IV. Non-Euclidean geometries and determination of distances
We are used to the Euclidean geometry to such an extend that we even take into account the fact that already in the beginning we often assume subconsciously that space is Euclidean. As to even start thinking that it might be otherwise, first, we have to know something about the non-Euclidean geometry.
Euclid created his geometries on a flat sheet of paper and he assumed that the sheet can be stretched out in all directions to the infinity. It could be sufficient when the Earth was a little smaller and people immediately would reach the conclusion that there are no infinite plane that every plane is curved, and eventually it will form a sphere. Our perception of space would be probably different.
On the surface of a sphere the geometry looks different than on the plane. Straight lines are not infinite, they get curved and eventually they form circles. We can construct various triangles or circles but the sum of angles in a triangle will be always more than 180 degrees and the circumference of a circle will be always less than 2πr. The difference will be bigger, the bigger the area of a triangle or circle is. Yet when we limit ourselves to small space on the surface of a sphere then the differences regarding the Euclidean geometry will be minor. According to our experience, a fairly accurate map of a small area of the Earth surface can be prepared. In other words, one can assume that such fraction of the surface of a sphere is flat.
When the Earth is concerned, one should pay attention to the differences between the non-Euclidean geometry in the sphere and the sphere described as a curved surface plunged into a three-dimensional Euclidean space.
In the Euclidean space spherical coordinates instead of orthogonal coordinates can be introduced and we still will be in the Euclidean space, which is non-curved. In a simple way, spherical coordinates can be replaced with the orthogonal ones, and the other way. The spherical coordinates are used in astronomy where deployment of objects we describe with the use of angles and distances estimated on the basis of the brightness of objects when adopting few additional assumptions. When space is the Euclidean space, the angles in degrees can be easily converted into radians and, for example, the diameter of the galaxy can be determined, provided we hold the knowledge of its distance and we measure the angle under which we see it. When space is curved, however, the diameter of the galaxy may be different than the computations indicate. In case of a spherical space the real diameter of the galaxy must be smaller. Can we observe something which would lead us to the conclusion that space is curved? Yes. We would see distant areas stretched out and we would image, for example, that galaxies rotate too fast in relation to their masses. And this is exactly what we observe. However, instead of accepting geometrical interpretation as evidence for a curved space, the dark matter was conceived…
At least few words should be also added about a hyperbolic geometry. Contrary to the Euclidean geometry, in which exactly one parallel straight line passes through the point outside the straight line, in the hyperbolic geometry there is an infinite number of such straight lines.
Further differences are, for example, that the sum of angles in a triangle here is less than 180 degrees and the circumference of a circle is more than 2πr. A saddle is frequently an example of a hyperbolic surface.
Hyperbolic geometry may also appear on a sphere provided that the radius of a sphere will be an imaginary number. (An imaginary number is a complex number which is squared and it gives a negative real value). Complex numbers are needed to define some phenomena of our world and it seems that they can also serve to describe the geometry of the Universe. We will return to this topic in the next chapter.
Because we “got used to” the Euclidean geometry we imagine curved surfaces in a non-curved high-dimensional space. Thus, it can be easier to picture it and describe. However, it does not have to be that way. The curvature of surface or space is determined by mutual distances of their points. In the Euclidean space the principle that is applied is that the distance of every two points x and y of coordinates x=(x1,x2,…,xn), y=(y1,y2,…,yn) can be defined by the formula
\(d = \sqrt {(y_1-x_1)^2+ ... +(y_n-x_n)^2}\),
Otherwise the space is not the Euclidean space. There are various possibilities and some of them we will discuss later.
The formula looks fairly complicated, basically it means that in the Euclidean space the generalised Pythagorean theorem applies. If we limit ourselves to the three-dimensional space with the axes x, y, z and we combine one from the points with the beginning of coordinates, the distance formula will be the following
\(d = \sqrt {x^2+y^2+z^2}\),
where x, y i z mean respective coordinates of the distant point. Such form is simple and more transparent. If necessary, one can go back to the general form of the formula at any moment.
In the light of the above considerations, in our analysis not only real numbers but also complex numbers which are not used in everday life, will be needed, some information within this respect may also be useful.