Non-euclidean geometry
Any formal system of geometry whose postulates and propositions differ in any matter from those established by Euclid in his treatise is called non-Euclidean geometry or non-Euclidean >Elements. There is not one system of non-Euclidean geometry, but many, although if the discussion is restricted to homogeneous spaces, in which the curvature of space is the same at each point, in which the points of space are indistinguishable, three can be distinguished. geometry formulations:
- La Euclidian geometry satisfies the five postulates of Euclides and has zero curvature (i.e. it is assumed in a flat space so the sum of the three inner angles of a triangle always gives 180°).
- La hyperbolic geometry It satisfies only the first four postulates of Euclides and has negative curvature (in this geometry, for example, the sum of the three inner angles of a triangle is less than 180°).
- La elliptical geometry It satisfies only the first four postulates of Euclides and has positive curvature (in this geometry, for example, the sum of the three inner angles of a triangle is greater than 180°).
All these are particular cases of Riemannian geometries, in which the curvature is constant, if we admit the possibility that the intrinsic curvature of the geometry varies from one point to another, we have a case of general Riemannian geometry, as it happens in the theory of general relativity where gravity causes an inhomogeneous curvature in space-time, the curvature being greater near mass concentrations, which is perceived as an attractive gravitational field.
History
The first example of non-Euclidean geometry was hyperbolic geometry, initially theorized by Immanuel Kant,[citation needed] later formalized (at the beginning of the 19th century) and independently by various authors, such as Carl Friedrich Gauss, Nikolai Lobachevski, János Bolyai, Eugenio Beltrami and Ferdinand Schweickard.
The developments of non-Euclidean geometries were gestated at the beginning with the aim of building explicit models in which Euclid's fifth postulate was not fulfilled.
Euclidean geometry had been developed by the Greeks and exposed by Euclid in the work The elements.
In his first published work, Thoughts on the true estimation of living forces (Gedanken von der wahren Schätzung der lebendigen Kräfte und Beurteilung der Beweise derer sich Herr von Leibniz und anderer Mechaniker in dieser Streitsache bedient haben, of 1746), Immanuel Kant considers spaces of more than three dimensions and affirms:
A science of all these possible kinds of space would certainly be the highest company that a finite understanding could undertake in the field of Geometry... If there are possible extensions with other dimensions, it is also very likely that God has brought them into existence, because their works have all the magnitude and variety they are capable of.
Those possible geometries that Kant envisions are what are today called Euclidean geometries of dimension greater than 3.
On the other hand, since ancient times it was considered that the fifth postulate of Euclid's book was not as evident as the other four because, when affirming that certain straight lines (the parallel ones) will not intersect when prolonging them indefinitely, he speaks of a somewhat abstract mental construction. That is why for many centuries unsuccessful attempts were made to prove it from the other four. At the beginning of the XIX century, an attempt was made to prove it by reductio ad absurdum, assuming that it is false and trying to obtain a contradiction. However, far from reaching an absurdity, it was found that there were coherent geometries different from the Euclidean. Thus, the first non-Euclidean geometry had been discovered (specifically, the first example that was achieved was a geometry called hyperbolic).
Constant curvature geometries
Hyperbolic Geometry
At the beginning of the XIX century, and independently, Gauss (1777-1855), Lobachevsky (1792-1856), János Bolyai and Ferdinand Schweickard managed to build hyperbolic geometry, from the attempt to deny Euclid's fifth postulate and try to obtain a contradiction. Instead of obtaining a contradiction, what they obtained was a curious geometry in which the three angles of a triangle added up to less than 180º sexagesimals (in Euclidean geometry the angles of any triangle always added up to exactly 180º).
The naturalness of this geometry was confirmed at the end of the century, when Beltrami showed that hyperbolic geometry coincides with the intrinsic geometry of a certain surface and Klein gave the projective interpretation of hyperbolic geometry. Both results prove that it is as consistent as Euclidean geometry (ie, if hyperbolic geometry leads to any contradiction, then Euclidean geometry does too).
Some claim that Gauss was the first to consider the possibility that the geometry of the Universe was not Euclidean. Knowing that in hyperbolic geometry the sum of the angles of any triangle is less than two right angles, he is said to have climbed to the top of three mountains with a theodolite, although the precision of his instruments was not enough to decide the question with such an experiment. However, others affirm that when he wrote that he was trying to correct the effects of possible curvatures, he was referring to correcting the effect of the terrestrial curvature in the cartographic studies that he was carrying out.
Elliptical geometry
Elliptical geometry is the second type of homogeneous non-Euclidean geometry, that is, where any point in space is indistinguishable from any other. A Riemann manifold of constant positive curvature is an example of elliptical geometry. A classical model of n-dimensional elliptical geometry is the n-sphere.
In elliptical geometry, geodesic lines have a role similar to straight lines in Euclidean geometry, with some important differences. Although the minimum possible distance between two points is given by a geodesic line, which are also lines of minimum curvature, Euclid's fifth postulate is not valid for elliptical geometry, since given a "straight line" From this geometry (that is, a geodesic line) and a point not contained in it, no geodesic can be drawn that does not intersect the first.
Euclidean geometry
Euclidean geometry is clearly an intermediate borderline case between elliptic geometry and hyperbolic geometry. In fact, Euclidean geometry is a geometry of zero curvature. It can be shown that any geometric space or Riemann manifold whose curvature is null is locally isometric to Euclidean space and therefore is a Euclidean space or identical to a portion of it.
Mathematical aspects
The spaces of constant curvature the Riemann curvature tensor is given in components by the following expression:
Rijkl=C(gilgjk− − gikgjl){displaystyle R_{ijkl}=C(g_{il}g_{jk}-g_{ik}g_{jl}),}
where gij{displaystyle g_{ij}} is the metric tensor expressed in any curviline coordinates. The Ricci tensor Rij{displaystyle R_{ij}} and the curvature S{displaystyle S} are respectively proportional to the metric tensor and the curvature:
Rij=(n− − 1)Cgij,S=n(n− − 1)C{displaystyle R_{ij}=(n-1)Cg_{ij},qquad S=n(n-1)}C
and where n{displaystyle n} is the dimension of space.
Another interesting aspect is that both in hyperbolic geometry, and in homogeneous elliptical geometry the whole space isometry group is a dimension Lie group n(n+1)/2{displaystyle n(n+1)/2}which coincides with the dimension of the isometry group of an Euclide dimension space n (although the three groups are different).
Non-constant curvature geometries
General Riemannian Geometry
At the suggestion of Gauss, Riemann's dissertation dealt with the hypothesis of Geometry. In his thesis, Riemann considers possible geometries that are infinitesimally (ie, in very small regions) Euclidean, the study of which is known today as Riemannian geometries. These geometries are generally non-homogeneous: some of the properties of the space may differ from one point to another, in particular the value of the curvature.
To study these geometries, Riemann introduced the curvature tensor formalism and showed that Euclidean geometry, hyperbolic geometry, and elliptic geometry are particular cases of Riemannian geometries, characterized by constant values of the curvature tensor. In a general Riemannin geometry, the curvature tensor will have variable values along different points of the geometry.
This makes the geometry not homogeneous, and allows us to distinguish some points from others. This is relevant in the theory of general relativity, since in principle it is possible to carry out experiments to measure distances and angles that make it possible to distinguish some points in space from others, as specified by numerous mental experiments imagined by Einstein and others in which a An experimenter locked in a box can perform experiments to decide the nature of the space-time around him.
Finally an interesting aspect of the riemannian geometry is that if the curvature is not constant then the isometry group of space has a strictly smaller dimension than n(n+1)/2{displaystyle n(n+1)/2} being n{displaystyle n} the dimension of space. Specifically according to the general relativity a space-time with a very irregular distribution of the matter could have a group of trivial isometric dimension 0.
Geometry of space-time and theory of relativity
Based on Riemann's ideas and results, around 1920 Einstein addressed the question of the geometric structure of the universe in his theory of general relativity. In it, he shows how the geometry of space-time has curvature, which is precisely what is observed as a gravitational field, and how, under the action of gravity, bodies follow the straightest possible lines within said geometry, lines that are They are called geodesics.
In addition, Einstein's equation states that for each observer, the mean curvature of space coincides, except for a constant factor, with the observed density, thus fulfilling Gauss's fantastic vision: the geometry unraveled by the Greeks is the infinitesimal structure of space; when generalizing said geometric structure, it has curvature.