Accelerations in special relativity (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time.
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.
Another useful formalism is four-acceleration, as its components can be connected in different inertial frames by a Lorentz transformation.
Equations for several forms of acceleration of bodies and their curved world lines follow from these formulas by integration.
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.
[3] For instance, equations of motion and acceleration transformations were developed in the papers of Hendrik Antoon Lorentz (1899, 1904),[H 1][H 2] Henri Poincaré (1905),[H 3][H 4] Albert Einstein (1905),[H 5] Max Planck (1906),[H 6] and four-acceleration, proper acceleration, hyperbolic motion, accelerating reference frames, Born rigidity, have been analyzed by Einstein (1907),[H 7] Hermann Minkowski (1907, 1908),[H 8][H 9] Max Born (1909),[H 10] Gustav Herglotz (1909),[H 11][H 12] Arnold Sommerfeld (1910),[H 13][H 14] von Laue (1911),[H 15][H 16] Friedrich Kottler (1912, 1914),[H 17] see section on history.
In accordance with both Newtonian mechanics and SR, three-acceleration or coordinate acceleration
in accordance with the Galilean transformation, therefore the three-acceleration derived from it is equal too in all inertial frames:[4] On the contrary in SR, both
:[5] In order to find out the transformation of three-acceleration, one has to differentiate the spatial coordinates
Starting from (1a), this procedure gives the transformation where the accelerations are parallel (x-direction) or perpendicular (y-, z-direction) to the velocity:[6][7][8][9][H 4][H 15] or starting from (1b) this procedure gives the result for the general case of arbitrary directions of velocities and accelerations:[10][11] This means, if there are two inertial frames
and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered, the expression is reduced to:[15][16] 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
, which gives in this case:[16][13][17] 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.
and when only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered, it follows:[12][19][18][H 1][H 2][H 14][H 12] Generalized by (1d) for arbitrary directions of
:[20][21][17] 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
and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered.
and only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity are considered[27][26][23][H 2][H 6] 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:[27][28][H 2] The relation (4b) between three-acceleration and three-force can also be obtained from the equation of motion[29][25][H 2][H 6] where
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
, or from the Lorentz transformed components of four-force, with the result:[29][30][24][H 3][H 15] Or generalized for arbitrary directions of
in a momentary inertial frame measured by a comoving spring balance can be called proper force.
Thus by (4e) where only accelerations parallel (x-direction) or perpendicular (y-, z-direction) to the velocity
, therefore (3a, 4c, 5a) can be summarized[37] By that, the apparent contradiction in the historical definitions of transverse mass
[38] Einstein (1905) described the relation between three-acceleration and proper force[H 5] while Lorentz (1899, 1904) and Planck (1906) described the relation between three-acceleration and three-force[H 2] 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).
by (3a) leads to the world line[12][18][19][25][41][42][H 10][H 15] The worldline corresponds to the hyperbolic equation
by (3a) can be seen as a centripetal acceleration,[13] leading to the worldline of a body in uniform rotation[43][44] where
[H 11][H 17] A body is called Born rigid if the spacetime distance between its infinitesimally separated worldlines or points remains constant during acceleration.
The proper reference frame established that way is closely related to Fermi coordinates.
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,[2] Pauli,[3] Miller,[49] Zahar,[50] Gourgoulhon,[48] and the historical sources in history of special relativity.