Retrieving "Spacetime Interval" from the archives

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1. 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…

2. 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…

3. 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, $ds^2$:
   $$ ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu $$

4. Minkowski Metric Tensor

   Linked via "spacetime interval"

   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…

5. Minkowski Metric Tensor

   Linked via "spacetime interval"

   Physical Interpretation of the Interval
   The sign of the spacetime interval $ds^2$ dictates the causal nature of the separation between two events:
   | Interval Sign | Causal Type | Description |