# History of Topics in Special Relativity/Three-acceleration

History of Topics in Special Relativity :
* Multiple chapter topics
* History of Lorentz transformations
* History of relativistic four vectors
* History of relativistic tensors
* Single page topics
* History of twin paradox
* History of time dilation
* History of three-acceleration transformation
* Müller's history of relativity

## 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 #Poincaré (1905/06), #Einstein (1907/08), #Abraham (1908), #Laue (1908), #Brill (1909), while the vector notation b) was given by #Tamaki (1913).

## History
### Poincaré (1905/06)

w: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)

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

### Abraham (1908)

w: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)

w: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)

w:Alexander von Brill wrote the transformation in which the primed frame moves in z-direction while the x-axis is perpendicular:
    ${\begin{matrix}{\frac {d^{2}x^{\prime }}{dt^{\prime 2}}}={\frac {d}{dt}}{\frac {\dot {x}}{k-kq{\dot {z}}}}\cdot {\frac {1}{\frac {dt'}{dt}}}={\frac {1}{k^{2}}}{\frac {{\ddot {x}}(1-q{\dot {z}})+q{\dot {x}}{\ddot {z}}}{(1-q{\dot {z}})^{3}}}\\{\frac {d^{2}z^{\prime }}{dt^{\prime 2}}}={\frac {d{\mathfrak {v}}_{z'}^{\prime }}{dt'}}={\frac {{\ddot {z}}{\sqrt {1-q^{2}}}^{3}}{(1-q{\dot {z}})^{3}}}\end{matrix}}$

### Tamaki (1913)

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