arXiv subjects

Vincenzo Catoni

Publications and source records attributed to Vincenzo Catoni.

3 recordsLinked to original sources

arXiv2005

Space-time trigonometry and formalization of the "Twin Paradox" for uniform and accelerated motions

The formal structure of the early Einstein's Special Relativity follows the axiomatic deductive method of Euclidean geometry. In this paper we show the deep-rooted relation between Euclidean and space-time geometries that are both linked to a two-dimensional number system: the complex and hyperbolic numbers, respectively. By studying the properties of these numbers together, pseudo-Euclidean trigonometry has been formalized with an axiomatic deductive method and this allows us to give a complete quantitative formalization of the twin paradox in a familiar "Euclidean" way for uniform motions as well as for accelerated ones.

physics.class-ph↗

arXiv2005

Hyperbolic trigonometry in two-dimensional space-time geometry

By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.

math-ph↗

arXiv2005

Lorentz surfaces with constant curvature and their physical interpretation

In recent years it has been recognized that the hyperbolic numbers (an extension of complex numbers, defined as z=x+h*y with h*h=1 and x,y real numbers) can be associated to space-time geometry as stated by the Lorentz transformations of special relativity. In this paper we show that as the complex numbers had allowed the most complete and conclusive mathematical formalization of the constant curvature surfaces in the Euclidean space, in the same way the hyperbolic numbers allow a representation of constant curvature surfaces with non-definite line elements (Lorentz surfaces). The results are obtained just as a consequence of the space-time symmetry stated by the Lorentz group, but, from a physical point of view, they give the right link between fields and curvature as postulated by general relativity. This mathematical formalization can open new ways for application in the studies of field theories.

math-ph↗