Fig. 3.21 The time-like geodesics on dS2. In this figure we show a family of geodesics parameterized by the value of the angular

momentum in the range {−0.5, 0.5}. The angle α is

instead fixed to the value

α = − π3

Equation (3.9.20) provides the analytic form of all time-like geodesics in dS2. The two integration constants are (the angular momentum) and the angle α.

It is instructive to visualize also the time geodesics as 3D curves that lye on the

(3.9.21), combined with the parametric description of the surface (3.9.2) yield the parametric form of the time-geodesics in 3D space:

3.9.1.3 Space-Like Geodesics

For space-like geodesics we have:

As one sees the difference with respect to the equation describing time-like orbits resides only in two signs. By means of algebraic substitutions completely analogous to those used in the previous case we obtain the parameterization of the space-like geodesics as 3D curves. We find

X = x

The sign-changes with respect to the time-like case have significant consequences. For a given value of the angular momentum the range of the X coordinate and hence of the x parameter is limited by:

Out of this range coordinates becomes imaginary as it is evident from (3.9.27). In Fig. 3.22 we display the shape of a family of space-like geodesics.

3.9.2The Riemannian Example of the Lobachevskij-Poincaré Plane

The second example we present of geodesic calculation is that relative to the hyperbolic upper plane model of Lobachevskij geometry found by Poincaré.

As many of my readers already know, the question of whether non-Euclidian geometries did or did not exist was a central issue of mathematical and philosophical thought for almost two-thousand years. The crucial question was whether the Vth postulate of Euclid about parallel lines was independent from the previous ones or not. Many mathematicians tried to demonstrate the Vth postulate and

Fig. 3.22 The space-like geodesics on dS2. In this figure we show a family of geodesics parameterized by the value of the angular momentum fixed at = 2 and

the angle α in the range

{− π3 , π3 }

typically came to erroneous or tautological conclusions since, as now we know, the Vth postulate is indeed independent and distinguishes Euclidian from other equally self-consistent geometries. The first attempt of a proof dates back to Posidonius of Rhodes (135–51 B.C.) as early as the first century B.C. This encyclopedic scholar, acclaimed as one of the most erudite man of his epoch, tried to modify the definition of parallelism in order to prove the postulate, but came to inconclusive and contradictory statements. In the modern era the most interesting and deepest attempt at the proof of the postulate is that of the Italian Jesuit

Giovanni Girolamo Saccheri (1667–1733). In his book Euclides ab omni naevo vindicatus, Saccheri tried to demonstrate the postulate with a reductio ad absurdum. So doing he actually proved a series of theorems in non-Euclidian geometry whose implications seemed so unnatural and remote from sensorial experience that Saccheri considered them absurd and flattered himself with the presumption of having proved the Vth postulate. The first to discover a consistent model of non-Euclidian geometry was probably Gauss around 1828. However he refrained from publishing his result since he did not wish to hear the screams of Boeotians. With this name he referred to the German philosophers of the time who, following Kant, considered Euclidian Geometry an a priori truth of human thought. Less influenced by post-Kantian philosophy in the remote town of Kazan of whose University he was for many years the rector, the Russian mathematician Nicolai Ivanovich Lobachevskij (1793–1856) discovered and formulated a consistent axiomatic set up of non-Euclidian geometry where the Vth postulated did not hold true and where the sum of internal angles of a triangle was less than π . An explicit model of Lobachevskij geometry was first created by Eugenio Beltrami (1836– 1900) by means of lines drawn on the hyperbolic surface known as the pseudosphere and then analytically realized by Henri Poincaré (1854–1912) some years later.

In 1882 Poincaré defined the following two-dimensional Riemannian manifold (M , g), where M is the upper plane: