History of Topics in Special Relativity/Four-acceleration

Overview
The four-acceleration follows by differentiation of the four-velocity $$U^{\mu}$$ of a particle with respect to the particle's proper time $$\tau$$. It can be represented as a function of three-velocity $$\mathbf{u}$$ and three-acceleration $$\mathbf{u}$$:


 * $$\begin{matrix}A^{\mu} & =\frac{dU^{\mu}}{d\tau} & =\left(\gamma_{u}^{4}\frac{\mathbf{a}\cdot\mathbf{u}}{c},\ \gamma_{u}^{2}\mathbf{a}+\gamma_{u}^{4}\frac{\left(\mathbf{a}\cdot\mathbf{u}\right)}{c^{2}}\mathbf{u}\right)\\

& (a) & (b) \end{matrix},\quad\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$.

and its inner product is equal to the proper acceleration


 * $$\begin{matrix}A^{\mu}A_{\mu} & =\gamma^{4}\left[\mathbf{a}^{2}+\gamma^{2}\left(\frac{\mathbf{u}\cdot\mathbf{a}}{c}\right)^{2}\right]=\left.\mathbf{a}_{0}\right.^{2}\\

& (c) \end{matrix}$$

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 acceleration becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-acceleration 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.

Minkowski (1907/08)
Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. In his lecture on December 1907, he didn't directly define a four-acceleraton vector, but used it implicitly in the definition of four-force and its density in terms of mass density $$\nu$$, mass m, four-velocity w:


 * $$\begin{matrix}\nu\frac{dw_{h}}{d\tau}=K+(w\overline{K})w\\

\nu\frac{d}{d\tau}\frac{dx}{d\tau}=X,\ \nu\frac{d}{d\tau}\frac{dy}{d\tau}=Y,\ \nu\frac{d}{d\tau}\frac{dz}{d\tau}=Z,\ \nu\frac{d}{d\tau}\frac{dt}{d\tau}=T\\ \hline m\frac{d}{d\tau}\frac{dx}{d\tau}=R_{x},\ m\frac{d}{d\tau}\frac{dy}{d\tau}=R_{y},\ m\frac{d}{d\tau}\frac{dz}{d\tau}=R_{z},\ m\frac{d}{d\tau}\frac{dt}{d\tau}=R_{t} \end{matrix}\text{ }$$

corresponding to (a).

In 1908, he denoted the derivative of the motion vector (four-velocity) with respect to proper time as "acceleration vector":


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

corresponding to (a).

Frank (1909)
Philipp Frank (1909) didn't explicitly mentions four-acceleration as vector, though he used its components while defining four-force (X,Y,Z,T):


 * $$\begin{align}\frac{d}{d\sigma}\frac{dt}{d\sigma} & =\frac{1}{\left(1-w^{2}\right)^{2}}\left(w_{x}\frac{dw_{x}}{dt}+w_{y}\frac{dw_{y}}{dt}+w_{z}\frac{dw_{z}}{dt}\right)\\

\frac{d^{2}x}{d\sigma^{2}} & =\frac{1}{1-w^{2}}\frac{dw_{x}}{dt}+\frac{w_{x}}{\left(1-w^{2}\right)^{2}}\left(w_{x}\frac{dw_{x}}{dt}+w_{y}\frac{dw_{y}}{dt}+w_{z}\frac{dw_{z}}{dt}\right)\\ & {\rm etc}. \end{align} $$

corresponding to (a, b).

Bateman (1909/10)
The first discussion in an English language paper of four-acceleration (even though in the broader context of spherical wave transformations), was given by Harry Bateman in a paper read 1909 and published 1910. He first defined four-velocity


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

from which he derived four-acceleration


 * $$\frac{dw_{1}}{ds}=\frac{\dot{w}_{x}}{1-w^{2}}+\frac{w_{x}(w\dot{w})}{\left(1-w^{2}\right)^{2}},\dots$$

equivalent to (a, b) as well as its inner product


 * $$\begin{align}\left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2} & =\frac{\dot{w}^{2}}{\left(1-w^{2}\right)^{2}}+\frac{(w\dot{w})^{2}}{\left(1-w^{2}\right)^{3}}+\frac{w^{2}(w\dot{w})^{2}}{\left(1-w^{2}\right)^{4}}-\frac{(w\dot{w})^{2}}{\left(1-w^{2}\right)^{4}}\\

& =\frac{\dot{w}^{2}}{\left(1-w^{2}\right)^{2}}+\frac{(w\dot{w})^{2}}{\left(1-w^{2}\right)^{3}} \end{align} $$

equivalent to (c). He also defined the four-jerk


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

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 acceleration” as a “1-vector”, its norm, and its relation to the “extended force”:


 * $$\begin{matrix}\mathbf{c}=\frac{d\mathbf{w}}{ds}=\frac{dx_{4}}{ds}\frac{d\mathbf{w}}{dx_{4}}=\frac{1}{1-v^{2}}\frac{d\mathbf{v}}{dx_{4}}+\frac{\mathbf{v}+\mathbf{k}_{4}}{\left(1-v^{2}\right)^{2}}v\frac{dv}{dx_{4}}\\

\mathbf{c}=\frac{1}{1-v^{2}}\frac{d\mathbf{v}}{dt}+\frac{\mathbf{v}+\mathbf{k}_{4}}{\left(1-v^{2}\right)^{2}}v\frac{dv}{dt}\\ \mathbf{c}=\frac{\mathbf{u}\frac{dv}{dt}}{\left(1-v^{2}\right)^{2}}+\frac{v\frac{d\mathbf{u}}{dt}}{1-v^{2}}+\frac{v\mathbf{k}_{4}\frac{dv}{dt}}{\left(1-v^{2}\right)^{2}}\\ \hline \begin{align}\sqrt{\mathbf{c}\cdot\mathbf{c}} & =\left[\frac{\left(\frac{dv}{dt}\right)^{2}}{\left(1-v^{2}\right)^{3/2}}+\frac{v^{2}\frac{d\mathbf{u}}{dt}\cdot\frac{d\mathbf{u}}{dt}}{\left(1-v^{2}\right)^{2}}\right]^{1/2}\\ & =\frac{1}{1-v^{2}}\left[\dot{\mathbf{v}}\dot{\cdot\mathbf{v}}+\frac{1}{1-v^{2}}\left(\mathbf{v}\dot{\cdot\mathbf{v}}\right)^{2}\right]^{1/2}\\ & =\frac{1}{\left(1-v^{2}\right)^{3/2}}\left[\dot{\mathbf{v}}\dot{\cdot\mathbf{v}}-\left(\mathbf{v}\times\dot{\mathbf{v}}\right)\cdot\left(\mathbf{v}\times\dot{\mathbf{v}}\right)\right]^{1/2} \end{align} \\ \hline m_{0}\mathbf{c}=\frac{dm_{0}\mathbf{w}}{ds}=\frac{dmv}{ds}\mathbf{k}_{1}+\frac{dm}{ds}\mathbf{k}_{4}=\frac{1}{\sqrt{1-v^{2}}}\left(\frac{dmv}{dt}\mathbf{k}_{1}+\frac{dm}{dt}\mathbf{k}_{4}\right) \end{matrix}$$

equivalent to (a,b).

Kottler (1912)
Friedrich Kottler defined four-acceleration in terms of four-velocity V as:


 * $$\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),\\ & \dot{\ \mathfrak{v}}=\mathfrak{\dot{\mathfrak{v}}_{\Vert}}+\dot{\mathfrak{v}}_{\bot} \end{align} $$

equivalent to (a,b). He related its inner product to curvature $$R_1$$ (in terms of Frenet-Serret formulas) and the “Minkowski acceleration” b:


 * $$\begin{matrix}\frac{c^{4}}{R_{1}^{2}}=\left(\frac{dV}{ds}\right)^{2}c^{4}=\sum\left(\frac{d^{2}x}{d\tau^{2}}\right)^{2}=\frac{\dot{\mathfrak{v}}_{\bot}^{2}}{\left(1-\mathfrak{v}^{2}/c^{2}\right)^{2}}+\frac{\dot{\mathfrak{v}}_{\Vert}^{2}}{\left(1-\mathfrak{v}^{2}/c^{2}\right)^{3}}\\

b=\sqrt{\sum_{\alpha=1}^{4}\left(\frac{d^{2}x^{(\alpha)}}{d\tau^{2}}\right)^{2}}=-c^{2}\sqrt{\sum_{\alpha=1}^{4}\left(\frac{d^{2}x^{(\alpha)}}{d\tau^{2}}\right)^{2}}=-\frac{c^{2}}{\mathrm{R}_{1}} \end{matrix}$$

equivalent to (c) and defined the four-jerk


 * $$-ic^{3}\frac{d^{2}V}{ds^{2}}=\frac{d^{3}x}{d\tau^{3}}=(\ddot{\mathfrak{v}},0)\frac{1}{\left(\sqrt{1-\mathfrak{v}^{2}/c^{2}}\right)^{3}}+(\ddot{\mathfrak{v}},0)\frac{3\frac{\mathfrak{v}\mathfrak{\dot{v}}}{c^{2}}}{\left(\sqrt{1-\mathfrak{v}^{2}/c^{2}}\right)^{5}}+(\mathfrak{v},ic)\left\{ \frac{\mathfrak{\dot{v}}^{2}/c^{2}+\frac{\mathfrak{v}\mathfrak{\ddot{v}}}{c^{2}}}{\left(\sqrt{1-\mathfrak{v}^{2}/c^{2}}\right)^{5}}+\frac{4\left(\frac{\mathfrak{v}\mathfrak{\dot{v}}}{c^{2}}\right)^{2}}{\left(\sqrt{1-\mathfrak{v}^{2}/c^{2}}\right)^{7}}\right\} $$

Von Laue (1912/13)
Max von Laue explicitly used the term “four-acceleration” (Viererbeschleunigung) for $$\dot{Y}$$ and defined its inner product, and its relation to four-force K as well:


 * $$\begin{matrix}\dot{Y}=\frac{dY}{d\tau},\quad|\dot{Y|}=\frac{1}{c}|\dot{\mathfrak{q}}^{0}|\\

K=m\frac{dY}{d\tau} \end{matrix}$$

corresponding to (a, c).

Silberstein (1914)
While Ludwik Silberstein used Biquaternions already in 1911, his first mention of the “acceleration-quaternion” Z was given in 1914. He also defined its conjugate, its Lorentz transformation, the relation of four-velocity Y, and its relation to four-force X:


 * $$\begin{matrix}Z=\frac{dY}{d\tau}=\frac{\iota}{c}\gamma\left(\mathbf{pa}'\right)+\epsilon\mathbf{a}\\

ZY_{c}+YZ_{c}=0\\ m\frac{dY}{d\tau}=mZ=X \end{matrix}$$

equivalent to (a,b).