History of Topics in Special Relativity/Four-velocity

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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 $𝐮$:

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

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:

$\begin{array}{c}{p}^{\prime },x^{\prime },y^{\prime },z^{\prime }\\ k^2{p}^2+x^2+y^2+z^2=k^2\\ k^{2}p{p}^{\prime }+x{x}^{\prime }+y{y}^{\prime }+z{z}^{\prime }=0\\ k^2{p}^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}+k^{2}p{p}^{″}+x{x}^{″}+y{y}^{″}+z{z}^{″}=0\\ v^2=k^2{p}^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\ v^2+k^{2}p{p}^{″}+x{x}^{″}+y{y}^{″}+z{z}^{″}=0\\ \frac{1}{2}\frac{d\left(v^2\right)}{dt}=k^2{p}^{\prime }{p}^{″}+x^{\prime }{x}^{″}+y^{\prime }{y}^{″}+z^{\prime }{z}^{″}\end{array}$

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-velocity as:

$k_0\xi ,k_0\eta ,k_0\zeta ,k_0$ with $k_0=\frac{1}{\sqrt{1-\sum {\xi }^2}}$

which is equivalent to (b) because

$[\xi ,\eta ,\zeta ]=\frac{𝐮}{c},\mathrm{\Sigma }{\xi }^2=\frac{𝐮\cdot 𝐮}{c^2}$

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):

$\begin{array}{c}{w}_1=\frac{{𝔴}_x}{\sqrt{1-{𝔴}^2}},w_2=\frac{{𝔴}_y}{\sqrt{1-{𝔴}^2}},w_3=\frac{{𝔴}_z}{\sqrt{1-{𝔴}^2}},w_4=\frac{i}{\sqrt{1-{𝔴}^2}}\\ w_1^2+w_2^2+w_3^2+w_4^2=-1\end{array}$

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

$\begin{array}{cccc}wF& =-\mathrm{\Phi }& & \text{electric rest force}\\ w{F}^{*}& =-i\mu \mathrm{\Psi }=\mu w{f}^{\ast }& & \text{magnetic rest force}\\ \mathrm{\Omega }& =iw[\mathrm{\Phi }\mathrm{\Psi }{]}^{\ast }& & \text{rest ray}\\ -w\bar{s}& ={\varrho }^{\prime }& & \text{rest density}\\ s+(w\bar{s})w& =-\sigma wF& & \text{rest current}\\ \nu \frac{d{w}_h}{d\tau }& =K+(w\bar{K})w& & \text{ponderomotive force density}\end{array}$

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 w:spherical wave transformations), was given by w: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:

$\begin{array}{c}\frac{d{w}_1}{ds}=\frac{{\dot{w}}_x}{1-w^2}+\frac{w_x(w\dot{w})}{{(1-w^2)}^2},\dots \\ \frac{d^2{w}_1}{d{s}^2}=\frac{{\ddot{w}}_x}{{(1-w^2)}^{\frac{1}{2}}}+\frac{3{\dot{w}}_x(w\dot{w})}{{(1-w^2)}^{\frac{1}{2}}}+\frac{w_x}{{(1-w^2)}^{\frac{1}{2}}}\{w\ddot{w}+\frac{3(w\dot{w}{)}^2}{1-w^2}+{\dot{w}}^2+\frac{(w\dot{w}{)}^2}{1-w^2}\},\dots \end{array}$

Ignatowski (1910)

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

$(\frac{𝔳}{\sqrt{1-n{𝔳}^2}},\frac{1}{\sqrt{1-n{𝔳}^2}})$

equivalent to (b).

Laue (1911)

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

$Y\Rightarrow \begin{array}{cccc}{Y}_x& =\frac{{𝔮}_x}{\sqrt{c^2-q^2}},& Y_y& =\frac{{𝔮}_y}{\sqrt{c^2-q^2}},\\ Y_z& =\frac{{𝔮}_z}{\sqrt{c^2-q^2}},& Y_l& =\frac{ic}{\sqrt{c^2-q^2}},(\text{or}{Y}_u=\frac{c}{\sqrt{c^2-q^2}}),\end{array}\Rightarrow |Y|=i$

which is equivalent to (b), which he used to formulate the four-potential $\mathrm{\Phi }$ of an arbitrarily moving point charge de, as well as the four-convection and four-conduction using four-current P:

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

and the vector products with electromagnetic tensor $𝔐$ and displacement tensor $𝔅$ and their duals:

$\begin{array}{cc}[][Y𝔅]& =\epsilon [Y𝔐]\\ [Y{𝔐}^{\ast }]& =\mu [Y{𝔅}^{\ast }]\end{array}$

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 $k^0$:

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

De Sitter (1911)

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

$\begin{array}{c}(\xi ),(\eta ),(\zeta ),(\kappa )\\ (\xi )=\frac{dx}{cd\tau },(\eta )=\frac{dy}{cd\tau },(\zeta )=\frac{dz}{cd\tau },(\kappa )=\frac{dct}{cd\tau }\\ \varphi {}^2=\xi {}^2+\eta {}^2+\zeta {}^2,(\varphi {)}^2=(\xi {)}^2+(\eta {)}^2+(\zeta {)}^2\\ (\kappa {)}^2-(\varphi {)}^2=1\\ (\frac{d\tau }{dt}=\sqrt{1-{\varphi }^2}=\frac{1}{(\kappa )},\frac{dt}{d\tau }=(\kappa )=\sqrt{1+(\varphi {)}^2})\end{array}$

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”:

$𝐰=\frac{1}{\sqrt{1-v^2}}({𝐤}_1\frac{d{x}_1}{d{x}_4}+{𝐤}_2\frac{d{x}_2}{d{x}_4}+{𝐤}_4)=\frac{𝐯+{𝐤}_4}{\sqrt{1-v^2}}$

equivalent to (a,b).

Kottler (1912)

w:Friedrich Kottler defined four-velocity as:

$\begin{array}{cccc}{V}^{(1)}& =\frac{{𝔳}_z}{ic}\frac{1}{\sqrt{1-{𝔳}^2/c^2}},& V^{(2)}& =\frac{{𝔳}_y}{ic}\frac{1}{\sqrt{1-{𝔳}^2/c^2}},\\ V^{(3)}& =\frac{{𝔳}_x}{ic}\frac{1}{\sqrt{1-{𝔳}^2/c^2}},& V^{(4)}& =\frac{1}{\sqrt{1-{𝔳}^2/c^2}},\end{array}[\sum _{\alpha =1}^4{\left(V^{(\alpha )}\right)}^2=1]$

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 $1/R_1$:

$\begin{array}{c}\frac{d{x}^{(\alpha )}}{ds}=c_1^{(\alpha )}=V^{(\alpha )},\frac{d^2{x}^{(\alpha )}}{d{s}^2}=\frac{d{c}_1^{(\alpha )}}{ds}=\frac{c_2^{(\alpha )}}{{\mathrm{R}}_1}=\frac{d{V}^{(\alpha )}}{ds},\alpha =1,2,3\\ \sum _{\alpha =1}^4{\left(\frac{d^2{x}^{(\alpha )}}{d{s}^2}\right)}^2={\left(\frac{1}{{\mathrm{R}}_1}\right)}^2={\left(\frac{dV}{ds}\right)}^2\end{array}$

and derived four-acceleration and four-jerk:

$\begin{array}{cc}-c^2\frac{dV}{ds}& =\frac{d^{2}x}{d{\tau }^2}=(\dot{𝔳},0)\frac{1}{1-{𝔳}^2/c^2}+(𝔳,ic)\frac{𝔳\dot{𝔳}/c^2}{{(1-{𝔳}^2/c^2)}^2}=\\ & =({\dot{𝔳}}_{\mathrm{\perp }},0)\frac{1}{1-{𝔳}^2/c^2}+({\dot{𝔳}}_{\mathfrak{\Vert }},0)\frac{1}{{(1-{𝔳}^2/c^2)}^2}+(0,\frac{i}{c}\frac{{\dot{𝔳}}_{\mathfrak{\Vert }}𝔳}{{(1-{𝔳}^2/c^2)}^2}),\\ -i{c}^3\frac{d^{2}V}{d{s}^2}=& \frac{d^{3}x}{d{\tau }^3}=(\ddot{𝔳},0)\frac{1}{{\left(\sqrt{1-{𝔳}^2/c^2}\right)}^3}+(\ddot{𝔳},0)\frac{3\frac{𝔳\dot{𝔳}}{c^2}}{{\left(\sqrt{1-{𝔳}^2/c^2}\right)}^5}+(𝔳,ic)\{\frac{{\dot{𝔳}}^2/c^2+\frac{𝔳\ddot{𝔳}}{c^2}}{{\left(\sqrt{1-{𝔳}^2/c^2}\right)}^5}+\frac{4{\left(\frac{𝔳\dot{𝔳}}{c^2}\right)}^2}{{\left(\sqrt{1-{𝔳}^2/c^2}\right)}^7}\}\\ & [\dot{𝔳}={\dot{𝔳}}_{\mathfrak{\Vert }}+{\dot{𝔳}}_{\mathrm{\perp }}]\end{array}$

Einstein (1912-14)

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

$\begin{array}{c}\left(G_{\mu }\right)=\left(\frac{d{x}_{\mu }}{\sqrt{-\sum d{x}_{\sigma }^2}}\right)\\ \begin{array}{cc}{G}_1& =\frac{{𝔮}_x}{\sqrt{c^2-q^2}}\\ & \dots \\ G_4& =\frac{ic}{\sqrt{c^2-q^2}}\end{array}\end{array}$

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

$\frac{{\overline{\eta }}_0}{c}\left(G_{\mu }\right)$

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:

$\begin{array}{c}Y=\frac{dq}{d\tau }={\gamma }_p[\iota c+𝐩]\\ Y{Y}_c=-c^2\\ Y^{\prime }=QYQ\\ Z=\frac{dY}{d\tau }\\ Z{Y}_c+Y{Z}_c=0\\ \frac{dmY}{d\tau }=X\end{array}$

equivalent to (a,b).

References

Mathematical

• {{#section:History of Topics in Special Relativity/mathsource|kil84}}

Relativity

• {{#section:History of Topics in Special Relativity/relsource|bate10elec}}
• {{#section:History of Topics in Special Relativity/relsource|sitter11grav}}
• {{#section:History of Topics in Special Relativity/relsource|einst12manu}}
• {{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
• {{#section:History of Topics in Special Relativity/relsource|kott12mink}}
• {{#section:History of Topics in Special Relativity/relsource|laue11prin}}
• {{#section:History of Topics in Special Relativity/relsource|laue13prin}}
• {{#section:History of Topics in Special Relativity/relsource|lewis12non}}
• {{#section:History of Topics in Special Relativity/relsource|mink07b}}
• {{#section:History of Topics in Special Relativity/relsource|mink08}}
• {{#section:History of Topics in Special Relativity/relsource|silber14quat}}