gravity#Newton's theory of gravitation

Gravity is the word used to describe both a fundamental physical interaction and the observed consequences of that interaction on macroscopic objects on Earth. Gravity is, by far, the weakest of the four fundamental interactions, approximately 10³⁸ times weaker than the strong interaction, 10³⁶ times weaker than the electromagnetic force, and 10²⁹ times weaker than the weak interaction. As a result, it has no significant influence at the level of subatomic particles.{{cite book |title=Scientific Development and Misconceptions Through the Ages: A Reference Guide |edition=illustrated |first1=Robert E. |last1=Krebs |publisher=Greenwood Publishing Group |year=1999 |isbn=978-0-313-30226-8 |page=[https://archive.org/details/scientificdevelo0000kreb/page/133 133] |url=https://archive.org/details/scientificdevelo0000kreb|url-access=registration }} However, gravity is the most significant interaction between objects at the macroscopic scale, and it determines the motion of planets, stars, galaxies, and even light.

Gravity, as the gravitational attraction at the surface of a planet or other celestial body,{{ citation | title = McGraw-Hill Dictionary of Scientific and Technical Terms | edition = 4th | location = New York | publisher = McGraw-Hill | year = 1989 | isbn = 0-07-045270-9 | ref = {{harvid|McGraw-Hill Dict|1989}} }} may also include the centrifugal force resulting from the planet's rotation {{Crossreference|text=(see {{slink||Earth's gravity}})|printworthy=1}}.

{{main|History of gravitational theory}}

The nature and mechanism of gravity were explored by a wide range of ancient scholars. In Greece, Aristotle believed that objects fell towards the Earth because the Earth was the center of the Universe and attracted all of the mass in the Universe towards it. He also thought that the speed of a falling object should increase with its weight, a conclusion that was later shown to be false.{{Cite web |last=Cappi |first=Alberto |title=The concept of gravity before Newton |url=http://www.cultureandcosmos.org/pdfs/16/Cappi_INSAPVII_Gravity_before_Newton.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.cultureandcosmos.org/pdfs/16/Cappi_INSAPVII_Gravity_before_Newton.pdf |archive-date=9 October 2022 |url-status=live |website=Culture and Cosmos}} While Aristotle's view was widely accepted throughout Ancient Greece, there were other thinkers such as Plutarch who correctly predicted that the attraction of gravity was not unique to the Earth.{{Cite journal |last1=Bakker |first1=Frederik |last2=Palmerino |first2=Carla Rita |date=1 June 2020 |title=Motion to the Center or Motion to the Whole? Plutarch's Views on Gravity and Their Influence on Galileo |url=https://www.journals.uchicago.edu/doi/abs/10.1086/709138 |journal=Isis |volume=111 |issue=2 |pages=217–238 |doi=10.1086/709138 |s2cid=219925047 |issn=0021-1753 |hdl=2066/219256 |hdl-access=free |access-date=2 May 2022 |archive-date=2 May 2022 |archive-url=https://web.archive.org/web/20220502172704/https://www.journals.uchicago.edu/doi/abs/10.1086/709138 |url-status=live }}

Although he did not understand gravity as a force, the ancient Greek philosopher Archimedes discovered the center of gravity of a triangle.{{cite book |last1=Neitz |first1=Reviel |url=https://books.google.com/books?id=ZC1MOaAkKnsC&pg=PT125 |title=The Archimedes Codex: Revealing The Secrets of the World's Greatest Palimpsest |last2=Noel |first2=William |date=13 October 2011 |publisher=Hachette UK |isbn=978-1-78022-198-4 |page=125 |access-date=10 April 2019 |archive-url=https://web.archive.org/web/20200107004958/https://books.google.com/books?id=ZC1MOaAkKnsC&pg=PT125 |archive-date=7 January 2020 |url-status=live}} He postulated that if two equal weights did not have the same center of gravity, the center of gravity of the two weights together would be in the middle of the line that joins their centers of gravity.{{cite book |author=Tuplin |first1=CJ |url=https://books.google.com/books?id=ajGkvOo0egwC&pg=PR11 |title=Science and Mathematics in Ancient Greek Culture |last2=Wolpert |first2=Lewis |publisher=Hachette UK |year=2002 |isbn=978-0-19-815248-4 |page=xi |access-date=10 April 2019 |archive-url=https://web.archive.org/web/20200117170945/https://books.google.com/books?id=ajGkvOo0egwC&pg=PR11 |archive-date=17 January 2020 |url-status=live}} Two centuries later, the Roman engineer and architect Vitruvius contended in his De architectura that gravity is not dependent on a substance's weight but rather on its "nature".{{Cite book | last = Vitruvius | first = Marcus Pollio | author-link = Marcus Vitruvius Pollio | editor = Alfred A. Howard | title = De Architectura libri decem | trans-title = Ten Books on Architecture | place = Harvard University, Cambridge | publisher = Harvard University Press | date = 1914 | chapter = 7 | page = 215 | chapter-url = http://www.gutenberg.org/files/20239/20239-h/29239-h.htm#Page_215 | others = Herbert Langford Warren, Nelson Robinson (illus), Morris Hicky Morgan | access-date = 10 April 2019 | archive-date = 13 October 2016 | archive-url = https://web.archive.org/web/20161013193438/http://www.gutenberg.org/files/20239/20239-h/29239-h.htm#Page_215 | url-status = live }} In the 6th century CE, the Byzantine Alexandrian scholar John Philoponus proposed the theory of impetus, which modifies Aristotle's theory that "continuation of motion depends on continued action of a force" by incorporating a causative force that diminishes over time.Philoponus' term for impetus is "ἑνέργεια ἀσώματος κινητική" ("incorporeal motive enérgeia"); see CAG XVII, [https://books.google.com/books?id=dVcqvVDiNVUC Ioannis Philoponi in Aristotelis Physicorum Libros Quinque Posteriores Commentaria] {{Webarchive|url=https://web.archive.org/web/20231222224140/https://books.google.com/books?id=dVcqvVDiNVUC |date=22 December 2023 }}, Walter de Gruyter, 1888, p. 642: "λέγω δὴ ὅτι ἑνέργειά τις ἀσώματος κινητικὴ ἑνδίδοται ὑπὸ τοῦ ῥιπτοῦντος τῷ ῥιπτουμένῳ [I say that impetus (incorporeal motive energy) is transferred from the thrower to the thrown]."

In 628 CE, the Indian mathematician and astronomer Brahmagupta proposed the idea that gravity is an attractive force that draws objects to the Earth and used the term gurutvākarṣaṇ to describe it.{{cite book |last1=Pickover |first1=Clifford |url=https://books.google.com/books?id=SQXcpvjcJBUC&pg=PA105 |title=Archimedes to Hawking: Laws of Science and the Great Minds Behind Them |date=16 April 2008 |publisher=Oxford University Press |isbn=9780199792689 |language=en |access-date=29 August 2017 |archive-url=https://web.archive.org/web/20170118060420/https://books.google.com/books?id=SQXcpvjcJBUC |archive-date=18 January 2017 |url-status=live}}{{rp|105}}{{cite book |last1=Bose |first1=Mainak Kumar |url=https://books.google.com/books?id=nbItAAAAMAAJ&q=gravity |title=Late classical India |publisher=A. Mukherjee & Co. |year=1988 |language=en |access-date=28 July 2021 |archive-url=https://web.archive.org/web/20210813203602/https://books.google.com/books?id=nbItAAAAMAAJ&q=gravity |archive-date=13 August 2021 |url-status=live}}{{cite book |last=Sen |first=Amartya |title=The Argumentative Indian |date=2005 |publisher=Allen Lane |isbn=978-0-7139-9687-6 |page=29}}

In the ancient Middle East, gravity was a topic of fierce debate. The Persian intellectual Al-Biruni believed that the force of gravity was not unique to the Earth, and he correctly assumed that other heavenly bodies should exert a gravitational attraction as well.{{cite book |last1=Starr |first1=S. Frederick |title=Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane |date=2015 |publisher=Princeton University Press |isbn=9780691165851 |page=260 |url=https://books.google.com/books?id=hWyYDwAAQBAJ&pg=PA260}} In contrast, Al-Khazini held the same position as Aristotle that all matter in the Universe is attracted to the center of the Earth.{{Cite encyclopedia|encyclopedia=Encyclopedia of the History of Arabic Science|editor-first=Rāshid|editor-last=Rushdī|date=1996|publisher=Psychology Press|isbn=9780415124119|first1=Mariam |last1=Rozhanskaya |first2=I. S. |last2=Levinova |title=Statics |volume=2 |pages=614–642}}

File:The Leaning Tower of Pisa SB.jpeg, where according to legend Galileo performed an experiment about the speed of falling objects]]

{{main|Scientific Revolution}}

In the mid-16th century, various European scientists experimentally disproved the Aristotelian notion that heavier objects fall at a faster rate.{{Cite book|last=Wallace|first=William A.|url=https://books.google.com/books?id=8GxQDwAAQBAJ&pg=PR21|title=Domingo de Soto and the Early Galileo: Essays on Intellectual History|publisher=Routledge|year=2018|isbn=978-1-351-15959-3|location=Abingdon, UK|pages=119, 121–22|language=en|orig-year=2004|access-date=4 August 2021|archive-date=16 June 2021|archive-url=https://web.archive.org/web/20210616043300/https://books.google.com/books?id=8GxQDwAAQBAJ&pg=PR21|url-status=live}} In particular, the Spanish Dominican priest Domingo de Soto wrote in 1551 that bodies in free fall uniformly accelerate. De Soto may have been influenced by earlier experiments conducted by other Dominican priests in Italy, including those by Benedetto Varchi, Francesco Beato, Luca Ghini, and Giovan Bellaso which contradicted Aristotle's teachings on the fall of bodies.

The mid-16th century Italian physicist Giambattista Benedetti published papers claiming that, due to specific gravity, objects made of the same material but with different masses would fall at the same speed.{{Cite journal| doi = 10.1086/349706| issn = 0021-1753| volume = 54| issue = 2| pages = 259–262| last = Drabkin| first = I. E.| title = Two Versions of G. B. Benedetti's Demonstratio Proportionum Motuum Localium| journal = Isis| year = 1963| jstor = 228543| s2cid = 144883728}} With the 1586 Delft tower experiment, the Flemish physicist Simon Stevin observed that two cannonballs of differing sizes and weights fell at the same rate when dropped from a tower.{{Cite book|url=https://books.google.com/books?id=YicuDwAAQBAJ&dq=delft+tower+experiment&pg=PA26|title=Ripples in Spacetime: Einstein, Gravitational Waves, and the Future of Astronomy|last=Schilling|first=Govert|date=31 July 2017|publisher=Harvard University Press|isbn=9780674971660|page=26|language=en|access-date=16 December 2021|archive-date=16 December 2021|archive-url=https://web.archive.org/web/20211216025328/https://books.google.com/books?id=YicuDwAAQBAJ&dq=delft+tower+experiment&pg=PA26|url-status=live}}

In the late 16th century, Galileo Galilei's careful measurements of balls rolling down inclines allowed him to firmly establish that gravitational acceleration is the same for all objects.Galileo (1638), Two New Sciences, First Day Salviati speaks: "If this were what Aristotle meant you would burden him with another error which would amount to a falsehood; because, since there is no such sheer height available on earth, it is clear that Aristotle could not have made the experiment; yet he wishes to give us the impression of his having performed it when he speaks of such an effect as one which we see."{{Cite book |last=Sobel |first=Dava |title=Galileo's daughter: a historical memoir of science, faith, and love |date=1993 |publisher=Walker |isbn=978-0-8027-1343-8 |location=New York}}{{rp|334}} Galileo postulated that air resistance is the reason that objects with a low density and high surface area fall more slowly in an atmosphere. In his 1638 work Two New Sciences Galileo proved that that the distance traveled by a falling object is proportional to the square of the time elapsed. His method was a form of graphical numerical integration since concepts of algebra and calculus were unknown at the time.{{cite book|last=Gillispie|first=Charles Coulston|url=https://archive.org/details/edgeofobjectivit00char/page/n13/mode/2up|title=The Edge of Objectivity: An Essay in the History of Scientific Ideas|publisher=Princeton University Press|year=1960|isbn=0-691-02350-6|pages=3–6|authorlink=Charles Coulston Gillispie}}{{rp|4}} This was later confirmed by Italian scientists Jesuits Grimaldi and Riccioli between 1640 and 1650. They also calculated the magnitude of the Earth's gravity by measuring the oscillations of a pendulum.J. L. Heilbron, Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics (Berkeley, California: University of California Press, 1979), p. 180.

Galileo also broke with incorrect ideas of Aristotelian philosophy by regarding inertia as persistence of motion, not a tendency to come to rest. By considering that the laws of physics appear identical on a moving ship to those on land, Galileo developed the concepts of reference frame and the principle of relativity.{{Cite book |last=Ferraro |first=Rafael |url=https://www.worldcat.org/title/141385334 |title=Einstein's space-time: an introduction to special and general relativity |date=2007 |publisher=Springer |isbn=978-0-387-69946-2 |location=New York |oclc=141385334}}{{rp|5}} These concepts would become central to Newton's mechanics, only to be transformed in Einstein's theory of gravity, the general theory of relativity.{{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}}{{rp|17}}

Johannes Kepler, in his 1609 book Astronomia nova described gravity as a mutual attraction, claiming that if the Earth and Moon were not held apart by some force they would come together. He recognized that mechanical forces cause action, creating a kind of celestial machine. On the other hand Kepler viewed the force of the Sun on the planets as magnetic and acting tangential to their orbits and he assumed with Aristotle that inertia meant objects tend to come to rest.{{Cite journal |last=Holton |first=Gerald |date=1956-05-01 |title=Johannes Kepler's Universe: Its Physics and Metaphysics |url=https://pubs.aip.org/ajp/article/24/5/340/1036024/Johannes-Kepler-s-Universe-Its-Physics-and |journal=American Journal of Physics |language=en |volume=24 |issue=5 |pages=340–351 |doi=10.1119/1.1934225 |bibcode=1956AmJPh..24..340H |issn=0002-9505}}Dijksterhuis, E. J. (1954). History of Gravity and Attraction before Newton. Cahiers d'Histoire Mondiale. Journal of World History. Cuadernos de Historia Mundial, 1(4), 839.{{rp|846}}

In 1666, Giovanni Alfonso Borelli avoided the key problems that limited Kepler. By Borelli's time the concept of inertia had its modern meaning as the tendency of objects to remain in uniform motion and he viewed the Sun as just another heavenly body. Borelli developed the idea of mechanical equilibrium, a balance between inertia and gravity. Newton cited Borelli's influence on his theory.{{rp|848}}

In 1657, Robert Hooke published his Micrographia, in which he hypothesized that the Moon must have its own gravity.{{Cite book |title=Out of the shadow of a giant: Hooke, Halley and the birth of British science |last1=Gribbin |last2=Gribbin |first1= John |first2=Mary |isbn=978-0-00-822059-4 |location=London |oclc=966239842 |year=2017 |publisher=William Collins |author-link=John Gribbin}}{{rp|57}} In a communication to the Royal Society in 1666 and his 1674 Gresham lecture, An Attempt to prove the Annual Motion of the Earth, Hooke took the important step of combining related hypothesis and then forming predictions based on the hypothesis.{{cite book |last=Stewart |first=Dugald |date=1816 |author-link=Dugald Stewart |title=Elements of the Philosophy of the Human Mind |volume= 2 |url=https://archive.org/details/b28041604/page/n5/mode/2up |page=[https://archive.org/details/b28041604/page/434/mode/2up 434] |publisher=Constable & Co; Cadell & Davies |location=Edinburgh; London }} He wrote:

{{blockquote|I will explain a system of the world very different from any yet received. It is founded on the following positions. 1. That all the heavenly bodies have not only a gravitation of their parts to their own proper centre, but that they also mutually attract each other within their spheres of action. 2. That all bodies having a simple motion, will continue to move in a straight line, unless continually deflected from it by some extraneous force, causing them to describe a circle, an ellipse, or some other curve. 3. That this attraction is so much the greater as the bodies are nearer. As to the proportion in which those forces diminish by an increase of distance, I own I have not discovered it....{{cite book |last=Hooke |first=Robert |date=1679 |title=Lectiones Cutlerianae, or A collection of lectures, physical, mechanical, geographical & astronomical : made before the Royal Society on several occasions at Gresham Colledge [i.e. College] : to which are added divers miscellaneous discourses |url=https://archive.org/details/LectionesCutler00Hook/page/n23/mode/2up}}{{sfnp|Hooke|1679|loc= An Attempt to prove the Annual Motion of the Earth, [https://archive.org/details/LectionesCutler00Hook/page/n23/mode/2up page 2, 3]}}}}

Hooke was an important communicator who helped reformulate the scientific enterprise.{{Cite journal |last=Guicciardini |first=Niccolò |date=2020-01-01 |title=On the invisibility and impact of Robert Hooke's theory of gravitation |url=https://www.degruyterbrill.com:443/document/doi/10.1515/opphil-2020-0131/html |journal=Open Philosophy |language=en |volume=3 |issue=1 |pages=266–282 |doi=10.1515/opphil-2020-0131 |issn=2543-8875|hdl=2434/746528 |hdl-access=free }} He was one of the first professional scientists and worked as the then-new Royal Society's curator of experiments for 40 years.{{Cite book |last=Purrington |first=Robert D. |title=The first professional scientist: Robert Hooke and the Royal Society of London |date=2009 |publisher=Birkhäuser |isbn=978-3-0346-0037-8 |series=Science networks. Historical studies |location=Basel, Switzerland Boston}} However his valuable insights remained hypotheses since he was unable to convert them in to a mathematical theory of gravity and work out the consequences.{{rp|853}} For this he turned to Newton, writing him a letter in 1679, outlining a model of planetary motion in a void or vacuum due to attractive action at a distance. This letter likely turned Newton's thinking in a new direction leading to his revolutionary work on gravity. When Newton reported his results in 1686, Hooke claimed the inverse square law portion was his "notion".

{{main|Newton's law of universal gravitation}}

File:Portrait of Sir Isaac Newton, 1689.jpg (1642–1727)]]

Before 1684, scientists including Christopher Wren, Robert Hooke and Edmund Halley determined that Kepler's third law, relating to planetary orbital periods, would prove the inverse square law if the orbits where circles. However the orbits were known to be ellipses. At Halley's suggestion, Newton tackled the problem and was able to prove that ellipses also proved the inverse square relation from Kepler's observations.{{rp|13}} In 1684, Isaac Newton sent a manuscript to Edmond Halley titled De motu corporum in gyrum ('On the motion of bodies in an orbit'), which provided a physical justification for Kepler's laws of planetary motion.{{cite book |last1=Sagan |first1=Carl |url=https://books.google.com/books?id=LhkoowKFaTsC |title=Comet |last2=Druyan |first2=Ann |publisher=Random House |year=1997 |isbn=978-0-3078-0105-0 |location=New York |pages=52–58 |author-link1=Carl Sagan |author-link2=Ann Druyan |access-date=5 August 2021 |archive-url=https://web.archive.org/web/20210615020250/https://books.google.com/books?id=LhkoowKFaTsC |archive-date=15 June 2021 |url-status=live |name-list-style=amp}} Halley was impressed by the manuscript and urged Newton to expand on it, and a few years later Newton published a groundbreaking book called Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy).

The revolutionary aspect of Newton's theory of gravity was the unification of Earth-bound observations of acceleration with celestial mechanics.{{rp|4}} In his book, Newton described gravitation as a universal force, and claimed that it operated on objects "according to the quantity of solid matter which they contain and propagates on all sides to immense distances always at the inverse square of the distances".{{Cite book |last=Newton |first=Isaac |author-link=Isaac Newton |title=The Principia, The Mathematical Principles of Natural Philosophy |date=1999 |publisher=University of California Press |location=Los Angeles |translator-last1=Cohen |translator-first1=I.B. |translator-last2=Whitman |translator-first2=A.}}{{rp|546}} This formulation had two important parts. First was equating inertial mass and gravitational mass. Newton's 2nd law defines force via $F=ma$ for inertial mass, his law of gravitational force uses the same mass. Newton did experiments with pendulums to verify this concept as best he could.{{rp|11}}

The second aspect of Newton's formulation was the inverse square of distance. This aspect was not new: the astronomer Ismaël Bullialdus proposed it around 1640. Seeking proof, Newton made quantitative analysis 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 Earth's gravity acts as if all its mass were concentrated at its center. That proof took him twenty years.{{rp|13}}

Newton's Principia was well received by the scientific community, and his law of gravitation quickly spread across the European world.{{Cite web |title=The Reception of Newton's Principia |url=http://physics.ucsc.edu/~michael/newtonreception6.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://physics.ucsc.edu/~michael/newtonreception6.pdf |archive-date=9 October 2022 |url-status=live |access-date=6 May 2022}} More than a century later, in 1821, his theory of gravitation rose to even greater prominence when it was used to predict the existence of Neptune. In that year, the French astronomer Alexis Bouvard used this theory to create a table modeling the orbit of Uranus, which was shown to differ significantly from the planet's actual trajectory. In order to explain this discrepancy, many astronomers speculated that there might be a large object beyond the orbit of Uranus which was disrupting its orbit. In 1846, the astronomers John Couch Adams and Urbain Le Verrier independently used Newton's law to predict Neptune's location in the night sky, and the planet was discovered there within a day.{{Cite web |title=This Month in Physics History |url=http://www.aps.org/publications/apsnews/202008/history.cfm |access-date=6 May 2022 |website=www.aps.org |language=en |archive-date=6 May 2022 |archive-url=https://web.archive.org/web/20220506231353/https://www.aps.org/publications/apsnews/202008/history.cfm |url-status=live }}{{Cite journal |last=McCrea |first=W. H. |date=1976 |title=The Royal Observatory and the Study of Gravitation |url=https://www.jstor.org/stable/531749 |journal=Notes and Records of the Royal Society of London |volume=30 |issue=2 |pages=133–140 |doi=10.1098/rsnr.1976.0010 |jstor=531749 |issn=0035-9149}}

Newton's formulation was later condensed into the inverse-square law:$$F\; =\; G\; \backslash frac\{m\_1\; m\_2\}\{r^2\},$$where {{mvar|F}} is the force, {{math|m₁}} and {{math|m₂}} 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 Newton's constant, Newton did not use this constant or formula, he only discussed proportionality. But this allowed him to come to an astounding conclusion we take for granted today: the gravity of the Earth on the Moon is the same as the gravity of the Earth on an apple:$$M\_\backslash text\{earth\}\; \backslash propto\; a\_\backslash text\{apple\}R\_\backslash text\{radius\; of\; earth\}^2\; =\; a\_\backslash text\{moon\}R\_\backslash text\{lunar\; orbit\}^2$$Using the values known at the time, Newton was able to verify this form of his law. The value of {{math|G}} was eventually measured by Henry Cavendish in 1797.{{Cite book |last=Zee |first=Anthony |title=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}}{{rp|31}}

{{main| History of general relativity}}

{{General relativity sidebar}}

Eventually, astronomers noticed an eccentricity in the orbit of the planet Mercury which could not be explained by Newton's theory: the perihelion of the orbit was increasing by about 42.98 arcseconds per century. The most obvious explanation for this discrepancy was an as-yet-undiscovered celestial body, such as a planet orbiting the Sun even closer than Mercury, but all efforts to find such a body turned out to be fruitless. In 1915, Albert Einstein developed a theory of general relativity which was able to accurately model Mercury's orbit.{{Cite journal |last=Nobil |first=Anna M. |date=March 1986 |title=The real value of Mercury's perihelion advance |journal=Nature |volume=320 |issue=6057 |pages=39–41 |bibcode=1986Natur.320...39N |doi=10.1038/320039a0 |s2cid=4325839 | issn=0028-0836}}

Einstein's theory brought two other ideas with independent histories into the physical theories of gravity: the principle of relativity and non-Euclidean geometry

The principle of relativity, introduced by Galileo and used as a foundational principle by Newton, lead to a long and fruitless search for a luminiferous aether after Maxwell's equations demonstrated that light propagated at a fixed speed independent of reference frame. In Newton's mechanics, velocities add: a cannon ball shot from a moving ship would travel with a trajectory which included the motion of the ship. Since light speed was fixed, it was assumed to travel in a fixed, absolute medium. Many experiments sought to reveal this medium but failed and in 1905 Einstein's special relativity theory showed the aether was not needed. Special relativity proposed that mechanics be reformulated to use the Lorentz transformation already applicable to light rather than the Galilean transformation adopted by Newton. Special relativity, as in special case, specifically did not cover gravity.{{rp|4}}

While relativity was associated with mechanics and thus gravity, the idea of altering geometry only joined the story of gravity once mechanics required the Lorentz transformations. Geometry was an ancient science that gradually broke free of Euclidean limitations when Carl Gauss discovered in the 1800s that surfaces in any number of dimensions could be characterized by a metric, a distance measurement along the shortest path between two points that reduces to Euclidean distance at infinitesimal separation. Gauss' student Bernhard Riemann developed this into a complete geometry by 1854. These geometries are locally flat but have global curvature.{{rp|4}}

In 1907, Einstein took his first step by using special relativity to create a new form of the equivalence principle. The equivalence of inertial mass and gravitational mass was a known empirical law. The {{mvar|m}} in Newton's first law, $F=ma$, has the same value as the {{mvar|m}} in Newton's law of gravity on Earth, $F=GMm/r^2$. In what he later described as "the happiest thought of my life" Einstein realized this meant that in free-fall, an accelerated coordinate system exists with no local gravitational field.{{Cite web |last1=Webb |first1=Joh |last2=Dougan |first2=Darren |date=23 November 2015 |title=Without Einstein it would have taken decades longer to understand gravity |url=https://phys.org/news/2015-11-einstein-decades-longer-gravity.html#:~:text=In%201907%2C%20Einstein%20had%20the,not%20feel%20his%20own%20weight. |access-date=21 May 2022 |archive-date=21 May 2022 |archive-url=https://web.archive.org/web/20220521182328/https://phys.org/news/2015-11-einstein-decades-longer-gravity.html#:~:text=In%201907%2C%20Einstein%20had%20the,not%20feel%20his%20own%20weight. |url-status=live }} Every description of gravity in any other coordinate system must transform to give no field in the free-fall case, a powerful invariance constraint on all theories of gravity.{{rp|20}}

Einstein's description of gravity was accepted by the majority of physicists for two reasons. First, by 1910 his special relativity was accepted in German physics and was spreading to other countries. Second, his theory explained experimental results like the perihelion of Mercury and the bending of light around the Sun better than Newton's theory.{{Cite journal |last=Brush |first=S. G. |date=1 January 1999 |title=Why was Relativity Accepted? |url=https://ui.adsabs.harvard.edu/abs/1999PhP.....1..184B |journal=Physics in Perspective |volume=1 |issue=2 |pages=184–214 |doi=10.1007/s000160050015 |bibcode=1999PhP.....1..184B |s2cid=51825180 |issn=1422-6944 |access-date=22 May 2022 |archive-date=8 April 2023 |archive-url=https://web.archive.org/web/20230408021700/https://ui.adsabs.harvard.edu/abs/1999PhP.....1..184B |url-status=live }}

In 1919, the British astrophysicist Arthur Eddington was able to confirm the predicted deflection of light during that year's solar eclipse.{{cite journal |last1=Dyson |first1=F. W. |author-link1=Frank Watson Dyson |last2=Eddington |first2=A. S. |author-link2=Arthur Eddington |last3=Davidson |first3=C. R. |date=1920 |title=A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919 |url=https://zenodo.org/record/1432106 |url-status=live |journal=Phil. Trans. Roy. Soc. A |volume=220 |issue=571–581 |pages=291–333 |bibcode=1920RSPTA.220..291D |doi=10.1098/rsta.1920.0009 |archive-url=https://web.archive.org/web/20200515065314/https://zenodo.org/record/1432106 |archive-date=15 May 2020 |access-date=1 July 2019 |doi-access=free}}. Quote, p. 332: "Thus the results of the expeditions to Sobral and Principe can leave little doubt that a deflection of light takes place in the neighbourhood of the sun and that it is of the amount demanded by Einstein's generalised theory of relativity, as attributable to the sun's gravitational field."{{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}}. Quote, p. 192: "About a dozen stars in all were studied, and yielded values 1.98 ± 0.11" and 1.61 ± 0.31", in substantial agreement with Einstein's prediction θ_☉ = 1.75"." Eddington measured starlight deflections twice those predicted by Newtonian corpuscular theory, in accordance with the predictions of general relativity. Although Eddington's analysis was later disputed, this experiment made Einstein famous almost overnight and caused general relativity to become widely accepted in the scientific community.{{Cite journal |last1=Gilmore |first1=Gerard |last2=Tausch-Pebody |first2=Gudrun |date=20 March 2022 |title=The 1919 eclipse results that verified general relativity and their later detractors: a story re-told |journal=Notes and Records: The Royal Society Journal of the History of Science |volume=76 |issue=1 |pages=155–180 |doi=10.1098/rsnr.2020.0040|s2cid=225075861 |doi-access=free |arxiv=2010.13744 }}

In 1959, American physicists Robert Pound and Glen Rebka performed an experiment in which they used gamma rays to confirm the prediction of gravitational time dilation. By sending the rays down a 74-foot tower and measuring their frequency at the bottom, the scientists confirmed that light is Doppler shifted as it moves towards a source of gravity. The observed shift also supports the idea that time runs more slowly in the presence of a gravitational field (many more wave crests pass in a given interval). If light moves outward from a strong source of gravity it will be observed with a redshift.{{Cite web |title=General Astronomy Addendum 10: Graviational Redshift and time dilation |url=https://homepage.physics.uiowa.edu/~rlm/mathcad/addendum%2010%20gravitational%20redshift%20and%20time%20dilation.htm |access-date=29 May 2022 |website=homepage.physics.uiowa.edu |archive-date=14 May 2022 |archive-url=https://web.archive.org/web/20220514063358/https://homepage.physics.uiowa.edu/~rlm/mathcad/addendum%2010%20gravitational%20redshift%20and%20time%20dilation.htm |url-status=live }} The time delay of light passing close to a massive object was first identified by Irwin I. Shapiro in 1964 in interplanetary spacecraft signals.{{Cite journal |last=Asada |first=Hideki |date=20 March 2008 |title=Gravitational time delay of light for various models of modified gravity |url=https://www.sciencedirect.com/science/article/pii/S0370269308001810 |journal=Physics Letters B |volume=661 |issue=2–3 |pages=78–81 |doi=10.1016/j.physletb.2008.02.006 |arxiv=0710.0477 |bibcode=2008PhLB..661...78A |s2cid=118365884 |language=en |access-date=29 May 2022 |archive-date=29 May 2022 |archive-url=https://web.archive.org/web/20220529140019/https://www.sciencedirect.com/science/article/pii/S0370269308001810 |url-status=live }}

In 1971, scientists discovered the first-ever black hole in the galaxy Cygnus. The black hole was detected because it was emitting bursts of x-rays as it consumed a smaller star, and it came to be known as Cygnus X-1.{{Cite web |title=The Fate of the First Black Hole |url=https://www.science.org/content/article/fate-first-black-hole |access-date=30 May 2022 |website=www.science.org |language=en |archive-date=31 May 2022 |archive-url=https://web.archive.org/web/20220531125138/https://www.science.org/content/article/fate-first-black-hole |url-status=live }} This discovery confirmed yet another prediction of general relativity, because Einstein's equations implied that light could not escape from a sufficiently large and compact object.{{Cite web |title=Black Holes Science Mission Directorate |url=https://webarchive.library.unt.edu/web/20170124200640/https://science.nasa.gov/astrophysics/focus-areas/black-holes |access-date=30 May 2022 |website=webarchive.library.unt.edu |archive-date=8 April 2023 |archive-url=https://web.archive.org/web/20230408021657/https://webarchive.library.unt.edu/web/20170124200640/https://science.nasa.gov/astrophysics/focus-areas/black-holes |url-status=live }}

Frame dragging, the idea that a rotating massive object should twist spacetime around it, was confirmed by Gravity Probe B results in 2011.{{cite web |url=http://www.nasa.gov/home/hqnews/2011/may/HQ_11-134_Gravity_Probe_B.html |title=NASA's Gravity Probe B Confirms Two Einstein Space-Time Theories |publisher=Nasa.gov |access-date=23 July 2013 |archive-date=22 May 2013 |archive-url=https://web.archive.org/web/20130522024606/http://www.nasa.gov/home/hqnews/2011/may/HQ_11-134_Gravity_Probe_B.html |url-status=live }}{{Cite web |title="Frame-Dragging" in Local Spacetime |url=https://einstein.stanford.edu/content/education/lithos/litho-fd.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://einstein.stanford.edu/content/education/lithos/litho-fd.pdf |archive-date=9 October 2022 |url-status=live |website=Stanford University}} In 2015, the LIGO observatory detected faint gravitational waves, the existence of which had been predicted by general relativity. Scientists believe that the waves emanated from a black hole merger that occurred 1.5 billion light-years away.{{Cite news |title=Gravitational Waves Detected 100 Years After Einstein's Prediction |url=https://www.ligo.caltech.edu/news/ligo20160211 |access-date=30 May 2022 |newspaper=Ligo Lab | Caltech |archive-date=27 May 2019 |archive-url=https://web.archive.org/web/20190527101043/https://www.ligo.caltech.edu/news/ligo20160211 |url-status=live }}

File:Falling ball.jpg

{{main|Gravity of Earth}}

Every planetary body (including the Earth) is surrounded by its own gravitational field, which can be conceptualized with Newtonian physics as exerting an attractive force on all objects. Assuming a spherically symmetrical planet, the strength of this field at any given point above the surface is proportional to the planetary body's mass and inversely proportional to the square of the distance from the center of the body.

File:Gravity action-reaction.gif

The strength of the gravitational field is numerically equal to the acceleration of objects under its influence.{{cite book |title=Companion to the History of Modern Science |first1=G.N. |last1=Cantor |first2=J.R.R. |last2=Christie |first3=M.J.S. |last3=Hodge |first4=R.C. |last4=Olby |publisher=Routledge |year=2006 |isbn=978-1-134-97751-2 |page=448 |url=https://books.google.com/books?id=gkJn6ciwYZsC&pg=PA448 |access-date=22 October 2017 |archive-date=17 January 2020 |archive-url=https://web.archive.org/web/20200117131121/https://books.google.com/books?id=gkJn6ciwYZsC&pg=PA448 |url-status=live }} The rate of acceleration of falling objects near the Earth's surface varies very slightly depending on latitude, surface features such as mountains and ridges, and perhaps unusually high or low sub-surface densities.{{Cite APOD|date = 15 December 2014|title = The Potsdam Gravity Potato|access-date = }} For purposes of weights and measures, a standard gravity value is defined by the International Bureau of Weights and Measures, under the International System of Units (SI).

The force of gravity experienced by objects on Earth's surface is the vector sum of two forces:{{cite book

|last1 = Hofmann-Wellenhof |first1 = B.

|last2 = Moritz |first2 = H.

|title = Physical Geodesy

|publisher = Springer

|edition = 2nd

|isbn = 978-3-211-33544-4

|year = 2006

|quote = § 2.1: "The total force acting on a body at rest on the earth's surface is the resultant of gravitational force and the centrifugal force of the earth's rotation and is called gravity.

}} (a) The gravitational attraction in accordance with Newton's universal law of gravitation, and (b) the centrifugal force, which results from the choice of an earthbound, rotating frame of reference. The force of gravity is weakest at the equator because of the centrifugal force caused by the Earth's rotation and because points on the equator are farthest from the center of the Earth. The force of gravity varies with latitude, and the resultant acceleration increases from about 9.780 m/s² at the Equator to about 9.832 m/s² at the poles.{{cite conference |last=Boynton |first=Richard |date=2001 |title=Precise Measurement of Mass |book-title=Sawe Paper No. 3147 |publisher=S.A.W.E., Inc. |location=Arlington, Texas |url=http://www.space-electronics.com/Literature/Precise_Measurement_of_Mass.PDF |access-date=22 December 2023 |archive-date=27 February 2007 |archive-url=https://web.archive.org/web/20070227132140/http://www.space-electronics.com/Literature/Precise_Measurement_of_Mass.PDF |url-status=dead }}{{cite web |url=http://curious.astro.cornell.edu/question.php?number=310 |title=Curious About Astronomy? |website= Cornell University |accessdate=22 December 2023 |archive-date=28 July 2013 |archiveurl=https://web.archive.org/web/20130728125707/http://curious.astro.cornell.edu/question.php?number=310}}

{{main|Gravity wave}}

Waves on oceans, lakes, and other bodies of water occur when the gravitational equilibrium at the surface of the water is disturbed by for example wind.{{Cite book |last=Young |first=I. R. |title=Wind generated ocean waves |date=1999 |publisher=Elsevier |isbn=978-0-08-043317-2 |edition=1st |series=Elsevier ocean engineering book series |location=Amsterdam ; New York}} Similar effects occur in the atmosphere where equilibrium is disturbed by thermal weather fronts or mountain ranges.{{Cite journal |last1=Fritts |first1=David C. |last2=Alexander |first2=M. Joan |date=March 2003 |title=Gravity wave dynamics and effects in the middle atmosphere |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2001RG000106 |journal=Reviews of Geophysics |language=en |volume=41 |issue=1 |page=1003 |doi=10.1029/2001RG000106 |bibcode=2003RvGeo..41.1003F |issn=8755-1209}}

{{main|Star formation}}

During star formation, gravitational attraction in a cloud of hydrogen gas competes with thermal gas pressure. As the gas density increases, the temperature rises, then the gas radiates energy, allowing additional gravitational condensation. If the mass of gas in the region is low, the process continues until a brown dwarf or gas-giant planet is produced. If more mass is available, the additional gravitational energy allows the central region to reach pressures sufficient for nuclear fusion, forming a star. In a star, again the gravitational attraction competes, with thermal and radiation pressure in hydrostatic equilibrium until the star's atomic fuel runs out. The next phase depends upon the total mass of the star. Very low mass stars slowly cool as white dwarf stars with a small core balancing gravitational attraction with electron degeneracy pressure. Stars with masses similar to the Sun go through a red giant phase before becoming white dwarf stars. Higher mass stars have complex core structures that burn helium and high atomic number elements ultimately producing an iron core. As their fuel runs out, these stars become unstable producing a supernova. The result can be a neutron star where gravitational attraction balances neutron degeneracy pressure or, for even higher masses, a black hole where gravity operates alone with such intensity that even light cannot escape.{{Cite book |last=Demtröder |first=Wolfgang |title=Astrophysics |date=2024 |chapter=Birth, Lifetime and Death of Stars |series=Undergraduate Lecture Notes in Physics |chapter-url=https://link.springer.com/10.1007/978-3-031-22135-4_5 |access-date=2025-05-04 |publisher=Springer Nature Switzerland |pages=121–175 |language=en |doi=10.1007/978-3-031-22135-4_5|isbn=978-3-031-22133-0 }}{{rp|121}}

{{Main|Gravitational wave}}

File:LIGO Hanford aerial 05.jpg Hanford Observatory located in Washington (state), United States, where gravitational waves were first observed in September 2015]]

General relativity predicts that energy can be transported out of a system through gravitational radiation also known as gravitational waves. The first indirect evidence for gravitational radiation was through measurements of the Hulse–Taylor binary in 1973. This system consists of a pulsar and neutron star in orbit around one another. Its orbital period has decreased since its initial discovery due to a loss of energy, which is consistent for the amount of energy loss due to gravitational radiation. This research was awarded the Nobel Prize in Physics in 1993.{{cite web |title=The Nobel Prize in Physics 1993 |publisher=Nobel Foundation |url=https://www.nobelprize.org/prizes/physics/1993/press-release/ |date=13 October 1993 |quote=for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation |access-date=22 December 2023 |archive-date=10 August 2018 |archive-url=https://web.archive.org/web/20180810182047/https://www.nobelprize.org/nobel_prizes/physics/laureates/1993/press.html |url-status=live }}

The first direct evidence for gravitational radiation was measured on 14 September 2015 by the LIGO detectors. The gravitational waves emitted during the collision of two black holes 1.3 billion light years from Earth were measured.{{Cite web|title = Gravitational waves: scientists announce 'we did it!'{{snd}}live|url = https://www.theguardian.com/science/across-the-universe/live/2016/feb/11/gravitational-wave-announcement-latest-physics-einstein-ligo-black-holes-live|website = the Guardian|date = 11 February 2016|access-date = 11 February 2016|first = Stuart|last = Clark|archive-date = 22 June 2018|archive-url = https://web.archive.org/web/20180622055957/https://www.theguardian.com/science/across-the-universe/live/2016/feb/11/gravitational-wave-announcement-latest-physics-einstein-ligo-black-holes-live|url-status = live}}{{cite journal |title=Einstein's gravitational waves found at last |journal=Nature News |url=http://www.nature.com/news/einstein-s-gravitational-waves-found-at-last-1.19361 |date=11 February 2016 |last1=Castelvecchi |first1=Davide |last2=Witze |first2=Witze |doi=10.1038/nature.2016.19361 |s2cid=182916902 |access-date=11 February 2016 |archive-date=12 February 2016 |archive-url=https://web.archive.org/web/20160212082216/http://www.nature.com/news/einstein-s-gravitational-waves-found-at-last-1.19361 |url-status=live |doi-access=free }} This observation confirms the theoretical predictions of Einstein and others that such waves exist. It also opens the way for practical observation and understanding of the nature of gravity and events in the Universe including the Big Bang.{{cite web|title=WHAT ARE GRAVITATIONAL WAVES AND WHY DO THEY MATTER?|date=13 January 2016 |url=http://www.popsci.com/whats-so-important-about-gravitational-waves|publisher=popsci.com|access-date=12 February 2016|archive-date=3 February 2016|archive-url=https://web.archive.org/web/20160203130600/http://www.popsci.com/whats-so-important-about-gravitational-waves|url-status=live}} Neutron star and black hole formation also create detectable amounts of gravitational radiation.{{cite journal |last1=Abbott |first1=B. P. |display-authors=etal. |collaboration=LIGO Scientific Collaboration & Virgo Collaboration |title=GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral |journal=Physical Review Letters |date=October 2017 |volume=119 |issue=16 |pages=161101 |doi=10.1103/PhysRevLett.119.161101 |pmid=29099225 |doi-access=free |arxiv=1710.05832 |url=http://www.ligo.org/detections/GW170817/paper/GW170817-PRLpublished.pdf |bibcode=2017PhRvL.119p1101A |access-date=28 September 2019 |archive-date=8 August 2018 |archive-url=https://web.archive.org/web/20180808012441/https://www.ligo.org/detections/GW170817/paper/GW170817-PRLpublished.pdf |url-status=live }} This research was awarded the Nobel Prize in Physics in 2017.{{cite web|title=Nobel prize in physics awarded for discovery of gravitational waves|url=https://www.theguardian.com/science/2017/oct/03/nobel-prize-physics-discovery-gravitational-waves-ligo|website=the Guardian|date=3 October 2017|access-date=3 October 2017|last1=Devlin|first1=Hanna|archive-date=3 October 2017|archive-url=https://web.archive.org/web/20171003102211/https://www.theguardian.com/science/2017/oct/03/nobel-prize-physics-discovery-gravitational-waves-ligo|url-status=live}}

{{main|Dark matter}}

At the cosmological scale, gravity is a dominant player. About 5/6 of the total mass in the universe consists of dark matter which interacts through gravity but not through electromagnetic interactions. The gravitation of clumps of dark matter known as dark matter halos attract hydrogen gas leading to stars and galaxies.{{Cite journal |last1=Wechsler |first1=Risa H. |last2=Tinker |first2=Jeremy L. |date=2018-09-14 |title=The Connection Between Galaxies and Their Dark Matter Halos |url=https://www.annualreviews.org/doi/10.1146/annurev-astro-081817-051756 |journal=Annual Review of Astronomy and Astrophysics |language=en |volume=56 |issue=1 |pages=435–487 |doi=10.1146/annurev-astro-081817-051756 |issn=0066-4146|arxiv=1804.03097 }}

{{main|Gravitational lensing}}

File:Einstein cross.jpg, four images of the same distant quasar around a foreground galaxy due to gravitational lensing – a single quasar is actually hidden behind a massive foreground object (a galaxy in this case)]]

Gravity acts on light and matter equally, meaning that a sufficiently massive object could warp light around it and create a gravitational lens. This phenomenon was first confirmed by observation in 1979 using the 2.1 meter telescope at Kitt Peak National Observatory in Arizona, which saw two mirror images of the same quasar whose light had been bent around the galaxy YGKOW G1.{{cite book |author1=Kar |first=Subal |url=https://books.google.com/books?id=IWFkEAAAQBAJ |title=Physics and Astrophysics: Glimpses of the Progress |publisher=CRC Press |year=2022 |isbn=978-1-000-55926-2 |edition=illustrated |page=106}} [https://books.google.com/books?id=IWFkEAAAQBAJ&pg=PT106 Extract of page 106].{{Cite web |title=Hubble, Hubble, Seeing Double! |url=https://www.nasa.gov/content/goddard/hubble-hubble-seeing-double/#.YpZyvYOZrRl |access-date=31 May 2022 |website=NASA |date=24 January 2014 |archive-date=25 May 2022 |archive-url=https://web.archive.org/web/20220525041837/https://www.nasa.gov/content/goddard/hubble-hubble-seeing-double/#.YpZyvYOZrRl |url-status=live }}

Many subsequent observations of gravitational lensing provide additional evidence for substantial amounts of dark matter around galaxies. Gravitational lenses do not focus like eyeglass lenses, but rather lead to annular shapes called Einstein rings.{{rp|370}}

{{Main|Speed of gravity}}

In December 2012, a research team in China announced that it had produced measurements of the phase lag of Earth tides during full and new moons which seem to prove that the speed of gravity is equal to the speed of light.[http://www.astrowatch.net/2012/12/chinese-scientists-find-evidence-for.html Chinese scientists find evidence for speed of gravity] {{Webarchive|url=https://web.archive.org/web/20130108083729/http://www.astrowatch.net/2012/12/chinese-scientists-find-evidence-for.html |date=8 January 2013 }}, astrowatch.com, 12/28/12. This means that if the Sun suddenly disappeared, the Earth would keep orbiting the vacant point normally for 8 minutes, which is the time light takes to travel that distance. The team's findings were released in Science Bulletin in February 2013.{{cite journal|last=TANG|first=Ke Yun|author2=HUA ChangCai |author3=WEN Wu |author4=CHI ShunLiang |author5=YOU QingYu |author6=YU Dan |title=Observational evidences for the speed of the gravity based on the Earth tide|journal=Chinese Science Bulletin|date=February 2013|volume=58|issue=4–5|pages=474–477|doi=10.1007/s11434-012-5603-3|bibcode=2013ChSBu..58..474T|doi-access=free}}

In October 2017, the LIGO and Virgo interferometer detectors received gravitational wave signals within 2 seconds of gamma ray satellites and optical telescopes seeing signals from the same direction. This confirmed that the speed of gravitational waves was the same as the speed of light.{{cite web|url=https://www.ligo.caltech.edu/page/press-release-gw170817|title=GW170817 Press Release|website=LIGO Lab – Caltech|access-date=24 October 2017|archive-date=17 October 2017|archive-url=https://web.archive.org/web/20171017010137/https://www.ligo.caltech.edu/page/press-release-gw170817|url-status=live}}

{{distinguish|Gravity anomaly}}

There are some observations that are not adequately accounted for, which may point to the need for better theories of gravity or perhaps be explained in other ways.

File:GalacticRotation2.svg.]]

• Galaxy rotation curves: Stars in galaxies follow a distribution of velocities where stars on the outskirts are moving faster than they should according to the observed distributions of luminous matter. Galaxies within galaxy clusters show a similar pattern. The pattern is considered strong evidence for dark matter, which would interact through gravitation but not electromagnetically; various modifications to Newtonian dynamics have also been proposed.{{Cite journal |last1=Sofue |first1=Yoshiaki |last2=Rubin |first2=Vera |date=2001-09-01 |title=Rotation Curves of Spiral Galaxies |url=https://www.annualreviews.org/content/journals/10.1146/annurev.astro.39.1.137 |journal=Annual Review of Astronomy and Astrophysics |language=en |volume=39 |pages=137–174 |doi=10.1146/annurev.astro.39.1.137 |issn=0066-4146|arxiv=astro-ph/0010594 |bibcode=2001ARA&A..39..137S }}
• Accelerated expansion: The expansion of the universe seems to be accelerating.{{Cite web |title=The Nobel Prize in Physics 2011 : Adam G. Riess Facts |url=https://www.nobelprize.org/prizes/physics/2011/riess/facts/ |access-date=19 March 2024 |website=NobelPrize.org |language=en-US |archive-date=28 May 2020 |archive-url=https://web.archive.org/web/20200528014511/https://www.nobelprize.org/prizes/physics/2011/riess/facts/ |url-status=live }} Dark energy has been proposed to explain this.{{Cite web |title=What is Dark Energy? Inside our accelerating, expanding Universe |url=https://science.nasa.gov/universe/the-universe-is-expanding-faster-these-days-and-dark-energy-is-responsible-so-what-is-dark-energy/ |access-date=19 March 2024 |website=science.nasa.gov |date=5 February 2024 |language=en |archive-date=19 March 2024 |archive-url=https://web.archive.org/web/20240319153930/https://science.nasa.gov/universe/the-universe-is-expanding-faster-these-days-and-dark-energy-is-responsible-so-what-is-dark-energy/ |url-status=live }}
• Flyby anomaly: Various spacecraft have experienced greater acceleration than expected during gravity assist maneuvers.{{Cite journal |last1=Anderson |first1=John D. |last2=Campbell |first2=James K. |last3=Ekelund |first3=John E. |last4=Ellis |first4=Jordan |last5=Jordan |first5=James F. |date=3 March 2008 |title=Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth |url=https://link.aps.org/doi/10.1103/PhysRevLett.100.091102 |journal=Physical Review Letters |language=en |volume=100 |issue=9 |page=091102 |doi=10.1103/PhysRevLett.100.091102 |pmid=18352689 |bibcode=2008PhRvL.100i1102A |issn=0031-9007}} The Pioneer anomaly has been shown to be explained by thermal recoil due to the distant sun radiation on one side of the space craft.{{Cite journal |last1=Turyshev |first1=Slava G. |last2=Toth |first2=Viktor T. |last3=Kinsella |first3=Gary |last4=Lee |first4=Siu-Chun |last5=Lok |first5=Shing M. |last6=Ellis |first6=Jordan |date=12 June 2012 |title=Support for the Thermal Origin of the Pioneer Anomaly |url=https://link.aps.org/doi/10.1103/PhysRevLett.108.241101 |journal=Physical Review Letters |volume=108 |issue=24 |pages=241101 |doi=10.1103/PhysRevLett.108.241101|pmid=23004253 |arxiv=1204.2507 |bibcode=2012PhRvL.108x1101T }}{{Cite journal |last=Iorio |first=Lorenzo |date=May 2015 |title=Gravitational anomalies in the solar system? |url=https://www.worldscientific.com/doi/abs/10.1142/S0218271815300153 |journal=International Journal of Modern Physics D |language=en |volume=24 |issue=6 |pages=1530015–1530343 |doi=10.1142/S0218271815300153 |issn=0218-2718|arxiv=1412.7673 |bibcode=2015IJMPD..2430015I }}

{{see also | Introduction to general relativity}}

In modern physics, general relativity is considered the most successful theory of gravitation.{{Cite book |last=Stephani |first=Hans |title=Exact Solutions to Einstein's Field Equations |year=2003 |isbn=978-0-521-46136-8 |pages=1 |publisher=Cambridge University Press |language=en}} Physicists continue to work to find solutions to the Einstein field equations that form the basis of general relativity and continue to test the theory, finding excellent agreement in all cases.{{cite web

| title = Einstein's general relativity theory is questioned but still stands for now

| work = Science News

| publisher = Science Daily

| date = 25 July 2019

| url = https://www.sciencedaily.com/releases/2019/07/190725150408.htm

| doi =

| accessdate = 11 August 2024}}{{cite web

| last = Lea

| first = Robert

| title = Einstein's greatest theory just passed its most rigorous test yet

| website = Scientific American

| publisher = Springer Nature America, Inc.

| date = 15 September 2022

| url = https://www.scientificamerican.com/article/einsteins-greatest-theory-just-passed-its-most-rigorous-test-yet/

| format =

| doi =

| accessdate = 11 August 2024}}{{rp|p.9}}

Any theory of gravity must conform to the requirements of special relativity and experimental observations. Newton's theory of gravity assumes action at a distance and therefore cannot be reconciled with special relativity. The simplest generalization of Newton's approach would be a scalar theory with the gravitational potential represented by a single number in a 4 dimensional spacetime. However this type of theory fails to predict gravitational redshift or the deviation of light by matter and gives values for the precession of Mercury which are incorrect. A vector field theory predicts negative energy gravitational waves so it also fails. Furthermore, no theory without curvature in spacetime can be consistent with special relativity. The simplest theory consistent with special relativity and the well-studied observations is general relativity.{{Cite journal |last1=Debono |first1=Ivan |last2=Smoot |first2=George |date=2016-09-28 |title=General Relativity and Cosmology: Unsolved Questions and Future Directions |journal=Universe |language=en |volume=2 |issue=4 |pages=23 |doi=10.3390/universe2040023 |doi-access=free |arxiv=1609.09781 |bibcode=2016Univ....2...23D |issn=2218-1997}}

Unlike Newton's formula with one parameter, {{math|G}}, force in general relativity is terms of 10 numbers formed in to a metric tensor.{{rp|70}}In general relativity the effects of gravitation are described in different ways in different frames of reference. In a free-falling or co-moving coordinate system, an object travels in a straight line. In other coordinate systems, the object accelerates and thus is seen to move under a force. The path in spacetime (not 3D space) taken by a free-falling object is called a geodesic and the length of that path as measured by time in the objects frame is the shortest (or rarely the longest) one. Consequently the effect of gravity can be described as curving spacetime. In a weak stationary gravitational field, general relativity reduces to Newton's equations. The corrections introduced by general relativity on Earth are on the order of 1 part in a billion.{{rp|77}}

{{main| Einstein field equations}}

The Einstein field equations are a system of 10 partial differential equations which describe how matter affects the curvature of spacetime. The system is may be expressed in the form

$$G\_\{\backslash mu\; \backslash nu\}\; +\; \backslash Lambda\; g\_\{\backslash mu\; \backslash nu\}\; =\; \backslash kappa\; T\_\{\backslash mu\; \backslash nu\},$$

where {{mvar|G{{sub|μν}}}} is the Einstein tensor, {{mvar|g{{sub|μν}}}} is the metric tensor, {{mvar|T{{sub|μν}}}} is the stress–energy tensor, {{math|Λ}} is the cosmological constant, $G$ is the Newtonian constant of gravitation and $c$ is the speed of light.{{Cite web |title=Einstein Field Equations (General Relativity) |url=https://warwick.ac.uk/fac/sci/physics/intranet/pendulum/generalrelativity/ |access-date=24 May 2022 |website=University of Warwick |language=en |archive-date=25 May 2022 |archive-url=https://web.archive.org/web/20220525140036/https://warwick.ac.uk/fac/sci/physics/intranet/pendulum/generalrelativity/ |url-status=live }} The constant $\backslash kappa\; =\; \backslash frac\{8\backslash pi\; G\}\{c^4\}$ is referred to as the Einstein gravitational constant.{{Cite web |title=How to understand Einstein's equation for general relativity |url=https://bigthink.com/starts-with-a-bang/einstein-general-theory-relativity-equation/ |access-date=24 May 2022 |website=Big Think |date=15 September 2021 |language=en-US |archive-date=26 May 2022 |archive-url=https://web.archive.org/web/20220526023430/https://bigthink.com/starts-with-a-bang/einstein-general-theory-relativity-equation/ |url-status=live }}

{{main|Solutions of the Einstein field equations}}

The non-linear second-order Einstein field equations are extremely complex and have been solved in only a few special cases.{{Cite web |last=Siegel |first=Ethan |title=This Is Why Scientists Will Never Exactly Solve General Relativity |url=https://www.forbes.com/sites/startswithabang/2019/12/04/this-is-why-scientists-will-never-exactly-solve-general-relativity/ |access-date=27 May 2022 |website=Forbes |language=en |archive-date=27 May 2022 |archive-url=https://web.archive.org/web/20220527212804/https://www.forbes.com/sites/startswithabang/2019/12/04/this-is-why-scientists-will-never-exactly-solve-general-relativity/ |url-status=live }} These cases however has been transformational in our understanding of the cosmos. Several solutions are the basis for understanding black holes and for our modern model of the evolution of the universe since the Big Bang.{{Cite book |author=Longair |first=Malcolm S. |author-link=Malcolm Longair |url=http://link.springer.com/10.1007/978-3-540-73478-9 |title=Galaxy Formation |date=2008 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-73477-2 |series=Astronomy and Astrophysics Library |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-540-73478-9}}{{rp|227}}

{{main | Tests of general relativity}}

File:1919 eclipse positive.jpg provided one of the first opportunities to test the predictions of general relativity.]]

Testing the predictions of general relativity has historically been difficult, because they are almost identical to the predictions of Newtonian gravity for small energies and masses.{{Cite web |title=Testing General Relativity |url=https://asd.gsfc.nasa.gov/blueshift/index.php/2015/11/27/testing-general-relativity/ |access-date=29 May 2022 |website=NASA Blueshift |language=en-US |archive-date=16 May 2022 |archive-url=https://web.archive.org/web/20220516115115/https://asd.gsfc.nasa.gov/blueshift/index.php/2015/11/27/testing-general-relativity/ |url-status=live }} A wide range of experiments provided support of general relativity.{{cite book

| last = Will

| first = Clifford M.

| title = Theory and Experiment in Gravitational Physics

| publisher = Cambridge Univ. Press

| date = 2018

| location =

| language =

| url = https://books.google.com/books?id=gf1uDwAAQBAJ

| archive-url=

| archive-date=

| doi =

| id =

| isbn = 9781107117440

| mr =

| zbl =

| jfm =}}{{rp|p.1–9}}{{Cite journal |last=Lindley |first=David |date=12 July 2005 |title=The Weight of Light |url=https://physics.aps.org/story/v16/st1 |journal=Physics |language=en |volume=16 |access-date=22 May 2022 |archive-date=25 May 2022 |archive-url=https://web.archive.org/web/20220525201415/https://physics.aps.org/story/v16/st1 |url-status=live }}{{Cite web |title=Hafele-Keating Experiment |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/airtim.html |access-date=22 May 2022 |website=hyperphysics.phy-astr.gsu.edu |archive-date=18 April 2017 |archive-url=https://web.archive.org/web/20170418005731/http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/airtim.html |url-status=live }}{{Cite web |title=How the 1919 Solar Eclipse Made Einstein the World's Most Famous Scientist |url=https://www.discovermagazine.com/the-sciences/how-the-1919-solar-eclipse-made-einstein-the-worlds-most-famous-scientist |access-date=22 May 2022 |website=Discover Magazine |language=en |archive-date=22 May 2022 |archive-url=https://web.archive.org/web/20220522141013/https://www.discovermagazine.com/the-sciences/how-the-1919-solar-eclipse-made-einstein-the-worlds-most-famous-scientist |url-status=live }}{{Cite web |title=At Long Last, Gravity Probe B Satellite Proves Einstein Right |url=https://www.science.org/content/article/long-last-gravity-probe-b-satellite-proves-einstein-right |access-date=22 May 2022 |website=www.science.org |language=en |archive-date=22 May 2022 |archive-url=https://web.archive.org/web/20220522141013/https://www.science.org/content/article/long-last-gravity-probe-b-satellite-proves-einstein-right |url-status=live }} Today, Einstein's theory of relativity is used for all gravitational calculations where absolute precision is desired, although Newton's inverse-square law is accurate enough for virtually all ordinary calculations.{{rp|79}}{{cite book

| last = Hassani

| first = Sadri

| title = From Atoms to Galaxies: A conceptual physics approach to scientific awareness

| publisher = CRC Press

| date = 2010

| location =

| pages = 131

| language =

| url = https://books.google.com/books?id=oypZ_a9pqdsC&pg=PA131

| archive-url=

| archive-date=

| doi =

| id =

| isbn = 9781439808504

| mr =

| zbl =

| jfm =}}

{{Main|Graviton|Quantum gravity}}

Despite its success in predicting the effects of gravity at large scales, general relativity is ultimately incompatible with quantum mechanics. This is because general relativity describes gravity as a smooth, continuous distortion of spacetime, while quantum mechanics holds that all forces arise from the exchange of discrete particles known as quanta. This contradiction is especially vexing to physicists because the other three fundamental forces (strong force, weak force and electromagnetism) were reconciled with a quantum framework decades ago.{{Cite web |title=Gravity Probe B – Special & General Relativity Questions and Answers |url=https://einstein.stanford.edu/content/relativity/a11758.html#:~:text=Quantum%20mechanics%20is%20incompatible%20with,exchange%20of%20well-defined%20quanta. |access-date=1 August 2022 |website=einstein.stanford.edu |archive-date=6 June 2022 |archive-url=https://web.archive.org/web/20220606161408/https://einstein.stanford.edu/content/relativity/a11758.html#:~:text=Quantum%20mechanics%20is%20incompatible%20with,exchange%20of%20well-defined%20quanta. |url-status=live }} As a result, researchers have begun to search for a theory that could unite both gravity and quantum mechanics under a more general framework.{{Cite book |last1=Huggett |first1=Nick |title=Beyond Spacetime: The Foundations of Quantum Gravity |last2=Matsubara |first2=Keizo |last3=Wüthrich |first3=Christian |publisher=Cambridge University Press |year=2020 |isbn=9781108655705 |pages=6 |language=en}}

One path is to describe gravity in the framework of quantum field theory (QFT), which has been successful to accurately describe the other fundamental interactions. The electromagnetic force arises from an exchange of virtual photons, where the QFT description of gravity is that there is an exchange of virtual gravitons.{{cite book |last1=Feynman |first1=R. P. |url=https://archive.org/details/feynmanlectureso0000feyn_g4q1 |title=Feynman lectures on gravitation |last2=Morinigo |first2=F. B. |last3=Wagner |first3=W. G. |last4=Hatfield |first4=B. |date=1995 |publisher=Addison-Wesley |isbn=978-0-201-62734-3 |url-access=registration}}{{cite book | author=Zee, A. |title=Quantum Field Theory in a Nutshell | publisher = Princeton University Press | date=2003 | isbn=978-0-691-01019-9}} This description reproduces general relativity in the classical limit. However, this approach fails at short distances of the order of the Planck length,{{cite book | author=Randall, Lisa | title=Warped Passages: Unraveling the Universe's Hidden Dimensions | publisher=Ecco | date=2005 | isbn=978-0-06-053108-9 | url=https://archive.org/details/warpedpassagesun00rand_1 }} where a more complete theory of quantum gravity (or a new approach to quantum mechanics) is required.

{{Main|Alternatives to general relativity}}

General relativity has withstood many tests over a large range of mass and size scales.{{cite journal | last=Will | first=Clifford M. | title=The Confrontation between General Relativity and Experiment | journal=Living Reviews in Relativity | volume=17 | issue=1 | date=2014-12-01 | issn=2367-3613 | doi=10.12942/lrr-2014-4 | pages=4 | pmid=28179848 | pmc=5255900 | arxiv=1403.7377 | bibcode=2014LRR....17....4W | doi-access=free}}{{cite arXiv |eprint=1705.04397v1|last1= Asmodelle|first1= E.|title= Tests of General Relativity: A Review|class= physics.class-ph|year= 2017}} When applied to interpret astronomical observations, cosmological models based on general relativity introduce two components to the universe,{{cite book | last=Ryden | first=Barbara Sue | title=Introduction to cosmology | publisher=Cambridge University Press | publication-place=Cambridge | date=2017 | isbn=978-1-316-65108-7 | page=}} dark matter{{cite journal | last1=Garrett | first1=Katherine | last2=Duda | first2=Gintaras | title=Dark Matter: A Primer | journal=Advances in Astronomy | volume=2011 | date=2011 | issn=1687-7969 | doi=10.1155/2011/968283 | doi-access=free | pages=1–22| arxiv=1006.2483 | bibcode=2011AdAst2011E...8G }} and dark energy,{{cite journal | last1=Li | first1=Miao | last2=Li | first2=Xiao-Dong | last3=Wang | first3=Shuang | last4=Wang | first4=Yi | title=Dark energy: A brief review | journal=Frontiers of Physics | volume=8 | issue=6 | date=2013 | issn=2095-0462 | doi=10.1007/s11467-013-0300-5 | pages=828–846| arxiv=1209.0922 | bibcode=2013FrPhy...8..828L }} the nature of which is currently an unsolved problem in physics. The many successful, high precision predictions of the standard model of cosmology has led astrophysicists to conclude it and thus general relativity will be the basis for future progress.{{Cite journal |last=Turner |first=Michael S. |date=2022-09-26 |title=The Road to Precision Cosmology |url=https://www.annualreviews.org/content/journals/10.1146/annurev-nucl-111119-041046 |journal=Annual Review of Nuclear and Particle Science |language=en |volume=72 |issue=2022 |pages=1–35 |doi=10.1146/annurev-nucl-111119-041046 |issn=0163-8998|arxiv=2201.04741 |bibcode=2022ARNPS..72....1T }}{{Cite journal |last1=Abdalla |first1=Elcio |last2=Abellán |first2=Guillermo Franco |last3=Aboubrahim |first3=Amin |last4=Agnello |first4=Adriano |last5=Akarsu |first5=Özgür |last6=Akrami |first6=Yashar |last7=Alestas |first7=George |last8=Aloni |first8=Daniel |last9=Amendola |first9=Luca |last10=Anchordoqui |first10=Luis A. |last11=Anderson |first11=Richard I. |last12=Arendse |first12=Nikki |last13=Asgari |first13=Marika |last14=Ballardini |first14=Mario |last15=Barger |first15=Vernon |date=2022-06-01 |title=Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies |url=https://linkinghub.elsevier.com/retrieve/pii/S2214404822000179 |journal=Journal of High Energy Astrophysics |volume=34 |pages=49–211 |doi=10.1016/j.jheap.2022.04.002 |issn=2214-4048|arxiv=2203.06142 |bibcode=2022JHEAp..34...49A }} However, dark matter is not supported by the standard model of particle physics, physical models for dark energy do not match cosmological data, and some cosmological observations are inconsistent. These issues have led to the study of alternative theories of gravity.{{cite web |author=Cooper |first=Keith |date=6 February 2024 |title=Cosmic combat: delving into the battle between dark matter and modified gravity |url=https://physicsworld.com/a/cosmic-combat-delving-into-the-battle-between-dark-matter-and-modified-gravity |publisher=physicsworld}}

{{cols|colwidth=30em}}

• {{Annotated link |Anti-gravity}}
• {{Annotated link |Artificial gravity}}
• {{Annotated link |Equations for a falling body}}
• {{Annotated link |Escape velocity}}
• {{Annotated link |Atmospheric escape}}
• {{Annotated link |Gauss's law for gravity}}
• {{Annotated link |Gravitational potential}}
• {{Annotated link |Gravitational biology}}
• {{Annotated link |Newton's laws of motion}}
• {{Annotated link |Standard gravitational parameter}}
• {{Annotated link |Weightlessness}}

{{colend}}

{{clear}}

{{Reflist|30em}}

• {{cite book |first=Isaac |author-link=Isaac Newton |last=Newton |translator=Cohen |translator-first=I. Bernard |title=The Principia : mathematical principles of natural philosophy |contribution=A Guide to Newton's Principia |contributor=I. Bernard Cohen |publisher=University of California Press |date=1999 |orig-date=1687 |isbn=9780520088160 |oclc=313895715}}
• {{cite book |last1=Halliday |first1=David |author-link1=David Halliday (physicist) |last2=Resnick |first2=Robert |last3=Krane |first3=Kenneth S. |title=Physics v. 1 |location=New York |publisher=John Wiley & Sons |date=2001 |isbn=978-0-471-32057-9}}
• {{cite book | last = Serway | first = Raymond A. | author2 = Jewett, John W. | title = Physics for Scientists and Engineers | edition = 6th | publisher = Brooks/Cole | date = 2004 | isbn = 978-0-534-40842-8 | url = https://archive.org/details/physicssciengv2p00serw }}
• {{cite book | last = Tipler | first = Paul | title = Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics | edition = 5th | publisher = W.H. Freeman | date = 2004 | isbn = 978-0-7167-0809-4 }}
• {{cite book |author=Thorne |first1=Kip S. |author-link1=Kip Thorne |last2=Misner |first2=Charles W. |author-link2=Charles W. Misner |last3=Wheeler |first3=John Archibald |author-link3=John Archibald Wheeler |title=Gravitation |publisher=W.H. Freeman |date=1973 |isbn=978-0-7167-0344-0}}
• {{cite news

|title=Everything you thought you knew about gravity is wrong

|first=Richard

|last=Panek

|date=2 August 2019

|newspaper=The Washington Post

|url=https://www.washingtonpost.com/outlook/everything-you-thought-you-knew-about-gravity-is-wrong/2019/08/01/627f3696-a723-11e9-a3a6-ab670962db05_story.html}}

{{sister project links|d=y|wikt=gravity|v=Gravitation|b=Physics Study Guide/Gravity|s=1911 Encyclopædia Britannica/Gravitation|c=category:Gravitation|n=no|q=Gravity|m=no|mw=no|species=no}}

• [https://feynmanlectures.caltech.edu/I_07.html The Feynman Lectures on Physics Vol. I Ch. 7: The Theory of Gravitation]
• {{springer|title=Gravitation|id=p/g045040}}
• {{springer|title=Gravitation, theory of|id=p/g045050}}

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