Gauss' law for gravity
   HOME

TheInfoList



OR:

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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 Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
(
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
) of the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
over any closed surface is equal to the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
enclosed. Gauss's law for gravity is often more convenient to work from than is Newton's law. The form of Gauss's law for gravity is mathematically similar to
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
for
electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
, one of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
. 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 Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
. This is because both Newton's law and Coulomb's law describe
inverse-square In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...
interaction in a 3-dimensional space.


Qualitative statement of the law

The
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
g (also called
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
) 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 In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
of the gravitational field over a closed surface, analogous to how
magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ( ...
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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
.''


Integral form

The integral form of Gauss's law for gravity states: where * (also written \oint_) 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, whose magnitude is the area of an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
piece of the surface ∂''V'', and whose direction is the outward-pointing
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
(see
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
for more details), *g is the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
, *''G'' is the universal
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, and *''M'' is the total mass enclosed within the surface ∂''V''. The left-hand side of this equation is called the
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
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 In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
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 where \nabla\cdot denotes
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
, ''G'' is the universal
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, and ''ρ'' is the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
at each point.


Relation to the integral form

The two forms of Gauss's law for gravity are mathematically equivalent. The
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
states: \oint_\mathbf\cdot d \mathbf = \int_V\nabla\cdot\mathbf\,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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
vector field defined on a neighborhood of ''V''. Given also that M = \int_\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\ dV = -4 \pi G\int_\rho\ dV which can be rewritten: \int_V(\nabla\cdot\mathbf)\ dV = \int_ (-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 = -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(\mathbf) = -\frac \mathbf where *er is the radial
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
, *''r'' is the radius, , r, . *''M'' is the mass of the particle, which is assumed to be a point mass located at the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
. A proof using vector calculus is shown in the box below. It is mathematically identical to the proof of
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
(in
electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
) starting from
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
.


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 cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
of g (see Helmholtz decomposition). In addition to Gauss's law, the assumption is used that g is
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
(has zero curl), as gravity is a conservative force: :\nabla \times \mathbf = 0 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:


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 In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of a scalar potential, called the gravitational potential: \mathbf=-\nabla\phi. Then the differential form of Gauss's law for gravity becomes
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
: \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, ), and Poisson's equation becomes (see Del in cylindrical and spherical coordinates): \frac\frac\left(r^2\, \frac\right) = 4\pi G \rho(r) while the gravitational field is: \mathbf(\mathbf) = -\mathbf\frac. 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 .


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 A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of a 3- ...
for more details on how these derivations are done. Three such applications are as follows:


Bouguer plate

We can conclude (by using a " Gaussian pillbox") that for an infinite, flat plate (
Bouguer plate In geodesy and geophysics, the Bouguer anomaly (named after Pierre Bouguer) is a gravity anomaly, corrected for the height at which it is measured and the attraction of terrain. The height correction alone gives a free-air gravity anomaly. Defini ...
) 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 plateThe mechanics problem solver, by Fogiel, pp 535–536
/ref> (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 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 A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of a 3- ...
) 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

In the case of a spherically symmetric mass distribution we can conclude (by using a spherical
Gaussian surface A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of a 3- ...
) that the field strength at a distance ''r'' from the center is inward with a magnitude of ''G''/''r''2 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 In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy. Isaac Newton proved the shell the ...
for this direct derivation.


Derivation from Lagrangian

The Lagrangian density for Newtonian gravity is \mathcal(\mathbf,t) = - \rho(\mathbf,t) \phi(\mathbf,t) - (\nabla \phi(\mathbf,t))^2 Applying Hamilton's principle to this Lagrangian, the result is Gauss's law for gravity: 4 \pi G \rho (\mathbf,t) = \nabla^2 \phi(\mathbf,t). See
Lagrangian (field theory) Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for details.


See also

* Carl Friedrich Gauss *
Divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
*
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
for electricity * Gauss's law for magnetism *
Vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
*
Integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
*
Flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
*
Gaussian surface A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of a 3- ...


References


Further reading

*For usage of the term "Gauss's law for gravity" see, for example, {{Carl Friedrich Gauss Gravity Theories of gravity Vector calculus
Gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
Newtonian gravity