Let $M$ be a compact smooth manifold without boundary, and let $g_0 \in \Gamma(S^2T^*M)$ be a smooth Riemannian metric on $M$. Set $\bar{g} := g_0$. Define the DeTurck vector field associated to a time-dependent Riemannian metric $g(t)$ and the background metric $\bar{g}$ by