Gauss’s flux theorem for gravity (also known as Gauss’s law for gravity) in differential form says that
In this note I want to show that one can get quite far towards deriving Gauss’s law for gravity without knowing Newton’s law of universal gravitation, but not all the way. To explore this, suppose that all we know is that the gravitational force depends on mass and radial distance:
Here, is an unspecified function, is a mass which can be taken as being located at the origin, is the radial distance from the origin, and is a radial unit vector.
Now we imagine a closed spherical surface of radius centered at the origin. The total flux of the gravitational field over the closed surface is
= (this explains where the comes from).
The total flux is independent of so to eliminate we must have where is some unspecified function of , and therefore
By the divergence theorem we can write this as
and therefore differentiating both sides with respect to we get
If we set we see that this is *nearly* Gauss’s law:
We only need Newton’s law to tell us that at this final stage.
The link with the scalar potential comes through which gives us
(a well known type of partial differential equation known as Poisson’s equation).