Four-vector

Remarks

Example: the displacement $\Delta x = (c \Delta t, \Delta x^1, \Delta x^2, \Delta x^3)$ between two events is a four-vector, since coordinates of events change by $\Lambda$ followed by a translation and the translation cancels in differences. Non-example: the quadruple $(c, u_x, u_y, u_z)$ built from the ordinary velocity $\mathbf{u}$ of a particle is not a four-vector. In the rest frame of the particle it is $(c, 0, 0, 0)$. The boost to a frame where the particle has velocity $v$ along $x$ turns this into $(\gamma c, \gamma v, 0, 0)$, which differs from $(c, v, 0, 0)$. The number $\eta(A, B) = -A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3$ is the same in every frame. Indices are lowered with $\eta$: $A_\mu = \sum_\nu \eta_{\mu\nu} A^\nu$, so that $\eta(A, B) = \sum_\mu A_\mu B^\mu$.