general relativity - Is the description of the gravitational field as a vector field and a tensor field compatible?

By electric or magnetic fields we mean the vector fields $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ respectively. But a gravitational field in Newtonian theory is a vector field that $\vec{g}(\vec{r})$ that obeys $\nabla\times\vec{g}=0$ and $\nabla\cdot\vec{g}=4\pi G\rho_{m}$. But in Einstein's theory we describe gravity as a tensor field $g_{\mu\nu}(x)$. How are these two descriptions of gravity, one in terms of the vector field $\vec{g}(\vec{r})$ and in other in terms of the tensor field $g_{\mu\nu}(x)$, compatible with each other in terms of the number of degrees of freedom? Thanks!