Fix $t \in I$ and work in any coordinate chart $(U,x)$ on $M$. Let $\nabla$ be the Levi-Civita connection of $g(t)$, let $\Delta = g^{ab}\nabla_a\nabla_b$ denote the rough Laplacian on covariant $2$-tensors, and raise indices using $g(t)$. Then on $U$,