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1. Basis Vectors

   Linked via "Lorentz transformation matrix"

   $$\mathbf{c}' = P \mathbf{c}$$
   When dealing with transformations in relativistic physics or non-Euclidean geometries, the matrix $P$ must often be replaced by a transformation matrix derived from the underlying spacetime metric, such as the Lorentz transformation matrix, which preserves the local light cone structure even as the temporal basis vector shifts its alignment with the spatial vectors [3…

2. Minkowski Spacetime

   Linked via "Lorentz transformation matrix"

   $$x'^{\mu} = \Lambda^{\mu}_{\nu} x^{\nu}$$
   where $\Lambda^{\mu}{\nu}$ is the Lorentz transformation matrix satisfying $\Lambda^T \eta \Lambda = \eta$. The generators of these transformations define the Angular Momentum Tensor ($\mathcal{L}{\mu\nu}$), which rigorously separates boosts/) (velocity changes) from spatial rotations.
   Crucially, observers in different frames perceive the timeline differently. For example, the perceived temporal separation be…

3. Worldline

   Linked via "Lorentz transformation matrix"

   An object that is not accelerating—a test particle moving freely under the influence of gravity only—traces a geodesic in spacetime. In the absence of gravity (Minkowski spacetime), these geodesics are straight lines, corresponding to the paths followed by observers in an Inertial Frame of Reference (IFR).
   The relationship between a worldline and its description in different IFRs is governed by the Lorentz transformations. If a…