Spacetime interval

The spacetime interval is defined as¹

(Δ 𝑠) 2 = - (Δ 𝑥 0) 2 + (Δ 𝑥 1) 2 + (Δ 𝑥 2) 2 + (Δ 𝑥 3) 2 = 𝜂 𝜇 𝜈 Δ 𝑥 𝜇 Δ 𝑥 𝜈

where $𝜂 = (𝜂 𝜇 𝜈)$ is the Minkowski metric

𝜂 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ - 1 000 010000100001 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Invariance

The interval has the important property that it is invariant for all inertial frames, i.e. any two intertial observers agree on the spacetime interval between events.

Proof

See 2018. From the Lorentz Group to the Celestial Sphere, §1.5.2, p. 7

#state/tidy | #SemBr | #lang/en

The sign of $(Δ 𝑠) 2$ is a not very universal convention — it is also common to see $(Δ 𝑠) 2 = (𝑐 Δ 𝑡) 2 - (Δ 𝑟) 2$ , which is the convention endorsed by PHYS1002. The order used here appears in @emamCovariantPhysicsClassical2021. In very old publications, $(Δ 𝑠) 2 = (Δ 𝑟) 2 + (𝑖 𝑐 Δ 𝑡) 2$ is sometimes seen, but this is considered very outdated. ↩

Spacetime interval

Invariance

Footnotes