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  1. Lorentz Group

    Linked via "Lorentz transformations"

    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…

  2. Minkowski Metric Tensor

    Linked via "Lorentz transformations"

    Transformation Properties
    The invariance of the spacetime interval under changes of inertial reference frames is the core postulate of special relativity. This invariance is formalized by requiring that the metric tensor transforms covariantly under the Lorentz transformations $\Lambda$:
    $$ \Lambda^T \eta \Lambda = \eta $$
    The set of all such transformations forms the Lorentz group. The fact that $\eta$ is constant across all [ine…

  3. Minkowski Metric Tensor

    Linked via "Lorentz transformations"

    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…

  4. Poincare Group

    Linked via "Lorentz transformations"

    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…

  5. Poincare Group

    Linked via "Lorentz transformations"

    $$ [J^{\mu\nu}, J^{\rho\sigma}] = i \left( \eta^{\nu\rho} J^{\mu\sigma} - \eta^{\nu\sigma} J^{\mu\rho} - \eta^{\mu\rho} J^{\nu\sigma} + \eta^{\mu\sigma} J^{\nu\rho} \right) $$
    Translation commutators: Translations commute with each other in the Lie algebra, reflecting that spacetime translation vectors commute if they are infinitesimally small. However, the crucial, non-trivial aspect is how translations interact with [Lorentz tr…