VIII. A few words about space
Most people probably imagine the concept of space as a three-dimensional Euclidean space. For a long time no one could think that it might be otherwise. Why? Because such a model constitutes good estimation of physical space for short macroscopic distances we usually encounter in life.
However, the fact that physical space may not be the Euclidean space is proved by the effect of the curvature of space created by mass and also the examinations of very short distances where quantum phenomena occur.
If not the Euclidean space, then what instead? In mathematics the definition of space can be provided more generally as a collection of elements with an additional structure. Some of them can significantly differ comparing to what we usually understand as the concept of space. When we want to create a model corresponding to the real space we need to have a possibility to determine distances in it. What we need is a metric space in which the metrics is defined, that is, a function which specifies distances between every pair of elements of such collection (provided additional conditions are satisfied). Elements of such space are called points. Such space does have to be neither finite, nor coherent. But whether the real space is coherent? Our perception of a coherent space is based on sight which misleads us. When we watch a film we see fluent motion not discrete shots. In objects we see we do not also distinguish swirling atoms. When we describe reality with the use of equations, then we create a model of a coherent space. However, when we use cellular automata, space can be modelled by a complete grid of cells. One should always pay attention to the fact that any model is only an estimation of reality and it does not contain all its properties and the properties it includes are not precisely defined. We also sometimes can interpret wrongly the obtained results.
We may, for example, create a model of space in which the distance between two points A and B of the following coordinates A=(x1,y1,z1), B=(x2,y2,z2) is defined by the formula d = | x2 - x1 |. One may see that the distance can be zero not only for identical points, it is also sufficient when the first coordinates are identical. Therefore, it is not about the metrics in its essential meaning, but it allows to call the defined distance pseudo-metrics and examine the properties of such space. We can establish the distance between various points and analyse not precise results because such defined distance can be short also for two points characterised by the very different the second and the third coordinates. It can be observed that in comparison to the Euclidean space, distances are sometimes significantly shortened. It may be believed that in such space distances are truly shorter and the mathematical model describes it correctly, or another interpretation may be employed. For example, the entire space can be imagined as built from planes parallel to the axis x and the estimated distance in such case does not mean the distance between two points but the distance between two planes in which points are located. When we estimated something, what needs to be always taken into the consideration, is actually the question what really we estimated.
Now let us analyse what properties the real space has. People used to imagine it as a static scene in which various objects move. The fact that space is not an absolutely independent scene was already discovered by Newton. He understood that it cannot be stated whether two events occurred at the same location when they happened at different moments. That is because no state at rest is established. When we, for example, throw a ball and catch it in the same place, is it really the same place? We believe that yes. However, if someone observed us from the Cosmos he would notice that the Earth moved and rotated and we are in a completely different location in space. A similar phenomenon is with the entire galaxy that moves and rotates and therefore there is no static point which wouldn’t move.
At this stage one could already come to the conclusion that time and space can be replaced as they are interchangeable. However, instead of the expression ‘time flows’ one should say ‘we flow in time’. Provided that I am sitting in one place and I think that I do not move in space, I can state that I move only in time. For the observer in the Cosmos I also move in space. In other words, what for one observer is moving in time, for the other may be perceived as moving in space.
If we assume that time and space are somehow interchangeable and we believe that our Universe exists only within finite time, than all indicates that space should be also finite.
Anyway we see that time and space somehow are correlated and should be examined simultaneously. Therefore, introduction to the concept of space-time appeared to be very useful.