# Gauss law in gravitation

Is it possible to use Gauss's law of electromagnetism, (The net electric flux through any closed surface is equal to $1⁄\epsilon$ times the net electric charge enclosed within that surface.) to calculate the gravitational field at point by making certain changes, i.e, by replacing electric flux with gravitational flux, $1⁄\epsilon$ with $1/(4\pi\,G)$, and charge with mass?

• See e.g. Wikipedia. Mar 7 '16 at 16:30

Yes, you can use Gauss's law for gravity.

$$\nabla\cdot\vec{g} = 4 \pi\, G\, \rho$$

or

$$\oint \vec{g}\cdot\mathrm{d}\vec{a} = 4 \pi\, G\, M_\mathrm{enc}$$

where $\vec{g}$ is the gravitational field (equivalently, acceleration due to gravity), $\rho$ is mass density, and $M_\mathrm{enc}$ is the total mass enclosed by the Gaussian surface.

When you make the comparison to Gauss's law for electric fields, you can see how the constants work out the way that they do:

$$E = \frac{1}{4\pi\, \epsilon_0} \frac{Q}{r^2}, \quad\quad g = G\, \frac{M}{r^2},$$

so $1/\epsilon_0\rightarrow 4\pi\, G$.

One common use for Gauss's law for gravity is to determine the gravitational field strength at a given depth inside the Earth. It is very similar to the calculation for the electric field inside a charged, insulating sphere.

• In my original post I messed up the constants... fixed Mar 7 '16 at 17:39
• Indeed the close match between flux of the field in Einstein's treatment to Newton's for a spherically symmetric weak field can be demonstrated using this Gauss' Law approach. Nov 26 '18 at 18:23

Gauss Law for Gravity basically says that the total gravitational flux emanating from a sphere enclosing the Earth is $$4 \pi G M$$.

Now divide this by the total surface of the sphere $$4 \pi R^2$$ with $$R$$ the radius of the Earth.

The result is $$\frac{GM}{R^2}$$ giving the gravitational flux density. If you calculate the numerical result you get $$9.81\mathrm{m/s^2}$$.