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 ${\displaystyle \tau }$, 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 ${\displaystyle \mathbf {u} }$:

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

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]

${\displaystyle {\begin{matrix}p',x',y',z'\\\hline k^{2}p^{2}+x^{2}+y^{2}+z^{2}=k^{2}\\k^{2}pp'+xx'+yy'+zz'=0\\k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}+k^{2}pp''+xx''+yy''+zz''=0\\\hline v^{2}=k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\v^{2}+k^{2}pp''+xx''+yy''+zz''=0\\{\frac {1}{2}}{\frac {d\left(v^{2}\right)}{dt}}=k^{2}p'p''+x'x''+y'y''+z'z''\end{matrix}}}$

If the Gaussian curvature ${\displaystyle 1/k^{2}}$ (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 ${\displaystyle k^{2}=-c^{2}}$ 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]

${\displaystyle k_{0}\xi ,\quad k_{0}\eta ,\quad k_{0}\zeta ,\quad k_{0}}$ with ${\displaystyle k_{0}={\frac {1}{\sqrt {1-\sum \xi ^{2}}}}}$

which is equivalent to (b) because

${\displaystyle \left[\xi ,\eta ,\zeta \right]={\frac {\mathbf {u} }{c}},\ \Sigma \xi ^{2}={\frac {\mathbf {u} \cdot \mathbf {u} }{c^{2}}}}$

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]

${\displaystyle {\begin{matrix}w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}\\w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}=-1\end{matrix}}{\text{ }}}$

which is equivalent to (b), which he used to form further four-vectors using electromagnetic quantities etc.:

{\displaystyle {\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]

${\displaystyle {\dot {x}},{\dot {y}},{\dot {z}},{\dot {t}}}$

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]

${\displaystyle w_{1}={\frac {w_{x}}{\sqrt {1-w^{2}}}},\ w_{2}={\frac {w_{y}}{\sqrt {1-w^{2}}}},\ w_{3}={\frac {w_{z}}{\sqrt {1-w^{2}}}},\ w_{4}={\frac {1}{\sqrt {1-w^{2}}}},}$

equivalent to (b), from which he derived four-acceleration and four-jerk:

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

Ignatowski (1910)

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

${\displaystyle \left({\frac {\mathfrak {v}}{\sqrt {1-n{\mathfrak {v}}^{2}}}},\ {\frac {1}{\sqrt {1-n{\mathfrak {v}}^{2}}}}\right)}$

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]

{\displaystyle 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 ${\displaystyle \Phi }$ of an arbitrarily moving point charge de, as well as the four-convection and four-conduction using four-current P:[R 7]

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

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

{\displaystyle {\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 ${\displaystyle {\dot {Y}}}$, four force K in terms of four-acceleration, and material four-current M (i.e. four-momentum density) using rest mass density ${\displaystyle k^{0}}$:[R 9]

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

De Sitter (1911)

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

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

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]

${\displaystyle \mathbf {w} ={\frac {1}{\sqrt {1-v^{2}}}}\left(\mathbf {k} _{1}{\frac {dx_{1}}{dx_{4}}}+\mathbf {k} _{2}{\frac {dx_{2}}{dx_{4}}}+\mathbf {k} _{4}\right)={\frac {\mathbf {v} +\mathbf {k} _{4}}{\sqrt {1-v^{2}}}}}$

equivalent to (a,b).

Kottler (1912)

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

{\displaystyle {\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 ${\displaystyle 1/R_{1}}$:[R 13]

${\displaystyle {\begin{matrix}{\frac {dx^{(\alpha )}}{ds}}=c_{1}^{(\alpha )}=V^{(\alpha )},\ {\frac {d^{2}x^{(\alpha )}}{ds^{2}}}={\frac {dc_{1}^{(\alpha )}}{ds}}={\frac {c_{2}^{(\alpha )}}{\mathrm {R} _{1}}}={\frac {dV^{(\alpha )}}{ds}},\quad \alpha =1,2,3\\\sum _{\alpha =1}^{4}\left({\frac {d^{2}x^{(\alpha )}}{ds^{2}}}\right)^{2}=\left({\frac {1}{\mathrm {R} _{1}}}\right)^{2}=\left({\frac {dV}{ds}}\right)^{2}\end{matrix}}}$

and derived four-acceleration and four-jerk:[R 14]

{\displaystyle {\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]

{\displaystyle {\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]

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

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]

${\displaystyle {\begin{matrix}Y={\frac {dq}{d\tau }}=\gamma _{p}[\iota c+\mathbf {p} ]\\YY_{c}=-c^{2}\\Y'=QYQ\\Z={\frac {dY}{d\tau }}\\ZY_{c}+YZ_{c}=0\\{\frac {dmY}{d\tau }}=X\end{matrix}}}$

equivalent to (a,b).

References

Mathematical
1. Killing (1884/5), p. 5
• Killing, W. (1885) [1884], "Die Mechanik in den Nicht-Euklidischen Raumformen", Journal für die Reine und Angewandte Mathematik, 98: 1–48
Relativity
1. Poincaré (1905/06), p. 173
2. Minkowski (1907/8), p. 64–65,73
3. Minkowski (1908), p. 84
4. Bateman (1909/10), p. 253f
5. Ignatowski (1910), p. 23
6. Laue (1911), p. 63
7. Laue (1911), p. 102, 119
8. Laue (1911), p. 124
9. Laue (1913), p. 69–70, 80, 230
10. De Sitter (1911), p. 392
11. Lewis/Wilson (1912), p. 443f
12. Kottler (1912), p. 1663
13. Kottler (1912), p. 1707
14. Kottler (1912), p. 1663
15. Einstein (1912), p. 84
16. Einstein (1912), p. 97
17. Silberstein (1914), p. 183
• Minkowski, Hermann (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88