History of Topics in Special Relativity/Four-acceleration

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).