{{Short description|Classical statement of gravity as force}}
{{Classical mechanics|expanded=core}}

'''Newton's law of universal gravitation''' describes [[gravity]] as a [[force]]: any [[particle]] attracts any other particle with a force [[Proportionality (mathematics)#Direct proportionality|proportional]] to the product of their masses and [[Proportionality (mathematics)#Inverse proportionality|inversely proportional]] to the square of the distance between their centers of mass. Separated, spherically symmetrical objects attract and are attracted [[Shell theorem|as if all their mass were concentrated at their centers]]. The publication of the law has become known as the "[[Unification (physics)#Unification of gravity on Earth with astronomical behaviors|first great unification]]", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.<ref>{{Cite journal |last1=Freedman |first1=Daniel Z. |last2=van Nieuwenhuizen |first2=Peter |date=1978 |title=Supergravity and the Unification of the Laws of Physics |url=https://www.jstor.org/stable/24955642 |journal=Scientific American |volume=238 |issue=2 |pages=126–143 |doi=10.1038/scientificamerican0278-126 |jstor=24955642 |bibcode=1978SciAm.238b.126F |issn=0036-8733}}</ref><ref>{{cite book |first=Klaus |last=Mainzer |title=Symmetries of Nature: A Handbook for Philosophy of Nature and Science |url=https://books.google.com/books?id=QekhAAAAQBAJ&pg=PA8 |date=2 December 2013 |publisher=Walter de Gruyter |isbn=978-3-11-088693-1 |pages=8ff }}</ref>

This is a general [[physical law]] derived from [[empirical observation]]s by what [[Isaac Newton]] called ''[[inductive reasoning]]''.<ref>Isaac Newton: "In [experimental] philosophy particular propositions are inferred from the phenomena and afterwards rendered general by induction": ''[[Philosophiae Naturalis Principia Mathematica|Principia]]'', Book 3, ''General Scholium'', at p.&nbsp;392 in Volume 2 of Andrew Motte's English translation published 1729.</ref> It is a part of [[classical mechanics]] and was formulated in Newton's work ''[[Philosophiæ Naturalis Principia Mathematica]]'' (Latin for 'Mathematical Principles of Natural Philosophy' (the ''Principia'')), first published on 5 July 1687. 

The equation for universal gravitation thus takes the form:
<math display="block">F=G\frac{m_1m_2}{r^2},</math>
where ''F'' is the gravitational force acting between two objects, ''m''<sub>1</sub> and ''m''<sub>2</sub> are the masses of the objects, ''r'' is the distance between the [[Center of mass|centers of mass]], and ''G'' is the [[gravitational constant]], ({{physconst|G|round=3|ref=no}}).

The first test of Newton's law of gravitation between masses in the laboratory was the [[Cavendish experiment]] conducted by the British scientist [[Henry Cavendish]] in 1798.<ref name="The Michell-Cavendish Experiment">Hodges, Laurent. [http://www.public.iastate.edu/~lhodges/Michell.htm "The Michell–Cavendish Experiment"]. Indiana State University.</ref> It took place 111 years after the publication of Newton's ''Principia'' and approximately 71 years after his death.

Newton's law of gravitation resembles [[Coulomb's law]] of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are [[inverse-square law]]s, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has charge in place of mass and a different constant.

Newton's law was later superseded by [[Albert Einstein]]'s theory of [[general relativity]], but the universality of the gravitational constant is intact and the law still continues to be used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when there is a need for extreme accuracy, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at small distances (such as [[Mercury (planet)|Mercury]]'s orbit around the [[Sun]]).

== History ==
{{Main|History of gravitational theory}}

Before Newton's law of gravity, there were many theories explaining gravity. Philosophers made observations about things falling down − and developed theories why they do – as early as [[Aristotle]] who thought that rocks fall to the ground because seeking the ground was an essential part of their nature.<ref>{{cite web |last1=McShea |first1=Daniel W |last2=Babcock |first2=Gunnar O |title=Elusive but everywhere |url=https://aeon.co/essays/a-new-field-theory-reveals-the-hidden-forces-that-guide-us |date=November 4, 2024 |access-date=November 30, 2024 |publisher=[[Aeon (magazine)|Aeon]]}}</ref>

Around 1600, the [[scientific method]] began to take root. [[René Descartes]] started over with a more fundamental view, developing ideas of matter and action independent of theology. [[Galileo Galilei]] wrote about experimental measurements of falling and rolling objects. [[Johannes Kepler]]'s [[Kepler's laws of planetary motion| laws of planetary motion]] summarized [[Tycho Brahe]]'s astronomical observations.<ref name=Hesse2005/>{{rp|132}}

Around 1666, [[Isaac Newton]] developed the idea that Kepler's laws must also apply to the orbit of the Moon around the Earth and then to all objects on Earth. The analysis required assuming that the gravitation force acted as if all of the mass of the Earth were concentrated at its center, an unproven conjecture at that time. His calculations of the Moon orbit time was within 16% of the known value. By 1680, new values for the diameter of the Earth improved his orbit time to within 1.6%, but more importantly Newton had found a proof of his earlier conjecture.<ref name="Feather">{{cite book |last1=Feather |first1=Norman |url=https://archive.org/details/introductiontoph0000feat |title=An Introduction to the Physics of Mass Length and Time |date=1959 |publisher=Edinburgh University Press |isbn=978-1-135-64613-4 |lccn=60051995}}</ref>{{rp|201}}

In 1687, Newton published his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' which combined his [[Newton's laws of motion|laws of motion]] with new mathematical analysis to explain Kepler's empirical results.<ref name=Hesse2005>{{Cite book |last=Hesse |first=Mary B. |title=Forces and fields: the concept of action at a distance in the history of physics |date=2005 |publisher=Dover |isbn=978-0-486-44240-2 |location=Mineola, New York}}</ref>{{rp|134}} Newton's formulation was later condensed into the inverse-square law:<math display="block">F = G \frac{m_1 m_2}{r^2}, </math>where {{mvar|F}} is the force, {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}} are the masses of the objects interacting, {{mvar|r}} is the distance between the centers of the masses and {{math|''G''}} is the [[gravitational constant]] {{physconst|G|after=.|round=3}} While {{math|''G''}} is also called [[Gravitational constant|Newton's constant]], Newton did not use this constant or formula, he only discussed proportionality.<ref name="Zee-2013"/>{{rp|31}} That was sufficient to show that the gravity of the Earth on the Moon is the same as the gravity of the Earth on an apple:<math display="block">M_\text{earth} \propto a_\text{apple}R_\text{radius of earth}^2 = a_\text{moon}R_\text{lunar orbit}^2 </math> Using the values known at the time, Newton was able to verify this form of his law. The value of {{math|''G''}} was eventually [[Cavendish experiment|measured]] by [[Henry Cavendish]] in 1797.<ref name="Zee-2013">{{Cite book |last=Zee |first=Anthony |title=Einstein Gravity in a Nutshell |title-link=Einstein Gravity in a Nutshell |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14558-7 |edition=1 |series=In a Nutshell Series |location=Princeton}}</ref>{{rp|31}} 

Newton made quantitative analysis based on this formula around 1665, considering the period and distance of the Moon's orbit and considering the timing of objects falling on Earth. Newton did not publish these results at the time because he could not prove that the [[Shell theorem|Earth's gravity acts as if all its mass were concentrated at its center]]. That proof took him twenty years.<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=978-0-471-92567-5 |author-link=Steven Weinberg |url-access=registration}}</ref>{{rp|13}} When Newton presented Book 1 of the unpublished text in April 1686 to the [[Royal Society]], [[Robert Hooke]] made a [[Hooke–Newton inverse square law controversy|claim that Newton had obtained the inverse square law from him]], ultimately a frivolous accusation.<ref name=Feather/>{{rp|204}}

=== Newton's "causes hitherto unknown" ===
{{Main | Action at a distance}}
While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" that his equations implied. In 1692, in his third letter to Bentley, he wrote: "That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it."<ref>{{cite web |title=Original letter from Isaac Newton to Richard Bentley (Normalized) |url=https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00258 |access-date=2025-03-25 |website=www.newtonproject.ox.ac.uk }}</ref><ref>{{cite book |last=Newton |first=Isaac |title=Four letters from Sir Isaac Newton to Doctor Bentley: containing some arguments in proof of a deity |date=1756 |publisher=Printed for R. and J. Dodsley |url=https://archive.org/details/fourlettersfroms00newt/page/26/mode/2up?q=no+man }}</ref>{{rp|26}}

Newton's 1713 ''[[General Scholium]]'' in the second edition of ''Principia'' explains his model of gravity, translated in this case by [[Samuel Clarke]]:
{{blockquote|I have explained the Phænomena <!-- Please do not correct the spelling or capitalization in this quote --> of the Heavens and the Sea, by the Force of Gravity; but the Cause of Gravity I have not yet assigned. It is a Force arising from some Cause, which reaches to the very Centers of the Sun and Planets, without any diminution of its Force: And it acts, not proportionally to the Surfaces of the Particles it acts upon, as Mechanical Causes use to do; but proportionally to the Quantity of Solid Matter: And its Action reaches every way to immense Distances, decreasing always in a duplicate ratio of the Distances. But the Cause of these Properties of Gravity, I have not yet found deducible from Phænomena: And Hypotheses I make not.<ref name=Cohen-1962/>{{rp|383}}}} 
The last sentence is Newton's famous<ref name=Cohen-1962>{{cite journal |last=Cohen |first=I. Bernard |date=1962 |title=The First English Version of Newton's Hypotheses non fingo |url=https://www.jstor.org/stable/227788 |access-date=2025-03-26 |journal=Isis |volume=53 |issue=3 |pages=379–388 |doi=10.1086/349598 |jstor=227788 }}</ref> and highly debated<ref>{{cite book |url=https://www.degruyter.com/document/doi/10.3138/9781442632783/html |title=The Methodological Heritage of Newton |date=1970-12-31 |publisher=University of Toronto Press |isbn=978-1-4426-3278-3 |editor-last=Butts |editor-first=Robert E. |doi=10.3138/9781442632783 |editor-last2=Davis |editor-first2=John W. }}</ref> [[Latin]] phrase [[Hypotheses non fingo]]. In other translations it comes out "I feign no hypotheses".<ref>{{cite book |last=Westfall |first=Richard S. |title=The Construction of Modern Science: Mechanisms and Mechanics |publisher=[[Cambridge University Press]] |year=1978 |isbn=0-521-21863-2 |lccn=77084001 |ol=4567428M }}</ref>

Newton made two predictions based on his gravitational theory. 
In 1774 the [[Schiehallion experiment]] compared the gravitational attraction of the Earth to that of Schiehallion mountain, confirming one of these predictions. The second prediction concerned the mutual attraction of two massive spheres, confirmed by [[Henry Cavendish]] in 1798.<ref name="Sillitto">{{cite web|url=http://www.sillittopages.co.uk/schie/schie90.html|title=Maskelyne on Schiehallion: A Lecture to The Royal Philosophical Society of Glasgow|last=Sillitto|first=R.M.|date=31 October 1990|access-date=28 December 2008|archive-date=18 April 2019|archive-url=https://web.archive.org/web/20190418011810/http://www.sillittopages.co.uk/schie/schie90.html|url-status=dead}}</ref>

== Modern form ==
In modern language, the law states the following:
{| style="border-top:1px solid #aaa; border-bottom:1px solid #aaa; margin-bottom:1em"
|style="padding:0.1em 2em 0 0.5em"| <span style="white-space:nowrap">Every [[Point mass|point]] [[mass]] attracts every single other point mass by a [[force]] acting along</span> the [[Line (mathematics)|line]] intersecting both points. The force is [[Proportionality (mathematics)|proportional]] to the two masses and [[Proportionality (mathematics)#Inverse proportionality|inversely proportional]] to the [[Square (algebra)|square]] of the distance between them:<ref name="Newton1"/>
|rowspan="2" style="width:99%; vertical-align:top"| [[File:NewtonsLawOfUniversalGravitation.svg|left|Diagram of two masses attracting one another|class=skin-invert-image]]
|-
|style="padding:0 2em 0.1em 0.5em; white-space:nowrap"|
<math display="block">F = G \frac{m_1 m_2}{r^2}\ </math>
where
* ''F'' is the force between the objects;
* ''G'' is the [[Newtonian constant of gravitation]] ({{physconst|G|round=3|ref=no}});
* ''m''<sub>1</sub> is the mass of the first object;
* ''m''<sub>2</sub> is the mass of the second object;
* ''r'' is the distance between the centers of the masses.
|}
[[File:Gravity Big G Measurements NIST.png|thumb|upright=2.0|Error plot showing experimental values for ''G'']]

Assuming [[International System of Units|SI units]], ''F'' is measured in [[newton (units)|newton]]s (N), ''m''<sub>1</sub> and ''m''<sub>2</sub> in [[kilogram]]s (kg), ''r'' in meters (m), and the constant ''G'' is {{physconst|G|after=.}} The value of the constant ''G'' was first accurately determined from the results of the [[Cavendish experiment]] conducted by the [[United Kingdom|British]] scientist [[Henry Cavendish]] in 1798, although Cavendish did not himself calculate a numerical value for ''G''.<ref name="The Michell-Cavendish Experiment"/> This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's ''Principia'' and 71 years after Newton's death, so none of Newton's calculations could use the value of ''G''; instead he could only calculate a force relative to another force.

== Bodies with spatial extent ==
[[File:Earth-G-force.png|thumb|right|Gravitational field strength within the Earth]]
[[File:Gravity field near earth.gif|thumb|Gravity field near the surface of the Earth – an object is shown accelerating toward the surface]]

If the bodies in question have spatial extent (as opposed to being point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses that constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails [[integral|integrating]] the force (in vector form, see below) over the extents of the two [[Physical body|bodies]].

In this way, it can be shown that an object with a spherically symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its center.<ref name=Newton1>Proposition 75, Theorem 35: p. 956 – I.Bernard Cohen and Anne Whitman, translators: [[Isaac Newton]], ''The Principia'': [[Mathematical Principles of Natural Philosophy]]. Preceded by ''A Guide to Newton's Principia'', by I.Bernard Cohen. University of California Press 1999 {{ISBN|0-520-08816-6}} {{ISBN|0-520-08817-4}}</ref> (This is not generally true for bodies that are not spherically symmetrical.)

For points ''inside'' a spherically symmetric distribution of matter, Newton's [[shell theorem]] can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance ''r''<sub>0</sub> from the center of the mass distribution:<ref>{{cite web |url=http://farside.ph.utexas.edu/teaching/336k/lectures/node109.html |title=Rotational Flattening |website=farside.ph.utexas.edu }}</ref>
* The portion of the mass that is located at radii {{nowrap|''r'' < ''r''<sub>0</sub>}} causes the same force at the radius ''r''<sub>0</sub> as if all of the mass enclosed within a sphere of radius ''r''<sub>0</sub> was concentrated at the center of the mass distribution (as noted above).
* The portion of the mass that is located at radii {{nowrap|''r'' > ''r''<sub>0</sub>}} exerts ''no net'' gravitational force at the radius ''r''<sub>0</sub> from the center. That is, the individual gravitational forces exerted on a point at radius ''r''<sub>0</sub> by the elements of the mass outside the radius ''r''<sub>0</sub> cancel each other.

As a consequence, for example, within a shell of uniform thickness and density there is ''no net'' gravitational acceleration anywhere within the hollow sphere.

== Vector form ==
[[File:Gravitymacroscopic.svg|thumb|Gravity field surrounding Earth from a macroscopic perspective]]

Newton's law of universal gravitation can be written as a [[vector (geometry)|vector]] [[equation]] to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.
<math display="block">
\mathbf{F}_{21} =
- G {m_1m_2\over {|\mathbf r_{21}|}^2}\hat\mathbf r_{21} = - G {m_1m_2\over {|\mathbf r_{21}|}^3}\mathbf r_{21}
</math>
where
* '''F'''<sub>21</sub> is the force applied on body 2 exerted by body 1,
* ''G'' is the [[gravitational constant]],
* ''m''<sub>1</sub> and ''m''<sub>2</sub> are respectively the masses of bodies 1 and 2,
* '''r'''<sub>21</sub> = '''r'''<sub>2</sub> − '''r'''<sub>1</sub> is the [[displacement vector]] between bodies 1 and 2, and
* <math> \hat\mathbf r_{21} \ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf{r_2 - r_1}}{|\mathbf {r_2 - r_1}|} </math> is the [[unit vector]] from body 1 to body 2.<ref>The vector difference '''r'''<sub>2</sub> − '''r'''<sub>1</sub> points from object 1 to object 2. See Fig. 11–6. of [https://feynmanlectures.caltech.edu/I_11.html#Ch11-S5 The Feynman Lectures on Physics, Volume I], equation (9.19) of [https://feynmanlectures.caltech.edu/I_09.html#Ch9-S7 The Feynman Lectures on Physics, Volume I] and {{slink|Euclidean vector#Addition and subtraction}}</ref>

It can be seen that the vector form of the equation is the same as the [[scalar (physics)|scalar]] form given earlier, except that '''F''' is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that '''F'''<sub>12</sub> = −'''F'''<sub>21</sub>.

== Gravity field ==
<!-- This section is linked from [[Center of mass]] -->
{{main|Gravitational field}}
{{Unsourced section|date=March 2025}}

[[File:Gravitational Field.gif|thumb|Computer generated simulation of two bodies orbiting each other with plotted gravitational field]]
The '''gravitational field''' is a [[vector field]] that describes the gravitational force that would be applied on an object in any given point in space, per unit mass. It is actually equal to the [[gravitational acceleration]] at that point.

It is a generalisation of the vector form, which becomes particularly useful if more than two objects are involved (such as a rocket between the Earth and the Moon). For two objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write '''r''' instead of '''r'''<sub>12</sub> and ''m'' instead of ''m''<sub>2</sub> and define the gravitational field '''g'''('''r''') as:<ref>{{Cite web |last=Hilst |first=Robert |date=2004 |title=essentials2.dvi |url=https://ocw.mit.edu/courses/12-201-essentials-of-geophysics-fall-2004/7fa24d336366b74c52adb48ae6c8cf6f_ch2.pdf |url-status=live |website=MIT OpenCourseWare}}</ref>
<math display="block">\mathbf g(\mathbf r) =
- G {m_1 \over {{\vert \mathbf{r} \vert}^2}}
\, \mathbf{\hat{r}}
</math>
so that we can write:
<math display="block">\mathbf{F}( \mathbf r) = m \mathbf g(\mathbf r). </math>

This formulation is dependent on the objects causing the field. The field has the dimension of acceleration; in the [[SI]], its unit is m/s<sup>2</sup>.

Gravitational fields are also [[Conservative field|conservative]]; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field ''V''('''r''') such that
<math display="block"> \mathbf{g}(\mathbf{r}) = - \nabla V( \mathbf r).</math>

If ''m''<sub>1</sub> is a point mass or the mass of a sphere with homogeneous mass distribution, the force field '''g'''('''r''') outside the sphere is isotropic, i.e., depends only on the distance ''r'' from the center of the sphere. In that case
<math display="block"> V(r) = -G\frac{m_1}{r}. </math>

As per [[Gauss's law for gravity|Gauss's law]], field in a symmetric body can be found by the mathematical equation:
{{block indent|{{oiint
| intsubscpt = <math>\partial V</math>
| integrand = <math>\mathbf{g(r)}\cdot d\mathbf{A} = -4\pi G M_\text{enc},</math>
}}}}
where <math>\partial V</math> is a closed surface and <math>M_\text{enc}</math> is the mass enclosed by the surface.

Hence, for a hollow sphere of radius <math>R</math> and total mass <math>M</math>,
<math display="block">|\mathbf{g(r)}| = \begin{cases}
  0, & \text{if } r < R \\
\\
  \dfrac{GM}{r^2}, & \text{if } r \ge R
\end{cases}
</math>

For a uniform solid sphere of radius <math>R</math> and total mass <math>M</math>,
<math display="block">|\mathbf{g(r)}| = \begin{cases}
  \dfrac{GM r}{R^3}, & \text{if } r < R \\
\\
  \dfrac{GM}{r^2}, & \text{if } r \ge R
\end{cases}
</math>

== Limitations ==

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities <math>\phi / c^{2}</math> and <math>(v/c)^2</math> are both much less than one, where <math>\phi</math> is the [[gravitational potential]], <math>v</math> is the velocity of the objects being studied, and <math>c</math> is the [[speed of light]] in vacuum.<ref>
{{cite book
 | last1 = Misner | first1 = Charles W. | author1-link = Charles W. Misner
 | last2 = Thorne | first2 = Kip S. | author2-link = Kip Thorne
 | last3 = Wheeler | first3 = John Archibald | author3-link = John Archibald Wheeler
 | title = Gravitation | place= New York | publisher = W. H. Freeman and Company
 | date = 1973 | isbn = 978-0-7167-0344-0 |page=1049
}}</ref> For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since
<math display="block">\frac{\phi}{c^2}=\frac{GM_\mathrm{sun}}{r_\mathrm{orbit}c^2} \sim 10^{-8},
\quad \left(\frac{v_\mathrm{Earth}}{c}\right)^2=\left(\frac{2\pi r_\mathrm{orbit}}{(1\ \mathrm{yr})c}\right)^2 \sim 10^{-8} ,</math>
where <math>r_\text{orbit} </math> is the radius of the Earth's orbit around the Sun.

In situations where either dimensionless parameter is large, then [[general relativity]] must be used to describe the system. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity.

=== Observations conflicting with Newton's formula ===
* Newton's theory does not fully explain the [[Apsidal precession|precession of the perihelion]] of the orbits of the planets, especially that of Mercury, which was detected long after the life of Newton.<ref>[[Max Born]] (1924), ''Einstein's Theory of Relativity'' (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and the Earth.)</ref> There is a 43 [[arcsecond]] per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th century.
* The predicted angular [[Gravitational lens|deflection of light rays by gravity]] (treated as particles travelling at the expected speed) that is calculated by using Newton's theory is only one-half of the deflection that is observed by astronomers.{{citation needed|reason=The literature has much controversy on this.|date=June 2020}} Calculations using general relativity are in much closer agreement with the astronomical observations.
* In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey both Newton's law of universal gravitation and general relativity. Astrophysicists, however, explain this marked phenomenon by assuming the presence of large amounts of [[dark matter]].

=== Einstein's solution ===
{{Spacetime|cTopic=Relation to gravity}}
The first two conflicts with observations above were explained by Einstein's theory of [[general relativity]], in which gravitation is a manifestation of [[curved spacetime]] instead of being due to a force propagated between bodies. In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. This allowed a description of the motions of light and mass that was consistent with all available observations. In general relativity, the gravitational force is a [[fictitious force]] resulting from the [[curvature of spacetime]], because the [[gravitational acceleration]] of a body in [[free fall]] is due to its [[world line]] being a [[geodesic]] of [[spacetime]].

== Extensions ==

In recent years, quests for non-inverse square terms in the law of gravity have been carried out by [[neutron interferometry]].<ref>{{cite journal | doi=10.1103/PhysRevC.75.015501 |title = Neutron interferometric method to provide improved constraints on non-Newtonian gravity at the nanometer scale | journal=Physical Review C | volume=75 | issue=1 | article-number=015501 |year = 2007 |last1 = Greene |first1 = Geoffrey L. | last2=Gudkov | first2=Vladimir | bibcode=2007PhRvC..75a5501G | arxiv=hep-ph/0608346 |s2cid = 39665455 }}</ref>

== Solutions ==
The problem of predicting the motion of ''n'' objects subject to gravity is known as the [[n-body problem|''n''-body problem]]. The [[two-body problem]] has been completely solved, but for more bodies the solution is in general [[Chaos (Mathematics)|chaotic]] and can only be obtained [[Numerical analysis|numerically]]. The most-studied case is the [[three-body problem]], for which several solutions for particular cases are known, for example those giving rise to the [[Lagrange points]].{{cn|date=November 2025}}

== See also ==
{{Portal|Physics}}
* {{Annotated link|Bentley's paradox}}
* {{Annotated link|Gauss's law for gravity}}
* {{Annotated link|Jordan and Einstein frames}}
* {{Annotated link|Kepler orbit}}
* {{Annotated link|Newton's cannonball}}
* {{Annotated link|Newton's laws of motion}}
* {{Annotated link|Social gravity}}
* {{Annotated link|Static forces and virtual-particle exchange}}

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

== External links ==
* {{Commons category-inline}}
* {{YouTube |5C5_dOEyAfk |Feather and Hammer Drop on Moon}}
* [https://web.archive.org/web/20090817212723/http://www.pythia.com.ar/?id=gravlaw Newton's Law of Universal Gravitation Javascript calculator]

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