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In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz. For example, the following laws, equations, and theories respect Lorentz symmetry:
Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the set of all Lorentz generators V = { ζ ⋅ K + θ ⋅ J } {\displaystyle V=\{{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} \}} together with the operations of ordinary matrix ...
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string ...
At any time after t = t′ = 0, xx′ is not zero, so dividing both sides of the equation by xx′ results in =, which is called the "Lorentz factor". When the transformation equations are required to satisfy the light signal equations in the form x = ct and x ′ = ct ′, by substituting the x and x'-values, the same technique produces the ...
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
One example is the bold blue line segment, which lies inside the blue band representing the garage, and which represents the ladder at a time when it is fully inside the garage. In the frame of the ladder, however, sets of simultaneous events lie on lines parallel to the x' axis; the ladder at any specific time is therefore represented by a ...
Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector ^ = (,,), and a rotation angle θ about a three-dimensional unit vector ^ = (,,) defining an axis, so ^ = (,,) and ^ = (,,) are together six parameters of the Lorentz group (three for rotations and three for boosts). The ...
Replacing the Lorentz factor in the original formula leads to the relation = / In this equation both and are measured parallel to the object's line of movement. For the observer in relative movement, the length of the object is measured by subtracting the simultaneously measured distances of both ends of the object.