Gravitational potential sign conventions ($\mathbf{g}=-\nabla \Phi =\nabla \phi ,\phi =-\Phi$)

The usual Newtonian gravitational potential satisfies $\mathbf{g}=-\nabla \Phi$ and ${\nabla }^2\Phi =4\pi G\rho$. An equally consistent convention sets $\phi =-\Phi$, giving $\mathbf{g}=\nabla \phi$ and ${\nabla }^2\phi =-4\pi G\rho$. With zero potential at infinity, a point mass has $\Phi =-GM/r$ and $\phi =GM/r$. Both conventions describe the same inward gravitational field and the same Gauss's law for gravity. Only the signs of the potential and its differences change. The gravitational potential energy per unit test mass is $\Phi$, or $-\phi$ in the alternative convention. A formula for a potential difference must therefore be read together with its relation to the gravitational field.