Acceleration (special relativity)

Accelerations in special relativity (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which also leads to more complex definitions of "acceleration". SR as the theory of flat Minkowski spacetime remains valid in the presence of accelerations, because general relativity (GR) is only required when there is curvature of spacetime caused by the energy-momentum tensor (which is mainly determined by mass). However, since the amount of spacetime curvature is not particularly high on Earth or its vicinity, SR remains valid for most practical purposes, such as experiments in particle accelerators.
One can derive transformation formulas for ordinary accelerations in three spatial dimensions (three-acceleration or coordinate acceleration) as measured in an external inertial frame of reference, as well as for the special case of proper acceleration measured by a comoving accelerometer. Another useful formalism is four-acceleration, as its components can be connected in different inertial frames by a Lorentz transformation. Also equations of motion can be formulated which connect acceleration and force. Equations for several forms of acceleration of bodies and their curved world lines follow from these formulas by integration. Well known special cases are hyperbolic motion for constant longitudinal proper acceleration or uniform circular motion. Eventually, it is also possible to describe these phenomena in accelerated frames in the context of special relativity, see Proper reference frame (flat spacetime). In such frames, effects arise which are analogous to homogeneous gravitational fields, which have some formal similarities to the real, inhomogeneous gravitational fields of curved spacetime in general relativity. In the case of hyperbolic motion one can use Rindler coordinates, in the case of uniform circular motion one can use Born coordinates.
Concerning the historical development, relativistic equations containing accelerations can already be found in the early years of relativity, as summarized in early textbooks by Max von Laue (1911, 1921) or Wolfgang Pauli (1921). For instance, equations of motion and acceleration transformations were developed in the papers of Hendrik Antoon Lorentz (1899, 1904), Henri Poincaré (1905), Albert Einstein (1905), Max Planck (1906), and four-acceleration, proper acceleration, hyperbolic motion, accelerating reference frames, Born rigidity, have been analyzed by Einstein (1907), Hermann Minkowski (1907, 1908), Max Born (1909), Gustav Herglotz (1909), Arnold Sommerfeld (1910), von Laue (1911), Friedrich Kottler (1912, 1914), see section on history.
Three-acceleration.
In accordance with both Newtonian mechanics and SR, three-acceleration or coordinate acceleration formula_1 is the first derivative of velocity formula_2 with respect to coordinate time or the second derivative of the location formula_3 with respect to coordinate time:
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 formula_5 in accordance with the Galilean transformation, therefore the three-acceleration derived from it is equal too in all inertial frames:
On the contrary in SR, both formula_7 and formula_8 depend on the Lorentz transformation, therefore also three-acceleration formula_9 and its components vary in different inertial frames. When the relative velocity between the frames is directed in the x-direction by formula_10 with formula_11 as Lorentz factor, the Lorentz transformation has the form
or for arbitrary velocities formula_12 of magnitude formula_13:
\left(\gamma_{v}-1\right)-t\gamma_{v}\right]\\
t^{\prime} & =\gamma_{v}\left(t-\frac{\mathbf{r\cdot v}}{c^2}\right)
 & \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)
In order to find out the transformation of three-acceleration, one has to differentiate the spatial coordinates formula_7 and formula_15 of the Lorentz transformation with respect to formula_8 and formula_17, from which the transformation of three-velocity (also called velocity-addition formula) between formula_18 and formula_19 follows, and eventually by another differentiation with respect to formula_8 and formula_17 the transformation of three-acceleration between formula_9 and formula_23 follows. Starting from (), this procedure gives the transformation where the accelerations are parallel (x-direction) or perpendicular (y-, z-direction) to the velocity:
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}}\\
 & \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}}\\
or starting from () this procedure gives the result for the general case of arbitrary directions of velocities and accelerations:
This means, if there are two inertial frames formula_24 and formula_25 with relative velocity formula_26, then in formula_24 the acceleration formula_9 of an object with momentary velocity formula_18 is measured, while in formula_25 the same object has an acceleration formula_23 and has the momentary velocity formula_19. 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.
Four-acceleration.
If four-vectors are used instead of three-vectors, namely formula_33 as four-position and formula_34 as four-velocity, then the four-acceleration formula_35 of an object is obtained by differentiation with respect to proper time formula_36 instead of coordinate time:
 & =\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)
where formula_9 is the object's three-acceleration and formula_18 its momentary three-velocity of magnitude formula_39 with the corresponding Lorentz factor formula_40. If only the spatial part is considered, and when the velocity is directed in the x-direction by formula_41 and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered, the expression is reduced to:
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 formula_43 and formula_44 in two inertial frames with relative speed formula_45 are connected by a Lorentz transformation analogous to (, ). Another property of four-vectors is the invariance of the inner product formula_46 or its magnitude formula_47, which gives in this case:
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 formula_48 in these frames can be directly measured by an accelerometer, and is called proper acceleration or rest acceleration. The relation of formula_49 in a momentary inertial frame formula_25 and formula_9 measured in an external inertial frame formula_24 follows from (, ) with formula_53, formula_54, formula_55 and formula_56. So in terms of (), when the velocity is directed in the x-direction by formula_57 and when only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered, it follows:
a_{y}^{0} & =\frac{a_{y}}{1-\frac{u^{2}}{c^{2}}}\\
 & \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)
 & \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)
Generalized by () for arbitrary directions of formula_18 of magnitude formula_39:
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 formula_25, in which formula_62 and by formula_63 it follows formula_64:
Thus the magnitude of four-acceleration corresponds to the magnitude of proper acceleration. By combining this with (), an alternative method for the determination of the connection between formula_49 in formula_25 and formula_9 in formula_24 is given, namely
from which () follows again when the velocity is directed in the x-direction by formula_41 and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered.
Acceleration and force.
Assuming constant mass formula_71, the four-force formula_72 as a function of three-force formula_73 is related to four-acceleration () by formula_74, thus:
The relation between three-force and three-acceleration for arbitrary directions of the velocity is thus
\mathbf{a} & =\frac{1}{m\gamma}\left(\mathbf{f}-\frac{(\mathbf{f}\cdot\mathbf{u})\mathbf{u}}{c^{2}}\right)
When the velocity is directed in the x-direction by formula_41 and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered
f_{y} & =\frac{ma_{y}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\\
 & \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}}}\\
 & \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)
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:
The relation () between three-acceleration and three-force can also be obtained from the equation of motion
where formula_78 is the three-momentum. The corresponding transformation of three-force between formula_73 in formula_24 and formula_81 in formula_25 (when the relative velocity between the frames is directed in the x-direction by formula_10 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 formula_18, formula_9, formula_86, formula_87, or from the Lorentz transformed components of four-force, with the result:
(\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)}\\
 & \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)}\\
Or generalized for arbitrary directions of formula_18, as well as formula_26 with magnitude formula_13:
Proper acceleration and proper force.
The force formula_91 in a momentary inertial frame measured by a comoving spring balance can be called proper force. It follows from (, ) by setting formula_92 and formula_54 as well as formula_55 and formula_56. Thus by () where only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity formula_57 are considered:
 & \begin{align}f_{x} & =f_{x}^{0}\\
f_{y} & =f_{y}^{0}\sqrt{1-\frac{u^{2}}{c^{2}}}\\
 & \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)
Generalized by () for arbitrary directions of formula_18 of magnitude formula_39:
Since in momentary inertial frames one has four-force formula_100 and four-acceleration formula_101, equation () produces the Newtonian relation formula_102, therefore (, , ) can be summarized
By that, the apparent contradiction in the historical definitions of transverse mass formula_103 can be explained. Einstein (1905) described the relation between three-acceleration and proper force
while Lorentz (1899, 1904) and Planck (1906) described the relation between three-acceleration and three-force
Curved world lines.
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: 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 () for proper acceleration:
a) Hyperbolic motion: The constant, longitudinal proper acceleration formula_106 by () leads to the world line
 & \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)
The worldline corresponds to the hyperbolic equation formula_107, 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 formula_108 by () can be seen as a centripetal acceleration, leading to the worldline of a body in uniform rotation
where formula_109 is the tangential speed, formula_110 is the orbital radius, formula_111 is the angular velocity as a function of coordinate time, and formula_112 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. In particular, it can be shown that hyperbolic motion and uniform circular motion are special cases of motions having constant curvatures and torsions, 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.
Accelerated reference frames.
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. 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.
History.
For further information see von Laue, Pauli, Miller, Zahar, Gourgoulhon, and the historical sources in history of special relativity.