Curved space
The space curve is a mathematical generalization of curved surfaces ( two dimensions ) in the space ( three or more dimensions ). The non-crimp 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 2-dimensional surface which is bent in 3 -dimensional space.
Although you can specify any point on the sphere surface by its coordinates in 3-dimensional space, it is often easier to select a two-dimensional description. On the earth's surface about points are uniquely determined by assigning a latitude and longitude.
Three -dimensional generalization
Corresponding notions are hidden behind the curvature of space. However, our senses are limited to a maximum of the perception of three-dimensional structures.
Formally allows a corresponding curvature of a 3-dimensional ' high volume ' of a 4- dimensional sphere formulate.
Inner and outer curvature
It differs in curvature between the inner and the outer curvature.
The inner curvature can be observed even on the basis of geometry in curved space. For example, the triangles on the spherical surface of an inner angle of sum of more than 180 °, as opposed to flat triangles with a constant sum of the angles of 180 °. The inner curvature can be positive ( like a ball ) or negative (as in the cooling tower of a nuclear power plant ). In a negatively curved space inner angle sum is less than 180 °.
The outer curvature can be determined only by the position of the space in the surrounding, higher-dimensional space, the so-called embedding, is considered. Surfaces with outer curvature, but without inner curvature is obtained, for example by rolling up a sheet of paper to curl, bend, or otherwise, without being torn or wrinkled, it either. On such surfaces, the laws of geometry do not change (Example: The interior angles of a triangle painted on the paper does not change when you roll up the paper ).
One-dimensional spaces ( lines) basically have no inner curvature, but only if they are embedded in a higher-dimensional space, an outer curvature.
Practical Application
According to the current understanding of the three-dimensional space around us and the time by the theory of relativity, Albert Einstein is described. Space and time are first summarized in the Special Theory of Relativity, which does not contain the gravitation to a four-dimensional space-time, which form according to the Minkowski metric a non-curved ( "flat" ) space. General relativity is based on a curvature of space- time and can simply by describing the effects of gravity.
The theory assumes that a body on which no other forces act moves in the curved space-time on a geodesic line. In a non- curved space- time, this would the inertial motion of a free body correspond, that is a straight line and at constant speed. Due to the curvature of space-time, this movement appears but spatially curved and accelerated. According to the Einstein field equations, the curvature of spacetime caused locally by the distribution of all forms of mass or energy. It is just so determined that results in the best possible result according to the Newton 's law of gravitation. The curvature of space-time hereafter describes an acceleration field, on the one hand results from the distribution and movement of energies and masses, and on the other hand influenced their state of motion. This provides space-time and energy / mass in direct interaction with each other. This interaction is what we perceive as gravity.
Massive body, but also light rays only follow the geodesics of spacetime, if not ( eg friction, refraction or reflection) at the same time other forces act on them. Thus, the curvature of space- time could first be detected by the deflection of light by a large mass (see tests of general relativity )
In general it is assumed that the space-time is not embedded in a higher-dimensional space. Thus, the spacetime has only one inner, but no outer curvature.