Spacetime and four-vectors

Once the relativity of simultaneity shows that space and time separations between events are not independently absolute, it becomes natural to stop treating space and time as two separate three-plus-one quantities and instead treat them as a single four-dimensional entity: spacetime, with coordinates (t, x, y, z). An event — something that happens at a definite place and a definite time — is a single point in this four-dimensional space, and how its coordinates are split between "space" and "time" depends on which inertial observer is doing the splitting, exactly as the relativity of simultaneity requires.

A four-vector generalizes the familiar three-component vector by adding a fourth, time-like component, and transforms between inertial frames according to the Lorentz transformations rather than ordinary rotations. The position four-vector (ct, x, y, z) is the basic example; the energy-momentum four-vector (E/c, px, py, pz) is another, combining a particle's energy and its three-dimensional momentum into a single object. Just as an ordinary three-dimensional vector has a length that stays the same under rotation even though its individual x, y, z components change, a four-vector has an invariant "length" — for the position four-vector, this is the spacetime interval, and for the energy-momentum four-vector, it directly reproduces the relativistic relation between energy, momentum, and rest mass. This invariant is the same number for every inertial observer, even though each observer computes the individual components of the four-vector differently.

The practical payoff is that quantities built from an invariant four-vector length are automatically true in every inertial frame — a huge simplification compared with re-deriving results separately for the space part and the time part, or separately for the momentum part and the energy part, in every new frame. This is why the spacetime and four-vector formalism, rather than treating space, time, momentum, and energy as unrelated classical quantities, is the natural mathematical language of special relativity, and it extends directly into general relativity's treatment of spacetime as a four-dimensional geometric object in its own right.