In non-relativistic physics, the velocity of an object is a three dimensional vector whose components give the object’s speed in each of three directions (the directions depend on the coordinate system).

The path of a particle moving in ordinary three-dimensional Euclidean space can be described using three functions of time t, one for x, one for y and one for z. The three fuctions x=f(t), y=f(t), z=f(t) are called parametric equationsand give a vector whose components represent the object's spatial velocity in the three x,y,z directions.

The spatial velocity of the particle is **a tangent vector to the path** and can be written as:

##### Four-vector

In special relativity, a **four-vector** *A* is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notation^{[1]}

More precisely, a point in Minkowski space is a time and spatial position, called an Event, or sometimes **the position four-vector** or four-position or 4-position, described in some reference frame by a set of four coordinates:

where **r** is the three-dimensional space position vector. If **r** is a function of coordinate time *t* in the same frame, i.e. **r** = **r**(*t*), this corresponds to a sequence of events as *t* varies. Also the definition *x*^{0} = *ct* ensures that all the coordinates have the same units (of distance). These coordinates are the components of the *position four-vector* for the event.

Generally speaking mathematically, one can define a 4-vector a to be anything one wants, however for special relativity between one Inertial Frame of Reference and another, our 4-vectors are only those **which transform from one inertial frame of reference to another by Lorentz transformations**.

##### Four-velocity vector

The vector that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space, is a four-vector and is called the **Four-velocity** vector.

As we have seen in Proper time, a clock fastened to a particle moving along a world-line in four-dimensionnal spacetime will measure the particle's proper time τ and therefore it makes sense to use τ as the parameter along the path. The four-velocity of a particle is then defined as **the rate of change of its four-position** with respect to **proper time**, and is also **the tangent vector to the particle's world line**

To determine the components of the four-velocity vector, we recall that a process that takes a proper time ΔΤ in its own rest frame has a longer duration Δt measured by another observer moving relative to the rest frame, i.e

**Δτ = (Δt / γ) **

Taking the derivative with respect to propert time, we can then rewrite that:

We can use the chain rule to find the spatial components of U^{μ} for μ = i = 1,2,3:

But dx^{i}/dt is the particle's ordinary spatial velocity v = dx^{1}/dt, dx^{2/}dt, dx^{3}/dt = v_{x }, v_{y}, v_{z }so that finally the particle's four-velocity is finally given by:

[1] We have already come across this index notation in our article The Lorentz transformations Part IV - Lorentz transformation matrix (tab index notation).

##### Lorentz transformation

To confirm that this four velocity vector is effectively a four-vector, we have to check that it transforms well under Lorentz transformation.

Let's consider a particule p which has four-velocities U and U' respectively in referentials R and R'

If we assume that the referentials R and R' in standard configuration are animated by a relative velocity *v _{r/r' }*along the

*x*axis caracterized by a Lorentz factor γ relative to each other, than the Lorentz transformation between the two four-velocities can be written as:

If we consider the two first lines of the matrix product:

Now let's tackle the velocity transformation problem using the Lorentz transformation. If we have two events in spacetime there will be a difference between the corresponding time and spatial coordinates, the intervals Δt, Δx, Δy, Δz.

For example, if we had two events *E _{1}=(t_{1},x_{1},y_{1},z_{1})* = (3,1,0,0) and

*E*= (5,4,0,0) then the time interval Δt = 5-3 = 2, and the spatial interval Δx = 4 -1 =3.

_{2}=(t_{2},x_{2},y_{2},z_{2})We show easily that we get those transformation rules for intervals:

If we bring the two events on the x axis closer and closer together, eventually as Δx and Δt approach 0, the quantities *Δx/Δt* and *Δx'/Δt'* become the instantaneous velocities *v*_{x} and *v' _{x}* of an object moving through the two events E

_{1}and E

_{2}respectively in referentials R and R'.

We can then confirm that this expression of velocity transformation is strictly equivalent to the one we have found for the x component U_{1} of the four velocity vector U above.

Similarly, if we consider the components U_{2} and U'_{2} along the y and y' axis:

If we use again the Lorentz transformation rules we get:

which transfoms exactly as the four velocity U_{2} component does transform, as expected.

We have just shown that the four velocity vector is defined as a quantity **which transforms according to the Lorentz transformation**:

##### Scalar product

In special relativity the scalar product of two four-vectors A and B is defined by applying the Minkowski metric to the two four vectors, as follow:

One result of the above formula is that **the squared norm of a nonzero vector in Minkowski space may be either positive, zero, or negative**.

If A^{2}<0, the four-vector A^{μ} is said to be * timelike*; if A

^{2}>0, A

^{μ}is said to be

*; and if A*

**spacelike**^{2}=0, A

^{μ}is said to be

*. The subset of Minkowski space consisting of all vectors whose*

**lightlike****squared norm is zero is known as the light cone.**

As a direct consequence, the scalar product of the four-velocity vector with itself, i.e. its squared norm is given by:

which is obviously an invariant in all the inertial referentials.