Gauss's law for gravity

In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's law for gravity is often more convenient to work from than Newton's law.^[1]

This article is about Gauss's law concerning the gravitational field. For analogous laws concerning different fields, see Gauss's law and Gauss's law for magnetism. For Gauss's theorem, a mathematical theorem relevant to all of these laws, see Divergence theorem.

The form of Gauss's law for gravity is mathematically similar to Gauss's law for electrostatics, one of Maxwell's equations. Gauss's law for gravity has the same mathematical relation to Newton's law that Gauss's law for electrostatics bears to Coulomb's law. This is because both Newton's law and Coulomb's law describe inverse-square interaction in a 3-dimensional space.

$\oiint$ $\scriptstyle \partial V$ (also written $\oint _{\partial V}$ ) denotes a surface integral over a closed surface,

∂V is any closed surface (the boundary of an arbitrary volume V),

dA is a , whose magnitude is the area of an infinitesimal piece of the surface ∂V, and whose direction is the outward-pointing surface normal (see surface integral for more details),

vector

g is the ,

gravitational field

G is the universal , and

gravitational constant

M is the total mass enclosed within the surface ∂V.

The integral form of Gauss's law for gravity states:

where

The left-hand side of this equation is called the flux of the gravitational field. Note that according to the law it is always negative (or zero), and never positive. This can be contrasted with Gauss's law for electricity, where the flux can be either positive or negative. The difference is because charge can be either positive or negative, while mass can only be positive.

Relation to Newton's law[edit]

Deriving Gauss's law from Newton's law[edit]

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:

For usage of the term "Gauss's law for gravity" see, for example, Moody, M. V.; Paik, H. J. (1 March 1993). "Gauss's law test of gravity at short range". Physical Review Letters. 70 (9): 1195–1198. :1993PhRvL..70.1195M. doi:10.1103/PhysRevLett.70.1195. PMID 10054315.