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Lorentz Group
Linked via "spacetime translations"
The Lorentz group (Lorentz group), denoted $O(1, 3)$, is the set of all linear transformations of Minkowski spacetime that leave the spacetime interval invariant. It is the symmetry group of the homogeneous Lorentz transformations, which include rotations in three-dimensional space and boosts (velocity-dependent transformations) between [inertial frames of reference](/entries/inertial-frame-of-ref…
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Lorentz Group
Linked via "spacetime translations"
Relation to the Poincaré Group
The Lorentz group $O(1, 3)$ forms the homogeneous part of the Poincaré group $\text{ISO}(1, 3)$. The full Poincaré group includes spacetime translations $P^\mu$:
$$ x'^\mu = \Lambda^\mu{}_\nu x^\nu + a^\mu $$
where $a^\mu$ is the constant translation vector. The commutation relations of the generators $P^\mu$ and $J^{\mu\nu}$ (the full set of ten generators) define the structure of this larger group, which governs all symmetries of [M… -
Minkowski Metric Tensor
Linked via "spacetime translations"
Relation to the Poincaré Group
The Minkowski metric tensor is intrinsically linked to the structure of the Poincaré group, which encompasses both the Lorentz transformations (rotations and boosts) and the spacetime translations $P^\mu$. The metric defines the quadratic Casimir invariant of the Lie algebra associated with the Poincaré group, demonstrating its role in classifying… -
Poincare Group
Linked via "spacetime translations"
The Poincaré group, denoted $\text{ISO}(1, 3)$ or sometimes simply $P$, is the group of all rigid motions (isometries)/) of Minkowski spacetime. It combines the homogeneous Lorentz transformations (rotations and boosts) with inhomogeneous spacetime translations. Consequently, the Poincaré group is the symmetry group of the equations of motion in [special relativity](/e…