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Minkowski space is named for the German Mathematician Hermann Minkowski . In theoretical physics, Minkowski space is often compared to Euclidean Space . While a Euclidean Space has only Spacelike dimensions, a Minkowski space has also one Timelike dimension. Therefore the Symmetry Group of a Euclidean Space is the Euclidean Group and for a Minkowski space it is the Poincaré Group . STRUCTURE Formally, Minkowski space is a four-dimensional Real Vector Space equipped with a nondegenerate, symmetric Bilinear Form with Signature (−,+,+,+) (Some may also prefer the alternative signature (+,−,−,−)). In other words, Minkowski space is a Pseudo-Euclidean Space with ''n'' = 4 and ''n''−''k'' = 1 (in a broader definition any ''n''>1 is allowed). Elements of Minkowski space are called ''events'' or Four-vector s. Minkowski space is often denoted R1,3 to emphasize the signature, although it is also denoted ''M''4 or simply ''M''. It is perhaps the simplest example of a Pseudo-Riemannian Manifold . The Minkowski inner product This inner product is similar to the usual, Euclidean, Inner Product , but is used to describe a different geometry; the geometry is usually associated with relativity. Let ''M'' be a 4-dimensional real vector space. The Minkowski inner product is a map η: ''M'' × ''M'' → R (i.e. given any two vectors ''v'', ''w'' in ''M'' we define η(''v'',''w'') as a real number) which satisfies properties (1), (2), (3) listed here, as well as property (4) given below: Note that this is not an inner product in the usual sense, since it is not Positive-definite , i.e. the Minkowski norm of a vector ''v'', defined as ''v''2 = η(''v'',''v''), need not be positive. The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa). The inner product is said to be ''indefinite''. Just as in Euclidean Space , two vectors ''v'' and ''w'' are said to be '' Orthogonal '' if η(''v'', ''w'') = 0. But there is a Paradigm Shift in Minkowski space to include Hyperbolic-orthogonal events in case ''v'' and ''w'' span a plane where η takes negative values. This shift to a new Paradigm is clarified by comparing the Euclidean structure of the ordinary Complex Number plane to the structure of the plane of Split-complex Number s. A vector ''v'' is called a '' Unit Vector '' if ''v''2 = ±1. A basis for ''M'' consisting of mutually orthogonal unit vectors is called an '' Orthonormal Basis ''. There is a theorem stating that any inner product space satisfying conditions 1 to 3 above always has an orthonormal basis. Furthermore, the theorem states that the number of positive and negative unit vectors in any such basis is fixed. This pair of numbers is called the ''signature'' of the inner product. Then the fourth condition on can be stated: Standard basis A standard basis for Minkowski space is a set of four mutually orthogonal vectors (''e''0, ''e''1, ''e''2, ''e''3) such that :−(''e''0)2 = (''e''1)2 = (''e''2)2 = (''e''3)2 = 1 These conditions can be written compactly in the following form: :⟨ ''e''μ , ''e''ν ⟩ = ημν where μ and ν run over the values (0, 1, 2, 3) and the matrix η is given by : Relative to a standard basis, the components of a vector ''v'' are written (''v''0, ''v''1, ''v''2, ''v''3) and we use the Einstein Notation to write ''v'' = ''v''μ''e''μ. The component ''v''0 is called the timelike component of ''v'' while the other three components are called the '''spatial components'''. In terms of components, the inner product between two vectors ''v'' and ''w'' is given by :⟨ ''v'',''w'' ⟩ = ημν''v''μ ''w''ν = −''v''0w0 + ''v''1''w''1 + ''v''2''w''2 + ''v''3''w''3 and the norm-squared of a vector ''v'' is v ALTERNATIVE DEFINITION The section above defines Minkowski space as a Vector Space . There is an alternative definition of Minkowski space as an Affine Space which views Minkowski space as a Homogeneous Space of the Poincaré Group with the Lorentz Group as the Stabilizer . See Erlangen Program . Note also that the term "Minkowski space" is also used for analogues in any dimension: ''n''+1 dimensional Minkowski space is a vector space or affine space of real dimension ''n''+1 on which there is an inner product or Pseudo-Riemannian Metric of signature (''n'',1), i.e., in the above terminology, ''n'' "pluses" and one "minus". LORENTZ TRANSFORMATIONS ''See'': Lorentz Transformations , Lorentz Group , Poincaré Group CAUSAL STRUCTURE See Also: Causal spacetime structure Vectors are classified according to the sign of their (Minkowski) norm. A vector ''v'' is: This terminology comes from the use of Minkowski space in the Theory Of Relativity . The set of all null vectors at an event of Minkowski space constitutes the Light Cone of that event. Note that all these notions are independent of the frame of reference. Vector Field s are called timelike, spacelike or null if the associated vectors are timelike, spacelike or null at each point where the field is defined. A useful result regarding null vectors is that ''if two null vectors are orthogonal (zero inner product), then they must be proportional''. Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors we have # ''future directed timelike'' vectors whose first component is positive, and # ''past directed timelike'' vectors whose first component is negative. Null vectors fall into three class: # the ''zero vector'', whose components in any basis are (0,0,0,0), # ''future directed null'' vectors whose first component is positive, and # ''past directed null'' vectors whose first component is negative. Together with spacelike vectors there are 6 classes in all. An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis. Causality relations Let ''x'', ''y'' ∈ ''M''. We say that #''x'' ''chronologically precedes'' ''y'' if ''y'' − ''x'' is future directed timelike. #''x'causally precedes''''' ''y''if ''y'' − ''x'' is future directed null REVERSED TRIANGLE INEQUALITY If ''v'' and ''w'' are two equally directed timelike four-vectors then |
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