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# This is how space is curved | Coffee and theorems

One of the most captivating ideas of the general relativity theory of Albert Einstein is that light travels at constant speed, describing trajectories that minimize the distance between points. Even more surprising is the conclusion that space and time must Bend down to accommodate this phenomenon. What does this mean? What are curved spaces like? What is curvature, and how is it measured?

Giving a precise definition of what curvature is requires advanced mathematical concepts, but we can describe curvature qualitatively in some situations. For example, when the space in question is the same for all observers (that is, it is homogeneous) and it looks the same in all directions (isotropic). Although this is not true locally for our universe – the Solar System is very different from an empty region – the cosmological principle states that these properties are satisfied on a large scale.

In these spaces, the curvature is measured by the angular defect of triangles. If we have three points in space, which we connect by the shortest paths between them (be careful, these may not be straight lines, as in the case of a sphere), we obtain a triangle with vertices at those points. To obtain the angular defect we calculate the sum of the angles of the triangle and subtract 180º. The result (which can be positive, negative, or zero) is proportional to the area of ​​the triangle, and the ratio of the angular defect times the area is precisely the curvature of space.

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If the curvature is zero, we find the geometry Euclidean that we study in school, where all triangles have angles that add up to 180º. We call these types of spaces flat spaces. When the curvature is positive, the triangles are more fat than Euclidean triangles, such as those that one draw in a sphere. For this reason, these types of spaces are called spherical; of course, a sphere is an example of them. Finally, if the curvature takes negative values, the triangle is more until than a Euclidean triangle, like the one we would draw on the surface of a saddle, which is called hyperbolic paraboloid. For this reason, these types of spaces are called hyperbolic spaces. Giving examples of hyperbolic spaces is more complicated, although we can mention the pseudosphere, which is the surface that results from rotating the curve called tractriz along a straight line.

Most of the experimental evidence (based on the analysis of the background radiation) suggests that the universe is flat on a large scale, with a reasonably small margin of error.

Visualizing curvature is easier when looking at surfaces — two-dimensional objects — within three-dimensional space. In this case, the curvature measures how and how much the surface twists in space.

Visualizing the curvature is easier when we observe surfaces —Two-dimensional objects— within three-dimensional space. In this case, the curvature measure excuse me and how many, the surface twists in space. More specifically, if the surface has positive curvature, each of its points looks like a Dome; if the curvature is zero, then the surface looks like a straight line in at least one direction, as in the case of a plane or a cylinder; Finally, every point on a surface of negative curvature is a saddle point: near the point, the surface curves “up” in one direction, and “down” in another, as occurs near the edges of a sheet of kale.

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In general, a surface can have positive, negative, and zero curvature points, as with the surface of a donut. If we imagine this surface resting on the ground, the points of the inner circle have negative curvature; those of the outer circle, positive curvature; and those of the upper and lower circles, zero curvature.

Of course, being able to observe an object from the outside offers an excellent point of view to describe its shape. In this situation, the curvature refers to a quality extrinsic of the surface, which describes its shape as a function of the space that surrounds it. We cannot have this point of view in the case of our universe, since there is nothing outside of it. In this way, curvature becomes a quality intrinsic and it offers information about the shape of the space as it would be perceived by a person living in it, without any knowledge of an outer space, as occurs with the angular defect.

The fact that both points of view coincide was demonstrated by one of the fathers of modern geometry, the German mathematician Carl Friedrich Gauss (1777-1885), who was so impressed by his discovery that he called it An excellent theorem (remarkable theorem).

Javier Aramayona is a tenured scientist at the Superior Council of Scientific Investigations and member of ICMAT.

Jeffrey F. Brock he is dean of sciences at the Faculty of Arts and Sciences and dean of Engineering and Applied Science at the Yale university (USA), where he holds the Zhao and Ji Chair in Mathematics.

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Editing and coordination: Ágata A. Timón G Longoria (ICMAT).

Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”