# Gravitational flux and divergence theorem

checksix
Hi. I've been reading PF for quite a while and have decided to ask my first question. Please be gentle. (I'm a retired computer programmer, not a student)...

I've been learning Gauss' divergence theorem and I understand what "flux density" is when considering things like fluid transport or particle streams but I'm having trouble understanding what it means when talking about gravitational fields. I wonder if somebody can straighten me out.

Here's what I know:

For water flowing in a pipe, for example, the field under consideration would be called "volumetric flux density" and would have units of, say, liters per second per square meter. The "thing" that's flowing would be "water molecules".

For particles emitted by a radiation source, for example, the field under consideration would be called "particle intensity" and would have units of, say, particles per second per square meter. The "thing" that's flowing would be "particles".

Now here's where I have trouble:

For gravity the field would be called "gravitational field strength" or "gravitational flux density" and would have units of newtons per kilogram. But this looks completely different from the water and particle cases: there are no units indicating "something flowing per unit time per unit area".

I could play with the units to make it look like the water and particle cases:

N / kg = (kg m / s^2) / kg = m / s^2 = (m^3 / s) / s / m^2

So now I have something that looks like a "flux density" - I have "something" per second per unit area.

But what is this "something" that is "flowing" ? It has units of (m^3 / s) which is a volumetric flow. Does this "something" have a name? Thanks for reading this far. Hope someone can enlighten me.

Homework Helper
The gravitational (more correctly gravitostatic) case and the electrostatic case have the same description, because essentially the laws that govern them are the same.

The flux of the gravitational intensity doesn't have a <material> description, because it uses the classical concept of <field> rather than particles/substances. The concept of field is a rather abstract one and could be used to account for the description of interactions between particles (ideally point-particles).

So the flux of a gravitostatic field is nothing but the <amount> of field intensity generated by a mass distribution inside a volume V.

The fundamental equation states that the flux of the gravitational field generated by a mass distribution inside a volume V is equal to the mass generating the field.

$$\oint\oint_{\Sigma} \vec{\Gamma}\left(\vec{r}\right) \cdot d\vec{\sigma} = m_{V_{\Sigma}}$$

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checksix
Thanks, that helped. So I guess, in a sense, it is the field itself that is "flowing". Or maybe not. Perhaps it's best not to press too hard on the fluid analogy and just trust the formalism.