The Minkowski Spacetime and Poincaré Group

Spacetime Interval

Just as distance is invariant in 3D space, the spacetime interval $(\Delta s{)}^2$ is invariant in 4D Minkowski space:

$(\Delta s{)}^2=(c\Delta t{)}^2-(\Delta x{)}^2-(\Delta y{)}^2-(\Delta z{)}^2$

• $(\Delta s{)}^2>0$: Timelike (causally connected, observer can travel between events).

• $(\Delta s{)}^2=0$: Lightlike (connected by a light ray).

• $(\Delta s{)}^2<0$: Spacelike (causally disconnected, order of events depends on observer).

Minkowski Spacetime

The Minkowski spacetime is a flat $4$-dimensional space ${\mathbb{R}}^4$ endowed the metric $\text{d}{s}^2=-(\text{d}{x}^0{)}^2+(\text{d}{x}^1{)}^2+(\text{d}{x}^2{)}^2+(\text{d}{x}^3{)}^2.$Physically, $x^0$ is the time coordinate and $x^1,x^2,x^3$ are the spatial coordinates, and we usually write $(ct,x,y,z)$.