Herglotz-Noether theorem

Born rigidity is a concept in special relativity. 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), who gave a detailed description of the case of constant proper acceleration which he called hyperbolic motion. When subsequent authors such as Paul Ehrenfest (1909) tried to incorporate rotational motions as well, it became clear that Born rigidity is a very restrictive sense of rigidity, leading to the Herglotz-Noether theorem, according to which there are severe restrictions on rotational Born rigid motions. It was formulated by Gustav Herglotz (1909, who classified all forms of rotational motions) and in a less general way by Fritz Noether (1909). As a result, Born (1910) and others gave alternative, less restrictive definitions of rigidity.

Born rigidity is satisfied if the orthogonal spacetime distance between infinitesimally separated curves or worldlines is constant, 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. Born rigidity is a constraint on the motion of an extended body, achieved by careful application of forces to different parts of the body. A body rigid in itself would violate special relativity, as its speed of sound would be infinite.

A classification of all possible Born rigid motions can be obtained using the Herglotz-Noether theorem. This theorem states, that all irrotational Born rigid motions (class A) consist of hyperplanes rigidly moving through spacetime, while any rotational Born rigid motion (class B) must be isometric Killing motions. This implies that a Born rigid body only has three degrees of freedom. Thus a body can be brought in a Born rigid way from rest into any translational motion, but it cannot be brought in a Born rigid way from rest into rotational motion.

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