Acceleration (special relativity)

In accordance with both Newtonian mechanics and SR, three-acceleration or coordinate acceleration $\backslash mathbf\{a\}=\backslash left(a\_\{x\},\backslash \; a\_\{y\},\backslash \; a\_\{z\}\backslash right)$ is the first derivative of velocity $\backslash mathbf\{u\}=\backslash left(u\_\{x\},\backslash \; u\_\{y\},\backslash \; u\_\{z\}\backslash right)$ with respect to coordinate time or the second derivative of the location $\backslash mathbf\{r\}=\backslash left(x,\backslash \; y,\backslash \; z\backslash right)$ with respect to coordinate time:

:$\backslash mathbf\{a\}=\backslash frac\{d\backslash mathbf\{u\}\}\{dt\}=\backslash frac\{d^\{2\}\backslash mathbf\{r\}\}\{dt^\{2\}\}$.

However, the theories sharply differ in their predictions in terms of the relation between three-accelerations measured in different inertial frames. In Newtonian mechanics, time is absolute by $t\text{'}=t$ in accordance with the Galilean transformation, therefore the three-acceleration derived from it is equal too in all inertial frames:Sexl & Schmidt (1979), p. 116

:$\backslash mathbf\{a\}=\backslash mathbf\{a\}\text{'}$.

On the contrary in SR, both $\backslash mathbf\{r\}$ and $t$ depend on the Lorentz transformation, therefore also three-acceleration $\backslash mathbf\{a\}$ and its components vary in different inertial frames. When the relative velocity between the frames is directed in the x-direction by $v=v\_\{x\}$ with $\backslash gamma\_\{v\}=1/\backslash sqrt\{1-v^\{2\}/c^\{2\}\}$ as Lorentz factor, the Lorentz transformation has the form

{{NumBlk|:|$\backslash begin\{array\}\{c|c\}$

\begin{align}x' & =\gamma_{v}(x-vt)\\

y' & =y\\

z' & =z\\

t^{\prime} & =\gamma_{v}\left(t-\frac{v}{c^2}x\right)

\end{align}

& \begin{align}x & =\gamma_{v}(x'+vt')\\

y & =y'\\

z & =z'\\

t & =\gamma_{v}\left(t'+\frac{v}{c^2}x'\right)

\end{align}

\end{array}|{{EquationRef|1a}}}}

or for arbitrary velocities $\backslash mathbf\{v\}=\backslash left(v\_\{x\},\backslash \; v\_\{y\},\backslash \; v\_\{z\}\backslash right)$ of magnitude $|\backslash mathbf\{v\}|=v$:Møller (1955), p. 41

{{NumBlk|:|$\backslash begin\{array\}\{c|c\}$

\begin{align}\mathbf{r}' & =\mathbf{r}+\mathbf{v}\left[\frac{\left(\mathbf{r\cdot v}\right)}{v^{2}}\left(\gamma_{v}-1\right)-t\gamma_{v}\right]\\

t^{\prime} & =\gamma_{v}\left(t-\frac{\mathbf{r\cdot v}}{c^2}\right)

\end{align}

& \begin{align}\mathbf{r} & =\mathbf{r}'+\mathbf{v}\left[\frac{\left(\mathbf{r'\cdot v}\right)}{v^{2}}\left(\gamma_{v}-1\right)+t'\gamma_{v}\right]\\

t & =\gamma_{v}\left(t'+\frac{\mathbf{r'\cdot v}}{c^2}\right)

\end{align}

\end{array}|{{EquationRef|1b}}}}

In order to find out the transformation of three-acceleration, one has to differentiate the spatial coordinates $\backslash mathbf\{r\}$ and $\backslash mathbf\{r\}\text{'}$ of the Lorentz transformation with respect to $t$ and $t\text{'}$, from which the transformation of three-velocity (also called velocity-addition formula) between $\backslash mathbf\{u\}$ and $\backslash mathbf\{u\}\text{'}$ follows, and eventually by another differentiation with respect to $t$ and $t\text{'}$ the transformation of three-acceleration between $\backslash mathbf\{a\}$ and $\backslash mathbf\{a\}\text{'}$ follows. Starting from ({{equationNote|1a}}), this procedure gives the transformation where the accelerations are parallel (x-direction) or perpendicular (y-, z-direction) to the velocity:Tolman (1917), p. 48French (1968), p. 148Zahar (1989), p. 232Freund (2008), p. 96

{{NumBlk|:|$\backslash begin\{array\}\{c|c\}$

\begin{align}a_{x}^{\prime} & =\frac{a_{x}}{\gamma_{v}^{3}\left(1-\frac{u_{x}v}{c^{2}}\right)^{3}}\\

a_{y}^{\prime} & =\frac{a_{y}}{\gamma_{v}^{2}\left(1-\frac{u_{x}v}{c^{2}}\right)^{2}}+\frac{a_{x}\frac{u_{y}v}{c^{2}}}{\gamma_{v}^{2}\left(1-\frac{u_{x}v}{c^{2}}\right)^{3}}\\

a_{z}^{\prime} & =\frac{a_{z}}{\gamma_{v}^{2}\left(1-\frac{u_{x}v}{c^{2}}\right)^{2}}+\frac{a_{x}\frac{u_{z}v}{c^{2}}}{\gamma_{v}^{2}\left(1-\frac{u_{x}v}{c^{2}}\right)^{3}}

\end{align}

& \begin{align}a_{x} & =\frac{a_{x}^{\prime}}{\gamma_{v}^{3}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)^{3}}\\

a_{y} & =\frac{a_{y}^{\prime}}{\gamma_{v}^{2}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)^{2}}-\frac{a_{x}^{\prime}\frac{u_{y}^{\prime}v}{c^{2}}}{\gamma_{v}^{2}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)^{3}}\\

a_{z} & =\frac{a_{z}^{\prime}}{\gamma_{v}^{2}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)^{2}}-\frac{a_{x}^{\prime}\frac{u_{z}^{\prime}v}{c^{2}}}{\gamma_{v}^{2}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)^{3}}

\end{align}

\end{array}|{{EquationRef|1c}}}}

or starting from ({{equationNote|1b}}) this procedure gives the result for the general case of arbitrary directions of velocities and accelerations:Kopeikin & Efroimsky & Kaplan (2011), p. 141Rahaman (2014), p. 77

{{NumBlk|:|

\mathbf{a} & =\frac{\mathbf{a}'}{\gamma_{v}^{2}\left(1+\frac{\mathbf{v\cdot u}'}{c^{2}}\right)^{2}}-\frac{\mathbf{(a'\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma_{v}^{3}\left(1+\frac{\mathbf{v\cdot u}'}{c^{2}}\right)^{3}}-\frac{\mathbf{(a'\cdot v)u}'}{c^{2}\gamma_{v}^{2}\left(1+\frac{\mathbf{v\cdot u}'}{c^{2}}\right)^{3}}

\end{align}

|{{EquationRef|1d}}}}

This means, if there are two inertial frames $S$ and $S\text{'}$ with relative velocity $\backslash mathbf\{v\}$, then in $S$ the acceleration $\backslash mathbf\{a\}$ of an object with momentary velocity $\backslash mathbf\{u\}$ is measured, while in $S\text{'}$ the same object has an acceleration $\backslash mathbf\{a\}\text{'}$ and has the momentary velocity $\backslash mathbf\{u\}\text{'}$. As with the velocity addition formulas, also these acceleration transformations guarantee that the resultant speed of the accelerated object can never reach or surpass the speed of light.

{{Main|Four-acceleration}}

If four-vectors are used instead of three-vectors, namely $\backslash mathbf\{R\}$ as four-position and $\backslash mathbf\{U\}$ as four-velocity, then the four-acceleration $\backslash mathbf\{A\}=\backslash left(A\_\{t\},\backslash \; A\_\{x\},\backslash \; A\_\{y\},\backslash \; A\_\{z\}\backslash right)=\backslash left(A\_\{t\},\backslash \; \backslash mathbf\{A\}\_\{r\}\backslash right)$ of an object is obtained by differentiation with respect to proper time $\backslash mathbf\{\backslash tau\}$ instead of coordinate time:Pauli (1921), p. 627Freund (2008), pp. 267-268Ashtekar & Petkov (2014), p. 53

{{NumBlk|:|

& =\left(\gamma^{4}\frac{\mathbf{u}\cdot\mathbf{a}}{c},\ \gamma^{4}\frac{(\mathbf{a}\cdot\mathbf{u})\mathbf{u}}{c^{2}}+\gamma^{2}\mathbf{a}\right)

\end{align}

|{{EquationRef|2a}}}}

where $\backslash mathbf\{a\}$ is the object's three-acceleration and $\backslash mathbf\{u\}$ its momentary three-velocity of magnitude $|\backslash mathbf\{u\}|=u$ with the corresponding Lorentz factor $\backslash gamma=1/\backslash sqrt\{1-u^\{2\}/c^\{2\}\}$. If only the spatial part is considered, and when the velocity is directed in the x-direction by $u=u\_\{x\}$ and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered, the expression is reduced to:Sexl & Schmidt (1979), p. 198, Solution to example 16.1Ferraro (2007), p. 178

:$\backslash mathbf\{A\}\_\{r\}=\backslash mathbf\{a\}\backslash left(\backslash gamma^\{4\},\backslash \; \backslash gamma^\{2\},\backslash \; \backslash gamma^\{2\}\backslash right)$

Unlike the three-acceleration previously discussed, it is not necessary to derive a new transformation for four-acceleration, because as with all four-vectors, the components of $\backslash mathbf\{A\}$ and $\backslash mathbf\{A\}\text{'}$ in two inertial frames with relative speed $v$ are connected by a Lorentz transformation analogous to ({{equationNote|1a}}, {{equationNote|1b}}). Another property of four-vectors is the invariance of the inner product $\backslash mathbf\{A\}^\{2\}=-A\_\{t\}^\{2\}+\backslash mathbf\{A\}\_\{r\}^\{2\}$ or its magnitude $|\backslash mathbf\{A\}|=\backslash sqrt\{\backslash mathbf\{A\}^\{2\}\}$, which gives in this case:Kopeikin & Efroimsky & Kaplan (2011), p. 137

{{NumBlk|:|$|\backslash mathbf\{A\}\text{'}|=|\backslash mathbf\{A\}|=\backslash sqrt\{\backslash gamma^\{4\}\backslash left[\backslash mathbf\{a\}^\{2\}+\backslash gamma^\{2\}\backslash left(\backslash frac\{\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{a\}\}\{c\}\backslash right)^\{2\}\backslash right]\}$.|{{EquationRef|2b}}}}

{{Main|Proper acceleration}}

In infinitesimal small durations there is always one inertial frame, which momentarily has the same velocity as the accelerated body, and in which the Lorentz transformation holds. The corresponding three-acceleration $\backslash mathbf\{a\}^\{0\}=\backslash left(a\_\{x\}^\{0\},\backslash \; a\_\{y\}^\{0\},\backslash \; a\_\{z\}^\{0\}\backslash right)$ in these frames can be directly measured by an accelerometer, and is called proper accelerationRindler (1977), pp. 49-50 or rest acceleration.von Laue (1921), pp. 88-89 The relation of $\backslash mathbf\{a\}^\{0\}$ in a momentary inertial frame $S\text{'}$ and $\backslash mathbf\{a\}$ measured in an external inertial frame $S$ follows from ({{equationNote|1c}}, {{equationNote|1d}}) with $\backslash mathbf\{a\}\text{'}=\backslash mathbf\{a\}^\{0\}$, $\backslash mathbf\{u\}\text{'}=0$, $\backslash mathbf\{u\}=\backslash mathbf\{v\}$ and $\backslash gamma=\backslash gamma\_\{v\}$. So in terms of ({{equationNote|1c}}), when the velocity is directed in the x-direction by $u=u\_\{x\}=v=v\_\{x\}$ and when only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered, it follows:

{{NumBlk|:|$\backslash begin\{array\}\{c|c|cc\}$

\begin{align}a_{x}^{0} & =\frac{a_{x}}{\left(1-\frac{u^{2}}{c^{2}}\right)^{3/2}}\\

a_{y}^{0} & =\frac{a_{y}}{1-\frac{u^{2}}{c^{2}}}\\

a_{z}^{0} & =\frac{a_{z}}{1-\frac{u^{2}}{c^{2}}}

\end{align}

& \begin{align}a_{x} & =a_{x}^{0}\left(1-\frac{u^{2}}{c^{2}}\right)^{3/2}\\

a_{y} & =a_{y}^{0}\left(1-\frac{u^{2}}{c^{2}}\right)\\

a_{z} & =a_{z}^{0}\left(1-\frac{u^{2}}{c^{2}}\right)

\end{align}

& \text{or} & \begin{align}\mathbf{a}^{0} & =\mathbf{a}\left(\gamma^{3},\ \gamma^{2},\ \gamma^{2}\right)\\

\mathbf{a} & =\mathbf{\mathbf{a}}^{0}\left(\frac{1}{\gamma^{3}},\ \frac{1}{\gamma^{2}},\ \frac{1}{\gamma^{2}}\right)

\end{align}

\end{array}|{{EquationRef|3a}}}}

Generalized by ({{equationNote|1d}}) for arbitrary directions of $\backslash mathbf\{u\}$ of magnitude $|\backslash mathbf\{u\}|=u$:Rebhan (1999), p. 775Nikolić (2000), eq. 10

:

\mathbf{a} & =\frac{1}{\gamma^{2}}\left[\mathbf{a}^{0}-\frac{(\mathbf{a}^{0}\cdot\mathbf{u})\mathbf{u}}{u^{2}}\left(1-\frac{1}{\gamma}\right)\right]

\end{align}

There is also a close relationship to the magnitude of four-acceleration: As it is invariant, it can be determined in the momentary inertial frame $S\text{'}$, in which $\backslash mathbf\{A\}\_\{r\}^\{\backslash prime\}=\backslash mathbf\{a\}^\{0\}$ and by $dt\text{'}/d\backslash tau=1$ it follows $d^\{2\}t\text{'}/d\backslash tau^\{2\}=A\_\{t\}^\{\backslash prime\}=0$:Rindler (1977), p. 67

{{NumBlk|:|$|\backslash mathbf\{A\}\text{'}|=\backslash sqrt\{0+\backslash left.\backslash mathbf\{a\}^\{0\}\backslash right.^\{2\}\}=|\backslash mathbf\{a\}^\{0\}|$.|{{EquationRef|3b}}}}

Thus the magnitude of four-acceleration corresponds to the magnitude of proper acceleration. By combining this with ({{equationNote|2b}}), an alternative method for the determination of the connection between $\backslash mathbf\{a\}^\{0\}$ in $S\text{'}$ and $\backslash mathbf\{a\}$ in $S$ is given, namely

:$|\backslash mathbf\{a\}^\{0\}|=|\backslash mathbf\{A\}|=\backslash sqrt\{\backslash gamma^\{4\}\backslash left[\backslash mathbf\{a\}^\{2\}+\backslash gamma^\{2\}\backslash left(\backslash frac\{\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{a\}\}\{c\}\backslash right)^\{2\}\backslash right]\}$

from which ({{equationNote|3a}}) follows again when the velocity is directed in the x-direction by $u=u\_\{x\}$ and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered.

{{Further|Relativistic mechanics#Force|Mass in special relativity}}

Assuming constant mass $m$, the four-force $\backslash mathbf\{F\}$ as a function of three-force $\backslash mathbf\{f\}$ is related to four-acceleration ({{equationNote|2a}}) by $\backslash mathbf\{F\}=m\backslash mathbf\{A\}$, thus:Freund (2008), p. 276

{{NumBlk|:|$\backslash mathbf\{F\}=\backslash left(\backslash gamma\backslash frac\{\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}\}\{c\},\backslash \; \backslash gamma\backslash mathbf\{f\}\backslash right)=m\backslash mathbf\{A\}=m\backslash left(\backslash gamma^\{4\}\backslash left(\backslash frac\{\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{a\}\}\{c\}\backslash right),\backslash \; \backslash gamma^\{4\}\backslash left(\backslash frac\{\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{a\}\}\{c^\{2\}\}\backslash right)\backslash mathbf\{u\}+\backslash gamma^\{2\}\backslash mathbf\{a\}\backslash right)$|{{equationRef|4a}}}}

The relation between three-force and three-acceleration for arbitrary directions of the velocity is thusRindler (1977), pp. 89-90Sexl & Schmidt (1979), solution of example 16.2, p. 198

{{NumBlk|:|

\mathbf{a} & =\frac{1}{m\gamma}\left(\mathbf{f}-\frac{(\mathbf{f}\cdot\mathbf{u})\mathbf{u}}{c^{2}}\right)

\end{align}

|{{equationRef|4b}}}}

When the velocity is directed in the x-direction by $u=u\_\{x\}$ and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered

{{NumBlk|:|$\backslash begin\{array\}\{c|c|cc\}$

\begin{align}f_{x} & =\frac{ma_{x}}{\left(1-\frac{u^{2}}{c^{2}}\right)^{3/2}}\\

f_{y} & =\frac{ma_{y}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\\

f_{z} & =\frac{ma_{z}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}

\end{align}

& \begin{align}a_{x} & =\frac{f_{x}}{m}\left(1-\frac{u^{2}}{c^{2}}\right)^{3/2}\\

a_{y} & =\frac{f_{y}}{m}\sqrt{1-\frac{u^{2}}{c^{2}}}\\

a_{z} & =\frac{f_{z}}{m}\sqrt{1-\frac{u^{2}}{c^{2}}}

\end{align}

& \text{or} & \begin{align}\mathbf{f} & =m\mathbf{a}\left(\gamma^{3},\ \gamma,\ \gamma\right)\\

\mathbf{a} & =\frac{\mathbf{f}}{m}\left(\frac{1}{\gamma^{3}},\ \frac{1}{\gamma},\ \frac{1}{\gamma}\right)

\end{align}

\end{array}|{{EquationRef|4c}}}}

Therefore, the Newtonian definition of mass as the ratio of three-force and three-acceleration is disadvantageous in SR, because such a mass would depend both on velocity and direction. Consequently, the following mass definitions used in older textbooks are not used anymore:von Laue (1921), p. 210Pauli (1921), p. 635

:$m\_\{\backslash Vert\}=\backslash frac\{f\_\{x\}\}\{a\_\{x\}\}=m\backslash gamma^\{3\}$ as "longitudinal mass",

:$m\_\{\backslash perp\}=\backslash frac\{f\_\{y\}\}\{a\_\{y\}\}=\backslash frac\{f\_\{z\}\}\{a\_\{z\}\}=m\backslash gamma$ as "transverse mass".

The relation ({{equationNote|4b}}) between three-acceleration and three-force can also be obtained from the equation of motionTolman (1917), pp. 73-74Møller (1955), pp. 74-75

{{NumBlk|:|$\backslash mathbf\{f\}=\backslash frac\{d\backslash mathbf\{p\}\}\{dt\}=\backslash frac\{d(m\backslash gamma\backslash mathbf\{u\})\}\{dt\}=\backslash frac\{d(m\backslash gamma)\}\{dt\}\backslash mathbf\{u\}+m\backslash gamma\backslash frac\{d\backslash mathbf\{u\}\}\{dt\}=m\backslash gamma^\{3\}\backslash left(\backslash frac\{(\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{u\})\backslash mathbf\{u\}\}\{c^\{2\}\}\backslash right)+m\backslash gamma\backslash mathbf\{a\}$|{{equationRef|4d}}}}

where $\backslash mathbf\{p\}$ is the three-momentum. The corresponding transformation of three-force between $\backslash mathbf\{f\}$ in $S$ and $\backslash mathbf\{f\}\text{'}$ in $S\text{'}$ (when the relative velocity between the frames is directed in the x-direction by $v=v\_\{x\}$ and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered) follows by substitution of the relevant transformation formulas for $\backslash mathbf\{u\}$, $\backslash mathbf\{a\}$, $m\backslash gamma$, $d(m\backslash gamma)/dt$, or from the Lorentz transformed components of four-force, with the result:von Laue (1921), p. 113

{{NumBlk|:|$\backslash begin\{array\}\{c|c\}$

\begin{align}f_{x}^{\prime} & =\frac{f_{x}-\frac{v}{c^{2}}(\mathbf{f}\cdot\mathbf{u})}{1-\frac{u_{x}v}{c^{2}}}\\

f_{y}^{\prime} & =\frac{f_{y}}{\gamma_{v}\left(1-\frac{u_{x}v}{c^{2}}\right)}\\

f_{z}^{\prime} & =\frac{f_{z}}{\gamma_{v}\left(1-\frac{u_{x}v}{c^{2}}\right)}

\end{align}

& \begin{align}f_{x} & =\frac{f_{x}^{\prime}+\frac{v}{c^{2}}(\mathbf{f}^{\prime}\cdot\mathbf{u}^{\prime})}{1+\frac{u_{x}^{\prime}v}{c^{2}}}\\

f_{y} & =\frac{f_{y}^{\prime}}{\gamma_{v}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)}\\

f_{z} & =\frac{f_{z}^{\prime}}{\gamma_{v}\left(1+\frac{u_{x}^{\prime}v}{c^{2}}\right)}

\end{align}

\end{array}|{{equationRef|4e}}}}

Or generalized for arbitrary directions of $\backslash mathbf\{u\}$, as well as $\backslash mathbf\{v\}$ with magnitude $|\backslash mathbf\{v\}|=v$:Møller (1955), p. 73Kopeikin & Efroimsky & Kaplan (2011), p. 173

{{NumBlk|:|

\mathbf{f} & =\frac{\frac{\mathbf{f}'}{\gamma_{v}}+\left\{ (\mathbf{f'\cdot u}')\frac{v^{2}}{c^{2}}+(\mathbf{f'\cdot v})\left(1-\frac{1}{\gamma_{v}}\right)\right\} \frac{\mathbf{v}}{v^{2}}}{1+\frac{\mathbf{v\cdot u'}}{c^{2}}}

\end{align}

|{{equationRef|4f}}}}

The force $\backslash mathbf\{f\}^\{0\}$ in a momentary inertial frame measured by a comoving spring balance can be called proper force.Shadowitz (1968), p. 101Pfeffer & Nir (2012), p. 115, "In the special case in which the particle is momentarily at rest relative to the observer S, the force he measures will be the proper force". It follows from ({{equationNote|4e}}, {{equationNote|4f}}) by setting $\backslash mathbf\{f\}\text{'}=\backslash mathbf\{f\}^\{0\}$ and $\backslash mathbf\{u\}\text{'}=0$ as well as $\backslash mathbf\{u\}=\backslash mathbf\{v\}$ and $\backslash gamma=\backslash gamma\_\{v\}$. Thus by ({{equationNote|4e}}) where only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity $u=u\_\{x\}=v=v\_\{x\}$ are considered:

{{NumBlk|:|$\backslash begin\{array\}\{c|c|cc\}$

\begin{align}f_{x}^{0} & =f_{x}\\

f_{y}^{0} & =\frac{f_{y}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\\

f_{z}^{0} & =\frac{f_{z}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}

\end{align}

& \begin{align}f_{x} & =f_{x}^{0}\\

f_{y} & =f_{y}^{0}\sqrt{1-\frac{u^{2}}{c^{2}}}\\

f_{z} & =f_{z}^{0}\sqrt{1-\frac{u^{2}}{c^{2}}}

\end{align}

& \text{or} & \begin{align}\mathbf{f}^{0} & =\mathbf{f}\left(1,\ \gamma,\ \gamma\right)\\

\mathbf{f} & =\mathbf{f}^{0}\left(1,\ \frac{1}{\gamma},\ \frac{1}{\gamma}\right)

\end{align}

\end{array}|{{EquationRef|5a}}}}

Generalized by ({{equationNote|4f}}) for arbitrary directions of $\backslash mathbf\{u\}$ of magnitude $|\backslash mathbf\{u\}|=u$:Møller (1955), p. 74Rebhan (1999), p. 818

:

\mathbf{f} & =\frac{\mathbf{f}^{0}}{\gamma}+\frac{(\mathbf{f}^{0}\cdot\mathbf{u})\mathbf{u}}{u^{2}}\left(1-\frac{1}{\gamma}\right)

\end{align}

Since in momentary inertial frames one has four-force $\backslash mathbf\{F\}=\backslash left(0,\backslash ,\backslash mathbf\{f\}^\{0\}\backslash right)$ and four-acceleration $\backslash mathbf\{A\}=\backslash left(0,\backslash ,\backslash mathbf\{a\}^\{0\}\backslash right)$, equation ({{equationNote|4a}}) produces the Newtonian relation $\backslash mathbf\{f\}^\{0\}=m\backslash mathbf\{a\}^\{0\}$, therefore ({{equationNote|3a}}, {{equationNote|4c}}, {{equationNote|5a}}) can be summarizedsee Lorentz's 1904-equations and Einstein's 1905-equations in section on history

{{NumBlk|:|

\mathbf{f} & =\mathbf{f}^{0}\left(1,\ \frac{1}{\gamma},\ \frac{1}{\gamma}\right)=m\mathbf{a}^{0}\left(1,\ \frac{1}{\gamma},\ \frac{1}{\gamma}\right)=m\mathbf{a}\left(\gamma^{3},\ \gamma,\ \gamma\right)

\end{align}

|{{equationRef|5b}}}}

By that, the apparent contradiction in the historical definitions of transverse mass $m\_\{\backslash perp\}$ can be explained.Mathpages (see external links), "Transverse Mass in Einstein's Electrodynamics", eq. 2,3 Einstein (1905) described the relation between three-acceleration and proper force

:$m\_\{\backslash perp\backslash \; \backslash mathrm\{Einstein\}\}=\backslash frac\{f\_\{y\}^\{0\}\}\{a\_\{y\}\}=\backslash frac\{f\_\{z\}^\{0\}\}\{a\_\{z\}\}=m\backslash gamma^\{2\}$,

while Lorentz (1899, 1904) and Planck (1906) described the relation between three-acceleration and three-force

:$m\_\{\backslash perp\backslash \; \backslash mathrm\{Lorentz\}\}=\backslash frac\{f\_\{y\}\}\{a\_\{y\}\}=\backslash frac\{f\_\{z\}\}\{a\_\{z\}\}=m\backslash gamma$.

{{Further|Proper reference frame (flat spacetime)#Spacetime Frenet-Serret equations|label1=Spacetime Frenet-Serret equations|Hyperbolic motion (relativity)|label2=Hyperbolic motion|Born rigidity}}

By integration of the equations of motion one obtains the curved world lines of accelerated bodies corresponding to a sequence of momentary inertial frames (here, the expression "curved" is related to the form of the worldlines in Minkowski diagrams, which should not be confused with "curved" spacetime of general relativity). In connection with this, the so-called clock hypothesis of clock postulate has to be considered:Rindler (1977), p. 43Koks (2006), section 7.1 The proper time of comoving clocks is independent of acceleration, that is, the time dilation of these clocks as seen in an external inertial frame only depends on its relative velocity with respect to that frame. Two simple cases of curved world lines are now provided by integration of equation ({{equationNote|3a}}) for proper acceleration:

a) Hyperbolic motion: The constant, longitudinal proper acceleration $\backslash alpha=a\_\{x\}^\{0\}=a\_\{x\}\backslash gamma^\{3\}$ by ({{equationNote|3a}}) leads to the world lineFraundorf (2012), section IV-BPhysicsFAQ (2016), see external links.

{{NumBlk|:|

& \tau(t)=\frac{c}{\alpha}\ln\left(\sqrt{1+\left(\frac{\alpha t}{c}\right)^{2}}+\frac{\alpha t}{c}\right),\quad x(t)=\frac{c^{2}}{\alpha}\left(\sqrt{1+\left(\frac{\alpha t}{c}\right)^{2}}-1\right)

\end{align}

|{{equationRef|6a}}}}

The worldline corresponds to the hyperbolic equation $c^\{4\}/\backslash alpha^\{2\}=\backslash left(x+c^\{2\}/\backslash alpha\backslash right)^\{2\}-c^\{2\}t^\{2\}$, from which the name hyperbolic motion is derived. These equations are often used for the calculation of various scenarios of the twin paradox or Bell's spaceship paradox, or in relation to space travel using constant acceleration.

b) The constant, transverse proper acceleration $a\_\{y\}^\{0\}=a\_\{y\}\backslash gamma^\{2\}$ by ({{equationNote|3a}}) can be seen as a centripetal acceleration, leading to the worldline of a body in uniform rotationPauri & Vallisneri (2000), eq. 13Bini & Lusanna & Mashhoon (2005), eq. 28,29

{{NumBlk|:|

y & =r\sin\Omega_{0}t=r\sin\Omega\tau\\

z & =z\\

t & =\gamma\tau=\frac{\tau}{\sqrt{1-\left(\frac{r\Omega_{0}}{c}\right)^{2}}}=\tau\sqrt{1+\left(\frac{r\Omega}{c}\right)^{2}}

\end{align}|{{equationRef|6b}}}}

where $v=r\backslash Omega\_\{0\}$ is the tangential speed, $r$ is the orbital radius, $\backslash Omega\_\{0\}$ is the angular velocity as a function of coordinate time, and $\backslash Omega=\backslash gamma\backslash Omega\_\{0\}$ as the proper angular velocity.

A classification of curved worldlines can be obtained by using the differential geometry of triple curves, which can be expressed by spacetime Frenet-Serret formulas.Synge (1966) In particular, it can be shown that hyperbolic motion and uniform circular motion are special cases of motions having constant curvatures and torsions,Pauri & Vallisneri (2000), Appendix A satisfying the condition of Born rigidity. A body is called Born rigid if the spacetime distance between its infinitesimally separated worldlines or points remains constant during acceleration.

{{Main|Proper reference frame (flat spacetime)|Rindler coordinates|Born coordinates}}

Instead of inertial frames, these accelerated motions and curved worldlines can also be described using accelerated or curvilinear coordinates. The proper reference frame established that way is closely related to Fermi coordinates.Misner & Thorne & Wheeler (1973), Section 6Gourgoulhon (2013), entire book For instance, the coordinates for an hyperbolically accelerated reference frame are sometimes called Rindler coordinates, or those of a uniformly rotating reference frame are called rotating cylindrical coordinates (or sometimes Born coordinates). In terms of the equivalence principle, the effects arising in these accelerated frames are analogous to effects in a homogeneous, fictitious gravitational field. In this way it can be seen, that the employment of accelerating frames in SR produces important mathematical relations, which (when further developed) play a fundamental role in the description of real, inhomogeneous gravitational fields in terms of curved spacetime in general relativity.

For further information see von Laue, Pauli, Miller,Miller (1981) Zahar,Zahar (1989) Gourgoulhon, and the historical sources in history of special relativity.

;1899{{colon}}: Hendrik Lorentz derived the correct (up to a certain factor $\backslash epsilon$) relations for accelerations, forces and masses between a resting electrostatic systems of particles $S\_\{0\}$ (in a stationary aether), and a system $S$ emerging from it by adding a translation, with $k$ as the Lorentz factor:

::$\backslash frac\{1\}\{\backslash epsilon^\{2\}\}$, $\backslash frac\{1\}\{k\backslash epsilon^\{2\}\}$, $\backslash frac\{1\}\{k\backslash epsilon^\{2\}\}$ for $\backslash mathbf\{f\}/\backslash mathbf\{f\}^\{0\}$ by ({{equationNote|5a}});

::$\backslash frac\{1\}\{k^\{3\}\backslash epsilon\}$, $\backslash frac\{1\}\{k^\{2\}\backslash epsilon\}$, $\backslash frac\{1\}\{k^\{2\}\backslash epsilon\}$ for $\backslash mathbf\{a\}/\backslash mathbf\{a\}^\{0\}$ by ({{equationNote|3a}});

::$\backslash frac\{k^\{3\}\}\{\backslash epsilon\}$, $\backslash frac\{k\}\{\backslash epsilon\}$, $\backslash frac\{k\}\{\backslash epsilon\}$ for $\backslash mathbf\{f\}/(m\backslash mathbf\{a\})$, thus longitudinal and transverse mass by ({{equationNote|4c}});

:Lorentz explained that he has no means of determining the value of $\backslash epsilon$. If he had set $\backslash epsilon=1$, his expressions would have assumed the exact relativistic form.

;1904{{colon}}: Lorentz derived the previous relations in a more detailed way, namely with respect to the properties of particles resting in the system $\backslash Sigma\text{'}$ and the moving system $\backslash Sigma$, with the new auxiliary variable $l$ equal to $1/\backslash epsilon$ compared to the one in 1899, thus:

::$\backslash mathfrak\{F\}(\backslash Sigma)=\backslash left(l^\{2\},\backslash \; \backslash frac\{l^\{2\}\}\{k\},\backslash \; \backslash frac\{l^\{2\}\}\{k\}\backslash right)\backslash mathfrak\{F\}(\backslash Sigma\text{'})$ for $\backslash mathbf\{f\}$ as a function of $\backslash mathbf\{f\}^\{0\}$ by ({{equationNote|5a}});

::$m\backslash mathfrak\{j\}(\backslash Sigma)=\backslash left(l^\{2\},\backslash \; \backslash frac\{l^\{2\}\}\{k\},\backslash \; \backslash frac\{l^\{2\}\}\{k\}\backslash right)m\backslash mathfrak\{j\}(\backslash Sigma\text{'})$ for $m\backslash mathbf\{a\}$ as a function of $m\backslash mathbf\{a\}^\{0\}$ by ({{equationNote|5b}});

::$\backslash mathfrak\{j\}(\backslash Sigma)=\backslash left(\backslash frac\{l\}\{k^\{3\}\},\backslash \; \backslash frac\{l\}\{k^\{2\}\},\backslash \; \backslash frac\{l\}\{k^\{2\}\}\backslash right)\backslash mathfrak\{j\}(\backslash Sigma\text{'})$ for $\backslash mathbf\{a\}$ as a function of $\backslash mathbf\{a\}^\{0\}$ by ({{equationNote|3a}});

::$m(\backslash Sigma)=\backslash left(k^\{3\}l,\backslash \; kl,\backslash \; kl\backslash right)m(\backslash Sigma\text{'})$ for longitudinal and transverse mass as a function of the rest mass by ({{equationNote|4c}}, {{equationNote|5b}}).

:This time, Lorentz could show that $l=1$, by which his formulas assume the exact relativistic form. He also formulated the equation of motion

::$\{\backslash displaystyle\; \backslash mathfrak\{F\}=\backslash frac\{d\backslash mathfrak\{G\}\}\{dt\}\}$ with $\{\backslash displaystyle\; \backslash mathfrak\{G\}=\backslash frac\{e^\{2\}\}\{6\backslash pi\; c^\{2\}R\}kl\backslash mathfrak\{w\}\}$

:which corresponds to ({{equationNote|4d}}) with $\backslash mathbf\{f\}=\backslash frac\{d\backslash mathbf\{p\}\}\{dt\}=\backslash frac\{d(m\backslash gamma\backslash mathbf\{u\})\}\{dt\}$, with $l=1$, $\backslash mathfrak\{F\}=\backslash mathbf\{f\}$, $\backslash mathfrak\{G\}=\backslash mathbf\{p\}$, $\backslash mathfrak\{w\}=\backslash mathbf\{u\}$, $k=\backslash gamma$, and $e^\{2\}/(6\backslash pi\; c^\{2\}R)=m$ as electromagnetic rest mass. Furthermore, he argued, that these formulas should not only hold for forces and masses of electrically charged particles, but for other processes as well so that the earth's motion through the aether remains undetectable.

;1905{{colon}}: Henri Poincaré introduced the transformation of three-force ({{equationNote|4e}}):

::$X\_\{1\}^\{\backslash prime\}=\backslash frac\{k\}\{l^\{3\}\}\backslash frac\{\backslash rho\}\{\backslash rho^\{\backslash prime\}\}\backslash left(X\_\{1\}+\backslash epsilon\backslash Sigma\; X\_\{1\}\backslash xi\backslash right),\backslash quad\; Y\_\{1\}^\{\backslash prime\}=\backslash frac\{\backslash rho\}\{\backslash rho^\{\backslash prime\}\}\backslash frac\{Y\_\{1\}\}\{l^\{3\}\},\backslash quad\; Z\_\{1\}^\{\backslash prime\}=\backslash frac\{\backslash rho\}\{\backslash rho^\{\backslash prime\}\}\backslash frac\{Z\_\{1\}\}\{l^\{3\}\}$

:with $\backslash frac\{\backslash rho\}\{\backslash rho^\{\backslash prime\}\}=\backslash frac\{k\}\{l^\{3\}\}(1+\backslash epsilon\backslash xi)$, and $k$ as the Lorentz factor, $\backslash rho$ the charge density. Or in modern notation: $\backslash epsilon=v$, $\backslash xi=u\_\{x\}$, $\backslash left(X\_\{1\},\backslash \; Y\_\{1\},\backslash \; Z\_\{1\}\backslash right)=\backslash mathbf\{f\}$, and $\backslash Sigma\; X\_\{1\}\backslash xi=\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}$. As Lorentz, he set $l=1$.

;1905{{colon}}: Albert Einstein derived the equations of motions on the basis of his special theory of relativity, which represent the relation between equally valid inertial frames without the action of a mechanical aether. Einstein concluded, that in a momentary inertial frame $k$ the equations of motion retain their Newtonian form:

::$\backslash mu\backslash frac\{d^\{2\}\backslash xi\}\{d\backslash tau^\{2\}\}=\backslash epsilon\; X\text{'},\backslash quad\backslash mu\backslash frac\{d^\{2\}\backslash eta\}\{d\backslash tau^\{2\}\}=\backslash epsilon\; Y\text{'},\backslash quad\backslash mu\backslash frac\{d^\{2\}\backslash zeta\}\{d\backslash tau^\{2\}\}=\backslash epsilon\; Z\text{'}$.

:This corresponds to $\backslash mathbf\{f\}^\{0\}=m\backslash mathbf\{a\}^\{0\}$, because $\backslash mu=m$ and $\backslash left(\backslash frac\{d^\{2\}\backslash xi\}\{d\backslash tau^\{2\}\},\backslash \; \backslash frac\{d^\{2\}\backslash eta\}\{d\backslash tau^\{2\}\},\backslash \; \backslash frac\{d^\{2\}\backslash zeta\}\{d\backslash tau^\{2\}\}\backslash right)=\backslash mathbf\{a\}^\{0\}$ and $\backslash left(\backslash epsilon\; X\text{'},\backslash \; \backslash epsilon\; Y\text{'},\backslash \; \backslash epsilon\; Z\text{'}\backslash right)=\backslash mathbf\{f\}^\{0\}$. By transformation into a relatively moving system $K$ he obtained the equations for the electrical and magnetic components observed in that frame:

::$\backslash frac\{d^\{2\}x\}\{dt^\{2\}\}=\backslash frac\{\backslash epsilon\}\{\backslash mu\}\backslash frac\{1\}\{\backslash beta^\{3\}\}X,\backslash quad\backslash frac\{d^\{2\}y\}\{dt^\{2\}\}=\backslash frac\{\backslash epsilon\}\{\backslash mu\}\backslash frac\{1\}\{\backslash beta\}\backslash left(Y-\backslash frac\{v\}\{V\}N\backslash right),\backslash quad\backslash frac\{d^\{2\}z\}\{dt^\{2\}\}=\backslash frac\{\backslash epsilon\}\{\backslash mu\}\backslash frac\{1\}\{\backslash beta\}\backslash left(Z+\backslash frac\{v\}\{V\}M\backslash right)$.

:This corresponds to ({{equationNote|4c}}) with $\backslash mathbf\{a\}=\backslash frac\{\backslash mathbf\{f\}\}\{m\}\backslash left(\backslash frac\{1\}\{\backslash gamma^\{3\}\},\backslash \; \backslash frac\{1\}\{\backslash gamma\},\backslash \; \backslash frac\{1\}\{\backslash gamma\}\backslash right)$, because $\backslash mu=m$ and $\backslash left(\backslash frac\{d^\{2\}x\}\{dt^\{2\}\},\backslash \; \backslash frac\{d^\{2\}y\}\{dt^\{2\}\},\backslash \; \backslash frac\{d^\{2\}z\}\{dt^\{2\}\}\backslash right)=\backslash mathbf\{a\}$ and $\backslash left[\backslash epsilon\; X,\backslash \; \backslash epsilon\backslash left(Y-\backslash frac\{v\}\{V\}N\backslash right),\backslash \; \backslash epsilon\backslash left(Z+\backslash frac\{v\}\{V\}M\backslash right)\backslash right]=\backslash mathbf\{f\}$ and $\backslash beta=\backslash gamma$. Consequently, Einstein determined the longitudinal and transverse mass, even though he related it to the force $\backslash left(\backslash epsilon\; X\text{'},\backslash \; \backslash epsilon\; Y\text{'},\backslash \; \backslash epsilon\; Z\text{'}\backslash right)=\backslash mathbf\{f\}^\{0\}$ in the momentary rest frame measured by a comoving spring balance, and to the three-acceleration $\backslash mathbf\{a\}$ in system $K$:

::$\backslash begin\{array\}\{c|c\}$

\begin{align}\mu\beta^{3}\frac{d^{2}x}{dt^{2}} & =\epsilon X=\epsilon X'\\

\mu\beta^{2}\frac{d^{2}y}{dt^{2}} & =\epsilon\beta\left(Y-\frac{v}{V}N\right)=\epsilon Y'\\

\mu\beta^{2}\frac{d^{2}z}{dt^{2}} & =\epsilon\beta\left(Z+\frac{v}{V}M\right)=\epsilon Z'

\end{align}

& \begin{align}\frac{\mu}{\left(\sqrt{1-\left(\frac{v}{V}\right)^{2}}\right)^{3}} & \ \text{longitudinal mass}\\

\\

\frac{\mu}{1-\left(\frac{v}{V}\right)^{2}} & \ \text{transverse mass}

\end{align}

\end{array}

:This corresponds to ({{equationNote|5b}}) with $m\backslash mathbf\{a\}\backslash left(\backslash gamma^\{3\},\backslash \; \backslash gamma^\{2\},\backslash \; \backslash gamma^\{2\}\backslash right)=\backslash mathbf\{f\}\backslash left(1,\backslash \; \backslash gamma,\backslash \; \backslash gamma\backslash right)=\backslash mathbf\{f\}^\{0\}$.

;1905{{colon}}: Poincaré introduces the transformation of three-acceleration ({{equationNote|1c}}):

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

:where $\backslash left(\backslash xi,\backslash \; \backslash eta,\backslash \; \backslash zeta\backslash right)=\backslash mathbf\{u\}$ as well as $k=\backslash gamma$ and $\backslash epsilon=v$ and $\backslash mu=1+\backslash xi\backslash epsilon=1+u\_\{x\}v$.

:Furthermore, he introduced the four-force in the form:

::$k\_\{0\}X\_\{1\},\backslash quad\; k\_\{0\}Y\_\{1\},\backslash quad\; k\_\{0\}Z\_\{1\},\backslash quad\; k\_\{0\}T\_\{1\}$

:where $k\_\{0\}=\backslash gamma\_\{0\}$ and $\backslash left(X\_\{1\},\backslash \; Y\_\{1\},\backslash \; Z\_\{1\}\backslash right)=\backslash mathbf\{f\}$ and $T\_\{1\}=\backslash Sigma\; X\_\{1\}\backslash xi=\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}$.

;1906{{colon}}: Max Planck derived the equation of motion

::$\backslash frac\{m\backslash ddot\{x\}\}\{\backslash sqrt\{1-\backslash frac\{q^\{2\}\}\{c^\{2\}\}\}\}=e\backslash mathfrak\{E\}\_\{x\}-\backslash frac\{e\backslash dot\{x\}\}\{c^\{2\}\}\backslash left(\backslash dot\{x\}\backslash mathfrak\{E\}\_\{x\}+\backslash dot\{y\}\backslash mathfrak\{E\}\_\{y\}+\backslash dot\{z\}\backslash mathfrak\{E\}\_\{z\}\backslash right)+\backslash frac\{e\}\{c\}\backslash left(\backslash dot\{y\}\backslash mathfrak\{H\}\_\{z\}-\backslash dot\{z\}\backslash mathfrak\{H\}\_\{y\}\backslash right)\backslash \; \backslash text\{etc.\}$

:with

::$e\backslash left(\backslash dot\{x\}\backslash mathfrak\{E\}\_\{x\}+\backslash dot\{y\}\backslash mathfrak\{E\}\_\{y\}+\backslash dot\{z\}\backslash mathfrak\{E\}\_\{z\}\backslash right)=\backslash frac\{m\backslash left(\backslash dot\{x\}\backslash ddot\{x\}+\backslash dot\{y\}\backslash ddot\{y\}+\backslash dot\{z\}\backslash ddot\{z\}\backslash right)\}\{\backslash left(1-\backslash frac\{q^\{2\}\}\{c^\{2\}\}\backslash right)^\{3/2\}\}$ and $e\backslash mathfrak\{E\}\_\{x\}+\backslash frac\{e\}\{c\}\backslash left(\backslash dot\{y\}\backslash mathfrak\{H\}\_\{z\}-\backslash dot\{z\}\backslash mathfrak\{H\}\_\{y\}\backslash right)=X\backslash \; \backslash text\{etc.\}$

:and

::$\backslash frac\{d\}\{dt\}\backslash left\backslash \{\; \backslash frac\{m\backslash dot\{x\}\}\{\backslash sqrt\{1-\backslash frac\{q^\{2\}\}\{c^\{2\}\}\}\}\backslash right\backslash \}\; =X\backslash \; \backslash text\{etc.\}$

:The equations correspond to ({{equationNote|4d}}) with

::$\backslash mathbf\{f\}=\backslash frac\{d\backslash mathbf\{p\}\}\{dt\}=\backslash frac\{d(m\backslash gamma\backslash mathbf\{u\})\}\{dt\}=m\backslash gamma^\{3\}\backslash left(\backslash frac\{(\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{u\})\backslash mathbf\{u\}\}\{c^\{2\}\}\backslash right)+m\backslash gamma\backslash mathbf\{a\}$, with $X=f\_\{x\}$ and $q=v$ and $\backslash dot\{x\}\backslash ddot\{x\}+\backslash dot\{y\}\backslash ddot\{y\}+\backslash dot\{z\}\backslash ddot\{z\}=\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{a\}$, in agreement with those given by Lorentz (1904).

;1907{{colon}}: Einstein analyzed a uniformly accelerated reference frame and obtained formulas for coordinate dependent time dilation and speed of light, analogous to those given by Kottler-Møller-Rindler coordinates.

;1907{{colon}}: Hermann Minkowski defined the relation between the four-force (which he called the moving force) and the four acceleration

::$m\backslash frac\{d\}\{d\backslash tau\}\backslash frac\{dx\}\{d\backslash tau\}=R\_\{x\},\backslash quad\; m\backslash frac\{d\}\{d\backslash tau\}\backslash frac\{dy\}\{d\backslash tau\}=R\_\{y\},\backslash quad\; m\backslash frac\{d\}\{d\backslash tau\}\backslash frac\{dz\}\{d\backslash tau\}=R\_\{z\},\backslash quad\; m\backslash frac\{d\}\{d\backslash tau\}\backslash frac\{dt\}\{d\backslash tau\}=R\_\{t\}$

:corresponding to $m\backslash mathbf\{A\}=\backslash mathbf\{F\}$.

;1908{{colon}}: Minkowski denotes the second derivative $x,y,z,t$ with respect to proper time as "acceleration vector" (four-acceleration). He showed, that its magnitude at an arbitrary point $P$ of the worldline is $c^\{2\}/\backslash varrho$, where $\backslash varrho$ is the magnitude of a vector directed from the center of the corresponding "curvature hyperbola" ({{langx|de|Krümmungshyperbel}}) to $P$.

;1909{{colon}}: Max Born denotes the motion with constant magnitude of Minkowski's acceleration vector as "hyperbolic motion" ({{langx|de|Hyperbelbewegung}}), in the course of his study of rigidly accelerated motion. He set $p=dx/d\backslash tau$ (now called proper velocity) and $q=-dt/d\backslash tau=\backslash sqrt\{1+p^\{2\}/c^\{2\}\}$ as Lorentz factor and $\backslash tau$ as proper time, with the transformation equations

::$x=-q\backslash xi,\backslash quad\; y=\backslash eta,\backslash quad\; z=\backslash zeta,\backslash quad\; t=\backslash frac\{p\}\{c^\{2\}\}\backslash xi$.

:which corresponds to ({{equationNote|6a}}) with $\backslash xi=c^\{2\}/\backslash alpha$ and $p=c\backslash sinh(\backslash alpha\backslash tau/c)$. Eliminating $p$ Born derived the hyperbolic equation $x^\{2\}-c^\{2\}t^\{2\}=\backslash xi^\{2\}$, and defined the magnitude of acceleration as $b=c^\{2\}/\backslash xi$. He also noticed that his transformation can be used to transform into a "hyperbolically accelerated reference system" ({{langx|de|hyperbolisch beschleunigtes Bezugsystem}}).

;1909{{colon}}: Gustav Herglotz extends Born's investigation to all possible cases of rigidly accelerated motion, including uniform rotation.

;1910{{colon}}: Arnold Sommerfeld brought Born's formulas for hyperbolic motion in a more concise form with $l=ict$ as the imaginary time variable and $\backslash varphi$ as an imaginary angle:

::$x=r\backslash cos\backslash varphi,\backslash quad\; y=y\text{'},\backslash quad\; z=z\text{'},\backslash quad\; l=r\backslash sin\backslash varphi$

:He noted that when $r,y,z$ are variable and $\backslash varphi$ is constant, they describe the worldline of a charged body in hyperbolic motion. But if $r,y,z$ are constant and $\backslash varphi$ is variable, they denote the transformation into its rest frame.

;1911{{colon}}: Sommerfeld explicitly used the expression "proper acceleration" ({{langx|de|Eigenbeschleunigung}}) for the quantity $\backslash dot\{v\}\_\{0\}$ in $\backslash dot\{v\}=\backslash dot\{v\}\_\{0\}\backslash left(1-\backslash beta^\{2\}\backslash right)^\{3/2\}$, which corresponds to ({{equationNote|3a}}), as the acceleration in the momentary inertial frame.

;1911{{colon}}: Herglotz explicitly used the expression "rest acceleration" ({{langx|de|Ruhbeschleunigung}}) instead of proper acceleration. He wrote it in the form $\backslash gamma\_\{l\}^\{0\}=\backslash beta^\{3\}\backslash gamma\_\{l\}$ and $\backslash gamma\_\{t\}^\{0\}=\backslash beta^\{2\}\backslash gamma\_\{t\}$ which corresponds to ({{equationNote|3a}}), where $\backslash beta$ is the Lorentz factor and $\backslash gamma\_\{l\}^\{0\}$ or $\backslash gamma\_\{t\}^\{0\}$ are the longitudinal and transverse components of rest acceleration.

;1911{{colon}}: Max von Laue derived in the first edition of his monograph "Das Relativitätsprinzip" the transformation for three-acceleration by differentiation of the velocity addition

::

:equivalent to ({{equationNote|1c}}) as well as to Poincaré (1905/6). From that he derived the transformation of rest acceleration (equivalent to {{equationNote|3a}}), and eventually the formulas for hyperbolic motion which corresponds to ({{equationNote|6a}}):

::$\backslash pm\backslash mathfrak\{q\}\_\{x\}=\backslash pm\backslash frac\{dx\}\{dt\}=\backslash frac\{cbt\}\{\backslash sqrt\{c^\{2\}+b^\{2\}t^\{2\}\}\},\backslash quad\backslash pm\backslash left(x-x\_\{0\}\backslash right)=\backslash frac\{c\}\{b\}\backslash sqrt\{c^\{2\}+b^\{2\}t^\{2\}\},$

:thus

::$x^\{2\}-c^\{2\}t^\{2\}=x^\{2\}-u^\{2\}=c^\{4\}/b^\{2\},\backslash quad\; y=\backslash eta,\backslash quad\; z=\backslash zeta$,

:and the transformation into a hyperbolic reference system with imaginary angle $\backslash varphi$:

::$\backslash begin\{array\}\{c|c\}$

\begin{align}X & =R\cos\varphi\\

L & =R\sin\varphi

\end{align}

& \begin{align}R^{2} & =X^{2}+L^{2}\\

\tan\varphi & =\frac{L}{X}

\end{align}

\end{array}.

:He also wrote the transformation of three-force as

::

:equivalent to ({{equationNote|4e}}) as well as to Poincaré (1905).

;1912–1914{{colon}}: Friedrich Kottler obtained general covariance of Maxwell's equations, and used four-dimensional Frenet-Serret formulas to analyze the Born rigid motions given by Herglotz (1909). He also obtained the proper reference frames for hyperbolic motion and uniform circular motion.

;1913{{colon}}: von Laue replaced in the second edition of his book the transformation of three-acceleration by Minkowski's acceleration vector for which he coined the name "four-acceleration" ({{langx|de|Viererbeschleunigung}}), defined by $\backslash dot\{Y\}=\backslash frac\{dY\}\{d\backslash tau\}$ with $Y$ as four-velocity. He showed, that the magnitude of four-acceleration corresponds to the rest acceleration $\backslash dot\{\backslash mathfrak\{q\}\}^\{0\}$ by

::$|\backslash dot\{Y|\}=\backslash frac\{1\}\{c\}|\backslash dot\{\backslash mathfrak\{q\}\}^\{0\}|$,

:which corresponds to ({{equationNote|3b}}). Subsequently, he derived the same formulas as in 1911 for the transformation of rest acceleration and hyperbolic motion, and the hyperbolic reference frame.

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{{Cite journal|author=Born, Max|year=1909|title=Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips|trans-title=Wikisource translation: The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity|journal=Annalen der Physik|volume=335|issue=11|pages=1–56 |doi=10.1002/andp.19093351102|bibcode=1909AnP...335....1B|url=https://zenodo.org/record/1424151}}

{{Cite journal|author=Einstein, Albert|year=1905|title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik|volume=322|issue=10 |pages=891–921|doi=10.1002/andp.19053221004|url=http://sedici.unlp.edu.ar/handle/10915/2786|bibcode=1905AnP...322..891E|doi-access=free}}; See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].

{{Citation|author=Einstein, Albert|year=1908

|orig-year=1907|title=Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen|journal=Jahrbuch der Radioaktivität und Elektronik

|volume=4|pages=411–462|bibcode = 1908JRE.....4..411E|url=http://www.soso.ch/wissen/hist/SRT/E-1907.pdf}}; English translation [http://einsteinpapers.press.princeton.edu/vol2-trans/266 On the relativity principle and the conclusions drawn from it] at Einstein paper project.

{{Cite journal|author=Herglotz, G|year=1910|orig-year=1909|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper |trans-title=Wikisource translation: On bodies that are to be designated as "rigid" from the standpoint of the relativity principle|journal=Annalen der Physik|volume=336|issue=2|pages=393–415|doi=10.1002/andp.19103360208|bibcode=1910AnP...336..393H|url=https://zenodo.org/record/1424161}}

{{Cite journal|author=Herglotz, G.|year=1911|title=Über die Mechanik des deformierbaren Körpers vom Standpunkte der Relativitätstheorie|journal=Annalen der Physik|volume=341|issue=13|pages=493–533|doi=10.1002/andp.19113411303|url=http://gallica.bnf.fr/ark:/12148/bpt6k153397.image.f509|bibcode=1911AnP...341..493H}}

{{Cite book|author=Laue, Max von|year=1911|title=Das Relativitätsprinzip|publisher=Vieweg |location=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}

{{Cite book|author=Laue, Max von|year=1913|title=Das Relativitätsprinzip|edition=2. Ausgabe|publisher=Vieweg |location=Braunschweig}}

{{Cite journal|author=Lorentz, Hendrik Antoon|year=1899|title=Simplified Theory of Electrical and Optical Phenomena in Moving Systems|journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences|volume=1|pages=427–442|title-link=s:en:Simplified Theory of Electrical and Optical Phenomena in Moving Systems|bibcode=1898KNAB....1..427L}}

{{Cite journal|author=Lorentz, Hendrik Antoon|year=1904|title=Electromagnetic phenomena in a system moving with any velocity smaller than that of light|journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences|volume=6|pages=809–831|title-link=s:Electromagnetic phenomena|bibcode=1903KNAB....6..809L}}

{{Cite journal|author=Kottler, Friedrich|year=1912|title=Über die Raumzeitlinien der Minkowski'schen Welt|trans-title=Wikisource translation: On the spacetime lines of a Minkowski world|journal=Wiener Sitzungsberichte 2a|volume=121|pages=1659–1759|hdl=2027/mdp.39015051107277}}

{{Cite journal|author=Kottler, Friedrich|year=1914a|title=Relativitätsprinzip und beschleunigte Bewegung|journal=Annalen der Physik|volume=349|issue=13|pages=701–748|url=http://gallica.bnf.fr/ark:/12148/bpt6k15347v.image.f737|doi=10.1002/andp.19143491303|bibcode=1914AnP...349..701K}}

{{Cite journal|author=Kottler, Friedrich|year=1914b|title=Fallende Bezugssysteme vom Standpunkte des Relativitätsprinzips|journal=Annalen der Physik|volume=350|issue=20|pages=481–516|url=http://gallica.bnf.fr/ark:/12148/bpt6k153486.image.f494|doi=10.1002/andp.19143502003|bibcode=1914AnP...350..481K}}

{{Citation

|author=Minkowski, Hermann

|year=1908

|orig-year=1907

|title=Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern

|trans-title=Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies

|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse

|pages=53–111|title-link=s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern

}}

{{Cite journal|author=Minkowski, Hermann|year=1909|orig-year=1908|title=Raum und Zeit. Vortrag, gehalten auf der 80. Naturforscher-Versammlung zu Köln am 21. September 1908.|trans-title=Wikisource translation: Space and Time|journal=Jahresbericht der Deutschen Mathematiker-Vereinigung|location=Leipzig|title-link=s:Raum und Zeit (Minkowski)}}

{{Cite journal|author=Planck, Max|year=1906|title=Das Prinzip der Relativität und die Grundgleichungen der Mechanik |trans-title=Wikisource translation: The Principle of Relativity and the Fundamental Equations of Mechanics|journal=Verhandlungen Deutsche Physikalische Gesellschaft|volume=8|pages=136–141}}

{{Cite journal|author=Poincaré, Henri|year=1905|title=Sur la dynamique de l'électron |trans-title=Wikisource translation: On the Dynamics of the Electron|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=140|pages=1504–1508|title-link=s:fr:Sur la dynamique de l'électron (juin)}}

{{Cite journal|author=Poincaré, Henri|year=1906|orig-year=1905|title=Sur la dynamique de l'électron |trans-title=Wikisource translation: On the Dynamics of the Electron|journal=Rendiconti del Circolo Matematico di Palermo|volume=21|pages=129–176|title-link=s:fr:Sur la dynamique de l'électron (juillet)|doi=10.1007/BF03013466|hdl=2027/uiug.30112063899089|bibcode=1906RCMP...21..129P|s2cid=120211823}}

{{Cite journal |author=Sommerfeld, Arnold |year=1910 |title=Zur Relativitätstheorie II: Vierdimensionale Vektoranalysis|trans-title=Wikisource translation: On the Theory of Relativity II: Four-dimensional Vector Analysis |journal=Annalen der Physik |volume=338 |issue=14 |pages=649–689|doi=10.1002/andp.19103381402|bibcode=1910AnP...338..649S|url=https://zenodo.org/record/1424179 }}

{{Cite journal|author=Sommerfeld, Arnold|year=1911|title=Über die Struktur der gamma-Strahlen|journal=Sitzungsberichte der Mathematematisch-physikalischen Klasse der K. B. Akademie der Wissenschaften zu München|issue=1|pages=1–60 |url=http://publikationen.badw.de/003395686}}

• Mathpages: [http://www.mathpages.com/home/kmath674/kmath674.htm Transverse Mass in Einstein's Electrodynamics], [http://www.mathpages.com/rr/s2-09/2-09.htm Accelerated Travels], [http://www.mathpages.com/home/kmath422/kmath422.htm Born Rigidity, Acceleration, and Inertia], [http://www.mathpages.com/home/kmath528/kmath528.htm Does A Uniformly Accelerating Charge Radiate?]
• Physics FAQ: [http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html Acceleration in Special Relativity], [http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html The Relativistic Rocket]

Category:Special relativity

Special relativity