Is there a special name for this vector field?

Question

The gravitational field of a point mass can be shown to be solenoidal (i.e. its divergence is zero) at all points except where the point mass actually is. This implies that there exists a vector field $\vec{F}$ whose curl is the point mass's gravitational field. I know in magnetostatics, due to Gauss's law for magnetism (i.e. $\vec{\nabla}\cdot\vec{B}=0)$, there exists a (not necessarily unique) vector field $\vec{A}$ called the magnetic vector potential such that $\vec{\nabla}\times{\vec{A}}=\vec{B}$. My question is: is there a similar name for that vector field $\vec{F}$ such that $\vec{\nabla}\times{\vec{F}}=\vec{g}$, where $\vec{g}$ is the gravitational field? And how come I don't think I ever really see this vector field get used? I mean, since the gravitational field of a point mass is also irrotational everywhere (except where the point mass is), it is quite common to be using a scalar field $\Phi$ called the gravitational potential such that $-\vec{\nabla}\Phi=\vec{g}$.

Answer 1

The gravitational field is not a solenoidal field. See the definition. The difference between the magnetic field and the gravitational field is that the magnetic field is source-free everywhere, while the gravitational field (just like the electric field) ist only source-free almost everywhere. While this might seem a minor difference, it is actually of topological relevance: the magnetic field lines are closed (they do not end at a source), while the gravitational and electric field lines begin and end at sources.

Of course you can represent the gravitational field (just like the electric field) locally (i.e. in a compact, simply connected region that does not contain sources) by a curl even though it is not a solenoidal field. But for what purpose? That might be the reason why you don't see it: because there is no use case for it.

Answer 2

In order to describe any vector field, you need to know its divergence and curl in order to have it uniquely determined. When we are dealing with gravitational field $$ \vec{g}=-Gm\frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^3} $$ One can find that its divergence is $$ \nabla·\vec{g}=-4\pi G\rho_m $$ This divergence is $4\pi G\rho_m$ at mass sources and $0$ everywhere else. You can check this by calculating divergence in spherical coordinates and by integrating it over a sphere of constant radius.

Its curl, though, is exactly equal to zero (check it!) and that is precisely the reason why we can write the field as the gradient of a potential function $$ \nabla\times\vec{g}=0 \to \vec{g}=-\nabla\phi $$ This kind of fields are called conservative or irrotational. Thus, we have the field completely defined by its divergence and curl.