In Bryce DeWitt's Lectures on Gravitation, in eq. 2.7 on page 25 when he describes the rigid motion of a continuum he states $$x^\mu(\xi,\tau)=x^\mu(0,\sigma)+\xi^in^\mu_i(\sigma)\,\,\,(i=1,2,3) \, .$$
In this equation $\xi$ are the material coordinates and $x^\mu(0,\tau)$ is the origin of the labels $\xi^i$, $\sigma$ is the proper time for the particle taken as origin for material coordinates and $\tau$ is the proper time for the particle with labels $\xi$. I don't understand why he assumes that as motion takes place the coordinates $\xi$ do not change assuming that the three bases $n^\mu_i(\sigma)$ vary properly. I know this assumption is correct in Newtonian mechanics because you can imagine a rigid triad moving with the object and respect to it the object is static but this seem to be impossible according to relativity.