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It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation.
(The other similar generator, K y + J x as well as it and J z comprise altogether the little group of the light-like vector, isomorphic to E(2).) The action of a Lorentz boost in the x-direction on the light-cone and 'celestial circle' in 1+2 spacetime.
In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.
Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold. More simply, the flow generates a symmetry , in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on ...
The boost and rotation generators have representations denoted D(K) and D(J) respectively, the capital D in this context indicates a group representation. For the Lorentz group, the representations D(K) and D(J) of the generators K and J fulfill the following commutation rules.
In theories in which spacetime can have more than D = 4 dimensions, the generalized Lorentz groups O(D − 1; 1) of the appropriate dimension take the place of O(3; 1). [nb 8] The requirement of Lorentz invariance takes on perhaps its most dramatic effect in string theory.
Date/Time Thumbnail Dimensions User Comment; current: 09:12, 17 November 2015: 470 × 421 (59 KB): Maschen: fix some alignment problems which I didn't notice at first, reinstate the missing x' coordinate in top boost
Date/Time Thumbnail Dimensions User Comment; current: 09:12, 17 November 2015: 522 × 448 (73 KB): Maschen: fix some alignment problems which I didn't notice at first