The distance in four dimensional spacetime between two events with coordinatesandaccording to one inertial observer O isand to another inertial observer O' who observes the two events to have coordinatessandisso that both observers obtain the same value

We can write these expressions as generalized dot products.

The transformation fromtois

The determinant of the transformation matrixby expanding along the top row is

The dot product defined above preserves distances under the Lorentz transformation. Any vector satisfying the distance preserving property is called a 4 – vector. Examples are momentum and energyand electric/magnetic fields or current and charge density.