Given an asymptotically flat manifold $(M^{(n)},g)$ with ends $M_1,\ldots,M_\ell$, we commonly define the ADM mass of each $M_k$ by the following coordinate expression: $$m_{ADM}(M_k,g)=\lim_{\rho\to\infty}\frac{1}{2(n-1)\omega_{n-1}}\int_{S_\rho}\sum_{i,j=1}^{n}(g_{ij,i}-g_{ii,j})\frac{x^j}{|x|}\overline{d\mu}_{S_\rho},$$ where $S_\rho$ is the coordinate sphere of radius $\rho$, $\overline{d\mu}_{S_\rho}$ is the volume measure induced by the Euclidean background metric, and $\omega_{n-1}$ is the volume of the $(n-1)$-sphere. This definition can be found in most literature. Honestly, I haven't had hands-on experience of calculating this integral, but at least its integrand makes sense to me. By contrast, I have no idea what is going on in an equivalent definition. In his book Geometric Relativity, Dan A. Lee defines $$m_{ADM}(M_k,g)=\lim_{\rho\to\infty}\frac{1}{2(n-1)\omega_{n-1}}\int_{S_\rho}\left(\overline{\mathrm{div}}g-d(\overline{\mathrm{tr}}g)\right)(\overline{\nu})\ \overline{d\mu}_{S_\rho}.$$ As before, all barred quantities are computed using the Euclidean background metric, and $\overline{\nu}$ is the Euclidean unit outer normal. My question is, how do we compute $\overline{\mathrm{div}}g$, the divergence of $g$? It is completely insane for me to talk about the divergence of a quantity that is NOT a vector field. Is it really possible to define such entity?
Thank you.