Homotopy extension property

2014-05-25T20:16:04Z

76.181.79.110: /* Definition */ removed some ambiguity ~~~~ Dan

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Dirichlet character

2013-09-09T11:33:43Z

76.181.48.150: /* Modulus 6 */

In [[physics]], a '''gravitational field''' is a [[scientific model|model]] used to explain the influence 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 [[gravitation]]al phenomena, and is measured in [[newton (unit)|newtons]] per [[kilogram]] (N/kg). In its original concept, [[gravity]] was a [[force]] between point [[mass]]es. Following [[Isaac Newton|Newton]], [[Laplace]] attempted to model gravity as some kind of [[radiation]] field or [[fluid]], and since the 19th century explanations for gravity have usually been sought in terms of a field model, rather than a point attraction.

In a field model, rather than two particles attracting each other, the particles distort [[spacetime]] via their mass, and this distortion is what is perceived and measured as a "force". In such a model one states that matter moves in certain ways in response to the curvature of spacetime,<ref>{{cite book
|title=General relativity from A to B
|first1=Robert
|last1=Geroch
|publisher=University of Chicago Press
|year=1981
|isbn=0-226-28864-1
|page=181
|url=http://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181}}, [http://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181 Chapter 7, page 181]
</ref> and that there is either ''no gravitational force'',<ref>{{cite book
|title=Einstein's general theory of relativity: with modern applications in cosmology
|first1=Øyvind
|last1=Grøn
|first2=Sigbjørn
|last2=Hervik
|publisher=Springer Japan
|year=2007
|isbn=0-387-69199-5
|page=256
|url=http://books.google.com/books?id=IyJhCHAryuUC}}, [http://books.google.com/books?id=IyJhCHAryuUC&pg=PA256 Chapter 10, page 256]
</ref> or that gravity is a [[fictitious force]].<ref>{{cite book
|title=A short course in general relativity
|edition=3
|first1=J. Foster, J. D. Nightingale
|last1=J. Foster, J. D. Nightingale
|first2=J. Foster, J. D. Nightingale
|last2=J. Foster, J. D. Nightingale
|publisher=Springer Science & Business
|year=2006
|isbn=0-387-26078-1
|page=55
|url=http://books.google.com/books?id=wtoKZODmoVsC}}, [http://books.google.com/books?id=wtoKZODmoVsC&pg=PA55 Chapter 2, page 55]
</ref>

==Classical mechanics==

In [[classical mechanics]] as in [[physics]], the field is not [[reality|real]],{{clarification|date=April 2013}} but merely a [[Scientific modelling|model]] describing the effects of gravity. The field can be determined using [[Newton's law of universal gravitation]]. Determined in this way, the gravitational field '''g''' around a single particle of mass ''M'' is a [[vector field]] consisting at every point of a [[Vector (geometry)|vector]] pointing directly towards the particle. The magnitude of the field at every point is calculated applying the universal law, and represents the force per unit mass on any object at that point in space. Because the force field is conservative, there is a scalar potential energy per unit mass, ''Φ'', at each point in space associated with the force fields; this is called [[Potential energy#Gravitational potential energy|gravitational potential]].<ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8</ref> The gravitational field equation is<ref>Encyclopaedia of Physics, R.G. Lerner, G.L. Trigg, 2nd Edition, VHC Publishers, Hans Warlimont, Springer, 2005</ref>

:<math>\mathbf{g}=\frac{\mathbf{F}}{m}=-\frac{{\rm d}^2\mathbf{R}}{{\rm d}t^2}=-GM\frac{\mathbf{\hat{R}}}{|\mathbf{R}|^2}=-\nabla\Phi,</math>

where '''F''' is the [[gravitational force]], ''m'' is the mass of the [[test mass|test particle]], '''R''' is the position of the test particle, <math>\mathbf{\hat{R}}</math> is a [[unit vector]] in the direction of '''R''', ''t'' is [[time]], ''G'' is the [[gravitational constant]], and ∇ is the [[del operator]]

This includes [[Newton's law of gravitation]], and the relation between gravitational potential and field acceleration. Note that d2'''R'''/d''t''2 and '''F'''/''m'' are both equal to the [[gravitational acceleration]] '''g''' (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass<ref>Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1</ref>). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass [[density]] ''ρ'' of the attracting mass are:

:<math>-\nabla\cdot\mathbf{g}=\nabla^2\Phi=4\pi G\rho\!</math>

which contains [[Gauss' law for gravity]], and [[Poisson's equation#Newtonian gravity|Poisson's equation for gravity]]. Newton's and Gauss' law are mathematically equivalent, and are related by the [[divergence theorem]]. Poisson's equation is obtained by taking the [[divergence]] of both sides of the previous equation. These classical equations are [[differential equation|differential]] [[equations of motion]] for a test particle in the presence of a gravitational field, i.e. setting up and solving these equations allows the motion of a test mass to be determined and described.

The field around multiple particles is simply the [[Vector (geometry)#Addition and subtraction|vector sum]] of the fields around each individual particle. An object in such a field will experience a force that equals the vector sum of the forces it would feel in these individual fields. This is mathematically:<ref>Classical Mechanics (2nd Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, ISBN 07-084018-0. </ref>

:<math>\mathbf{g}_j^{\text{(net)}}=\sum_{i\ne j}\mathbf{g}_i =\frac{1}{m_j}\sum_{i\ne j}\mathbf{F}_i = -G\sum_{i\ne j}m_i\frac{\mathbf{\hat{R}}_{ij}}{{|\mathbf{R}_i-\mathbf{R}_j}|^2}=-\sum_{i \ne j}\nabla\Phi_i</math>

i.e. the gravitational field on mass ''mj'' is the sum of all gravitational fields due to all other masses ''mi'', except the mass ''mj'' itself. The unit vector <math>\mathbf{\hat{R}}_{ij}</math> is in the direction of {{nowrap|'''R'''''i'' − '''R'''''j''}}.

==General relativity==
In [[general relativity]] the gravitational field is determined by solving the [[Einstein field equations]],<ref> Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0 </ref>

:<math> \bold{G}=\frac{8\pi G}{c^4}\bold{T}.</math>

Here '''T''' is the [[stress–energy tensor]], '''G''' is the [[Einstein tensor]], and ''c'' is the [[speed of light]],

These equations are dependent on the distribution of matter and energy in a region of space, unlike Newtonian gravity, which is dependent only on the distribution of matter. The fields themselves in general relativity represent the [[General relativity#Spacetime as a curved Lorentzian manifold|curvature of spacetime]]. General relativity states that being in a region of curved space is [[equivalence principle#The Einstein equivalence principle|equivalent]] to [[acceleration|accelerating]] up the [[gradient]] of the field. By [[Newton's_laws_of_motion#Newton.27s second law|Newton's second law]], this will cause an object to experience a [[fictitious force]] if it is held still with respect to the field. This is why a person will feel himself pulled down by the force of gravity while standing still on the Earth's surface. In general the gravitational fields predicted by general relativity differ in their effects only slightly from those predicted by classical mechanics, but there are a number of easily verifiable [[General relativity#Predictions of general relativity|differences]], one of the most well known being the [[General relativity#bending of light|bending of light]] in such fields.

==See also==
*[[Classical mechanics]]
*[[Gravitation]]
*[[Gravitational potential]]
*[[Newton's law of universal gravitation]]
*[[Newton's laws of motion]]
*[[Potential energy]]
*[[Speed of gravity]]
*[[Tests of general relativity]]
*[[Defining equation (physics)]]

==Notes==
{{Reflist|2}}

[[Category:Theories of gravitation]]
[[Category:Gravitation]]
[[Category:Geodesy]]
[[Category:General relativity]]

[[de:Gravitation#Gravitationsfeld]]

Weyl law

2012-07-09T16:06:11Z

76.181.78.121: /* Generalizations */

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Homotopy extension property

Dirichlet character

Weyl law