What is promised is a debt, so we are busy today explaining what it means that the universe is flat. The task is not easy, perhaps because from school they do not teach us what alternatives there are and they present us with concepts taken from our sleeves that seem immovable truths, but that today we know that they are not.
First of all, what it does not mean: that the universe is flat has nothing to do with the number of dimensions. The adjective flat refers to the geometry of the universe, not the shape, so it also does not imply that the universe is shaped like a cookie, something that some hater commented on networks after our first article. The universe is said to be flat, but perhaps it would be best to say that its geometry is flat or euclidean, or that its curvature is zero. We therefore review our initial question: what does it mean to have plane geometry? What other geometries can the universe have (and we are not taught in school)?
The misinterpretations about the meaning of a “flat universe”, and even flat Earth, perhaps come from the fact that when we are little, at school, they do not tell us the whole truth. We could say that sometimes they lie to us. It has nothing to do with a conspiracy, they are more like white lies. It is impossible to present all our mathematical, physical or, what concerns us today, geometric knowledge to a child, so we start with the simplest concepts. This, in geometry, means that what is taught in schools is what we have been using in this field until the 19th century, when the mathematical theory about what is known as non-Euclidean geometries was developed. Those college concepts are pretty much the history of geometry, based on two-dimensional studies on a plane. Among these concepts we can mention, for example, that all right angles have 90º or that a single straight line can be drawn between two points on a plane.
We have already told you that it is a lie, said only like this, that the three angles of a triangle add up to 180º, as they teach us when we are very small. That is only true of certain geometries. But today we do not take triangles, but lines and right angles to describe the possible geometries of the universe (which are those that fulfill the Cosmological Principle), and we propose to answer questions. Imagining things in the four space-time dimensions of the universe is difficult, so we put examples in two dimensions, which are extensible to any more complicated space.
Imagine being in a city with a very square block distribution, all the streets intersect at 90º angles, all the blocks are square. We are not very used in Spain to a city of this type, the centers of historical cities are more chaotic, that they tell Toledo, but there are some (like Barcelona, for example, or the Salamanca district in Madrid). In a city like this, if we start to walk following straight lines through the streets (perpendicular), how many turns of 90º (the only thing allowed in that type of city) must we make at least to return to exactly the same position (and stay looking at the same direction)? The answer is four turns (with three turns we could get to the same place if we started from a corner, but we need one more turn to look in the same direction as at the beginning).
Now let’s forget about the imaginary city and say that we are on a surface. Is it always true that we have to make four 90º turns to get to the same place? The answer is … no, it is not true, that number four is not the only truth! At least not so for another type of geometry, which has positive curvature and is called spherical. Imagine that the surface we are on is that of a sphere, for example consider that the Earth is a perfect sphere and move through it. How many turns must they make to get to the same spot in straight lines?
Ahhhhh, but what is a straight line on a sphere? The “straight line”, a very Euclidean concept, becomes geodesic in non-Euclidean geometries. The geodesic in any geometry is a line that represents the shortest path between two points. The concept of geodesic It is essential to make a trip from Madrid to New York with the least possible fuel consumption (leaving aside drafts). We imagine that there are no flat-earth airplane pilots (at least, those who make transoceanic trips), because the geodesic on Earth is what is called a great circle, which is the circumference resulting from the intersection between a sphere and a plane that passes through the center of the sphere. A great circle is a curve on our typical flat maps, we usually see it on airplane screens. The shortest path between Madrid and New York, practically at the same latitude, is not a trajectory that always follows that latitude, but the only great circle that passes through the 2 cities, which implies traveling towards the northwest first, and then towards the southwest.
Let’s go back to the question of how to make right-angle turns in a sphere to get to the same place traveling in “straight lines” / geodesics. The answer can be three turns: we go through the equator, which is a great circle, we turn 90º to the north going up a meridian (another great circle) and reaching the pole, we turn 90 degrees going down another meridian, and when we reach the equator we turn another 90º, reaching the point where we left at some point. But we can also answer that a turn: we go through the equator (or another great circle), we travel half of it, we turn 90º and, by a meridian, first passing through the pole and continuing, we will arrive at the starting point (we will need one more turn to stay looking in the same direction). And the answer can also be zero! If we continue walking along the equator without turning, without separating from the same geodesic, we will be able to reach the same point.
The above described is not possible in a Euclidean space, we will never reach the starting point of our journey with less than four right-angle turns. Nor is it possible in another possible geometry of the universe, called hyperbolic, or with negative curvature (the opposite of spherical). The classic example of such a geometry, in 2 dimensions again, which is what is easy for us to visualize, is a horse saddle, a saddle in which all the points are analogous, anyone is surrounded by curves, ascents in one direction, downward at the perpendicular. In such a space, we can only walk through geodesics and return to the same place if we make at least 5 turns of 90º.
So what does it mean that the universe is flat? For it means, among other things, that an observer on Earth will never be able to see directly (with light coming through our eyes) what the Earth was like in the past. This would be possible in a spherical universe in which a ray of light, following a geodesic, could return to the same point in space (not in time), waiting long enough. In the case of the Earth and the universe we know, that would not be achieved before the Earth has disappeared, but in a small enough universe one could see its past! More properties of the flat universe in future installments, stay tuned and don’t forget to super vitaminize and mineralize yourself.
Pablo G. Pérez González He is a researcher at the Center for Astrobiology, dependent on the Higher Council for Scientific Research and the National Institute of Aerospace Technology (CAB / CSIC-INTA)
Cosmic Void It is a section in which our knowledge about the universe is presented in a qualitative and quantitative way. It is intended to explain the importance of understanding the cosmos not only from a scientific point of view but also from a philosophical, social and economic point of view. The name “cosmic vacuum” refers to the fact that the universe is and is, for the most part, empty, with less than 1 atom per cubic meter, despite the fact that in our environment, paradoxically, there are quintillion atoms per meter cubic, which invites us to reflect on our existence and the presence of life in the universe. The section is made up of Pablo G. Pérez González, researcher at the Center for Astrobiology; Patricia Sánchez Blázquez, Professor at the Complutense University of Madrid (UCM); and Eva Villaver, researcher at the Center for Astrobiology
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