It is defined so that the gravitational force experienced by a particle is equal to the mass of the particle multiplied by the gravitational field at that point.
where The left-hand side of this equation is called the flux of the gravitational field.
This can be contrasted with Gauss's law for electricity, where the flux can be either positive or negative.
The differential form of Gauss's law for gravity states
denotes divergence, G is the universal gravitational constant, and ρ is the mass density at each point.
The two forms of Gauss's law for gravity are mathematically equivalent.
where V is a closed region bounded by a simple closed oriented surface ∂V and dV is an infinitesimal piece of the volume V (see volume integral for more details).
we can apply the divergence theorem to the integral form of Gauss's law for gravity, which becomes:
This has to hold simultaneously for every possible volume V; the only way this can happen is if the integrands are equal.
Gauss's law for gravity can be derived from Newton's law of universal gravitation, which states that the gravitational field due to a point mass is:
[2] g(r), the gravitational field at r, can be calculated by adding up the contribution to g(r) due to every bit of mass in the universe (see superposition principle).
If we take the divergence of both sides of this equation with respect to r, and use the known theorem[2]
where δ(r) is the Dirac delta function, the result is
Using the "sifting property" of the Dirac delta function, we arrive at
which is the differential form of Gauss's law for gravity, as desired.
In addition to Gauss's law, the assumption is used that g is irrotational (has zero curl), as gravity is a conservative force: Even these are not enough: Boundary conditions on g are also necessary to prove Newton's law, such as the assumption that the field is zero infinitely far from a mass.
Apply this law to the situation where the volume V is a sphere of radius r centered on a point-mass M. It's reasonable to expect the gravitational field from a point mass to be spherically symmetric.
Plugging this in, and using the fact that ∂V is a spherical surface with constant r and area
Since the gravitational field has zero curl (equivalently, gravity is a conservative force) as mentioned above, it can be written as the gradient of a scalar potential, called the gravitational potential:
Then the differential form of Gauss's law for gravity becomes Poisson's equation:
In radially symmetric systems, the gravitational potential is a function of only one variable (namely,
), and Poisson's equation becomes (see Del in cylindrical and spherical coordinates):
Gauss's law can be used to easily derive the gravitational field in certain cases where a direct application of Newton's law would be more difficult (but not impossible).
More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times the difference in mass per unit area on either side of this z value.
In the case of an infinite uniform (in z) cylindrically symmetric mass distribution we can conclude (by using a cylindrical Gaussian surface) that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance (from the axis), regardless of any masses at a larger distance.
For example, inside an infinite uniform hollow cylinder, the field is zero.
In the case of a spherically symmetric mass distribution we can conclude (by using a spherical Gaussian surface) that the field strength at a distance r from the center is inward with a magnitude of G/r2 times only the total mass within a smaller distance than r. All the mass at a greater distance than r from the center has no resultant effect.
For example, a hollow sphere does not produce any net gravity inside.
Although this follows in one or two lines of algebra from Gauss's law for gravity, it took Isaac Newton several pages of cumbersome calculus to derive it directly using his law of gravity; see the article shell theorem for this direct derivation.
Applying Hamilton's principle to this Lagrangian, the result is Gauss's law for gravity: