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 , points at fixed proper distance have the metric in Rindler coordinatesTheir proper acceleration is . Thus a front at accelerates less than the rear; the proper distance stays even while the laboratory length changes.[ref]
Rigid body motion in special relativity by Jerrold Franklin. Section 3, equations 14–15, printed pp. 4–5
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.
A front worldline at fixed proper distance along a rear observer's simultaneity slice is parametrized by the displayed equations, with velocity and Lorentz factor evaluated at the rear event. The front time is locally future-oriented when , where is the rear signed proper acceleration. At a zero of that factor the rigid coordinate extension encounters an acceleration-horizon obstruction.
Two rockets maintaining fixed separation in an inertial laboratory frame do not preserve their comoving separation. A string connecting them is stretched rather than remaining unstrained; sufficiently large strain can break a real string. The contrast with Born rigidity follows from relativity of simultaneity and does not contradict Lorentz contraction.
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Born rigidity is a concept in the field of relativistic physics, particularly in the context of special relativity. It refers to the idea of an object's ability to maintain its shape and size while moving through spacetime without undergoing any deformation due to relativistic effects. The term comes from the work of Hermann Minkowski and is named after Max Born, who contributed significantly to the understanding of the topic.