Born rigidity, proposed by and later named after Max Born, 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 defining property of Born rigidity is locally constant distance in the co-moving frame for all points of the body in question. Thus this distance is Lorentz contracted 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.
It is a very restrictive sense of rigidity, as stated in the Herglotz-Noether theorem by Gustav Herglotz (1910) and Fritz Noether (1910). It states that a Born-rigid object has only three degrees of freedom, and its motion is in general definitely determined by the motion of one of its points. This leads to the Ehrenfest paradox, as it is impossible to put a disk from rest into rotation while maintaining its Born rigidity. Bell's spaceship paradox demonstrates that if there is a body at rest and then its two endpoints are accelerated with the same proper acceleration, then its Born rigidity will be broken. In general it was shown by Herglotz (1911), that a relativistic theory of elasticity can be based on the assumption, that stresses arise when the condition of Born rigidity is broken.
Several weaker substitutes have been proposed as rigidity conditions, such as by Noether (1910).