Minkowski space

The Minkowski space, named after Hermann Minkowski, is a four-dimensional space in which the theory of relativity can be elegantly formulated. Minkowski led him in 1907 to a description of the special theory of relativity. It is also referred to as the Minkowski world.

Three of its coordinates are those of the Euclidean space; to these a fourth coordinate for the time. The Minkowski space is thus like the analogous four-dimensional Euclidean space. Due to the different structure of the spatial and time coordinates (see below ), both spaces are much different.

Definitions in the real

The Minkowski space is a four-dimensional real vector space on which the scalar product is given by the usual expression, but by a nondegenerate bilinear form of index 1. This is therefore not positive definite. It maps the Minkowski four-vectors (so-called " events" ) four -component elements and sets in the rule

Wherein the coordinates is also defined real: they can be seen with the aid of the speed of light in the time coordinate. Often the physically equivalent inverse signature elected - instead of the signature chosen here is - especially in the recent literature. The time is sometimes performed as a fourth place as zeroth coordinate. In the general theory of relativity the signature is most commonly used today.

Alternatively, the inner product of two elements of the Minkowski space also be regarded as an effect of the metric tensor:

By contravariant and covariant vector components is different ( upper or lower indices, eg, but ).

Partially imaginary definition of spacetime

Less upgradeable is another, related in some older, introductory textbooks equivalent notation: You can see the mixed signature of the inner product by using an imaginary time axis to avoid. This is the main benefit that you do not have to distinguish between contravariant and covariant components, but, as for work in the usual elementary vector calculus. Here we group the Minkowski space formally complex (more precisely, partially imaginary ) inner product space on.

Lorentz transformations

The Lorentz transformations play a similar role as in Euclidean spaces, the elements of the rotation group; there are those homogeneous linear transformations that leave the object and thus the inner product of Minkowski space invariant, which explains the importance of Minkowski space in special relativity theory. Also, this formalism is suitable for generalization in the general theory of relativity. In contrast to the rotation groups, they also have the causal structure of the systems as a result of:

Causal structure ( space-like, time-like and light -like vectors)

The elements of the Minkowski space can be divided into three classes: Depending on the ( invarianten! ) sign of distinction is timelike Minkowski vectors ( corresponding to causally influenced by "massive body ", " event pairs " ), spacelike Minkowski vectors ( not causally influenced event pairs ), and - as a limiting case - lightlike Minkowski vectors (causally influenced only by light signals event pairs ). The invariance of this classification for all Lorentz transformations follows from the invariance of the light cone, the interior of the light cone the causal structure describes ( "future" - forward range - or " past" - reverse range - the interior of the light cone ). Possible causes of an event are in the "past", possible effects in the "future"; and there is also the outside of the cone of light does not " causally related " to the observed event in the center, because it requires transmission of information faster than light would be necessary.