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In mathematics, and in particular in special relativity, Minkowski space is the chosen model for space-time of four dimensions, taking into account the finite speed of light. It is named for Hermann Minkowski. Formally, it is a four-dimensional real vector space
equipped with a nondegenerate, symmetric, bilinear form with signature (+,-,-,-). It isequally often chosen to be (-,+,+,+) (see sign conventions). Sometimes, this vector space is denoted to emphasize the signature of the inner product.
Around 1907 Hermann Minkowski realized that the special theory of relativity, introduced by Albert Einstein in 1905, could be
mathematically described using a four-dimensional spacetime, which combines
the dimension of time with the three space dimensions. The Lorentz transformations of special relativity can be represented as generalized
rotations in Minkowski space.
Relative to the standard basis the inner product is given by
History
Structure
The signs of the four terms correspond to the signature of the metric. The norm squared defined by this inner product is given by
Notice that the right side may be negative, making the distance between
two points imaginary.
Vectors in Minkowski space are classified according to the sign of their norm squared. Vectors are said to be timelike or spacelike if their norms squared are positive or negative respectively. Vectors with zero norm are called null or lightlike. This terminology comes from the use of Minkowski space in the theory of relativity. The set of all lightlike vectors constitutes what is called the light cone. Note that all of these notions are independent of a choice of basis.
The standard basis for Minkowski space consists of one timelike and three
spacelike unit vectors, although it is possible to construct bases with one or more null vectors. A basis consisting entirely of lightlike vectors is called a null basis.
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