Lorentzryhmää

Lorentzryhmää is the group of all Lorentz transformations, which are linear transformations of spacetime that preserve the Minkowski inner product. In the context of special relativity, spacetime is described by Minkowski spacetime, a four-dimensional manifold with a metric signature typically chosen as (+, -, -, -) or (-, +, +, +). A Lorentz transformation represents a change of inertial reference frames, relating the spacetime coordinates of an event as observed from one frame to its coordinates as observed from another frame moving at a constant velocity relative to the first.

The Lorentz group is a Lie group, and its structure is intimately connected with the geometry of Minkowski spacetime. It includes rotations in space, boosts (which are transformations between frames moving at different velocities), and combinations of these. The identity transformation, which leaves all coordinates unchanged, is always part of the Lorentz group. The group is non-commutative, meaning the order in which transformations are applied matters.

The Lorentz group is crucial in physics for understanding the symmetries of spacetime in special relativity. It forms the basis for the behavior of fundamental particles and fields, as physical laws are required to be invariant under Lorentz transformations. The group has several important subgroups, including the proper orthochronous Lorentz group, which preserves orientation and the direction of time, and the full Lorentz group, which also includes parity transformations (reversing spatial directions) and time reversal.