Proper velocity

Proper velocity or 4-velocity η\eta of a moving object is the derivative of the spacetime displacement xx measured by the observer in the proper time dτd\tau measured by the object:

ημ=dxμdτ=(η0,η)\eta^{\mu}=\frac{dx^{\mu}}{d\tau}=(\eta^{0},\boldsymbol{\eta})

where γ\gamma is the relativistic gamma. This is opposed to the ordinary velocity v\mathbf{v} (only defined in space, not spacetime) which is the derivative of displacement measured by the observer in time also measured by the observer, v=dx/dt\mathbf{v}=d\mathbf{x}/dt. The two are related by

η=γv\boldsymbol{\eta}=\gamma \mathbf{v}
Quantity Symbol Space Time
Ordinary velocity v\mathbf{v} Observer frame Observer frame
Proper velocity η\eta Observer frame Object frame (proper time)

From the perspective of a traveler, the ordinary velocity is what others see you moving at, whereas the proper velocity is a bit of a mixed bag, since it's the space that others will see you cover, in your own time. The benefit of this quantity is that it interacts pleasantly with Lorentz transformations. The time component is given by the speed of light cc as

η0=dx0dτ=cdtdτ=c1−v2/c2=γc\eta^{0}=\frac{dx^{0}}{d\tau}=c \frac{dt}{d\tau}=\frac{c}{\sqrt{ 1- v^{2}/c^{2} }}=\gamma c

and the space components are

η=v1−v2/c2=γv\boldsymbol{\eta}=\frac{\mathbf{v}}{\sqrt{ 1-v^{2}/c^{2} }}=\gamma \mathbf{v}

When placed under Lorentz transformation, we get

ηˉμ=Λνμην\bar{\eta}^{\mu}=\Lambda_{\nu}^{\mu}\eta^{\nu}

By comparison, ordinary velocity must transform according to the Einstein velocity addition rule, which is much more verbose.