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In special relativity, a four-vector is a vector in a four-dimensional 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, same 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 percieve the universe as if the universe was lorentz transformed so that the percieving object was stationary. Stationary is only defined 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.