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Born rigidity is a restriction on a body's motion: infinitesimally neighboring material points keep the same proper distance when compared in their local instantaneous rest frame. It concerns the projected spatial strain of a four-velocity field. Equivalently, its expansion scalar and shear tensor of a timelike congruence vanish. A rotating motion need not admit one global simultaneity slice for the entire body, so this definition is local.

For rectilinear motion with constant positive rear proper acceleration $a_0$, points at fixed proper distance $\xi$ have the metric in Rindler coordinatesTheir proper acceleration is $a(\xi )=a_0/(1+a_0\xi /c^2)$. Thus a front at $\xi =L>0$ accelerates less than the rear; the proper distance stays $L$ even while the laboratory length changes.^[ref]

This explains the kinematic issue in Bell's spaceship paradox: prescribing equal accelerations and fixed laboratory separation is a different motion. Born rigidity does not describe an instantaneous material response to a push; stresses in a real body still propagate at a finite speed of sound.

Jerrold Franklin analyzed rigid motion and collisions in special relativity. His treatment distinguishes maintaining proper separations from maintaining the same coordinate separation at one observer's simultaneous times.^[ref]

For straight-line rigid acceleration, different points generally require different proper accelerations. A rule that simply gives every point the same acceleration in one inertial frame need not preserve the object's rest-frame length. The idealized kinematic prescription also does not provide a mechanism for instantaneous stress transmission through a real material.