Four-velocity

Time dilation

From time dilation, the differentials in coordinate time t and proper time τ are related by $${\displaystyle dt=\gamma (u)d\tau }$$ where the Lorentz factor, $${\displaystyle \gamma (u)={\frac {1}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\,,}$$ is a function of the Euclidean norm u of the 3d velocity vector ${\displaystyle {\vec {u}}}$: $${\displaystyle u=\left\|\ {\vec {u}}\ \right\|={\sqrt {\left(u^{1}\right)^{2}+\left(u^{2}\right)^{2}+\left(u^{3}\right)^{2}}}\,.}$$

Definition of the four-velocity

The four-velocity is the tangent four-vector of a timelike world line. The four-velocity ${\displaystyle \mathbf {U} }$ at any point of world line ${\displaystyle \mathbf {X} (\tau )}$ is defined as: $${\displaystyle \mathbf {U} ={\frac {d\mathbf {X} }{d\tau }}}$$ where ${\displaystyle \mathbf {X} }$ is the four-position and ${\displaystyle \tau }$ is the proper time.^[1]

The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined for tachyonic world lines, where the tangent vector is spacelike.

Components of the four-velocity

The relationship between the time t and the coordinate time x⁰ is defined by $${\displaystyle x^{0}=ct.}$$

Taking the derivative of this with respect to the proper time τ, we find the U^μ velocity component for μ = 0: $${\displaystyle U^{0}={\frac {dx^{0}}{d\tau }}={\frac {d(ct)}{d\tau }}=c{\frac {dt}{d\tau }}=c\gamma (u)}$$

and for the other 3 components to proper time we get the U^μ velocity component for μ = 1, 2, 3: $${\displaystyle U^{i}={\frac {dx^{i}}{d\tau }}={\frac {dx^{i}}{dt}}{\frac {dt}{d\tau }}={\frac {dx^{i}}{dt}}\gamma (u)=\gamma (u)u^{i}}$$ where we have used the chain rule and the relationships $${\displaystyle u^{i}={dx^{i} \over dt}\,,\quad {\frac {dt}{d\tau }}=\gamma (u)}$$

Thus, we find for the four-velocity ${\displaystyle \mathbf {U} }$: $${\displaystyle \mathbf {U} =\gamma {\begin{bmatrix}c\\{\vec {u}}\\\end{bmatrix}}.}$$

Written in standard four-vector notation this is: $${\displaystyle \mathbf {U} =\gamma \left(c,{\vec {u}}\right)=\left(\gamma c,\gamma {\vec {u}}\right)}$$ where ${\displaystyle \gamma c}$ is the temporal component and ${\displaystyle \gamma {\vec {u}}}$ is the spatial component.

In terms of the synchronized clocks and rulers associated with a particular slice of flat spacetime, the three spacelike components of four-velocity define a traveling object's proper velocity ${\displaystyle \gamma {\vec {u}}=d{\vec {x}}/d\tau }$ i.e. the rate at which distance is covered in the reference map frame per unit proper time elapsed on clocks traveling with the object.

Unlike most other four-vectors, the four-velocity has only 3 independent components ${\displaystyle u_{x},u_{y},u_{z}}$ instead of 4. The ${\displaystyle \gamma }$ factor is a function of the three-dimensional velocity ${\displaystyle {\vec {u}}}$.

When certain Lorentz scalars are multiplied by the four-velocity, one then gets new physical four-vectors that have 4 independent components.

For example:

• Four-momentum: $${\displaystyle \mathbf {P} =m_{o}\mathbf {U} =\gamma m_{o}\left(c,{\vec {u}}\right)=m\left(c,{\vec {u}}\right)=\left(mc,m{\vec {u}}\right)=\left(mc,{\vec {p}}\right)=\left({\frac {E}{c}},{\vec {p}}\right),}$$ where ${\displaystyle m_{o}}$ is the rest mass
• Four-current density: $${\displaystyle \mathbf {J} =\rho _{o}\mathbf {U} =\gamma \rho _{o}\left(c,{\vec {u}}\right)=\rho \left(c,{\vec {u}}\right)=\left(\rho c,\rho {\vec {u}}\right)=\left(\rho c,{\vec {j}}\right),}$$ where ${\displaystyle \rho _{o}}$ is the charge density

Effectively, the ${\displaystyle \gamma }$ factor combines with the Lorentz scalar term to make the 4th independent component $${\displaystyle m=\gamma m_{o}}$$ and $${\displaystyle \rho =\gamma \rho _{o}.}$$

Magnitude

Using the differential of the four-position in the rest frame, the magnitude of the four-velocity can be obtained by the Minkowski metric with signature (−, +, +, +): $${\displaystyle \left\|\mathbf {U} \right\|^{2}=\eta _{\mu \nu }U^{\mu }U^{\nu }=\eta _{\mu \nu }{\frac {dX^{\mu }}{d\tau }}{\frac {dX^{\nu }}{d\tau }}=-c^{2}\,,}$$ in short, the magnitude of the four-velocity for any object is always a fixed constant: $${\displaystyle \left\|\mathbf {U} \right\|^{2}=-c^{2}}$$

In a moving frame, the same norm is: $${\displaystyle \left\|\mathbf {U} \right\|^{2}={\gamma (u)}^{2}\left(-c^{2}+{\vec {u}}\cdot {\vec {u}}\right),}$$ so that: $${\displaystyle -c^{2}={\gamma (u)}^{2}\left(-c^{2}+{\vec {u}}\cdot {\vec {u}}\right),}$$

which reduces to the definition of the Lorentz factor.