# History of Topics in Special Relativity/Four-velocity

## Overview

The w:four-velocity is defined as the rate of change in the w:four-position of a particle with respect to the particle's w:proper time $\tau$ , while the derivative of four-velocity with respect to proper time is four-acceleration), and the product of four-velocity and mass is four-momentum. It can be represented as a function of three-velocity $\mathbf {u}$ :

${\begin{matrix}U^{\mu }&={\frac {\mathrm {d} X^{\mu }}{\mathrm {d} \tau }}&=\gamma \left(c,\ \mathbf {u} \right)\\&(a)&(b)\end{matrix}},\quad \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ .

## Historical notation

### Killing (1884/5)

w:Wilhelm Killing discussed Newtonian mechanics in non-Euclidean spaces by expressing coordinates (p,x,y,z), velocity v=(p',x',y',z'), acceleration (p”,x”,y”,z”) in terms of four components, obtaining the following relations:[M 1]

${\begin{matrix}p',x',y',z'\\\hline k^{2}p^{2}+x^{2}+y^{2}+z^{2}=k^{2}\\k^{2}pp'+xx'+yy'+zz'=0\\k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}+k^{2}pp''+xx''+yy''+zz''=0\\\hline v^{2}=k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\v^{2}+k^{2}pp''+xx''+yy''+zz''=0\\{\frac {1}{2}}{\frac {d\left(v^{2}\right)}{dt}}=k^{2}p'p''+x'x''+y'y''+z'z''\end{matrix}}$ If the Gaussian curvature $1/k^{2}$ (with k as radius of curvature) is negative the velocity becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-velocity in Minkowski space by setting $k^{2}=-c^{2}$ with c as speed of light. However, Killing obtained his results by differentiation with respect to Newtonian time t, not relativistic proper time, so his expressions aren't relativistic four-vectors in the first place, in particular they don't involve a limiting speed. Also the dot product of acceleration and velocity differs from the relativistic result.

### Poincaré (1905/6)

w:Henri Poincaré explicitly defined the four-force as:[R 1]

$k_{0}\xi ,\quad k_{0}\eta ,\quad k_{0}\zeta ,\quad k_{0}$ with $k_{0}={\frac {1}{\sqrt {1-\sum \xi ^{2}}}}$ which is equivalent to (b) because

$\left[\xi ,\eta ,\zeta \right]={\frac {\mathbf {u} }{c}},\ \Sigma \xi ^{2}={\frac {\mathbf {u} \cdot \mathbf {u} }{c^{2}}}$ ### Minkowski (1907/8)

w:Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. He defined the “velocity vector” (Geschwindigkeitsvektor or Raum-Zeit-Vektor Geschwindigkeit):[R 2]

${\begin{matrix}w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}\\w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}=-1\end{matrix}}{\text{ }}$ which is equivalent to (b), which he used to form further four-vectors using electromagnetic quantities etc.:

{\begin{aligned}wF&=-\Phi &&{\text{electric rest force}}\\wF^{*}&=-i\mu \Psi =\mu wf^{\ast }&&{\text{magnetic rest force}}\\\Omega &=iw[\Phi \Psi ]^{\ast }&&{\text{rest ray}}\\-w{\overline {s}}&=\varrho '&&{\text{rest density}}\\s+(w{\overline {s}})w&=-\sigma wF&&{\text{rest current}}\\\nu {\frac {dw_{h}}{d\tau }}&=K+(w{\overline {K}})w&&{\text{ponderomotive force density}}\end{aligned}} In 1908, he denoted the derivative of the position vector[R 3]

${\dot {x}},{\dot {y}},{\dot {z}},{\dot {t}}$ corresponding to (a). He went on to define the derivative of four-velocity with respect to proper time as "acceleration vector" (i.e. four-acceleration), and the product of four-velocity and mass as “momentum vector” (i.e. four-momentum).

### Bateman (1909/10)

The first discussion of four-velocity in an English language paper (even though in the broader context of w:spherical wave transformations), was given by w:Harry Bateman in a paper read 1909 and published 1910:[R 4]

$w_{1}={\frac {w_{x}}{\sqrt {1-w^{2}}}},\ w_{2}={\frac {w_{y}}{\sqrt {1-w^{2}}}},\ w_{3}={\frac {w_{z}}{\sqrt {1-w^{2}}}},\ w_{4}={\frac {1}{\sqrt {1-w^{2}}}},$ equivalent to (b), from which he derived four-acceleration and four-jerk:

${\begin{matrix}{\frac {dw_{1}}{ds}}={\frac {{\dot {w}}_{x}}{1-w^{2}}}+{\frac {w_{x}(w{\dot {w}})}{\left(1-w^{2}\right)^{2}}},\dots \\{\frac {d^{2}w_{1}}{ds^{2}}}={\frac {{\ddot {w}}_{x}}{\left(1-w^{2}\right)^{\frac {1}{2}}}}+{\frac {3{\dot {w}}_{x}(w{\dot {w}})}{\left(1-w^{2}\right)^{\frac {1}{2}}}}+{\frac {w_{x}}{\left(1-w^{2}\right)^{\frac {1}{2}}}}\left\{w{\ddot {w}}+{\frac {3(w{\dot {w}})^{2}}{1-w^{2}}}+{\dot {w}}^{2}+{\frac {(w{\dot {w}})^{2}}{1-w^{2}}}\right\},\dots \end{matrix}}$ ### Ignatowski (1910)

w:Wladimir Ignatowski defined the “vector of first kind”:[R 5]

$\left({\frac {\mathfrak {v}}{\sqrt {1-n{\mathfrak {v}}^{2}}}},\ {\frac {1}{\sqrt {1-n{\mathfrak {v}}^{2}}}}\right)$ equivalent to (b).

### Laue (1911)

In the first textbook on relativity, w:Max von Laue explicitly used the term “four-velocity” (Vierergeschwindigkeit) for Y:[R 6]

Y\Rightarrow {\begin{aligned}Y_{x}&={\frac {{\mathfrak {q}}_{x}}{\sqrt {c^{2}-q^{2}}}},&Y_{y}&={\frac {{\mathfrak {q}}_{y}}{\sqrt {c^{2}-q^{2}}}},\\Y_{z}&={\frac {{\mathfrak {q}}_{z}}{\sqrt {c^{2}-q^{2}}}},&Y_{l}&={\frac {ic}{\sqrt {c^{2}-q^{2}}}},\ \left({\text{or}}\ Y_{u}={\frac {c}{\sqrt {c^{2}-q^{2}}}}\right),\end{aligned}}\Rightarrow |Y|=i which is equivalent to (b), which he used to formulate the four-potential $\Phi$ of an arbitrarily moving point charge de, as well as the four-convection and four-conduction using four-current P:[R 7]

${\begin{matrix}\Phi ={\frac {de}{4\pi }}{\frac {Y}{(\Pi Y)}}\\K=-(YP)Y\\\Lambda =P+(YP)Y\end{matrix}}$ ,

and the vector products with electromagnetic tensor ${\mathfrak {M}}$ and displacement tensor ${\mathfrak {B}}$ and their duals:[R 8]

{\begin{aligned}[][Y{\mathfrak {B}}]&=\varepsilon [Y{\mathfrak {M}}]\\{}[Y{\mathfrak {M}}^{\ast }]&=\mu [Y{\mathfrak {B}}^{\ast }]\end{aligned}} In the second edition (1913), Laue used four-velocty Y in order to define the four-acceleration ${\dot {Y}}$ , four force K in terms of four-acceleration, and material four-current M (i.e. four-momentum density) using rest mass density $k^{0}$ :[R 9]

${\begin{matrix}{\dot {Y}}={\frac {dY}{d\tau }},\quad |{\dot {Y|}}={\frac {1}{c}}|{\dot {\mathfrak {q}}}^{0}|\\{\frac {d(mY)}{d\tau }}\Rightarrow m{\frac {dY}{d\tau }}=K\\M=k^{0}Y\end{matrix}}$ ,

### De Sitter (1911)

w:Willem De Sitter defined the four-velocity in terms of both velocity $\phi$ and w:Proper velocity $(\phi )$ as:[R 10]

${\begin{matrix}(\xi ),(\eta ),(\zeta ),(\kappa )\\\hline (\xi )={\frac {dx}{cd\tau }},\ (\eta )={\frac {dy}{cd\tau }},\ (\zeta )={\frac {dz}{cd\tau }},\ (\kappa )={\frac {dct}{cd\tau }}\\\phi {}^{2}=\xi {}^{2}+\eta {}^{2}+\zeta {}^{2},\quad (\phi )^{2}=(\xi )^{2}+(\eta )^{2}+(\zeta )^{2}\\(\kappa )^{2}-(\phi )^{2}=1\\\left({\frac {d\tau }{dt}}={\sqrt {1-\phi ^{2}}}={\frac {1}{(\kappa )}},\ {\frac {dt}{d\tau }}=(\kappa )={\sqrt {1+(\phi )^{2}}}\right)\end{matrix}}$ equivalent to (b).

### Wilson/Lewis (1912)

w:Gilbert Newton Lewis and w:Edwin Bidwell Wilson devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They defined the “extended velocity” as a “1-vector”:[R 11]

$\mathbf {w} ={\frac {1}{\sqrt {1-v^{2}}}}\left(\mathbf {k} _{1}{\frac {dx_{1}}{dx_{4}}}+\mathbf {k} _{2}{\frac {dx_{2}}{dx_{4}}}+\mathbf {k} _{4}\right)={\frac {\mathbf {v} +\mathbf {k} _{4}}{\sqrt {1-v^{2}}}}$ equivalent to (a,b).

### Kottler (1912)

w:Friedrich Kottler defined four-velocity as:[R 12]

{\begin{aligned}V^{(1)}&={\frac {{\mathfrak {v}}_{z}}{ic}}{\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},&V^{(2)}&={\frac {{\mathfrak {v}}_{y}}{ic}}{\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},\\V^{(3)}&={\frac {{\mathfrak {v}}_{x}}{ic}}{\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},&V^{(4)}&={\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},\end{aligned}}\quad \left[\sum _{\alpha =1}^{4}\left(V^{(\alpha )}\right)^{2}=1\right] equivalent to (a,b) from which he derived four-acceleration and four-jerk, and demonstrated its relation to the tangent c_{1}^{(\alpha)} in terms of Frenet-Serret formulas, its derivative with respect to proper time, and its relation to four-acceleration and curvature $1/R_{1}$ :[R 13]

${\begin{matrix}{\frac {dx^{(\alpha )}}{ds}}=c_{1}^{(\alpha )}=V^{(\alpha )},\ {\frac {d^{2}x^{(\alpha )}}{ds^{2}}}={\frac {dc_{1}^{(\alpha )}}{ds}}={\frac {c_{2}^{(\alpha )}}{\mathrm {R} _{1}}}={\frac {dV^{(\alpha )}}{ds}},\quad \alpha =1,2,3\\\sum _{\alpha =1}^{4}\left({\frac {d^{2}x^{(\alpha )}}{ds^{2}}}\right)^{2}=\left({\frac {1}{\mathrm {R} _{1}}}\right)^{2}=\left({\frac {dV}{ds}}\right)^{2}\end{matrix}}$ and derived four-acceleration and four-jerk:[R 14]

{\begin{aligned}-c^{2}{\frac {dV}{ds}}&={\frac {d^{2}x}{d\tau ^{2}}}=({\dot {\mathfrak {v}}},0){\frac {1}{1-{\mathfrak {v}}^{2}/c^{2}}}+({\mathfrak {v}},ic){\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}/c^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}=\\&=({\dot {\mathfrak {v}}}_{\bot },0){\frac {1}{1-{\mathfrak {v}}^{2}/c^{2}}}+({\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}},0){\frac {1}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}+\left(0,{\frac {i}{c}}{\frac {{\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}}{\mathfrak {v}}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}\right),\\-ic^{3}{\frac {d^{2}V}{ds^{2}}}=&{\frac {d^{3}x}{d\tau ^{3}}}=({\ddot {\mathfrak {v}}},0){\frac {1}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{3}}}+({\ddot {\mathfrak {v}}},0){\frac {3{\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}}{c^{2}}}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+({\mathfrak {v}},ic)\left\{{\frac {{\mathfrak {\dot {v}}}^{2}/c^{2}+{\frac {{\mathfrak {v}}{\mathfrak {\ddot {v}}}}{c^{2}}}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+{\frac {4\left({\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}}{c^{2}}}\right)^{2}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{7}}}\right\}\\&\left[{\dot {\mathfrak {v}}}={\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}}+{\dot {\mathfrak {v}}}_{\bot }\right]\end{aligned}} ### Einstein (1912-14)

In an unpublished manuscript on special relativity, written between 1912-1914, w:Albert Einstein defined four-velocity as:[R 15]

{\begin{matrix}\left(G_{\mu }\right)=\left({\frac {dx_{\mu }}{\sqrt {-\sum dx_{\sigma }^{2}}}}\right)\\\hline {\begin{aligned}G_{1}&={\frac {{\mathfrak {q}}_{x}}{\sqrt {c^{2}-q^{2}}}}\\&\dots \\G_{4}&={\frac {ic}{\sqrt {c^{2}-q^{2}}}}\end{aligned}}\end{matrix}} equivalent to (b) and used it to define four-momentum by multiplication with rest energy:[R 16]

${\frac {{\bar {\eta }}_{0}}{c}}\left(G_{\mu }\right)$ ### Silberstein (1914)

While w:Ludwik Silberstein used Biquaternions already in 1911, his first mention of the “velocity-quaternion” was given in 1914. He also defined its conjugate, its Lorentz transformation, the relation of four-acceleration Z, and its relation to the equation of motion as follows:[R 17]

${\begin{matrix}Y={\frac {dq}{d\tau }}=\gamma _{p}[\iota c+\mathbf {p} ]\\YY_{c}=-c^{2}\\Y'=QYQ\\Z={\frac {dY}{d\tau }}\\ZY_{c}+YZ_{c}=0\\{\frac {dmY}{d\tau }}=X\end{matrix}}$ equivalent to (a,b).