History of Topics in Special Relativity/Three-acceleration

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History of three-acceleration transformation

The Lorentz transformation of three-acceleration is given by

a) $\begin{array}{c}{a}_x^{\prime }=\frac{a_x}{{\gamma }^3{\mu }^3},a_y^{\prime }=\frac{a_y}{{\gamma }^2{\mu }^2}+\frac{a_x{u}_{y}v}{c^2{\gamma }^2{\mu }^3},a_z^{\prime }=\frac{a_z}{{\gamma }^2{\mu }^2}+\frac{a_x{u}_{z}v}{c^2{\gamma }^2{\mu }^3}\\ [\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},\mu =1-\frac{u_{x}v}{c^2}]\end{array}$

or in vector notation in arbitrary directions

b) $\begin{array}{c}{𝐚}^{\prime }=\frac{𝐚}{{\gamma }^2{\mu }^2}-\frac{\mathbf{(}𝐚\mathbf{\cdot }𝐯\mathbf{)}𝐯({\gamma }_v-1)}{v^2{\gamma }^3{\mu }^3}+\frac{\mathbf{(}𝐚\mathbf{\cdot }𝐯\mathbf{)}𝐮}{c^2{\gamma }^2{\mu }^3}\\ [\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},\mu =1-\frac{𝐯\mathbf{\cdot }𝐮}{c^2}]\end{array}$

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:

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

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

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):

$\frac{d^2{x}_0^{\prime }}{d{t}^{\prime 2}}=\frac{\frac{d}{dt}\left\{\frac{d{x}_0^{\prime }}{d{t}^{\prime }}\right\}}{\beta (1-\frac{v{x}_0^{\prime }}{c^2})}=\frac{1}{\beta }\frac{(1-\frac{v{\dot{x}}_0}{c^2}){\ddot{x}}_0+({\dot{x}}_0-v)\frac{v{\ddot{x}}_0}{c^2}}{(1-\frac{v{\dot{x}}_0}{c^2})}\text{etc.}$.

Abraham (1908)

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

$\begin{array}{cc}{\dot{𝔮}}_x^{\prime }& =\frac{{\dot{𝔮}}_x{\varkappa }^3}{{(1-\beta {𝔮}_x)}^3},\\ {\dot{𝔮}}_y^{\prime }& =\frac{{\dot{𝔮}}_y{\varkappa }^2}{{(1-\beta {𝔮}_x)}^2}+\frac{{𝔮}_y\beta {\dot{𝔮}}_x{\varkappa }^2}{{(1-\beta {𝔮}_x)}^3},(\varkappa =\sqrt{1-{\beta }^2})\\ {\dot{𝔮}}_z^{\prime }& =\frac{{\dot{𝔮}}_z{\varkappa }^2}{{(1-\beta {𝔮}_x)}^2}+\frac{{𝔮}_z\beta {\dot{𝔮}}_x{\varkappa }^2}{{(1-\beta {𝔮}_x)}^3},\end{array}$

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

$\begin{array}{c}{\dot{𝔮}}_x^{\prime }={\dot{𝔭}}_x,\varkappa {\dot{𝔮}}_y^{\prime }={\dot{𝔭}}_y,\varkappa {\dot{𝔮}}_z^{\prime }={\dot{𝔭}}_z\\ (\dot{𝔭}=\frac{\dot{𝔮}{\varkappa }^3}{{(1-\beta {𝔮}_x)}^2}+\frac{𝔮\beta {\dot{𝔮}}_x{\varkappa }^3}{{(1-\beta {𝔮}_x)}^3})\end{array}$

Laue (1908)

w:Max von Laue wrote the transformation in two dimensions x,y as follows:

$\begin{array}{cccc}{\dot{𝔮}}_x^{\prime }& ={\left(\frac{c\sqrt{c^2-v^2}}{c^2-v{𝔮}_x}\right)}^3{\dot{𝔮}}_x,& {\dot{𝔮}}_y^{\prime }& ={\left(\frac{c\sqrt{c^2-v^2}}{c^2-v{𝔮}_x}\right)}^2({\dot{𝔮}}_y+\frac{v{𝔮}_y{\dot{𝔮}}_x}{c^2-v{𝔮}_x}),\end{array}$

Brill (1909)

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

$\begin{array}{c}\frac{d^2{x}^{\prime }}{d{t}^{\prime 2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot \frac{1}{\frac{d{t}^{\prime }}{dt}}=\frac{1}{k^2}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z}{)}^3}\\ \frac{d^2{z}^{\prime }}{d{t}^{\prime 2}}=\frac{d{𝔳}_{z^{\prime }}^{\prime }}{d{t}^{\prime }}=\frac{\ddot{z}{\sqrt{1-q^2}}^3}{(1-q\dot{z}{)}^3}\end{array}$

Tamaki (1913)

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

${𝐚}^{\prime }=\frac{𝐚-\frac{1}{c^2}[𝐯[𝐯𝐪]]+\frac{1}{\beta }(1-\beta ){𝐯}_1\left({𝐯}_1𝐚\right)}{{\beta }^2{\{1-\frac{1}{c^2}(𝐯𝐪)\}}^3}$

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

$\begin{array}{cc}{𝐚}_v^{\prime }& =\frac{{𝐚}_v-\frac{1}{c^2}\beta {[𝐯[𝐯𝐪]]}_v}{{\beta }^3{\{1-\frac{1}{c^2}(𝐯𝐪)\}}^3}\\ {𝐚}_{\overline{v}}^{\prime }& =\frac{{𝐚}_{\overline{v}}-\frac{1}{c^2}{[𝐯[𝐯𝐪]]}_{\overline{v}}}{{\beta }^2{\{1-\frac{1}{c^2}(𝐯𝐪)\}}^3}\end{array}$

References

• {{#section:History of Topics in Special Relativity/relsource|abra08elek}}
• {{#section:History of Topics in Special Relativity/relsource|brill09}}
• {{#section:History of Topics in Special Relativity/relsource|einst07pri}}
• {{#section:History of Topics in Special Relativity/relsource|laue08}}
• {{#section:History of Topics in Special Relativity/relsource|poinc05b}}
• {{#section:History of Topics in Special Relativity/relsource|tamaki13b}}