Born rigidity

Remarks

The condition says that the distance between infinitesimally close world lines, measured orthogonally to $u$, does not change along the motion. Equivalently, the expansion and the shear of $u$ vanish, or the Lie derivative of $h$ along $u$ vanishes. Example: uniform motion $u = \gamma (c, v, 0, 0)$ with constant $v$ is Born rigid, since $\partial_\alpha u_\beta = 0$. Example: the world lines of fixed $\rho$ in Rindler coordinates form a Born rigid motion. Each is a hyperbolic motion with proper acceleration $c^2/\rho$. The metric $-\rho^2 d\eta^2 + d\rho^2 + dy^2 + dz^2$ does not depend on $\eta$, so the rest-space distances between these observers are constant. Non-example: a family of world lines that all have the same constant proper acceleration $\alpha$ along $x$, starting at rest at different points $x_0$, is not Born rigid. Their coordinate distance stays constant in the original frame, so the distance measured in their rest frames is $\gamma$ times it, which grows with the speed. The Rindler example shows that in a Born rigid accelerated rod the rear end must accelerate harder than the front.