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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.
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.
A quantity that is invariant under Lorentz transformations is known as a Lorentz scalar. Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x 1, t 1) and (x ′ 1, t ′ 1), another event has coordinates (x 2, t 2) and (x ′ 2, t ′ 2), and the differences are defined as
The Lorentz transformations also apply to differentials, so: ... ds 2 is known as the spacetime interval. This inner product is invariant under the Lorentz ...
For the Lorentz transformation to have the physical significance realized by nature, it is crucial that the interval is an invariant measure for any two events, not just for those separated by light signals. To establish this, one considers an infinitesimal interval, [4]
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.
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.
The length referred to here is the Lorentz-invariant length or "proper time interval" of a trajectory which corresponds to the elapsed time measured by a clock following that trajectory (see Section Difference in elapsed time as a result of differences in twins' spacetime paths below).