View source for Gauss's law for gravity

{{Use American English|date = February 2019}}
{{Short description|Restatement of Newton's law of universal gravitation}}
{{about|Gauss's law concerning the gravitational field|analogous laws concerning different fields|Gauss's law|and|Gauss's law for magnetism|Gauss's theorem, a mathematical theorem relevant to all of these laws|Divergence theorem}}

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.<ref>{{Cite web|url=https://physicscourses.colorado.edu/phys2210/phys2210_fa20/lecture/lec29-gauss-law-gravity/|title=Gauss's law and gravity}}</ref>

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 law|inverse-square]] interaction in a 3-dimensional space.

==Qualitative statement of the law==

{{Main|Gravitational field}}

The [[gravitational field]] '''g''' (also called [[gravitational 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 a [[surface integral]] of the gravitational field over a closed surface, analogous to how [[magnetic flux]] is a surface integral of the magnetic field.

Gauss's law for gravity states:

:''The gravitational flux through any [[closed surface]] is proportional to the enclosed [[mass]].''

==Integral form==

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

{{Equation box 1
|indent =:
|equation = {{oiint
| preintegral =
| intsubscpt = <math>\scriptstyle \partial V</math>
| integrand = <math>\mathbf{g}\cdot d\mathbf{A} = -4 \pi GM</math>}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where
*{{oiint | preintegral =
| intsubscpt = <math>\scriptstyle \partial V</math>
| integrand = }} (also written <math>\oint_{\partial V}</math>) denotes a surface integral over a closed surface,
*∂''V'' is any closed surface (the ''boundary'' of an arbitrary volume ''V''),
*''d'''''A''' is a [[Vector (geometric)|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]], and
*''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 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.

==Differential form==

The differential form of Gauss's law for gravity states

{{Equation box 1
|indent =:
|equation = <math>\nabla\cdot \mathbf{g} = -4\pi G\rho, </math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>\nabla\cdot</math> denotes [[divergence]], ''G'' is the universal [[gravitational constant]], and ''ρ'' is the [[density|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:
<math display="block">\oint_{\partial V}\mathbf{g}\cdot d \mathbf{A} = \int_V\nabla\cdot\mathbf{g}\,dV</math>
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
<math display="block">M = \int_{V}\rho\ dV</math>
we can apply the divergence theorem to the integral form of Gauss's law for gravity, which becomes:
<math display="block">\int_V\nabla\cdot\mathbf{g}\ dV = -4 \pi G\int_{V}\rho\ dV</math>
which can be rewritten:
<math display="block">\int_V(\nabla\cdot\mathbf{g})\ dV = \int_{V} (-4 \pi G\rho)\ dV.</math>
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
<math display="block">\nabla\cdot\mathbf{g} = -4\pi G \rho,</math>
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:
<math display="block">\mathbf{g}(\mathbf{r}) = -\frac{GM}{r^2} \mathbf{e_r}</math>
where
*'''e<sub>r</sub>''' 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 (mathematics)|origin]].

A proof using vector calculus is shown in the box below. It is mathematically identical to the proof of [[Gauss's law]] (in [[electrostatics]]) starting from [[Coulomb's law]].<ref name=Griffiths/>

{{math proof|title=Outline of proof
|proof=
'''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]]). To do this, we integrate over every point '''s''' in space, adding up the contribution to '''g'''('''r''') associated with the mass (if any) at '''s''', where this contribution is calculated by Newton's law. The result is:
<math display="block">\mathbf{g}(\mathbf{r}) = -G\int \rho(\mathbf{s})\frac{(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} d^3\mathbf{s}.</math>
(''d''<sup>3</sup>'''s''' stands for ''ds''<sub>''x''</sub>''ds''<sub>''y''</sub>''ds''<sub>''z''</sub>, each of which is integrated from −∞ to +∞.) If we take the divergence of both sides of this equation with respect to '''r''', and use the known theorem<ref name=Griffiths>See, for example, {{cite book | author=Griffiths, David J. | title=Introduction to Electrodynamics | edition=3rd | publisher=Prentice Hall | date=1998 | isbn=0-13-805326-X | page=[https://archive.org/details/introductiontoel00grif_0/page/50 50] | url-access=registration | url=https://archive.org/details/introductiontoel00grif_0/page/50 }}</ref>
<math display="block">\nabla \cdot \left(\frac{\mathbf{r}}{|\mathbf{r}|^3}\right) = 4\pi \delta(\mathbf{r})</math>
where ''δ''('''r''') is the [[Dirac delta function]], the result is
<math display="block">\nabla\cdot\mathbf{g}(\mathbf{r}) = -4\pi G\int \rho(\mathbf{s})\ \delta(\mathbf{r}-\mathbf{s})\ d^3\mathbf{s}.</math>
Using the "sifting<!--Note: This is not a typo, the word is really 'sifting' not 'shifting'--> property" of the Dirac delta function, we arrive at
<math display="block">\nabla\cdot\mathbf{g}(\mathbf{r}) = -4\pi G \rho(\mathbf{r})</math>
which is the differential form of Gauss's law for gravity, as desired.
}}

===Deriving Newton's law from Gauss's law and irrotationality===
It is impossible to mathematically prove Newton's law from Gauss's law ''alone'', because Gauss's law specifies the divergence of '''g''' but does not contain any information regarding the [[Curl (mathematics)|curl]] of '''g''' (see [[Helmholtz decomposition]]). In addition to Gauss's law, the assumption is used that '''g''' is [[conservative vector field|irrotational]] (has zero curl), as gravity is a [[conservative force]]:
:<math>\nabla \times \mathbf{g} = 0</math>
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.

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

{{math proof|title=Outline of proof
|proof=
Start with the integral form of Gauss's law:
<math display="block">\oint_{\partial V}\mathbf{g}\cdot d\mathbf{A} = -4 \pi GM.</math>
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. (We omit the proof for simplicity.) By making this assumption, '''g''' takes the following form:
<math display="block">\mathbf{g}(\mathbf{r}) = -g(r) \, \mathbf{e_r}</math>
(i.e., the direction of '''g''' is antiparallel 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 <math>4\pi r^2</math>,
: <math>g(r) \oint_{\partial V}(-\mathbf{e_r})\cdot d\mathbf{A} = -4 \pi GM</math>
: <math>g(r) \cdot (-4\pi r^2) = -4 \pi GM</math>
: <math>g(r) = \frac{GM}{r^2}</math>
: <math>\mathbf{g}(\mathbf{r}) = -GM\frac{\mathbf{e_r}}{r^2}</math>
which is Newton's law.
}}

==Poisson's equation and gravitational potential==
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 [[Potential energy#Gravitational potential energy|gravitational potential]]:
<math display="block">\mathbf{g}=-\nabla\phi.</math>
Then the differential form of Gauss's law for gravity becomes [[Poisson's equation]]:
<math display="block">\nabla^2\phi = 4\pi G\rho.</math>
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, <math>r=|\mathbf{r}|</math>), and Poisson's equation becomes (see [[Del in cylindrical and spherical coordinates]]):
<math display="block">\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\, \frac{\partial \phi}{\partial r}\right) = 4\pi G \rho(r)</math>
while the gravitational field is:
<math display="block">\mathbf{g}(\mathbf{r}) = -\mathbf{e_r}\frac{\partial \phi}{\partial r}.</math>

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 {{nowrap|1=''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|Bouguer plate}}

We can conclude (by using a "[[Gaussian surface|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 2''πG'' times the mass per unit area, independent of the distance to the plate<ref>[https://books.google.com/books?id=XVyD9pJpW-cC&pg=PA535 The mechanics problem solver, by Fogiel, pp&nbsp;535–536]</ref> (see also [[Physical geodesy#Gravity anomalies|gravity anomalies]]).

More generally, for a mass distribution with the density depending on one Cartesian coordinate ''z'' only, gravity for any ''z'' is 2''πG'' times the difference in mass per unit area on either side of this ''z'' value.

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

===Cylindrically symmetric mass distribution===

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 2''G''/''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.

===Spherically symmetric mass distribution===
{{Main|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''/''r''<sup>2</sup> 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. 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 article [[shell theorem]] for this direct derivation.

==Derivation from Lagrangian==
{{Main|Lagrangian (field theory)}}

The [[Lagrangian density]] for Newtonian gravity is
<math display="block">\mathcal{L}(\mathbf{x},t) = - \rho(\mathbf{x},t) \phi(\mathbf{x},t) - {1 \over 8 \pi G} (\nabla \phi(\mathbf{x},t))^2 </math>
Applying [[Hamilton's principle]] to this Lagrangian, the result is Gauss's law for gravity:
<math display="block">4 \pi G \rho (\mathbf{x},t) = \nabla^2 \phi(\mathbf{x},t). </math>
See [[Lagrangian (field theory)]] for details.

==See also==
{{Portal|Physics}}
{{div col|colwidth=22em}}
*[[Carl Friedrich Gauss]]
*[[Divergence theorem]]
*[[Gauss's law]] for electricity
*[[Gauss's law for magnetism]]
*[[Vector calculus]]
*[[Integral]]
*[[Flux]]
*[[Gaussian surface]]
{{div col end}}

==References==
{{reflist}}

==Further reading==
*For usage of the term "Gauss's law for gravity" see, for example, {{cite journal|last1=Moody|first1=M. V.|last2=Paik|first2=H. J.|title=Gauss's law test of gravity at short range|journal=Physical Review Letters|date=1 March 1993|volume=70|issue=9|pages=1195–1198|doi=10.1103/PhysRevLett.70.1195|pmid=10054315|bibcode=1993PhRvL..70.1195M}}

{{Theories of gravitation}}
{{Carl Friedrich Gauss}}

[[Category:Theories of gravity]]
[[Category:Vector calculus]]
[[Category:Carl Friedrich Gauss|Gravity]]
[[Category:Newtonian gravity]]