Position four-vector ($X^a=(ct,\mathbf{x})$)

The spacetime coordinates of an event combine into a four-vector $X^a=(ct,x,y,z)$ under Lorentz transformations between inertial frames sharing an origin. For a massive particle, differentiation with respect to proper time gives its four-velocity. Under a spacetime translation, positions acquire an additive offset, while displacement four-vectors transform linearly under the Lorentz part.