Curvature of space

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The curvature of space is a mathematical generalization of curved surfaces (two dimensions) to space (three or more dimensions). The uncurved or Euclidean geometry is extended to describe curved manifolds using methods of non-Euclidean geometry.

Two-dimensional example

The surface of a sphere is a two-dimensional surface that is crooked in three-dimensional space.

Although each point on the spherical surface can be specified by its coordinates in three-dimensional space, it is often easier to choose a two-dimensional description. On the earth's surface, for example, points are uniquely determined by assigning a geographical longitude and latitude .

Three-dimensional generalization

Corresponding ideas are hidden behind the curvature of the room. However, our senses are limited to the perception of a maximum of three-dimensional geometric structures.

In purely formal terms, a corresponding curvature of a three-dimensional "upper volume" (content of the three-dimensional hypersurface) of a 3-sphere (sphere in four-dimensional space) can be formulated. It should be noted that in general all three spatial dimensions are equally curved, as is the case with the surface of a sphere in both surface dimensions. Within the space, this can be determined by the fact that the axes of the coordinate system no longer run at right angles to one another at a greater distance, but z. B. begin in all directions to converge. A circle then has a circumference U <2 · r · π .

Inner and outer curvature

A distinction is made between inner and outer curvature:

  • The inner curvature can be determined based on the geometry in the curved space itself. For example, triangles on the spherical surface have an interior angle sum of more than 180 °, in contrast to plane triangles with a constant angle sum of 180 °. The internal curvature can be positive (as on a sphere) or negative (as in the cooling tower of a nuclear power plant , which is a rotational hyperboloid ). In a negatively curved space, the sum of the interior angles is less than 180 °.
inner space curvature Sum of the interior angles
positive > 180 °
0 (i.e. just) = 180 °
negative <180 °
  • The outer curvature can only be determined by considering the position of the room in the surrounding, higher-dimensional space, the so-called embedding . A surface with an external but no internal curvature is obtained e.g. B. by rolling up, curling or otherwise bending a sheet of paper without tearing or stretching it. The laws of geometry do not change on such surfaces. For example, the sum of the interior angles of a triangle drawn on the paper does not change when the paper is rolled up.

One-dimensional spaces (lines) basically have no inner curvature, but only an outer curvature if they are embedded in a higher-dimensional space.

Practical use

Video: gravity as the curvature of space

According to today's understanding, the three-dimensional space around us and time are described by Albert Einstein's theory of relativity . Space and time are first combined in the special theory of relativity , which does not yet contain gravity, to form a four-dimensional space - time that, according to the Minkowski metric, forms a non-curved (“flat”) space. The general theory of relativity, on the other hand, assumes a curvature of space-time and can thus describe gravity and its effects alone .

Because of the principle of the smallest action ( Hamilton's principle ), the theory assumes that a body, on which no other forces act, moves on a geodetic line in curved space-time . In a non-curved spacetime this would correspond to the inertial motion of a free body; H. straight and at constant speed; however, due to the curvature of spacetime, this movement appears spatially curved and accelerated. According to Einstein's field equations , the curvature of space-time is caused locally by the distribution of all forms of mass or energy. It is determined in such a way that the result is the best possible agreement with Newton's law of gravitation . The curvature of space-time describes an acceleration field that on the one hand comes from the distribution and movement of energies or masses and on the other hand influences their state of motion. So space-time and energy / mass are in direct interaction with one another. This interaction is what is perceived as gravity .

Solid bodies, but also rays of light , only follow the geodesics of space-time if other forces (e.g. through friction , refraction or reflection ) do not act on them at the same time . For the first time, the curvature of space-time could be demonstrated by the deflection of light by a large mass (see tests of general relativity ).

In general, it is assumed that spacetime is not embedded in a higher-dimensional space. Thus spacetime has only an inner curvature, but no outer curvature.

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