History of Topics in Special Relativity/Four-acceleration

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Overview

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

$\begin{array}{ccc}{A}^{\mu }& =\frac{d{U}^{\mu }}{d\tau }& =({\gamma }_u^4\frac{𝐚\cdot 𝐮}{c},{\gamma }_u^2𝐚+{\gamma }_u^4\frac{(𝐚\cdot 𝐮)}{c^2}𝐮)\\ & (a)& (b)\end{array},\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$.

and its inner product is equal to the proper acceleration

$\begin{array}{cc}{A}^{\mu }{A}_{\mu }& ={\gamma }^4[{𝐚}^2+{\gamma }^2{\left(\frac{𝐮\cdot 𝐚}{c}\right)}^2]={{𝐚}_0}^2\\ & (c)\end{array}$

Historical notation

Killing (1884/5)

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:

$\begin{array}{c}{p}^{″},x^{″},y^{″},z^{″}\\ k^2{p}^2+x^2+y^2+z^2=k^2\\ k^{2}p{p}^{\prime }+x{x}^{\prime }+y{y}^{\prime }+z{z}^{\prime }=0\\ k^2{p}^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}+k^{2}p{p}^{″}+x{x}^{″}+y{y}^{″}+z{z}^{″}=0\\ v^2=k^2{p}^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\ v^2+k^{2}p{p}^{″}+x{x}^{″}+y{y}^{″}+z{z}^{″}=0\\ \frac{1}{2}\frac{d\left(v^2\right)}{dt}=k^2{p}^{\prime }{p}^{″}+x^{\prime }{x}^{″}+y^{\prime }{y}^{″}+z^{\prime }{z}^{″}\end{array}$

If the Gaussian curvature $1/k^2$ (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 $k^2=-c^2$ 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.

Minkowski (1907/08)

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 $\nu$, mass m, four-velocity w:

$\begin{array}{c}\nu \frac{d{w}_h}{d\tau }=K+(w\bar{K})w\\ \nu \frac{d}{d\tau }\frac{dx}{d\tau }=X,\nu \frac{d}{d\tau }\frac{dy}{d\tau }=Y,\nu \frac{d}{d\tau }\frac{dz}{d\tau }=Z,\nu \frac{d}{d\tau }\frac{dt}{d\tau }=T\\ m\frac{d}{d\tau }\frac{dx}{d\tau }=R_x,m\frac{d}{d\tau }\frac{dy}{d\tau }=R_y,m\frac{d}{d\tau }\frac{dz}{d\tau }=R_z,m\frac{d}{d\tau }\frac{dt}{d\tau }=R_t\end{array}$

corresponding to (a).

In 1908, he denoted the derivative of the motion vector (four-velocity) with respect to proper time as "acceleration vector":

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

corresponding to (a).

Frank (1909)

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):

$\begin{array}{cc}\frac{d}{d\sigma }\frac{dt}{d\sigma }& =\frac{1}{{(1-w^2)}^2}(w_x\frac{d{w}_x}{dt}+w_y\frac{d{w}_y}{dt}+w_z\frac{d{w}_z}{dt})\\ \frac{d^{2}x}{d{\sigma }^2}& =\frac{1}{1-w^2}\frac{d{w}_x}{dt}+\frac{w_x}{{(1-w^2)}^2}(w_x\frac{d{w}_x}{dt}+w_y\frac{d{w}_y}{dt}+w_z\frac{d{w}_z}{dt})\\ & \mathrm{e}\mathrm{t}\mathrm{c}.\end{array}$

corresponding to (a, b).

Bateman (1909/10)

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

$w_1=\frac{w_x}{\sqrt{1-w^2}},w_2=\frac{w_y}{\sqrt{1-w^2}},w_3=\frac{w_z}{\sqrt{1-w^2}},w_4=\frac{1}{\sqrt{1-w^2}},$

from which he derived four-acceleration

$\frac{d{w}_1}{ds}=\frac{{\dot{w}}_x}{1-w^2}+\frac{w_x(w\dot{w})}{{(1-w^2)}^2},\dots$

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

$\begin{array}{cc}{\left(\frac{d{w}_1}{ds}\right)}^2+{\left(\frac{d{w}_2}{ds}\right)}^2+{\left(\frac{d{w}_3}{ds}\right)}^2-{\left(\frac{d{w}_4}{ds}\right)}^2& =\frac{{\dot{w}}^2}{{(1-w^2)}^2}+\frac{(w\dot{w}{)}^2}{{(1-w^2)}^3}+\frac{w^2(w\dot{w}{)}^2}{{(1-w^2)}^4}-\frac{(w\dot{w}{)}^2}{{(1-w^2)}^4}\\ & =\frac{{\dot{w}}^2}{{(1-w^2)}^2}+\frac{(w\dot{w}{)}^2}{{(1-w^2)}^3}\end{array}$

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

$\begin{array}{c}\frac{d^2{w}_1}{d{s}^2}=\frac{{\ddot{w}}_x}{{(1-w^2)}^{\frac{1}{2}}}+\frac{3{\dot{w}}_x(w\dot{w})}{{(1-w^2)}^{\frac{1}{2}}}+\frac{w_x}{{(1-w^2)}^{\frac{1}{2}}}\{w\ddot{w}+\frac{3(w\dot{w}{)}^2}{1-w^2}+{\dot{w}}^2+\frac{(w\dot{w}{)}^2}{1-w^2}\},\dots \end{array}$

Wilson/Lewis (1912)

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”:

$\begin{array}{c}𝐜=\frac{d𝐰}{ds}=\frac{d{x}_4}{ds}\frac{d𝐰}{d{x}_4}=\frac{1}{1-v^2}\frac{d𝐯}{d{x}_4}+\frac{𝐯+{𝐤}_4}{{(1-v^2)}^2}v\frac{dv}{d{x}_4}\\ 𝐜=\frac{1}{1-v^2}\frac{d𝐯}{dt}+\frac{𝐯+{𝐤}_4}{{(1-v^2)}^2}v\frac{dv}{dt}\\ 𝐜=\frac{𝐮\frac{dv}{dt}}{{(1-v^2)}^2}+\frac{v\frac{d𝐮}{dt}}{1-v^2}+\frac{v{𝐤}_4\frac{dv}{dt}}{{(1-v^2)}^2}\\ \begin{array}{cc}\sqrt{𝐜\cdot 𝐜}& ={[\frac{{\left(\frac{dv}{dt}\right)}^2}{{(1-v^2)}^{3/2}}+\frac{v^2\frac{d𝐮}{dt}\cdot \frac{d𝐮}{dt}}{{(1-v^2)}^2}]}^{1/2}\\ & =\frac{1}{1-v^2}{[\dot{𝐯}\dot{\cdot 𝐯}+\frac{1}{1-v^2}{\left(𝐯\dot{\cdot 𝐯}\right)}^2]}^{1/2}\\ & =\frac{1}{{(1-v^2)}^{3/2}}{[\dot{𝐯}\dot{\cdot 𝐯}-(𝐯\times \dot{𝐯})\cdot (𝐯\times \dot{𝐯})]}^{1/2}\end{array}\\ m_0𝐜=\frac{d{m}_0𝐰}{ds}=\frac{dmv}{ds}{𝐤}_1+\frac{dm}{ds}{𝐤}_4=\frac{1}{\sqrt{1-v^2}}(\frac{dmv}{dt}{𝐤}_1+\frac{dm}{dt}{𝐤}_4)\end{array}$

equivalent to (a,b).

Kottler (1912)

w:Friedrich Kottler defined four-acceleration in terms of four-velocity V as:

$\begin{array}{cc}-c^2\frac{dV}{ds}& =\frac{d^{2}x}{d{\tau }^2}=(\dot{𝔳},0)\frac{1}{1-{𝔳}^2/c^2}+(𝔳,ic)\frac{𝔳\dot{𝔳}/c^2}{{(1-{𝔳}^2/c^2)}^2}=\\ & =({\dot{𝔳}}_{\mathrm{\perp }},0)\frac{1}{1-{𝔳}^2/c^2}+({\dot{𝔳}}_{\mathfrak{\Vert }},0)\frac{1}{{(1-{𝔳}^2/c^2)}^2}+(0,\frac{i}{c}\frac{{\dot{𝔳}}_{\mathfrak{\Vert }}𝔳}{{(1-{𝔳}^2/c^2)}^2}),\\ & \dot{𝔳}={\dot{𝔳}}_{\mathfrak{\Vert }}+{\dot{𝔳}}_{\mathrm{\perp }}\end{array}$

equivalent to (a,b). He related its inner product to curvature $R_1$ (in terms of Frenet-Serret formulas) and the “Minkowski acceleration” b:

$\begin{array}{c}\frac{c^4}{R_1^2}={\left(\frac{dV}{ds}\right)}^2{c}^4=\sum {\left(\frac{d^{2}x}{d{\tau }^2}\right)}^2=\frac{{\dot{𝔳}}_{\mathrm{\perp }}^2}{{(1-{𝔳}^2/c^2)}^2}+\frac{{\dot{𝔳}}_{\Vert }^2}{{(1-{𝔳}^2/c^2)}^3}\\ b=\sqrt{\sum _{\alpha =1}^4{\left(\frac{d^2{x}^{(\alpha )}}{d{\tau }^2}\right)}^2}=-c^2\sqrt{\sum _{\alpha =1}^4{\left(\frac{d^2{x}^{(\alpha )}}{d{\tau }^2}\right)}^2}=-\frac{c^2}{{\mathrm{R}}_1}\end{array}$

equivalent to (c) and defined the four-jerk

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

Von Laue (1912/13)

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

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

corresponding to (a, c).

Silberstein (1914)

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:

$\begin{array}{c}Z=\frac{dY}{d\tau }=\frac{\iota }{c}\gamma \left({𝐩𝐚}^{\prime }\right)+ϵ𝐚\\ Z{Y}_c+Y{Z}_c=0\\ m\frac{dY}{d\tau }=mZ=X\end{array}$

equivalent to (a,b).

References

Mathematical

• {{#section:History of Topics in Special Relativity/mathsource|kil84}}

Relativity

• {{#section:History of Topics in Special Relativity/relsource|bate10elec}}
• {{#section:History of Topics in Special Relativity/relsource|einst12manu}}
• {{#section:History of Topics in Special Relativity/relsource|frank09a }}
• {{#section:History of Topics in Special Relativity/relsource|kott12mink}}
• {{#section:History of Topics in Special Relativity/relsource|laue13prin}}
• {{#section:History of Topics in Special Relativity/relsource|lewis12non}}
• {{#section:History of Topics in Special Relativity/relsource|mink07b}}
• {{#section:History of Topics in Special Relativity/relsource|mink08}}
• {{#section:History of Topics in Special Relativity/relsource|silber14quat}}