History of Topics in Special Relativity/Four-acceleration

The w:four-acceleration follows by differentiation of the four-velocity ${\displaystyle U^{\mu }}$ of a particle with respect to the particle's w:proper time ${\displaystyle \tau }$. It can be represented as a function of three-velocity ${\displaystyle \mathbf {u} }$ and three-acceleration ${\displaystyle \mathbf {u} }$:

${\displaystyle {\begin{matrix}A^{\mu }&={\frac {dU^{\mu }}{d\tau }}&=\left(\gamma _{u}^{4}{\frac {\mathbf {a} \cdot \mathbf {u} }{c}},\ \gamma _{u}^{2}\mathbf {a} +\gamma _{u}^{4}{\frac {\left(\mathbf {a} \cdot \mathbf {u} \right)}{c^{2}}}\mathbf {u} \right)\\&(a)&(b)\end{matrix}},\quad \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$.

and its inner product is equal to the proper acceleration

${\displaystyle {\begin{matrix}A^{\mu }A_{\mu }&=\gamma ^{4}\left[\mathbf {a} ^{2}+\gamma ^{2}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right)^{2}\right]=\left.\mathbf {a} _{0}\right.^{2}\\&(c)\end{matrix}}}$

w:Wilhelm Killing discussed Newtonian mechanics in non-Euclidean spaces by expressing coordinates (p,x,y,z), velocity v=(p',x',y',z'), acceleration (p”,x”,y”,z”) in terms of four components, obtaining the following relations:^{[M 1]}

${\displaystyle {\begin{matrix}p'',x'',y'',z''\\\hline k^{2}p^{2}+x^{2}+y^{2}+z^{2}=k^{2}\\k^{2}pp'+xx'+yy'+zz'=0\\k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}+k^{2}pp''+xx''+yy''+zz''=0\\\hline v^{2}=k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\v^{2}+k^{2}pp''+xx''+yy''+zz''=0\\{\frac {1}{2}}{\frac {d\left(v^{2}\right)}{dt}}=k^{2}p'p''+x'x''+y'y''+z'z''\end{matrix}}}$

If the Gaussian curvature ${\displaystyle 1/k^{2}}$ (with k as radius of curvature) is negative the acceleration becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-acceleration in Minkowski space by setting ${\displaystyle k^{2}=-c^{2}}$ with c as speed of light. However, Killing obtained his results by differentiation with respect to Newtonian time t, not relativistic proper time, so his expressions aren't relativistic four-vectors in the first place, in particular they don't involve a limiting speed. Also the dot product of acceleration and velocity differs from the relativistic result.

w:Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. In his lecture on December 1907, he didn't directly define a four-acceleraton vector, but used it implicitly in the definition of four-force and its density in terms of mass density ${\displaystyle \nu }$, mass m, four-velocity w:^{[R 1]}

${\displaystyle {\begin{matrix}\nu {\frac {dw_{h}}{d\tau }}=K+(w{\overline {K}})w\\\nu {\frac {d}{d\tau }}{\frac {dx}{d\tau }}=X,\ \nu {\frac {d}{d\tau }}{\frac {dy}{d\tau }}=Y,\ \nu {\frac {d}{d\tau }}{\frac {dz}{d\tau }}=Z,\ \nu {\frac {d}{d\tau }}{\frac {dt}{d\tau }}=T\\\hline m{\frac {d}{d\tau }}{\frac {dx}{d\tau }}=R_{x},\ m{\frac {d}{d\tau }}{\frac {dy}{d\tau }}=R_{y},\ m{\frac {d}{d\tau }}{\frac {dz}{d\tau }}=R_{z},\ m{\frac {d}{d\tau }}{\frac {dt}{d\tau }}=R_{t}\end{matrix}}{\text{ }}}$

corresponding to (a).

In 1908, he denoted the derivative of the motion vector (four-velocity) with respect to proper time as "acceleration vector":^{[R 2]}

${\displaystyle {\ddot {x}},{\ddot {y}},{\ddot {z}},{\ddot {t}}}$

corresponding to (a).

w:Philipp Frank (1909) didn't explicitly mentions four-acceleration as vector, though he used its components while defining four-force (X,Y,Z,T):^{[R 3]}

${\displaystyle {\begin{aligned}{\frac {d}{d\sigma }}{\frac {dt}{d\sigma }}&={\frac {1}{\left(1-w^{2}\right)^{2}}}\left(w_{x}{\frac {dw_{x}}{dt}}+w_{y}{\frac {dw_{y}}{dt}}+w_{z}{\frac {dw_{z}}{dt}}\right)\\{\frac {d^{2}x}{d\sigma ^{2}}}&={\frac {1}{1-w^{2}}}{\frac {dw_{x}}{dt}}+{\frac {w_{x}}{\left(1-w^{2}\right)^{2}}}\left(w_{x}{\frac {dw_{x}}{dt}}+w_{y}{\frac {dw_{y}}{dt}}+w_{z}{\frac {dw_{z}}{dt}}\right)\\&{\rm {etc}}.\end{aligned}}}$

corresponding to (a, b).

The first discussion in an English language paper of four-acceleration (even though in the broader context of w:spherical wave transformations), was given by w:Harry Bateman in a paper read 1909 and published 1910. He first defined four-velocity^{[R 4]}

${\displaystyle w_{1}={\frac {w_{x}}{\sqrt {1-w^{2}}}},\ w_{2}={\frac {w_{y}}{\sqrt {1-w^{2}}}},\ w_{3}={\frac {w_{z}}{\sqrt {1-w^{2}}}},\ w_{4}={\frac {1}{\sqrt {1-w^{2}}}},}$

from which he derived four-acceleration

${\displaystyle {\frac {dw_{1}}{ds}}={\frac {{\dot {w}}_{x}}{1-w^{2}}}+{\frac {w_{x}(w{\dot {w}})}{\left(1-w^{2}\right)^{2}}},\dots }$

equivalent to (a, b) as well as its inner product

${\displaystyle {\begin{aligned}\left({\frac {dw_{1}}{ds}}\right)^{2}+\left({\frac {dw_{2}}{ds}}\right)^{2}+\left({\frac {dw_{3}}{ds}}\right)^{2}-\left({\frac {dw_{4}}{ds}}\right)^{2}&={\frac {{\dot {w}}^{2}}{\left(1-w^{2}\right)^{2}}}+{\frac {(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{3}}}+{\frac {w^{2}(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{4}}}-{\frac {(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{4}}}\\&={\frac {{\dot {w}}^{2}}{\left(1-w^{2}\right)^{2}}}+{\frac {(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{3}}}\end{aligned}}}$

equivalent to (c). He also defined the four-jerk

${\displaystyle {\begin{matrix}{\frac {d^{2}w_{1}}{ds^{2}}}={\frac {{\ddot {w}}_{x}}{\left(1-w^{2}\right)^{\frac {1}{2}}}}+{\frac {3{\dot {w}}_{x}(w{\dot {w}})}{\left(1-w^{2}\right)^{\frac {1}{2}}}}+{\frac {w_{x}}{\left(1-w^{2}\right)^{\frac {1}{2}}}}\left\{w{\ddot {w}}+{\frac {3(w{\dot {w}})^{2}}{1-w^{2}}}+{\dot {w}}^{2}+{\frac {(w{\dot {w}})^{2}}{1-w^{2}}}\right\},\dots \end{matrix}}}$

w:Gilbert Newton Lewis and w:Edwin Bidwell Wilson devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They defined the “extended acceleration” as a “1-vector”, its norm, and its relation to the “extended force”:^{[R 5]}

${\displaystyle {\begin{matrix}\mathbf {c} ={\frac {d\mathbf {w} }{ds}}={\frac {dx_{4}}{ds}}{\frac {d\mathbf {w} }{dx_{4}}}={\frac {1}{1-v^{2}}}{\frac {d\mathbf {v} }{dx_{4}}}+{\frac {\mathbf {v} +\mathbf {k} _{4}}{\left(1-v^{2}\right)^{2}}}v{\frac {dv}{dx_{4}}}\\\mathbf {c} ={\frac {1}{1-v^{2}}}{\frac {d\mathbf {v} }{dt}}+{\frac {\mathbf {v} +\mathbf {k} _{4}}{\left(1-v^{2}\right)^{2}}}v{\frac {dv}{dt}}\\\mathbf {c} ={\frac {\mathbf {u} {\frac {dv}{dt}}}{\left(1-v^{2}\right)^{2}}}+{\frac {v{\frac {d\mathbf {u} }{dt}}}{1-v^{2}}}+{\frac {v\mathbf {k} _{4}{\frac {dv}{dt}}}{\left(1-v^{2}\right)^{2}}}\\\hline {\begin{aligned}{\sqrt {\mathbf {c} \cdot \mathbf {c} }}&=\left[{\frac {\left({\frac {dv}{dt}}\right)^{2}}{\left(1-v^{2}\right)^{3/2}}}+{\frac {v^{2}{\frac {d\mathbf {u} }{dt}}\cdot {\frac {d\mathbf {u} }{dt}}}{\left(1-v^{2}\right)^{2}}}\right]^{1/2}\\&={\frac {1}{1-v^{2}}}\left[{\dot {\mathbf {v} }}{\dot {\cdot \mathbf {v} }}+{\frac {1}{1-v^{2}}}\left(\mathbf {v} {\dot {\cdot \mathbf {v} }}\right)^{2}\right]^{1/2}\\&={\frac {1}{\left(1-v^{2}\right)^{3/2}}}\left[{\dot {\mathbf {v} }}{\dot {\cdot \mathbf {v} }}-\left(\mathbf {v} \times {\dot {\mathbf {v} }}\right)\cdot \left(\mathbf {v} \times {\dot {\mathbf {v} }}\right)\right]^{1/2}\end{aligned}}\\\hline m_{0}\mathbf {c} ={\frac {dm_{0}\mathbf {w} }{ds}}={\frac {dmv}{ds}}\mathbf {k} _{1}+{\frac {dm}{ds}}\mathbf {k} _{4}={\frac {1}{\sqrt {1-v^{2}}}}\left({\frac {dmv}{dt}}\mathbf {k} _{1}+{\frac {dm}{dt}}\mathbf {k} _{4}\right)\end{matrix}}}$

equivalent to (a,b).

w:Friedrich Kottler defined four-acceleration in terms of four-velocity V as:^{[R 6]}

${\displaystyle {\begin{aligned}-c^{2}{\frac {dV}{ds}}&={\frac {d^{2}x}{d\tau ^{2}}}=({\dot {\mathfrak {v}}},0){\frac {1}{1-{\mathfrak {v}}^{2}/c^{2}}}+({\mathfrak {v}},ic){\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}/c^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}=\\&=({\dot {\mathfrak {v}}}_{\bot },0){\frac {1}{1-{\mathfrak {v}}^{2}/c^{2}}}+({\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}},0){\frac {1}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}+\left(0,{\frac {i}{c}}{\frac {{\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}}{\mathfrak {v}}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}\right),\\&{\dot {\ {\mathfrak {v}}}}={\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}}+{\dot {\mathfrak {v}}}_{\bot }\end{aligned}}}$

equivalent to (a,b). He related its inner product to curvature ${\displaystyle R_{1}}$ (in terms of Frenet-Serret formulas) and the “Minkowski acceleration” b:^{[R 7]}

${\displaystyle {\begin{matrix}{\frac {c^{4}}{R_{1}^{2}}}=\left({\frac {dV}{ds}}\right)^{2}c^{4}=\sum \left({\frac {d^{2}x}{d\tau ^{2}}}\right)^{2}={\frac {{\dot {\mathfrak {v}}}_{\bot }^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}+{\frac {{\dot {\mathfrak {v}}}_{\Vert }^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{3}}}\\b={\sqrt {\sum _{\alpha =1}^{4}\left({\frac {d^{2}x^{(\alpha )}}{d\tau ^{2}}}\right)^{2}}}=-c^{2}{\sqrt {\sum _{\alpha =1}^{4}\left({\frac {d^{2}x^{(\alpha )}}{d\tau ^{2}}}\right)^{2}}}=-{\frac {c^{2}}{\mathrm {R} _{1}}}\end{matrix}}}$

equivalent to (c) and defined the four-jerk

${\displaystyle -ic^{3}{\frac {d^{2}V}{ds^{2}}}={\frac {d^{3}x}{d\tau ^{3}}}=({\ddot {\mathfrak {v}}},0){\frac {1}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{3}}}+({\ddot {\mathfrak {v}}},0){\frac {3{\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}}{c^{2}}}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+({\mathfrak {v}},ic)\left\{{\frac {{\mathfrak {\dot {v}}}^{2}/c^{2}+{\frac {{\mathfrak {v}}{\mathfrak {\ddot {v}}}}{c^{2}}}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+{\frac {4\left({\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}}{c^{2}}}\right)^{2}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{7}}}\right\}}$

w:Max von Laue explicitly used the term “four-acceleration” (Viererbeschleunigung) for ${\displaystyle {\dot {Y}}}$ and defined its inner product, and its relation to four-force K as well:^{[R 8]}

${\displaystyle {\begin{matrix}{\dot {Y}}={\frac {dY}{d\tau }},\quad |{\dot {Y|}}={\frac {1}{c}}|{\dot {\mathfrak {q}}}^{0}|\\K=m{\frac {dY}{d\tau }}\end{matrix}}}$

corresponding to (a, c).

While w:Ludwik Silberstein used Biquaternions already in 1911, his first mention of the “acceleration-quaternion” Z was given in 1914. He also defined its conjugate, its Lorentz transformation, the relation of four-velocity Y, and its relation to four-force X:^{[R 9]}

${\displaystyle {\begin{matrix}Z={\frac {dY}{d\tau }}={\frac {\iota }{c}}\gamma \left(\mathbf {pa} '\right)+\epsilon \mathbf {a} \\ZY_{c}+YZ_{c}=0\\m{\frac {dY}{d\tau }}=mZ=X\end{matrix}}}$

equivalent to (a,b).

Mathematical

• Killing, W. (1885) [1884], "Die Mechanik in den Nicht-Euklidischen Raumformen", Journal für die Reine und Angewandte Mathematik, 98: 1–48

Relativity

• Bateman, H. (1910) [1909], "The Transformation of the Electrodynamical Equations", Proceedings of the London Mathematical Society, 8: 223–264

See also the transcription The Transformation of the Electrodynamical Equations on English Wikisource

• Einstein, A. (1912–14), "Einstein's manuscript on the special theory of relativity", The collected papers of Albert Einstein, vol. 4, pp. 3–108{{citation}}: CS1 maint: date format (link)
• Frank, P. (1909), "Die Stellung des Relativitätsprinzips im System der Mechanik und Elektrodynamik", Wiener Sitzungsberichte IIa, 118: 373–446
• Kottler, F. (1912), "Über die Raumzeitlinien der Minkowski'schen Welt", Wiener Sitzungsberichte 2a, 121: 1659–1759

See also the transcription On the spacetime lines of a Minkowski world on English Wikisource

• Laue, M. v. (December 1912), Das Relativitätsprinzip (2. Edition), Braunschweig: Vieweg (published 1913)
• Lewis, G. N. & Wilson, E. B. (1912), "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences, 48: 387–507{{citation}}: CS1 maint: multiple names: authors list (link)
• Minkowski, H. (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111

See also the transcription Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern on German Wikisource

See also the transcription The Fundamental Equations for Electromagnetic Processes in Moving Bodies on English Wikisource

• Minkowski, H. (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88

See also the transcription Raum und Zeit on German Wikisource

See also the transcription Space and Time on English Wikisource

• Silberstein, L. (1914), The Theory of Relativity, London: Macmillan