It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics.
The concept was introduced by Max Born (1909),[1][2] who gave a detailed description of the case of constant proper acceleration which he called hyperbolic motion.
It was formulated by Gustav Herglotz (1909, who classified all forms of rotational motions)[4] and in a less general way by Fritz Noether (1909).
Born rigidity is satisfied if the orthogonal spacetime distance between infinitesimally separated curves or worldlines is constant,[7] or equivalently, if the length of the rigid body in momentary co-moving inertial frames measured by standard measuring rods (i.e. the proper length) is constant and is therefore subjected to Lorentz contraction in relatively moving frames.
A body that could maintain its own rigidity would violate special relativity, as its speed of sound would be infinite.
[9] It was shown by Herglotz (1911),[10] that a relativistic theory of elasticity can be based on the assumption, that stresses arise when the condition of Born rigidity is broken.
But if this phase is over and the centripetal acceleration becomes constant, the body can be uniformly rotating in agreement with Born rigidity.
Likewise, if it is now in uniform circular motion, this state cannot be changed without again breaking the Born rigidity of the body.
A classification of allowed, in particular rotational, Born rigid motions in flat Minkowski spacetime was given by Herglotz,[4] which was also studied by Friedrich Kottler (1912, 1914),[12] Georges Lemaître (1924),[13] Adriaan Fokker (1940),[14] George Salzmann & Abraham H. Taub (1954).
The resulting worldliness can be split into two classes: Herglotz defined this class in terms of equidistant curves which are the orthogonal trajectories of a family of hyperplanes, which also can be seen as solutions of a Riccati equation[15] (this was called "plane motion" by Salzmann & Taub[7] or "irrotational rigid motion" by Boyer[16][17]).
The general metric for these irrotational motions has been given by Herglotz, whose work was summarized with simplified notation by Lemaître (1924).
[19] Already Born (1909) pointed out that a rigid body in translational motion has a maximal spatial extension depending on its acceleration, given by the relation
[31] Worldlines of constant curvatures in flat spacetime were also studied by Kottler (1912),[12] Petrův (1964),[32] John Lighton Synge (1967, who called them timelike helices in flat spacetime),[33] or Letaw (1981, who called them stationary worldlines)[34] as the solutions of the Frenet–Serret formulas.
in the notation of Synge; see the following table) is the only Born rigid motion that belongs to both classes A and B.
Attempts to extend the concept of Born rigidity to general relativity have been made by Salzmann & Taub (1954),[7] C. Beresford Rayner (1959),[50] Pirani & Williams (1962),[30] Robert H. Boyer (1964).
[16] It was shown that the Herglotz–Noether theorem is not completely satisfied, because rigid rotating frames or congruences are possible which do not represent isometric Killing motions.
[30] Several weaker substitutes have also been proposed as rigidity conditions, such as by Noether (1909)[5] or Born (1910) himself.
[51] In contrast to the ordinary Born rigid congruence consisting of the "history of a spatial volume-filling set of points", they recover the six degrees of freedom of classical mechanics by using a quasilocal rigid frame by defining a congruence in terms of the "history of the set of points on the surface bounding a spatial volume".