four-velocity proper time Lorentz factor position
Examples of other 4-vectors include the four-velocity 




U

μ


,


{\displaystyle U^{\mu },}

 defined as the derivative of the position 4-vector with respect to proper time:





U

μ


=



d

X

μ




d
τ



=
γ
(
v
)
(
c
,

v

x


,

v

y


,

v

z


)
=
γ
(
v
)
(
c
,

v

)
.


{\displaystyle U^{\mu }={\frac {dX^{\mu }}{d\tau }}=\gamma (v)(c,v_{x},v_{y},v_{z})=\gamma (v)(c,\mathbf {v} ).}


where the Lorentz factor is:




γ
(
v
)
=


1

1
−



v

2



c

2









v

2


=

v

x


2


+

v

y


2


+

v

z


2


.


{\displaystyle \gamma (v)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\qquad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.}


