Inphysics,Gauss's law for gravity, also known asGauss's flux theorem for gravity, is a law of physics that is equivalent toNewton's law of universal gravitation. It is named afterCarl Friedrich Gauss. It states that theflux (surface integral) of thegravitational field over any closed surface is proportional to themass enclosed. Gauss's law for gravity is often more convenient to work from than Newton's law.^[1]

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

Thegravitational fieldg (also calledgravitational acceleration) is a vector field – a vector at each point of space (and time). 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.

Gravitational flux is asurface integral of the gravitational field over a closed surface, analogous to howmagnetic flux is a surface integral of the magnetic field.

Gauss's law for gravity states:

The gravitational flux through anyclosed surface is proportional to the enclosedmass.

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

where

• ${\displaystyle \mathrm{\partial }V}$ (also written${\displaystyle {\oint }_{\mathrm{\partial }V}}$) denotes a surface integral over a closed surface,
• ∂V is any closed surface (theboundary of an arbitrary volumeV),
• dA is avector, whose magnitude is the area of aninfinitesimal piece of the surface ∂V, and whose direction is the outward-pointingsurface normal (seesurface integral for more details),
• g is thegravitational field,
• G is the universalgravitational constant, and
• M is the total mass enclosed within the surface ∂V.

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

The differential form of Gauss's law for gravity states

where${\displaystyle \mathrm{\nabla }\cdot }$ denotesdivergence,G is the universalgravitational constant, andρ is themass density at each point.

The two forms of Gauss's law for gravity are mathematically equivalent. Thedivergence theorem states:$${\displaystyle {\oint }_{\mathrm{\partial }V}\mathbf{g}\cdot d\mathbf{A}={\int }_V\mathrm{\nabla }\cdot \mathbf{g}dV}$$whereV is a closed region bounded by a simple closed oriented surface ∂V anddV is an infinitesimal piece of the volumeV (seevolume integral for more details). The gravitational fieldg must be acontinuously differentiable vector field defined on a neighborhood ofV.

Given also that$${\displaystyle M={\int }_V\rho dV}$$we can apply the divergence theorem to the integral form of Gauss's law for gravity, which becomes:$${\displaystyle {\int }_V\mathrm{\nabla }\cdot \mathbf{g}dV=-4\pi G{\int }_V\rho dV}$$which can be rewritten:$${\displaystyle {\int }_V(\mathrm{\nabla }\cdot \mathbf{g})dV={\int }_V(-4\pi G\rho )dV.}$$This has to hold simultaneously for every possible volumeV; the only way this can happen is if the integrands are equal. Hence we arrive at$${\displaystyle \mathrm{\nabla }\cdot \mathbf{g}=-4\pi G\rho ,}$$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.

Gauss's law for gravity can be derived fromNewton's law of universal gravitation, which states that the gravitational field due to apoint mass is:$${\displaystyle \mathbf{g}(\mathbf{r})=-\frac{GM}{r^2}{\mathbf{e}}_{\mathbf{r}}}$$where

• e_r is the radialunit vector,
• r is the radius, |r|.
• M is the mass of the particle, which is assumed to be apoint mass located at theorigin.

A proof using vector calculus is shown in the box below. It is mathematically identical to the proof ofGauss's law (inelectrostatics) starting fromCoulomb's law.^[2]

It is impossible to mathematically prove Newton's law from Gauss's lawalone, because Gauss's law specifies the divergence ofg but does not contain any information regarding thecurl ofg (seeHelmholtz decomposition). In addition to Gauss's law, the assumption is used thatg isirrotational (has zero curl), as gravity is aconservative force:

${\displaystyle \mathrm{\nabla }\times \mathbf{g}=0}$

Even these are not enough: Boundary conditions ong are also necessary to prove Newton's law, such as the assumption that the field is zero infinitely far from a mass.

The proof of Newton's law from these assumptions is as follows:

Since the gravitational field has zero curl (equivalently, gravity is aconservative force) as mentioned above, it can be written as thegradient of ascalar potential, called thegravitational potential:$${\displaystyle \mathbf{g}=-\mathrm{\nabla }\varphi .}$$Then the differential form of Gauss's law for gravity becomesPoisson's equation:$${\displaystyle {\mathrm{\nabla }}^2\varphi =4\pi G\rho .}$$This provides an alternate means of calculating the gravitational potential and gravitational field. Although computingg via Poisson's equation is mathematically equivalent to computingg 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,${\displaystyle r=|\mathbf{r}|}$), and Poisson's equation becomes (seeDel in cylindrical and spherical coordinates):$${\displaystyle \frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }r}\right)=4\pi G\rho (r)}$$while the gravitational field is:$${\displaystyle \mathbf{g}(\mathbf{r})=-{\mathbf{e}}_{\mathbf{r}}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }r}.}$$

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 forr = 0.

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 articleGaussian surface for more details on how these derivations are done. Three such applications are as follows:

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^[3] (see alsogravity anomalies).

More generally, for a mass distribution with the density depending on one Cartesian coordinatez only, gravity for anyz is 2πG times the difference in mass per unit area on either side of thisz value.

In particular, a parallel combination of two parallel infinite plates of equal mass per unit area produces no gravitational field between them.

In the case of an infinite uniform (inz) cylindrically symmetric mass distribution we can conclude (by using a cylindricalGaussian surface) that the field strength at a distancer 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 sphericalGaussian surface) that the field strength at a distancer from the center is inward with a magnitude ofG/r² times only the total mass within a smaller distance thanr. All the mass at a greater distance thanr from the center has no resultant effect.

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 all masses not including the sphere, which can be inside and outside the sphere).

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 articleshell theorem for this direct derivation.

TheLagrangian density for Newtonian gravity is$${\displaystyle \mathcal{L}(\mathbf{x},t)=-\rho (\mathbf{x},t)\varphi (\mathbf{x},t)-\frac{1}{8\pi G}(\mathrm{\nabla }\varphi (\mathbf{x},t){)}^2}$$ApplyingHamilton's principle to this Lagrangian, the result is Gauss's law for gravity:$${\displaystyle 4\pi G\rho (\mathbf{x},t)={\mathrm{\nabla }}^2\varphi (\mathbf{x},t).}$$SeeLagrangian (field theory) for details.

• 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.Bibcode:1993PhRvL..70.1195M.doi:10.1103/PhysRevLett.70.1195.PMID 10054315.

• Theories of gravity
• Vector calculus
• Carl Friedrich Gauss
• Newtonian gravity

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