Is spacetime a real topological space

What is a room?

On the one hand, mathematical knowledge of spaces is enormous - on the other hand, mathematicians know almost nothing. Matthias Kreck from the Mathematical Institute of the University of Heidelberg looks at the development of the concept of space over the centuries. He explains what a room is, how you can recognize a room - and conveys in an understandable and impressive way that mathematics, which is all too often misunderstood, is an extremely exciting science.

The question "What is a room?" is probably as old as science itself. At least among the ancient Greeks, space is an issue with both mathematicians and philosophers. My specialty in mathematics is topology, i.e. the study of topos. We encounter the word "topos" for the first time in Aeschylus in the meaning of "place, area, region, district". A meaningful translation of topology, which in the narrower sense is a young science, is "theory of space".

 

At the time of the Greeks, the question would have been phrased a little differently, namely: What is space? When one speaks of space, one emphasizes that space is something that exists, is fixed, and certain independently of the human being. I quote from the book of the famous mathematician Herman Weyl "Raum, Zeit, Materie" from 1918: "The Greeks made space the subject of a science of the highest clarity and certainty. In ancient culture it was the idea of ​​pure science unfolded, geometry became one of the most powerful manifestations of the principle of the sovereignty of the spirit that animates that culture. " The clarity and certainty of the science of space is most impressively demonstrated by Euclidean geometry, in which a very simple and aesthetically satisfactory system of axioms is described, which defines a clear mathematical picture of a straight line, a plane, a space. For centuries, all sciences have been oriented towards geometry. It could - as Weyl writes - "be set up as the highest ideal of all sciences," to be practiced more geometrico. "With the end of the Middle Ages, skepticism grew and finally also employed the naive realism of things that existed independently of the observer Galileo, for example, emphasizes the subjectivity of sensory qualities. In the field of philosophy, it was Immanuel Kant who took a radical step with the insight that not only sensual qualities (for example colors) but also "space is only one." Form of our view "is (Weyl).

With Kant, space is transformed from a "thing-in-itself" that exists outside of us into a visual space. Of course, that doesn't mean that the space is completely arbitrary and that we can imagine it however we want. Our respective perception, our respective knowledge of space is permeated by an apparent a priori, which is reflected within mathematics in the great persuasiveness of geometrical sentences. But, looking back on the tremendous change in perception and knowledge of space, we must say our respective perception and knowledge. And when I speak of space, I should also assign the attribute "respective" to it and mean, when I speak briefly about space, the respective space. In this way space itself becomes a historical thing, something that changes with the way it is viewed over time.

On the basis of these considerations, the title of this article should perhaps be more correct: "What is a room today?", With the "today" aiming at the changing view and the "a" at the diversity of the view. Both terms underline the provisional nature of all spatial concepts.

The modern concept of space

In the following I present the mathematical concept of space, which came from Bernhard Riemann, and which was, so to speak, firmly connected with reality through the theory of relativity. Riemann's famous habilitation lecture from 1854 is entitled "On the hypotheses on which geometry is based". This is bold, because until shortly before Riemann, the building of the Euclidean geometry of wisdom seemed to be the last word and was shaken by the discovery of non-Euclidean geometries by János Bolyai, Carl Friedrich Gauß and Nikolaj Ivanovich Lobachefsky, but not really over in spirit thrown the heap. That Riemann speaks of space could be an indication that Riemann still adheres to the idea of ​​an absolute space, but the epistemological revolution indicated above is reflected in the word "hypotheses".

It is interesting to note that this bold step taken in mathematics was taken around 60 years later by Albert Einstein in physics, with Riemann's concept of space playing a central role in general relativity. Readers who have dealt with the theory of relativity will find astonishing parallels in the following presentations.

I proceed according to the following basic principle in deriving the concept of space. I focus on a partial aspect of the visual space which, according to our current knowledge, particularly corresponds to our view. Then I formulate two plausible-sounding postulates and then discuss what can be said about the space defined and "viewed" in this way.

The partial aspect is the local view of the room: Wherever I am in the room, I overlook a part of the room, which I can describe mathematically as follows: I take three folding rules, each of which are scaled from minus 1 to 1, and visualize them pairwise vertical ax cross. My view tells me that the open cube spanned by these three axes is mathematically described by three numbers (x1; x2; x3) between minus 1 and 1. Because a point in the cube spanned by the three axes is completely determined by the three projections along the planes spanned by two axes onto the third axis, which is perpendicular to it. The three numbers (x1; x2; x3) are called location coordinates.

It is very important for the following that the scale of my three folding rules does not play a role in this consideration, and also that they are perpendicular to one another is irrelevant. There is no reason to believe that I can even compare the scale or angles at different points in the room. It is only important that the area that I can survey contains a sub-area that I describe mathematically using an open cube. Now I come to the first postulate, what I would like to call the democracy or equality principle or, more mathematically speaking, the homogeneity postulate of space: The above local view should apply to every point in space. For example, there should be no place where the room looks like a plane, i.e. where it can be described by only two parameters. Although everyone here will confirm that this result is in harmony with his experience, it remains a postulate that is not covered by our elementary view. But it is a natural postulate.

Finally, I come to the second postulate, which I would like to call the smoothness requirement. Since this requirement is not very easy to specify, I would like to illustrate it with the clearer question: "What is a surface?". Think of the attempt to develop a mathematical concept of the surface of a solid body. For example, think of the earth's surface. In analogy to the above concept of space, it will be required that every point of a surface can be described mathematically by an axbox made up of two axes. Again one demands the principle of equality. In order to "get a grip" on an area, you need a group of axboxes, so that every point on the area is covered by at least one axbox.

We are familiar with something like this from school, namely from the world atlas. This contains the complete information of the earth's surface. Of course, certain places on earth appear several times in the atlas, for example Heidelberg on the map of Europe and on the map of Germany.

Now I come to the smoothness requirement. I drive a car through the area and look at the corresponding path on the Atlas. Then the smoothness requirement of my atlas is that the curve on the atlas is smooth, which mathematically means that it is differentiable, and clearly that one can speak of tangents to the path at any point. The decisive factor is that this smoothness can be seen in every map in the Atlas, i.e. just as in the map of Europe as it is in the map of Germany. So smoothness is a requirement of my atlas. An atlas is smooth if a curve that I am looking at on one map is smooth if and only if the corresponding curve is smooth again in all other maps where it appears.

This smoothness requirement, which we consider sensible for surfaces, can be formulated analogously for space, whereby the term “atlas” is reintroduced and for it, what means smooth. A precise definition would lead too far here. It should be noted, however, that only the smoothness postulate allows real processes to be conveniently mathematized, for example to set up the laws of motion.

We summarize: In Riemann's concept of space, a space is determined by the requirement that it can be described by three coordinates at every point, whereby a system of coordinates is selected, which allows us to speak of smoothness, i.e. in particular smooth curves in the room. In mathematics, such spaces are called three-dimensional manifolds. In this view of space, we started out from physics. This - like life in general - only becomes interesting by adding time and looking at matter. According to today's view, space and time themselves form a space that we imagine as a four-dimensional manifold. When the space-time manifold is filled with matter, forces come into play, which we imagine as fields. To address this would go too far here.

What we know about spaces and what we want to know

On the one hand, our mathematical knowledge of spaces is huge and, on the other hand, we know almost nothing. This sentence will surprise one or the other reader. I am often asked if you don't already know everything in mathematics. It's all about applying mathematics, and the mathematician's main work is now being done by the computer.

I want to underpin the statement that we know almost nothing with one of the most famous open mathematical questions, the Poincaré conjecture, which was established over 100 years ago. This assumption popped up here and there in the last year in the press, which otherwise hardly ever reports on mathematics. The reason, of course, is not that journalists took a serious interest in mathematics. The reason is that a wealthy American businessman pledged a million dollars to solve this conjecture. Something like that is worth reporting in this day and age.

We tried to explain what a room is. Now the follow-up question naturally follows: "How do you recognize a room?". The background to this question is that Riemann's concept of space, in contrast to Euclidean geometry, does not characterize a clear space. It defines a genus, so to speak, just as biologists have defined numerous genera. And just as biologists have gradually discovered new species within a genus, so mathematicians have discovered an enormous variety of mathematical spaces. And that raises the same question as in biology, namely how one can recognize from the simplest possible features what kind of room it is.

In 1895 Poincaré set up a feature by which one should recognize one of the most important spaces, namely the three-dimensional sphere. It takes a considerable amount of experience to imagine this fundamental space, and it is even more difficult to formulate Poincaré's characteristic, which I do not use at all. I just want to say something about the analogy for surfaces. There were times when the surface of the earth was imagined as an infinitely extended plane, so to speak as an area where you had to be careful not to go too far outwards, otherwise you would run the risk of falling down. This notion is outdated, we now know that the surface of the earth has the shape of a football, is mathematically a two-dimensional sphere.

In the case of physical space (without considering time as the fourth dimension) one can ask oneself analogously whether the naive idea that it is an infinitely extended cube (which one should also be careful not to come to the "end") or one Mathematically differently described space acts. The most obvious alternative, in analogy to the earth's surface, is the three-dimensional sphere.

The Poincaré conjecture now indicates features of how one can recognize the three-dimensional sphere. Despite tremendous efforts, it has not yet been possible to resolve this fundamental assumption. But there is a remarkable result by the American mathematician Richard Hamilton from 1982, which proves the Poincaré conjecture with an additional assumption. Even with this result, it is not possible to give a precise formulation within this framework. But at least one keyword should be mentioned under which one can imagine something.

Hamilton demands a curvature condition for the space, namely it should be positively curved in a certain sense. Curvature can best be imagined in the case of surfaces, and positively curved means that one looks at the tangential plane at every point and checks whether the surface in the vicinity of the point lies entirely on one side of the tangential plane. For example, the earth's surface is positively curved,

while this does not apply to the following area.

Everyone has heard that the term curvature plays a fundamental role in the theory of relativity. It is noteworthy that the concept of curvature is also of great importance when trying to recognize a space.

Pure and applied mathematics

While the great mathematicians of the past - one thinks of the probably greatest mathematician of all time, Carl Friedrich Gauß, who worked in all areas of mathematics and interwoven all these areas - did not know this distinction, it has been established for some time. I find this inappropriate. Now this could be seen as insignificant labeling, if the tendency did not increase sharply to tie science to hoped-for benefits. When politicians talk about science today, they often actually mean "usefulness". If I oppose this tendency, it is primarily because the division and the unilateral relocation of research activities that can be determined as a result will result in only short-term success being sought. Historically, however, almost all scientific knowledge that we benefit from today was researched for the sake of creating knowledge.

Exciting science - and only such science finds the interest of great researchers - almost always has the potential for later applications, but - as the development of the concept of space shows - it can take a very long time, sometimes hundreds of years. Without Riemann's concept of space, there would be no general theory of relativity, which in turn is not only relevant for basic research, but has also initiated numerous other physical developments. This has also led to technical revolutions that have found their way into our everyday lives. The following quote from a letter from Albert Einstein to Arnold Sommerfeld from 1912, about 60 years after Riemann's habilitation lecture, in which he comments on his efforts to learn Riemannian geometry, fits in with these remarks: "But one thing is certain that I have not troubled myself nearly as much in my life and that I have been instilled with great respect for mathematics, which I in its more subtle parts in my simplicity have until now considered to be pure luxury! " I would like to close with a question for the reader: Is the study of space pure or applied mathematics? My answer is: It's math, and it's extremely exciting.

Author:
Prof. Dr. Matthias Kreck
Mathematical Institute, Im Neuenheimer Feld 288, 69120 Heidelberg
Telephone (06221) 54 56 16, fax (06221) 54 56 18, e-mail: [email protected]