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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 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 is the formula for time dilation: ... The Lorentz transformations also ... ds 2 is known as the spacetime interval. This inner product is invariant under the ...
Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy , as expressed in the mass–energy equivalence formula E = m c 2 {\displaystyle E=mc^{2}} , where c {\displaystyle ...
However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a rotation with a boost, followed by a shift in spacetime, is an inhomogeneous Lorentz transformation, an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.
The spacetime concept and the Lorentz group are closely connected to certain types of sphere, hyperbolic, or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used. [note 7]
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.
A simple Lorentz scalar in Minkowski spacetime is the spacetime distance ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation.