Roberto Cannata

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.

Geometry of Minkowski space-time

Introduction.- Hyperbolic Numbers.- Geometrical Representation of Hyperbolic Numbers.- Trigonometry in the Hyperbolic (Minkowski) Plane.- Equilateral Hyperbolas and Triangles in the Hyperbolic Plane.- The Motions in Minkowski Space-Time (Twin Paradox).- Some Final Considerations.

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 + hy; h 2 = 1 x, y ∈ R, h > not ∈ R}) 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.

Some Final Considerations

The modern view of geometry, that we recall in the Appendix, has allowed us to formalize the “geometry of space–time” so that we can work in this plane as we usually do for Euclidean plane geometry. Otherwise the obtained mathematical system, following Euclidean geometry, combine the logical vision with the intuitive vision allowing us to agree with the following Einstein’s thought.

Equilateral Hyperbolas and Triangles in the Hyperbolic Plane

The equilateral hyperbolas, represented in the Minkowski space-time, hold the same properties of circles in Euclidean plane and satisfy similar theorems. At the same time equivalent relations to the ones in Euclidean plane between circles and triangles are obtained in hyperbolic plane between equilateral hyperbolas and triangles.

Trigonometry in the Hyperbolic (Minkowski) Plane

The correspondence between properties of complex numbers and Euclidean geometry allows to obtain an algebraic formalization of Euclidean geometry. Thanks to the equivalent properties between complex and hyperbolic numbers, the geometry of Minkowski space-time can be formalized in a similar algebraic way. Moreover, introducing two invariant quantities, the complete formalization of space-time trigonometry is obtained.

The Motions in Minkowski Space–Time (Twin Paradox)

All the curves in space-time plane, can be considered as a motion and the lengths of the time-like lines give the proper time. Therefore we can compare in a geometrical way, on different curves, the differences between proper times. Proper times are obtained by means of elementary mathematics for uniform, uniformly accelerated motions and their compositions; by means of differential calculus for the arbitrary time-like curves.

Geometrical Representation of Hyperbolic Numbers

A relevant property of Euclidean geometry is the Pythagorean distance between two points. From this definition the properties of analytical geometry follow. In a similar way the analytical geometry in Minkowski plane is introduced, starting from the invariant quantities of Special Relativity.