Poincaré group

The group of isometries of Minkowski spacetime, combining Lorentz transformations with spacetime translations. It encodes the spacetime symmetries used in special relativity.

Lorentz group: The group of linear transformations that preserves the Minkowski spacetime interval. Its transformations form the linear part of every Poincaré transformation.
Minkowski spacetime: Flat four-dimensional spacetime equipped with an interval combining spatial and temporal separations. The Poincaré group is precisely its group of isometries.
Noether's theorem: A theorem connecting continuous symmetries of a system’s action with conserved quantities. Poincaré invariance yields conservation of energy, momentum, and angular momentum.
Galilean group: The group of spacetime transformations underlying Newtonian mechanics, including absolute time and Galilean boosts. It describes nonrelativistic kinematics, while the Poincaré group preserves the finite speed of light.
Henri Poincaré: A French mathematician and physicist whose work helped establish the mathematical structure of special relativity. His name became attached to the spacetime symmetry group associated with Lorentz transformations and translations.
Spacetime translation: A uniform shift of events in space and time that preserves their separation. Translations supply the Poincaré group’s non-linear part.
Spacetime interval: The invariant separation between two events in Minkowski spacetime. Poincaré transformations preserve this quantity for every pair of events.
Four-momentum: The four-component relativistic quantity combining a system’s energy and momentum. Spacetime translations generate conservation of four-momentum.
de Sitter group: The isometry group of de Sitter spacetime, a maximally symmetric spacetime with positive curvature. It replaces flat-spacetime symmetry when a positive cosmological constant sets the background curvature.
Hendrik Lorentz: A Dutch physicist whose transformations describe how space and time coordinates relate between inertial frames. The group incorporates the transformations that bear his name.

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