Acceleration (special relativity)


Accelerations in special relativity 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 is only required when there is curvature of spacetime caused by the energy-momentum tensor. 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 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. 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 or Wolfgang Pauli. For instance, equations of motion and acceleration transformations were developed in the papers of Hendrik Antoon Lorentz, Henri Poincaré, Albert Einstein, Max Planck, and four-acceleration, proper acceleration, hyperbolic motion, accelerating reference frames, Born rigidity, have been analyzed by Einstein, Hermann Minkowski, Max Born, Gustav Herglotz, Arnold Sommerfeld, von Laue, Friedrich Kottler, see [|section on history].

Three-acceleration

In accordance with both Newtonian mechanics and SR, three-acceleration or coordinate acceleration is the first derivative of velocity with respect to coordinate time or the second derivative of the location 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 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 and depend on the Lorentz transformation, therefore also three-acceleration and its components vary in different inertial frames. When the relative velocity between the frames is directed in the x-direction by with as Lorentz factor, the Lorentz transformation has the form
or for arbitrary velocities of magnitude :
In order to find out the transformation of three-acceleration, one has to differentiate the spatial coordinates and of the Lorentz transformation with respect to and, from which the transformation of three-velocity between and follows, and eventually by another differentiation with respect to and the transformation of three-acceleration between and follows. Starting from, this procedure gives the transformation where the accelerations are parallel or perpendicular to the velocity:
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 and with relative velocity, then in the acceleration of an object with momentary velocity is measured, while in the same object is accelerated by and has the momentary velocity. 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 as four-position and as four-velocity, then the four-acceleration of an object is obtained by differentiation with respect to proper time instead of coordinate time:
where is the object's three-acceleration and its momentary three-velocity of magnitude with the corresponding Lorentz factor. If only the spatial part is considered, and when the velocity is directed in the x-direction by and only accelerations parallel or perpendicular 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 in two inertial frames with relative speed are connected by a Lorentz transformation. Thus replacing by in gives:
or replacing by in gives the transformation at arbitrary relative velocity :
On the other hand, the inner product with metric signature and consequently its magnitude is invariant, thus:

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 in these frames can be directly measured by an accelerometer, and is called proper acceleration or rest acceleration. The relation of in a momentary inertial frame and measured in an external inertial frame follows from with,, and. So in terms of, when the velocity is directed in the x-direction by and when only accelerations parallel or perpendicular to the velocity are considered, it follows:
Generalized by for arbitrary directions of of magnitude :
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, in which and by it follows :
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 in and in is given, namely
from which follows again when the velocity is directed in the x-direction by and only accelerations parallel or perpendicular to the velocity are considered.

Acceleration and force

The four-force as a function of three-force is given by. Four-force and four-acceleration and the invariant mass are furthermore related by, thus
The relation between three-force and three-acceleration for arbitrary directions of the velocity is thus
When the velocity is directed in the x-direction by and only accelerations parallel or perpendicular to the velocity are considered
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 is the three-momentum. The corresponding transformation of three-force between in and in or perpendicular follows by substitution of the relevant transformation formulas for,,,, or from the Lorentz transformed components of four-force, with the result:
Or generalized for arbitrary directions of, as well as with magnitude :

Proper acceleration and proper force

The force in a momentary inertial frame measured by a comoving spring balance can be called proper force. It follows from by setting and as well as and. Thus by where only accelerations parallel or perpendicular to the velocity are considered:
Generalized by for arbitrary directions of of magnitude :
Since in momentary inertial frames one has, the Newtonian relation holds, therefore can be summarized
By that, the apparent contradiction in the historical definitions of transverse mass can be explained. Einstein described the relation between three-acceleration and proper force
while Lorentz and Planck 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. 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 by leads to the world line
The worldline corresponds to the hyperbolic equation, 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 by can be seen as a centripetal acceleration, leading to the worldline of a body in uniform rotation
where is the tangential speed, is the orbital radius, is the angular velocity as a function of coordinate time, and 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. 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 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.
;1899: Hendrik Lorentz derived the correct relations for accelerations, forces and masses between a resting electrostatic systems of particles , and a system emerging from it by adding a translation, with as the Lorentz factor:
;1904: Lorentz derived the previous relations in a more detailed way, namely with respect to the properties of particles resting in the system and the moving system, with the new auxiliary variable equal to compared to the one in 1899, thus:
;1905: Henri Poincaré introduced the transformation of three-force :
;1905: 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 the equations of motion retain their Newtonian form:
;1905: Poincaré introduces the transformation of three-acceleration :
;1906: Max Planck derived the equation of motion
;1907: 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: Hermann Minkowski defined the relation between the four-force and the four acceleration
;1908: Minkowski denotes the second derivative with respect to proper time as "acceleration vector". He showed, that its magnitude at an arbitrary point of the worldline is, where is the magnitude of a vector directed from the center of the corresponding "curvature hyperbola" to.
;1909: Max Born denotes the motion with constant magnitude of Minkowski's acceleration vector as "hyperbolic motion", in the course of his study of rigidly accelerated motion. He set and as Lorentz factor and as proper time, with the transformation equations
;1909: Gustav Herglotz extends Born's investigation to all possible cases of rigidly accelerated motion, including uniform rotation.
;1910: Arnold Sommerfeld brought Born's formulas for hyperbolic motion in a more concise form with as the imaginary time variable and as an imaginary angle:
;1911: Sommerfeld explicitly used the expression "proper acceleration" for the quantity in, which corresponds to, as the acceleration in the momentary inertial frame.
;1911: Herglotz explicitly used the expression "rest acceleration" instead of proper acceleration. He wrote it in the form and which corresponds to, where is the Lorentz factor and or are the longitudinal and transverse components of rest acceleration.
;1911: Max von Laue derived in the first edition of his monograph "Das Relativitätsprinzip" the transformation for three-acceleration by differentiation of the velocity addition
;1912–1914: 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. He also obtained the proper reference frames for hyperbolic motion and uniform circular motion.
;1913: 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", defined by with as four-velocity. He showed, that the magnitude of four-acceleration corresponds to the rest acceleration by

Historical papers