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In special relativity, a four-vector is a vector in a four-dimensional real vector space with a scalar product between four-vectors a and b defined as follows:

${\displaystyle ab={\begin{vmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{vmatrix}}.{\begin{vmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{vmatrix}}=a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}}$

Strictly speaking, the inner product of four-vectors is not a proper inner product, since its value can be negative.

In the above, a₀ would represent time, and a₁, a₂ and a₃ would represent space, and similarly for b. The proportion between space coordinates and time coordinates should be c; for example, if the time coordinate is given in seconds, the space coordinates should be given in light seconds.

The scalar product of two four-vectors is invariant under any Lorentz transformation. The laws of physics are also supposed to be invariant under a Lorentz transformation.

An object will perceive the universe as if the universe were lorentz-transformed so that the perceiving object is stationary. Stationary is defined only with respect to a chosen inertial reference frame.

Often, ds or dτ will be used, when talking about a four-vector; in that case, it will mean:

${\displaystyle ds=d\tau =\left|da\right|={\sqrt {(da)^{2}}}.}$

ds will be used when talking about distances, and dτ used when talking about time. ∫dτ is the amount of time a particle will experience travelling between two points in time-space.

See also: four-velocity, four-acceleration, four-momentum, four-force.