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

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 expresson 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 by defining vector ${\mathfrak {\dot {p}}}$:

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

Laue (1908)

w:Max von Laue wrote the transformation for three-acceleration 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 formulas in terms of motions in z-direction:

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

Tamaki (1913)

w:Kajuro Tamaki was the first to write the formula in three-vector notation:

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

and went on to decompose the acceleration into two component vector, one in the direction of $\mathbf {v}$ and the other perpendicular to it:

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

References

• 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

• 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

• Sur la dynamique de l’électron on French Wikisource
• 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