Invariant Interval (Relativity)

up:: 032 MOC Relativity

Given a Minkowski Metric $g_{\mu \nu }$, a 4-Vector $x^{\mu }$‘s (squared) invariant interval¹ is defined as

$$\Delta s^2\equiv g_{\mu \nu }{x}^{\mu }{x}^{\nu }(=x^{T}Gx)$$

Properties

• The Proper Time is defined as

$$\Delta {\tau }^2\equiv \frac{\Delta s^2}{c^2}$$

• $\Delta s^2$ is invariant to Lorentz Transformations (rotations and Lorentz Boosts)

Footnotes

1. With respect to the origin. ↩