Poynting vector as the source of gravity

 $$u=\rho c^2,$$

so $$\nabla\cdot\textbf{g}=-\frac{4\pi G}{c^2}u.$$

Propose

$$\boxed{\nabla\times\textbf{g}=\alpha \textbf{S}}$$

Using

$$\frac{\partial u}{\partial t}=-\nabla\cdot\textbf{S}$$

we get a new source for \(\textbf{S}\):

$$\boxed{\textbf{S}=\frac{c^2}{4\pi G}\frac{\partial \textbf{g}}{\partial t}}$$