History of Topics in Special Relativity/Four-velocity

Overview
The four-velocity is defined as the rate of change in the four-position of a particle with respect to the particle's 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}}}}$$.

Killing (1884/5)
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:


 * $$\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^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}+k^{2}pp+xx+yy+zz=0\\ \hline v^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\ 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)
Henri Poincaré explicitly defined the four-velocity as:


 * $$k_{0}\xi,\quad k_{0}\eta,\quad k_{0}\zeta,\quad k_{0}$$ 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)
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):


 * $$\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{align}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{align} $$

In 1908, he denoted the derivative of the position vector


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

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 spherical wave transformations), was given by Harry Bateman in a paper read 1909 and published 1910:


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

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)
Wladimir Ignatowski defined the “vector of first kind”:


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

equivalent to (b).

Laue (1911)
In the first textbook on relativity, Max von Laue explicitly used the term “four-velocity” (Vierergeschwindigkeit) for Y:


 * $$Y\Rightarrow\begin{align}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{align} \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:


 * $$\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:


 * $$\begin{align}[][Y\mathfrak{B}] & =\varepsilon[Y\mathfrak{M}]\\{}

[Y\mathfrak{M}^{\ast}] & =\mu[Y\mathfrak{B}^{\ast}] \end{align} $$

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 $$k^0$$:


 * $$\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)
Willem De Sitter defined the four-velocity in terms of both velocity $$\phi$$ and Proper velocity $$(\phi)$$ as:


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


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

equivalent to (a,b).

Kottler (1912)
Friedrich Kottler defined four-velocity as:


 * $$\begin{align}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{align} \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 $$1/R_{1}$$:


 * $$\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:


 * $$\begin{align}-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{align} $$

Einstein (1912-14)
In an unpublished manuscript on special relativity, written between 1912-1914, Albert Einstein defined four-velocity as:


 * $$\begin{matrix}\left(G_{\mu}\right)=\left(\frac{dx_{\mu}}{\sqrt{-\sum dx_{\sigma}^{2}}}\right)\\

\hline \begin{align}G_{1} & =\frac{\mathfrak{q}_{x}}{\sqrt{c^{2}-q^{2}}}\\ & \dots\\ G_{4} & =\frac{ic}{\sqrt{c^{2}-q^{2}}} \end{align} \end{matrix}$$

equivalent to (b) and used it to define four-momentum by multiplication with rest energy:


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

Silberstein (1914)
While 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:


 * $$\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).