight is obtained regarding the Van Der Waals torque as well as Casimir attraction/repulsion mechanism., Comment: 24 Pages, 17 figure, 1 table

Using the essence of Feynman's path integral and the space-time geodesics, an infinity of differentiable paths that follow the geometry of a continuous geodesic are constructed, and a wave function is associated to each path as a probability amplitude identical to the Feynman's probability amplitude for each path. We prove that each probability amplitude obeys to the Schr\"odinger's equation for a non relativistic physical system moving in a time independent potential, starting from the Jacobi-Hamilton equation., Comment: 12 pages

In this paper, a geometrical interpretation of light diffraction is given using an infinity of fluctuating geodesics that represent paths of least time in an homogeneous space. Without using the wave theory, we provide a geometrical explanation of the deviation of light's overall direction from rectilinear when light encounters edges, apertures and screens, which reconciles light particle-like nature with the interference phenomena., Comment: 17 Pages, 16 Figures, registered in INPI with the numbers 511206/511149, 2014

In this paper we construct a discrete simulation of an expanding homogeneous and isotropic space-time that expands via expansion of its basic elements to figure out properties and characteristics of such a space-time and derive conclusions. We prove that in such an expanding space-time, the geodesics are curved and more precisely, they fluctuate on the boundaries of the expanding basic elements. The non existence of privileged expansion direction leads to the existence of an infinity of fluctuating geodesics between any two locations in this space-time, that provides a prediction of polarization in geometric optics, and a prediction of an earlier acceleration of the expansion as for the cosmic inflation model. This simulation is a case study and an example of space-time with variable topology using the principle of variation of topology via a transformation that creates holes., Comment: 24 pages, 7 figures in International Journal of Geometric Methods in Modern Physics, Vol.13 (2016)

In this paper the uncertainty principle is found via characteristics of continuous and nowhere differentiable functions. We prove that any physical system that has a continuous and nowhere differentiable position function is subject to an uncertainty in the simultaneous determination of values of its physical properties. The uncertainty in the simultaneous knowledge of the position deviation and the average rate of change of this deviation is found to be governed by a relation equivalent to the one discovered by Heisenberg in 1925. Conversely, we prove that any physical system with a continuous position function that is subject to an uncertainty relation must have a nowhere differentiable position function, which makes the set of continuous and nowhere differentiable functions a candidate for the quantum world., Comment: 15 pages, 1 figure, last version accepted for publication in Reports on Mathematical physics, July 2013

The incorporation of an adequate discrete expansion to the formalism of the special relativity that does not allow gravitational acceleration unravels unexplored phenomena. This extension takes into account consequences of a small variation of the limiting velocity for a large scale of time together with the space-time expansion. It determines a natural boundary between physical systems with high and low kinetic energy, provides a new insight for Earth revolution since its formation, indicates a new factor that maintains the earlier remnant heat inside Earth, indicates that the self-sustaining nuclear fission chain reaction that generates Earth major heat is increasing together with the space-time expansion, provides a new lead for the problem of galaxies missing mass, shows how time is entangled with space and expansion and leads to a new interpretation of its characteristics., Comment: 32 pages

We have built a new kind of manifolds which leads to an alternative new geometrical space. The study of the nowhere differentiable functions via a family of mean functions leads to a new characterization of this category of functions. A fluctuant manifold has been built with an appearance of a new structure on it at every scale, and we embedded into it an internal structure to transform its fluctuant geometry to a new fixed geometry. This approach leads us to what we call fractal manifold. The elements of this kind of manifold appear locally as tiny double strings, with an appearance of new structure at every step of approximation. We have obtained a variable dimensional space which is locally neither a continuum nor a discrete, but a mixture of both. Space acquires a variable geometry, it becomes explicitly dependent on the approximation process, and the geometry on it assumed to be characterized not only by curvature, but also by the appearance of new structure at every step of approximation., Comment: 30 page, A new mathematical object that describe a variable geometry

In this paper, we present a new cosmological model using fractal manifold. We prove that a space defined by this kind of manifold is an expanding space. This model provides us with consistent arguments pertaining to the relationship between variation of geometry and movement of matter. This study leads to the existence of new fundamental principles. A clear picture is then portrayed about the expansion of the universe represented by fractal manifold., Comment: 31 pages, no major change exept some correction and clarification added

In this paper, we prove the existence of two degrees of freedom that govern the movement of light in an expanding universe. The use of the fractal manifold model leads to reciprocal causality between variation of geometry and gravity, which both play a complementary role in the universe architecture. This study unravels new facts about the distribution of matter in the universe, and provides a new interpretation of Dark Matter and Dark Energy., Comment: This is an old version not published. The updated version of it was published in International Journal of Geometric Methods in Modern Physics and corresponds to arXiv:1302.1490, Infinity of Geodesics in an Homogeneous and Isotropic Expanding Space-Time