Pregeometry

We revisit the cosmological scenarios of Zatrikean pregeometry, focusing on the results of a decreasing maximum attainable speed c and using the assumption of c being equal to the expansion rate of the universe. This preliminary qualitative study leads to a universe that is almost flat, but has small angular inhomogenities. It also leads to high-z objects that are further away than expected in the standard model, removing the need for decceleration. Finally, the variable c solves the horizon problem.

We apply zatrikean pregeometry to polytropic stars, finding the corresponding zatrikean Lane-Emden equations, solving them numerically and presenting the results on the boundary conditions and the interior structure of the zatrikean polytropes.

We attempt a connection between thermodynamics and zatrikean pregeometry, i.e., a chess-like pregeometry. In zatrikean pregeometry space is represented by the abacus, a discrete chessboard-like structure consisting of a sufficiently large number of plaquettes called geobits. The particles move on the abacus from one geobit to the next following certain rules that resemble the game of chess. The sets of rules imposed on the motions of particles on the abacus are called premetrics. There is a variety of paths (called subabaces) leading from one geobit to another, and there is a class consisting of subabaces with the minimum number of geobits. These are called alyssoids (respectively, class of alyssoids) for the particular premetric, while those alyssoids with minimum length are called geodesics (respectively, class of geodesics) for the particular premetric. The so-called zatrikean geodesic was originally defined in G93 (Section 2) as the geodesic most closely following the line segment joining the two geobits. It is also called algorithmic geodesic since it is drawn with the assistance of four simple algorithms. This is a rectifiable curve; and a connection between rectifiable curves and thermodynamics is already available (DuPain, Kamae and Mendes-France 1986). Consequently, the so-called thermodynamic geodesic is defined as the particular member of the class of geodesics with maximum entropy. Since it does not necessarily correspond to the algorithmic geodesic, a new algorithm is devised that draws the geodesic with maximum entropy. Furthermore, the probability of each member of the class of geodesics can be determined as the difference of its entropy from the entropy of the thermodynamic geodesic.