Editing Gravitational field

{{short description|Vector field representing a mass's effect on surrounding space}}
{{Redirect|Gravity field|Earth's gravity field|Gravity of Earth}}
[[File:Earth-moon-field.svg|thumb|upright=1.5|Representation of the gravitational field of Earth and Moon combined (not to scale). Vector field (blue) and its associated scalar potential field (red). Point P between earth and moon is the [[point of equilibrium]].]]

In [[physics]], a '''gravitational field''' or '''gravitational acceleration field''' is a [[vector field|vector]] [[field (physics)|field]] used to explain the influences that a body extends into the space around itself.<ref>{{cite book |author-link=Richard Feynman |first=Richard |last=Feynman |title=The Feynman Lectures on Physics |volume=I |publisher=Addison Wesley Longman |year=1970 |isbn=978-0-201-02115-8 |url=https://feynmanlectures.caltech.edu/I_07.html }}</ref> A gravitational field is used to explain [[Gravity|gravitational]] phenomena, such as the ''[[gravitational force]] field'' exerted on another massive body. It has [[dimension (physics)|dimension]] of [[acceleration]] (L/T<sup>2</sup>) and it is measured in [[unit of measurement|units]] of [[newton (unit)|newtons]] per [[kilogram]] (N/kg) or, equivalently, in [[meter]]s per [[second]] squared (m/s<sup>2</sup>).

In its original concept, [[gravity]] was a [[force]] between point [[mass]]es. Following [[Isaac Newton]], [[Pierre-Simon Laplace]] attempted to model gravity as some kind of [[radiation]] field or [[fluid]],{{Cn|date=October 2024}} and since the 19th century, explanations for gravity in [[classical mechanics]] have usually been taught in terms of a field model, rather than a point attraction. It results from the [[spatial gradient]] of the [[gravitational potential field]].

In [[general relativity]], 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".{{citation needed|date=August 2020}} 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]] |date=1981 |isbn=978-0-226-28864-2 |page=181 |url=https://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181 }}</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 |date=2007 |isbn=978-0-387-69199-2 |page=256 |url=https://books.google.com/books?id=IyJhCHAryuUC&pg=PA256 }}</ref> or that gravity is a [[fictitious force]].<ref>{{cite book |title=A Short Course in General Relativity |edition=3 |first1=J. |last1=Foster |first2=J. D. |last2=Nightingale |publisher=Springer Science & Business |date=2006 |isbn=978-0-387-26078-5 |page=55 |url=https://books.google.com/books?id=wtoKZODmoVsC&pg=PA55 }}</ref>

Gravity is distinguished from other forces by its obedience to the [[equivalence principle]].

== Classical mechanics ==

In classical mechanics, a gravitational field is a physical quantity.<ref>{{cite book |author-link=Richard Feynman |first=Richard |last=Feynman |title=The Feynman Lectures on Physics |volume= II |publisher=Addison Wesley Longman |year=1970 |isbn=978-0-201-02115-8 |url=https://feynmanlectures.caltech.edu/II_01.html#Ch1-S2 |quote="A 'field' is any physical quantity which takes on different values at different points in space."}}</ref> A gravitational field can be defined using [[Newton's law of universal gravitation]]. Determined in this way, the gravitational field {{math|'''g'''}} around a single particle of mass {{math|''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 by 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, {{math|Φ}}, at each point in space associated with the force fields; this is called [[gravitational potential]].<ref>{{cite book|title=Dynamics and Relativity |first1=J. R. |last1=Forshaw |first2=A. G. |last2=Smith |publisher=Wiley |date=2009 |isbn=978-0-470-01460-8}}{{page needed|date=October 2017}}</ref> The gravitational field equation is<ref>{{cite book |url=https://archive.org/details/encyclopediaofph00lern |title=Encyclopaedia of Physics |date=1991 |publisher=[[Wiley-VCH]] |isbn=978-0-89573-752-6 |editor1-last=Lerner |editor1-first=R. G. |editor1-link=Rita G. Lerner |edition=2nd |editor2-last=Trigg |editor2-first=G. L. |url-access=registration}} p. 451</ref>
<math display="block">\mathbf{g}=\frac{\mathbf{F}}{m}=\frac{d^2\mathbf{R}}{dt^2}=-GM\frac{\mathbf{R}}{\left|\mathbf{R}\right|^3} = -\nabla\Phi ,</math>
where {{math|'''F'''}} is the [[gravitational force]], {{math|''m''}} is the mass of the [[test mass|test particle]], {{math|'''R'''}} is the radial vector of the test particle relative to the mass (or for Newton's second law of motion which is a time dependent function, a set of positions of test particles each occupying a particular point in space for the start of testing), {{math|''t''}} is [[time]], {{math|''G''}} is the [[gravitational constant]], and {{math|∇}} is the [[del operator]].

This includes Newton's law of universal gravitation, and the relation between gravitational potential and field acceleration. {{math|{{sfrac|d<sup>2</sup>'''R'''|d''t''<sup>2</sup>}}}} and {{math|{{sfrac|'''F'''|''m''}}}} are both equal to the [[gravitational acceleration]] {{math|'''g'''}} (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass<ref>{{cite book|title=Essential Principles of Physics |first1=P. M. |last1=Whelan |first2=M. J. |last2=Hodgeson |edition=2nd |date=1978 |publisher=John Murray |isbn=978-0-7195-3382-2}}{{page needed|date=October 2017}}</ref>). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass [[density]] {{math|''ρ''}} of the attracting mass is:
<math display="block">\nabla\cdot\mathbf{g}=-\nabla^2\Phi=-4\pi G\rho</math>
which contains [[Gauss's law for gravity]], and [[Poisson's equation#Newtonian gravity|Poisson's equation for gravity]]. Newton's law implies Gauss's law, but not vice versa; see ''[[Gauss's law for gravity#Relation to Newton's law|Relation between Gauss's and Newton's laws]]''.

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. A test particle in such a field will experience a force that equals the vector sum of the forces that it would experience in these individual fields. This is<ref>{{cite book|title=Classical Mechanics |edition=2nd |first=T. W. B. |last=Kibble |series=European Physics Series |publisher=[[McGraw Hill]] |location=UK |date=1973 |isbn=978-0-07-084018-8}}{{page needed|date=October 2017}}</ref>
<math display="block">\mathbf{g}
= \sum_{i}\mathbf{g}_i
= \frac{1}{m}\sum_{i}\mathbf{F}_i
= - G\sum_{i}m_i\frac{\mathbf{R}-\mathbf{R}_i}{\left|\mathbf{R}-\mathbf{R}_i\right|^3}
= - \sum_{i}\nabla\Phi_i ,</math>
i.e. the gravitational field on mass {{math|''m<sub>j</sub>''}} is the sum of all gravitational fields due to all other masses ''m''<sub>''i''</sub>, except the mass {{math|''m<sub>j</sub>''}} itself. {{math|'''R'''<sub>''i''</sub>}} is the position vector of the gravitating particle {{math|''i''}}, and {{math|'''R'''}} is that of the test particle.

== General relativity ==
{{See also|Gravitational acceleration#General relativity|Gravitational potential#General relativity}}
A freely moving particle in gravitational field has the equations of motion:
<math display="block">\frac{d^2x^\lambda}{d\tau^2} + \Gamma^\lambda_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0</math>
where <math>\tau</math> is the [[proper time]] for the particle, <math>\Gamma^\lambda_{\mu\nu}</math> are the [[Christoffel symbols]] and repeated indices are summed over.<ref name=Weinberg-1972/>{{rp|70}} 
The proper time can be expressed in terms of the [[metric tensor]]:
<math display="block">d\tau^2 = -g_{\mu\nu} dx^\mu dx^\nu. </math>
The field that determines the gravitational force is the [[Christoffel symbols]] and its derivatives, the [[metric tensor (general relativity)|metric tensor]] plays the role of the gravitational potential.<ref name=Weinberg-1972/>{{rp|73}}

In general relativity, the gravitational field is determined by solving the [[Einstein field equations]]<ref name=Weinberg-1972>{{cite book |last=Weinberg |first=Steven |url=https://archive.org/details/gravitationcosmo00stev_0 |title=Gravitation and cosmology |date=1972 |publisher=John Wiley & Sons |isbn=9780471925675 |author-link=Steven Weinberg |url-access=registration}}</ref>{{rp|157}}
<math display="block"> \mathbf{G} = \kappa \mathbf{T} ,</math>
where {{math|'''T'''}} is the [[stress–energy tensor]], {{math|'''G'''}} is the [[Einstein tensor]], and {{math|''κ''}} is the [[Einstein gravitational constant]].  The latter is defined as {{math|1=''κ'' = 8''πG''/''c''{{sup|4}}}}, where {{math|''G''}} is the [[Newtonian constant of gravitation]] and {{math|''c''}} is the [[speed of light]].

These equations are dependent on the distribution of matter, stress and momentum in a region of space, unlike Newtonian gravity, which is depends on only the distribution of matter.  The fields themselves in general relativity represent the curvature of spacetime.  General relativity states that being in a region of curved space is [[equivalence principle#Einstein equivalence principle|equivalent]] to [[acceleration|accelerating]] up the [[gradient]] of the field.  By [[Newton's laws of motion#Newton's 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#Consequences of Einstein's theory|differences]], one of the most well known being the [[General relativity#Light deflection and gravitational time delay|deflection of light]] in such fields.

== Embedding diagram ==
'''Embedding diagrams''' are three dimensional graphs commonly used to educationally illustrate gravitational potential by drawing gravitational potential fields as a gravitational topography, depicting the potentials as so-called '''gravitational wells''', [[sphere of influence (astrodynamics)|sphere of influence]].
 
== See also ==
{{div col|colwidth=20em}}
* [[Classical mechanics]]
* [[Entropic gravity]]
* [[Gravitation]]
* [[Gravitational energy]]
* [[Gravitational potential]]
* [[Gravitational wave]]
* [[Gravity map]]
* [[Newton's law of universal gravitation]]
* [[Newton's laws of motion]]
* [[Potential energy]]
* [[Specific force]]
* [[Speed of gravity]]
* [[Tests of general relativity]]
{{div col end}}

== References ==
{{reflist|30em}}

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[[Category:Theories of gravity]]
[[Category:Geodesy]]
[[Category:General relativity]]