History of Topics in Special Relativity/Three-acceleration

The Lorentz transformation of three-acceleration is given by

a) ${\displaystyle {\begin{matrix}a_{x}^{\prime }={\frac {a_{x}}{\gamma ^{3}\mu ^{3}}},\quad a_{y}^{\prime }={\frac {a_{y}}{\gamma ^{2}\mu ^{2}}}+{\frac {a_{x}u_{y}v}{c^{2}\gamma ^{2}\mu ^{3}}},\quad a_{z}^{\prime }={\frac {a_{z}}{\gamma ^{2}\mu ^{2}}}+{\frac {a_{x}u_{z}v}{c^{2}\gamma ^{2}\mu ^{3}}}\\\left[\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mu =1-{\frac {u_{x}v}{c^{2}}}\right]\end{matrix}}}$

or in vector notation in arbitrary directions

b) ${\displaystyle {\begin{matrix}\mathbf {a} '={\frac {\mathbf {a} }{\gamma ^{2}\mu ^{2}}}-{\frac {\mathbf {(a\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma ^{3}\mu ^{3}}}+{\frac {\mathbf {(a\cdot v)u} }{c^{2}\gamma ^{2}\mu ^{3}}}\\\left[\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mu =1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right]\end{matrix}}}$

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).

w:Henri Poincaré (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:^{[R 1]}

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

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

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):^{[R 2]}

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

w:Max Abraham derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:^{[R 3]}

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

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

${\displaystyle {\begin{matrix}{\mathfrak {\dot {q}}}_{x}^{\prime }={\mathfrak {\dot {p}}}_{x},\qquad \varkappa {\mathfrak {\dot {q}}}_{y}^{\prime }={\mathfrak {\dot {p}}}_{y},\qquad \varkappa {\mathfrak {\dot {q}}}_{z}^{\prime }={\mathfrak {\dot {p}}}_{z}\\\left({\mathfrak {\dot {p}}}={\frac {{\mathfrak {\dot {q}}}\varkappa ^{3}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{2}}}+{\frac {{\mathfrak {q}}\beta {\mathfrak {\dot {q}}}_{x}\varkappa ^{3}}{\left(1-\beta {\mathfrak {q}}_{x}\right)^{3}}}\right)\end{matrix}}}$

w:Max von Laue wrote the transformation in two dimensions x,y as follows:^{[R 4]}

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

w:Alexander von Brill wrote the transformation in which the primed frame moves in z-direction while the x-axis is perpendicular:^{[R 5]}

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

w:Kajuro Tamaki was the first to formulate the transformation as a single three-vector formula:^{[R 6]}

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

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

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

• Abraham, M. (1908), Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung; 2. Auflage, Leipzig: Teubner
• Brill, A. (1909), Vorlesungen zur Einführung in die Mechanik raumerfüllender Massen, Leipzig: B.G. Teubner
• Einstein, A. (1908) [1907], "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen (The Collected Papers of Albert Einstein Vol. 2)", Jahrbuch der Radioaktivität und Elektronik, 4: 411–462, Bibcode:1905AnP...322..891E, doi:10.1002/andp.19053221004. See also: English translation in "CPAE Vol. 2".
• Laue, M. v. (1908), "Die Wellenstrahlung einer bewegten Punktladung nach dem Relativitätsprinzip", Berichte der Deutschen Physikalischen Gesellschaft: 838–844

See also the transcription The Wave Radiation of a Moving Point Charge in Accordance with the Principle of Relativity on English Wikisource

• Poincaré, H. (1906) [1905], "Sur la dynamique de l'électron", Rendiconti del Circolo Matematico di Palermo, 21: 129–176

See also the transcription Sur la dynamique de l’électron on French Wikisource

See also the transcription On the Dynamics of the Electron on English Wikisource

• Tamaki, K. (1913), "On the symmetrical expressions of relations between the physical quantities in a stationary and moving system", Memoirs of the College of Science and Engineering Kyôto Imperial University, 5: 235–252