On the nonexistence of a Lobachevsky geometry model of an isotropic and homogeneous universe
Introduction
Can mathematics contribute to never ending discussions whether the geometry of our universe is globally elliptic, hyperbolic, or flat? What is the global structure of the universe? Assuming the validity of the Einstein cosmological principle, we give answers to these questions using several fundamental tools of differential geometry. In particular, we will demonstrate that the assumed homogeneity and isotropy of the universe necessarily leads to models which allow only hypersurfaces with a constant curvature and non-Lobachevsky geometry.
Recall that according to the Einstein cosmological principle, the universe is homogeneous and isotropic in the large-scale average [21], [22]. Astronomical observations indeed support that it is really homogeneous (with a uniform density) and isotropic with a resolution of about 108 light years and larger (see [19]). The universe thus seems to be the same at each point and there are no preferred directions, i.e. an observer is not able to distinguish one of his/her space directions from the other by any local physical measurements. Therefore, we will model this by a manifold having the same curvature at any point and in any direction (cf., e.g. [1], [13], [25]). The manifold will be imbedded into Euclidean space (or more generally, into a normed linear vector space isometric with this space) just to be able to define and investigate its curvature.
Note that locally the curvature of the universe can change rapidly due to gravitational influence of planets, stars, galaxies (cf. [15]). It even has singularities inside black holes (cf. [26]), in the big bang or in the big crunch (if it ever happens). In 1911, Einstein predicted that the trajectory of light rays is deflected by the sun (which was confirmed by astronomers during the total eclipse in 1919). This is because each light photon travels along a geodetic (the shortest trajectory between two points) in spacetime, which is curved in the neighbourhood of the sun. Around 1913, Einstein and Marcel Grossmann realized that the metric tensor describing the geometry of spacetime depends on the amount of gravitating matter in the region in question. In 1915, Einstein described a system of 10 nonlinear second order partial differential equations of hyperbolic type to model spacetime locally. The unknown gravitational potentials are entries of the 4×4 metric tensor, which is symmetric. The system is usually referred to as Einstein’s field equations (see, e.g. [5], [10], [14], [20], [28]) and describes the mutual relationship between matter and curvature.
In Fig. 1, we observe two examples of bending (deflection) of light in a close neighbourhood of stars. The three trajectories in (Fig. 1a) form a curved triangle. Notice that the sum of its angles satisfies the inequality which reminds of a famous assertion from Riemannian geometry. On the other hand, in Fig. 1b we see two stars of equal masses and three trajectories forming another curved triangle with α+β+γ<π. This case is reminiscent of the Lobachevsky geometry.
The above examples show that the universe locally may have different kinds of geometries with various curvatures. However, to find a “global curvature” of the universe, we have to consider very large scales, on which all local curvatures are averaged. This is like earth’s surface, whose curvature locally changes very much (due to mountains, valleys, saddle points, etc.), but whose global curvature is positive, and almost constant. We will, therefore, assume the validity of the cosmological principle, which has been verified not only in a close neighbourhood of our galaxy but also in very distant parts of the universe—such as the famous Hubble Deep Field or the Hubble Deep Field-South which are at least 1010 light years far. These fields have similar densities and isotropic distribution of galaxies even though they lie in almost opposite parts of the sky. There is also a remarkable evidence in the isotropy of the cosmic microwave background radiation [22] (see also [19, p. 760]). Recently, a great effort has been devoted to understanding γ-ray bursts, which seem to be distributed almost uniformly in the universe as well see, e.g. [11], [18]. The above arguments support the validity of the cosmological principle. Consequently, the assumptions on the homogeneity and isotropy are applied to the whole universe.
From now on we shall carefully distinguish between the two terms: universe and hypersurface (manifold). The first term will stand for the real universe, where we live, whereas the second term will denote an abstract mathematical object (defined in Section 2) which serves as a model for the universe. We shall see that mathematical models based on astronomical observations can help us in better understanding the global structure of the universe. Namely, in 3 Characterization of hypersurfaces whose all points are umbilic, 4 Conclusions for the global structure of the universe we show that any hypersurface (an “idealized” model of the universe for a fixed time) which is in accordance with the cosmological principle is either a perfectly flat hyperplane or the three-sphere in . Hence, the hyperbolic geometries are excluded. Models with variable time are discussed as well.
Section snippets
Preliminaries
According to [12], a topological n-dimensional manifold S is a metrizable space which is locally homeomorphic to the n-dimensional Euclidean space for a given n∈{1,2,3,…}. That is to say, every point of S has an open neighbourhood which is homeomorphic to an open set in . Therefore, a manifold looks almost the same at all points because it is locally homeomorphic to a Euclidean space.
There exist n-dimensional manifolds which cannot be imbedded into . For instance, the famous
Characterization of hypersurfaces whose all points are umbilic
Many books on non-Euclidean geometries present the following example to show that there are hypersurfaces with a constant Gaussian negative curvature to evoke a possibility of a hyperbolic universe. Take the parametric representation of tractrix (which is the evolvent of the catenary) in the (x1,x3) plane of , and consider the surface (called pseudosphere or tractoid [7], [23], [31]) obtained by
Conclusions for the global structure of the universe
It is not known whether every simply connected compact three-dimensional hypersurface in is homeomorphic to the three-sphere (this problem is called Poincaré’s conjecture). However, according to the Einstein’s cosmological principle, the curvature of the universe at a given constant time is the same at all points and all directions, which substantionally simplifies the corresponding model of the universe. Therefore, we take into account the following two assumptions:
- (A1)
S is a
Acknowledgments
The first author was supported by Grant no. 201/01/1200 of the Grant Agency of the Czech Republic. The authors wish to thank Jan H. Brandts, Vojtěch Pravda, and Alena Pravdová for fruitful discussions.
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