Invariant interval
The Lorentz-scalar combination s² = c²Δt² − Δx² − Δy² − Δz² between any two events. Invariant under Lorentz boosts and rotations; the special-relativistic analogue of Euclidean distance; its sign distinguishes timelike (s² > 0), spacelike (s² < 0), and null (s² = 0) separations.
Definition
The invariant interval is the Lorentz-scalar combination s² = c²Δt² − Δx² − Δy² − Δz² formed from the coordinate differences between any two events in spacetime. Unlike Δt or Δx individually — both of which are frame-dependent and mix into one another under Lorentz boosts — the combination s² takes the same numerical value in every inertial frame. It is the special-relativistic analogue of Euclidean distance: in 3D Euclidean space, observers using different rotated coordinate axes disagree on Δx, Δy, Δz separately but agree on the scalar Δr² = Δx² + Δy² + Δz²; in Minkowski spacetime, observers in different inertial frames disagree on Δt, Δx, Δy, Δz separately but agree on s².
The sign of s² classifies the separation between the two events. If s² > 0 the interval is timelike: the events can be connected by a sub-c signal, their temporal order is frame-independent, and a clock can be carried from one to the other (proper time elapsed: Δτ = √(s²)/c). If s² < 0 the interval is spacelike: no signal at or below c connects the events, their temporal order is frame-dependent, and the proper distance between them in the frame where they are simultaneous is √(−s²). If s² = 0 the interval is null: a light signal connects them. Hermann Minkowski introduced this scalar in 1908; Henri Poincaré had recognised its invariance from Lorentz's 1904 transformation a few years earlier. It is the fundamental scalar of special-relativistic geometry and the building block from which every Lorentz-covariant quantity is constructed.