History of Topics in Special Relativity/Three-acceleration

History of three-acceleration transformation
The Lorentz transformation of three-acceleration is given by


 * a) $$\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\quad a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\quad a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\

\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right] \end{matrix}$$

or in vector notation in arbitrary directions


 * b) $$\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\

\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right] \end{matrix}$$

Equations a) were given by, , , , , while the vector notation b) was given by.

Poincaré (1905/06)
Henri Poincaré (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:


 * $$\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}$$

where $$\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}$$, $$k=\gamma$$, $$\epsilon=v$$, $$\mu=1+\xi\epsilon=1+u_{x}v$$.

Einstein (1907/08)
Albert Einstein (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):


 * $$\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}$$.

Abraham (1908)
Max Abraham derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:


 * $$\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} & =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\

\mathfrak{\dot{q}}_{y}^{\prime} & =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\ \mathfrak{\dot{q}}_{z}^{\prime} & =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}, \end{aligned} $$

or simplified using three-vector $$\mathfrak{\dot{p}}$$:


 * $$\begin{matrix}\mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\qquad\varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\qquad\varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}\\

\left(\mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}\right) \end{matrix}$$

Laue (1908)
Max von Laue wrote the transformation in two dimensions x,y as follows:


 * $$\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} & =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, & \mathfrak{\dot{q}}_{y}^{\prime} & =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}

$$

Brill (1909)
Alexander von Brill wrote the transformation in which the primed frame moves in z-direction while the x-axis is perpendicular:


 * $$\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\

\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}} \end{matrix}$$

Tamaki (1913)
Kajuro Tamaki was the first to formulate the transformation as a single three-vector formula:


 * $$\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}$$

which he split into two parts: the first in the direction of $$\mathbf{v}$$ and the other one perpendicular to it:


 * $$\begin{aligned}\mathbf{a}_{v}^{\prime} & =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\

\mathbf{a}_{\bar{v}}^{\prime} & =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}} \end{aligned} $$