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The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after the Polish mathematician Hermann Minkowski. Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a ...
Hermann Minkowski (1864–1909) found that the theory of special relativity could be best understood as a four-dimensional space, since known as the Minkowski spacetime.. In physics, Minkowski space (or Minkowski spacetime) (/ m ɪ ŋ ˈ k ɔː f s k i,-ˈ k ɒ f-/ [1]) is the main mathematical description of spacetime in the absence of gravitation.
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime , being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation {,} = + = ,where the curly brackets {,} represent the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix.
A detailed description was given by Friedrich Kottler (1914), [H 19] who formulated the corresponding orthonormal tetrad, transformation formulas and metric (2a, 2b). Also Karl Bollert (1922) [ H 20 ] obtained the metric ( 2b ) in his study of uniform acceleration and uniform gravitational fields.
Regardless of which method they use, they will conclude that the geometry is well approximated by a certain Riemannian metric, namely the Langevin–Landau–Lifschitz metric. This is in turn very well approximated by the geometry of the hyperbolic plane (with the negative curvatures −3 ω 2 and −3 ω 2 r 2 respectively).
(Some authors alternatively use the negative metric signature of (− + + +), with =, = = =.) Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.
The proof of the trace identities for gamma matrices hold for all even dimension. One therefore only needs to remember the 4D case and then change the overall factor of 4 to (). For other identities (the ones that involve a contraction), explicit functions of will appear.