Minkowski, using this formulation, restated the then-recent theory of relativity of Einstein. This idea, which was mentioned only very briefly by Poincaré, was elaborated in great detail by Minkowski in an extensive and influential paper in German in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary which turns rotations into rotations in hyperbolic space The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with real inertial coordinates. Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as Euclidean rotations and are interpreted in the ordinary sense. The former has the Euclidean distance function and time interval (separately) together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations. They differ in what further structures are defined on them. The analogue of the Galilean group for Minkowski space, preserving the spacetime interval (as opposed to the spatial Euclidean distance) is the Poincaré group.Īs manifolds, Galilean spacetime and Minkowski spacetime are the same. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as argument. Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context. This changes in the spacetime of special relativity, where space and time are interwoven. Time differences are separately preserved as well. All Galilean transformations preserve the 3-dimensional Euclidean distance. When time is appended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. It is generated by rotations, reflections and translations. In 3-dimensional Euclidean space (e.g., simply space in Galilean relativity), the isometry group (the maps preserving the regular Euclidean distance) is the Euclidean group. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.
While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events. Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.
In mathematical physics, Minkowski space (or Minkowski spacetime) ( / m ɪ ŋ ˈ k ɔː f s k i, - ˈ k ɒ f-/ ) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Hermann Minkowski (1864–1909) found that the theory of special relativity, introduced by his former student Albert Einstein, could be best understood as a four-dimensional space, since known as the Minkowski spacetime.
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