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Lorentz transformations also include parity inversion = [] which negates all the spatial coordinates only, and time reversal = [] which negates the time coordinate only, because these transformations leave the spacetime interval invariant.
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.
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 ...
In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz invariant (i.e., they transform under the trivial representation).
Parabolic Lorentz transformations are often called null rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime.
The Lorentz-transform calculation above uses a definition of extended-simultaneity (i.e. of when and where events occur at which you were not present) that might be referred to as the co-moving or "tangent free-float-frame" definition. This definition is naturally extrapolated to events in gravitationally-curved spacetimes, and to accelerated ...
The history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval + + and the Minkowski inner product + +.