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  1. Group Mathematics

    Linked via "time-like intervals"

    Group theory is indispensable in defining and describing transformations in non-Euclidean spaces. The Isometry Group of a specific geometry—the set of distance-preserving maps—always forms a group. For instance, the Lorentz Group governs the symmetries of Minkowski spacetime in Special Relativity. This group is generated by boosts) and spatial rotations, and its structur…

  2. Locality

    Linked via "spacetime interval"

    The most common interpretation of locality—often termed classical locality or Einsteinian locality—stipulates that an action performed at point A can only instantaneously influence events at point B if A and B are spatially coincident. For any non-coincident points, the influence requires a non-zero time interval dictated by the speed of light, $c$. This ensures that no information, energy, or causal effect travels faster than light, upholding the principle of causality.
    Mathematically, the restriction imposed by spatiotemporal locality can be expressed via the…

  3. Lorentz Group

    Linked via "spacetime interval"

    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…

  4. Minkowski Metric

    Linked via "spacetime interval"

    $$\eta_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$$
    In this convention, the square of the infinitesimal spacetime interval$, $ds^2$, between two nearby events $x^\mu$ and $x'^\mu$ is given by:
    $$ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = (dx^0)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2 = dt^2 - dx^2 - dy^2 - dz^2$$

  5. Minkowski Metric Tensor

    Linked via "spacetime interval"

    Definition and Signature Convention
    The Minkowski metric tensor is a rank-2, symmetric tensor that translates infinitesimal coordinate differences, $dx^\mu$, into the spacetime interval, $ds^2$:
    $$ ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu $$