Gauss's law for gravity

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

In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics which is essentially equivalent to Newton's law of universal gravitation. Although Gauss's law for gravity is physically equivalent to Newton's law, there are many situations where Gauss's law for gravity offers a more convenient and simple way to do a calculation than Newton's law.

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 electricity 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.

Qualitative statement of the law

Main article: Gravitational field

Gauss's law for gravity states:

The gravitational flux through any closed surface is proportional to the enclosed mass.

Gauss's law has a simalar mathematical form in other physicical feilds, prominantly in electric fields (flux due to electric charge) and magnetic fields (net flux is zero due to no magnetic monopoles).

Integral form

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

$\scriptstyle \partial V$$\mathbf{g}\cdot d\mathbf{A} = -4 \pi GM$

where

$\scriptstyle \partial V$ dontes a surface integral over a closed surface, for simplicity $\oint_{\partial V}$ can be used (as done below in integral manipulations)

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

dA is a vector, 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),

g is the gravitational field,

G is the universal gravitational constant,

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

The left-hand side of this equation is called the flux of the gravitational field. Note that 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.

Differential form

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

$\nabla\cdot \mathbf{g} = -4\pi G\rho$

where

$\nabla\cdot$ denotes divergence, G is the universal gravitational constant, and ρ is the mass density at each point.

Relation to the integral form

The two forms of Gauss's law for gravity are mathematically equivalent. The divergence theorem states:

$\oint_{\part V}\mathbf{g}\cdot d \mathbf{A} = \int_V\nabla\cdot\mathbf{g}\ dV$

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). The gravitational field g must be a continuously differentiable vector field defined on a neighborhood of V.

Given also that

$M = \int_{V}\rho\ dV$

we can apply the divergence theorem to the integral form of Gauss's law for gravity, which becomes:

$\int_V\nabla\cdot\mathbf{g}\ dV = -4 \pi G\int_{V}\rho\ dV$

which can be rewritten:

$\int_V(\nabla\cdot\mathbf{g})\ dV = \int_{V} (-4 \pi G\rho)\ dV.$

This has to hold simultaneously for every possible volume V; the only way this can happen is if the integrands are equal. Hence we arrive at

$\nabla\cdot\mathbf{g} = -4\pi G \rho$

which is the differential form of Gauss's law for gravity.

It is possible to derive the integral form from the differential form using the reverse of this method.

Although the two forms are equivalent, one or the other might be more convenient to use in a particular computation.

Relation to Newton's law

Deriving Gauss's law from Newton's law

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:

$\mathbf{g}(\mathbf{r}) = -GM\frac{\mathbf{e_r}}{r^2}$

where

e_r is the radial unit vector,

r is the radius, |r|.

M is the mass of the particle, which is assumed to be a point mass located at the origin.

In this section, two alternative proofs of this fact are presented. The first proof uses integral calculus and visual notions about field lines, while the second proof uses differential calculus.

Proof using integral calculus and field lines

Special case: Spherical surface centered at a point mass

Outline of proof

Consider a spherical surface of radius r centered at a point-mass of mass M. The total flux of the gravitational field g over a closed surface ∂V, according to Newton's law, is: $\oint_{\part V}\mathbf{g}\cdot d\mathbf{A} = \oint_{\part V}-\frac{GM}{r^2}\ \mathbf{e_r}\cdot d\mathbf{A}$ However, the magnitude of the infinitesimal area element dA is just the area of the infinitesimal solid angle dΩ, given by $d\mathbf{A} = r^2\mathbf{e_r}\ d\Omega$ which gives us $\oint_{\part V}\mathbf{g}\cdot d\mathbf{A} = -GM\oint_{\part V}\frac{1}{r^2}\ \mathbf{e_r}\cdot r^2\mathbf{e_r}\ d\Omega$ By observing that er·er = 1, and that the integral of unity over a closed surface with respect to the solid angle is the surface area of a unit sphere, 4π, we arrive at $\oint_{\part V}\mathbf{g}\cdot d\mathbf{A} = -4 \pi GM$ which is the integral form of Gauss's law for gravity, for this special case.

General case

To move to the general case, we use the method of field lines. The gravitational field can be depicted via field lines, a set of lines or curves that follow the direction of the gravitational field. The magnitude of the field is required to be proportional to the density of field lines. Moreover, it can be shown that the flux of the field through a surface is proportional to the net number of field lines that pass through the surface (the term "net" means, specifically, the number that pass outward minus the number that pass inward).

Newton's law implies that the field lines will extend directly, radially inward towards the point mass in every direction. Moreover, the special case above shows that if we imagine a series of concentric spheres centered at the point mass, the same number of field lines will pass through each one. So in other words, the field lines all begin at infinity, and go directly inward towards the point mass, ending at the point mass, and coming in uniformly from all directions.

For any finite closed surface (not necessarily spherical) that encloses the point mass, each of the field lines will start at infinity outside the surface, pass through the surface at some point, and end at the point mass inside the surface. Therefore, the flux through the surface is a constant −4πGM, regardless of the shape of the surface, as long as the point mass is inside.

Likewise, for any finite closed surface that does not enclose the point mass, some of the field lines will pass into and then back out of the surface, and some field lines will not touch the surface at all. Regardless, the net flux through the surface is zero.

In every case, this is consistent with Gauss's law. To finish off the proof, we need to consider the case where there is more than one mass (or even infinitely many masses comprising a continuous distribution). The simplest way to handle this case is to say that both Newton's law and Gauss's law obey the superposition principle, so if Gauss's law is a consequence of Newton's law for a single mass, then it's a consequence of Newton's law for any number of masses. Alternatively, one can note that the net number of field lines that enter a surface equals the number of field lines that end on a mass inside the surface, which is proportional to the total mass inside the surface.

Proof using differential calculus

An alternative proof derives the differential form of Gauss's law for gravity from Newton's law of universal gravitation.

Outline of proof

Using the expression from Newton's law, we get the total field at r by using an integral to add up the field at r due to the mass at each other point in space with respect to an s coordinate system, to give $\mathbf{g}(\mathbf{r}) = -G\int_V \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} dV(\mathbf{s}).$ If we take the divergence of both sides of this equation with respect to r, and use the known theorem $\nabla \cdot \left(\frac{\mathbf{s}}{|\mathbf{s}|^3}\right) = 4\pi \delta(\mathbf{s})$ where δ(s) is the Dirac delta function, the result is $\nabla\cdot\mathbf{g}(\mathbf{r}) = -4\pi G\int_V \rho(\mathbf{s})\ \delta(\mathbf{r}-\mathbf{s})\ dV(\mathbf{s}).$ Using the "sifting property" of the Dirac delta function, we arrive at $\nabla\cdot\mathbf{g} = -4\pi G \rho$ which is the differential form of Gauss's law for gravity, as desired.

Deriving Newton's law from Gauss's law

Newton's law can be easily proven from Gauss's law, as follows.

Outline of proof

Start with the integral form of Gauss's law: $\oint_{\part V}\mathbf{g}\cdot d\mathbf{A} = -4 \pi GM.$ 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. By making this assumption, g takes the following form: $\mathbf{g}(\mathbf{r}) = g(r)\mathbf{e_r}$ (i.e., the direction of g is parallel to the direction of r, and the magnitude of g depends only on the magnitude, not direction, of r). Plugging this in, and using the fact that ∂V is a spherical surface with constant r and area $4\pi r^2$, $g(r)\oint_{\part V}\mathbf{e_r}\cdot d\mathbf{A} = -4 \pi GM$ $g(r)(4\pi r^2) = -4 \pi GM$ $g(r) = -GM/r^2$ $\mathbf{g}(\mathbf{r}) = -GM\frac{\mathbf{e_r}}{r^2}$ which is Newton's law.

Poisson's equation and gravitational potential

Since the gravitational field has zero curl (equivalently, gravity is a conservative force), it can be written as the gradient of a scalar potential, called the gravitational potential:

$\mathbf{g}=-\nabla\phi.$

Then the differential form of Gauss's law for gravity becomes Poisson's equation:

$\nabla^2\phi = 4\pi G\rho.$

This provides an alternate means of calculating the gravitational potential and gravitational field. Although computing g via Poisson's equation is mathematically equivalent to computing g directly from Gauss's law, one or the other approach may be an easier computation in a given situation.

In radially symmetric systems, the gravitational potential is a function of only one variable (namely, $r=|\mathbf{r}|$), and Poisson's equation becomes (see Del in cylindrical and spherical coordinates):

$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\, \frac{\partial \phi}{\partial r}\right) = 4\pi G \rho(r)$

while the gravitational field is:

$\mathbf{g}(\mathbf{r}) = -\mathbf{e_r}\frac{\partial \phi}{\partial r}.$

When solving the equation it should be taken into account that in the case of finite densities ∂ϕ/∂r has to be continuous at boundaries (discontinuities of the density), and zero for r = 0.

Applications

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). See the article Gaussian surface for more details on how these derivations are done. Three such applications are as follows:

Bouguer plate

Main article: Bouguer plate

We can conclude (by using a "Gaussian pillbox") that for an infinite, flat plate (Bouguer plate) of any finite thickness, the gravitational field outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate (see also gravity anomalies).

More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times (the mass per unit area above z, minus the mass per unit area below z).

In particular, a combination of two equal parallel infinite plates does not produce any gravity inside.

Cylindrically symmetric mass distribution

In the case of an infinite 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 hollow cylinder, the field is zero.

Spherically symmetric mass distribution

Main article: Shell theorem

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 can be ignored.

For example, a hollow sphere does not produce any net gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the resultant field is that of any masses inside and outside the sphere only).

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.

See also

• Carl Friedrich Gauss
• Divergence theorem
• Gauss's law for electricity
• Gauss's law for magnetism
• Vector calculus
• Integration
• Flux
• Gaussian surface

References

• For usage of the term "Gauss's law for gravity" see, for example, this article.