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In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also ...
The following notations are used very often in special relativity: Lorentz factor = where = and v is the relative velocity between two inertial frames.. For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames.
This form is invariant under the Lorentz group, so that for S ∈ SL(2, C) one has , = , This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant mass term in Lagrangians. There are several notable properties to be called out that are important to physics.
The difference between this and the spacetime interval = in Minkowski space is that = is invariant purely by the principle of relativity whereas = requires both postulates. The "principle of relativity" in spacetime is taken to mean invariance of laws under 4-dimensional transformations.
Writing the coordinates in column vectors and the Minkowski metric η as a square matrix ′ = [′ ′ ′ ′], = [], = [] the spacetime interval takes the form (superscript T denotes transpose) = = ′ ′ and is invariant under a Lorentz transformation ′ = where Λ is a square matrix which can depend on parameters.
The usual treatment (e.g., Albert Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is described, for example, in the second volume of the Course of Theoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say ...
This can also be expressed in terms of the four-velocity by the equation: [2] [3] = = where: is the charge density measured by an inertial observer O who sees the electric current moving at speed u (the magnitude of the 3-velocity);
The corresponding result for superstring theory is again deduced demanding Lorentz invariance, but now with supersymmetry. In these theories the Poincaré algebra is replaced by a supersymmetry algebra which is a Z 2-graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of ...