Hammer and feather drop: astronaut DavidScott (from mission Apollo 15) on theMoon enacting the legend of Galileo'sgravity experiment. (1.38 MB, ogg/Theoraformat).

Gravity

Gravity (from Latin gravitas 'weight'[1]), or gravitation, is anatural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light[2]—are broughttoward (or gravitate toward) one another. On Earth, gravity givesweight to physical objects, and the Moon's gravity causes theocean tides. The gravitational attraction of the original gaseousmatter present in the Universe caused it to begin coalescing andforming stars and caused the stars to group together into galaxies,so gravity is responsible for many of the large-scale structures inthe Universe. Gravity has an infinite range, although its effectsbecome increasingly weaker as objects get further away.

Gravity is most accurately described by the general theory ofrelativity (proposed by Albert Einstein in 1915), which describesgravity not as a force, but as a consequence of masses movingalong geodesic lines in a curved spacetime caused by the unevendistribution of mass. The most extreme example of this curvatureof spacetime is a black hole, from which nothing—not even light—can escape once past the black hole's event horizon.[3] However, for most applications, gravity is wellapproximated by Newton's law of universal gravitation, which describes gravity as a force causing any twobodies to be attracted toward each other, with magnitude proportional to the product of their masses andinversely proportional to the square of the distance between them.

Gravity is the weakest of the four fundamental interactions of physics, approximately 1038 times weaker thanthe strong interaction, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weakinteraction. As a consequence, it has no significant influence at the level of subatomic particles.[4] In contrast,it is the dominant interaction at the macroscopic scale, and is the cause of the formation, shape and trajectory(orbit) of astronomical bodies.

Current models of particle physics imply that the earliest instance of gravity in the Universe, possibly in theform of quantum gravity, supergravity or a gravitational singularity, along with ordinary space and time,developed during the Planck epoch (up to 10−43 seconds after the birth of the Universe), possibly from aprimeval state, such as a false vacuum, quantum vacuum or virtual particle, in a currently unknown manner.[5]

Attempts to develop a theory of gravity consistent with quantum mechanics, a quantum gravity theory, whichwould allow gravity to be united in a common mathematical framework (a theory of everything) with the otherthree fundamental interactions of physics, are a current area of research.

History of gravitational theoryAncient worldScientific revolutionNewton's theory of gravitationEquivalence principle

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Contents

General relativitySolutionsTests

Gravity and quantum mechanics

SpecificsEarth's gravityEquations for a falling body near the surface of the EarthGravity and astronomyGravitational radiationSpeed of gravity

Anomalies and discrepanciesAlternative theories

Historical alternative theoriesModern alternative theories

See alsoFootnotesReferencesFurther readingExternal links

The ancient Greek philosopher Archimedes discovered the center of gravity of a triangle.[6] He also postulatedthat if two equal weights did not have the same center of gravity, the center of gravity of the two weightstogether would be in the middle of the line that joins their centers of gravity.[7]

The Roman architect and engineer Vitruvius in De Architectura postulated that gravity of an object did notdepend on weight but its "nature".[8]

In ancient India, Aryabhata first identified the force to explain why objects are not thrown outward as the earthrotates. Brahmagupta described gravity as an attractive force and used the term "gurutvaakarshan" forgravity.[9][10]

Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and early 17thcenturies. In his famous (though possibly apocryphal[11]) experiment dropping balls from the Tower of Pisa,and later with careful measurements of balls rolling down inclines, Galileo showed that gravitationalacceleration is the same for all objects. This was a major departure from Aristotle's belief that heavier objectshave a higher gravitational acceleration.[12] Galileo postulated air resistance as the reason that objects with lessmass fall more slowly in an atmosphere. Galileo's work set the stage for the formulation of Newton's theory ofgravity.[13]

History of gravitational theory

Ancient world

Scientific revolution

English physicist andmathematician, Sir Isaac Newton(1642–1727)

In 1687, English mathematician Sir Isaac Newton published Principia,which hypothesizes the inverse-square law of universal gravitation. In hisown words, "I deduced that the forces which keep the planets in theirorbs must [be] reciprocally as the squares of their distances from thecenters about which they revolve: and thereby compared the forcerequisite to keep the Moon in her Orb with the force of gravity at thesurface of the Earth; and found them answer pretty nearly."[14] Theequation is the following:

Where F is the force, m1 and m2 are the masses of the objects interacting,r is the distance between the centers of the masses and G is thegravitational constant.

Newton's theory enjoyed its greatest success when it was used to predictthe existence of Neptune based on motions of Uranus that could not be

accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain LeVerrier predicted the general position of the planet, and Le Verrier's calculations are what led Johann GottfriedGalle to the discovery of Neptune.

A discrepancy in Mercury's orbit pointed out flaws in Newton's theory. By the end of the 19th century, it wasknown that its orbit showed slight perturbations that could not be accounted for entirely under Newton'stheory, but all searches for another perturbing body (such as a planet orbiting the Sun even closer thanMercury) had been fruitless. The issue was resolved in 1915 by Albert Einstein's new theory of generalrelativity, which accounted for the small discrepancy in Mercury's orbit. This discrepancy was the advance inthe perihelion of Mercury of 42.98 arcseconds per century.[15]

Although Newton's theory has been superseded by Albert Einstein's general relativity, most modern non-relativistic gravitational calculations are still made using Newton's theory because it is simpler to work withand it gives sufficiently accurate results for most applications involving sufficiently small masses, speeds andenergies.

The equivalence principle, explored by a succession of researchers including Galileo, Loránd Eötvös, andEinstein, expresses the idea that all objects fall in the same way, and that the effects of gravity areindistinguishable from certain aspects of acceleration and deceleration. The simplest way to test the weakequivalence principle is to drop two objects of different masses or compositions in a vacuum and see whetherthey hit the ground at the same time. Such experiments demonstrate that all objects fall at the same rate whenother forces (such as air resistance and electromagnetic effects) are negligible. More sophisticated tests use atorsion balance of a type invented by Eötvös. Satellite experiments, for example STEP, are planned for moreaccurate experiments in space.[16]

Formulations of the equivalence principle include:

The weak equivalence principle: The trajectory of a point mass in a gravitational field dependsonly on its initial position and velocity, and is independent of its composition.[17]

Newton's theory of gravitation

Equivalence principle

Two-dimensional analogy of spacetime distortiongenerated by the mass of an object. Matterchanges the geometry of spacetime, this (curved)geometry being interpreted as gravity. White linesdo not represent the curvature of space but insteadrepresent the coordinate system imposed on thecurved spacetime, which would be rectilinear in aflat spacetime.

The Einsteinian equivalence principle: The outcome of any local non-gravitational experimentin a freely falling laboratory is independent of the velocity of the laboratory and its location inspacetime.[18]

The strong equivalence principle requiring both of the above.

In general relativity, the effects of gravitation are ascribedto spacetime curvature instead of a force. The startingpoint for general relativity is the equivalence principle,which equates free fall with inertial motion and describesfree-falling inertial objects as being accelerated relative tonon-inertial observers on the ground.[19][20] InNewtonian physics, however, no such acceleration canoccur unless at least one of the objects is being operatedon by a force.

Einstein proposed that spacetime is curved by matter, andthat free-falling objects are moving along locally straightpaths in curved spacetime. These straight paths are calledgeodesics. Like Newton's first law of motion, Einstein'stheory states that if a force is applied on an object, itwould deviate from a geodesic. For instance, we are nolonger following geodesics while standing because themechanical resistance of the Earth exerts an upward forceon us, and we are non-inertial on the ground as a result.This explains why moving along the geodesics inspacetime is considered inertial.

Einstein discovered the field equations of general relativity, which relate the presence of matter and thecurvature of spacetime and are named after him. The Einstein field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor ofspacetime. A metric tensor describes a geometry of spacetime. The geodesic paths for a spacetime arecalculated from the metric tensor.

Notable solutions of the Einstein field equations include:

The Schwarzschild solution, which describes spacetime surrounding a spherically symmetricnon-rotating uncharged massive object. For compact enough objects, this solution generated ablack hole with a central singularity. For radial distances from the center which are muchgreater than the Schwarzschild radius, the accelerations predicted by the Schwarzschildsolution are practically identical to those predicted by Newton's theory of gravity.The Reissner-Nordström solution, in which the central object has an electrical charge. Forcharges with a geometrized length which are less than the geometrized length of the mass ofthe object, this solution produces black holes with double event horizons.The Kerr solution for rotating massive objects. This solution also produces black holes withmultiple event horizons.The Kerr-Newman solution for charged, rotating massive objects. This solution also producesblack holes with multiple event horizons.

General relativity

Solutions

The cosmological Friedmann-Lemaître-Robertson-Walker solution, which predicts theexpansion of the Universe.

The tests of general relativity included the following:[21]

General relativity accounts for the anomalous perihelion precession of Mercury.[22]

The prediction that time runs slower at lower potentials (gravitational time dilation) has beenconfirmed by the Pound–Rebka experiment (1959), the Hafele–Keating experiment, and theGPS.The prediction of the deflection of light was first confirmed by Arthur Stanley Eddington from hisobservations during the Solar eclipse of 29 May 1919.[23][24] Eddington measured starlightdeflections twice those predicted by Newtonian corpuscular theory, in accordance with thepredictions of general relativity. However, his interpretation of the results was later disputed.[25]

More recent tests using radio interferometric measurements of quasars passing behind the Sunhave more accurately and consistently confirmed the deflection of light to the degree predictedby general relativity.[26] See also gravitational lens.The time delay of light passing close to a massive object was first identified by Irwin I. Shapiroin 1964 in interplanetary spacecraft signals.Gravitational radiation has been indirectly confirmed through studies of binary pulsars. On 11February 2016, the LIGO and Virgo collaborations announced the first observation of agravitational wave.Alexander Friedmann in 1922 found that Einstein equations have non-stationary solutions(even in the presence of the cosmological constant). In 1927 Georges Lemaître showed thatstatic solutions of the Einstein equations, which are possible in the presence of thecosmological constant, are unstable, and therefore the static Universe envisioned by Einsteincould not exist. Later, in 1931, Einstein himself agreed with the results of Friedmann andLemaître. Thus general relativity predicted that the Universe had to be non-static—it had toeither expand or contract. The expansion of the Universe discovered by Edwin Hubble in 1929confirmed this prediction.[27]

The theory's prediction of frame dragging was consistent with the recent Gravity Probe Bresults.[28]

General relativity predicts that light should lose its energy when traveling away from massivebodies through gravitational redshift. This was verified on earth and in the solar system around1960.

An open question is whether it is possible to describe the small-scale interactions of gravity with the sameframework as quantum mechanics. General relativity describes large-scale bulk properties whereas quantummechanics is the framework to describe the smallest scale interactions of matter. Without modifications theseframeworks are incompatible.[29]

One path is to describe gravity in the framework of quantum field theory, which has been successful toaccurately describe the other fundamental interactions. The electromagnetic force arises from an exchange ofvirtual photons, where the QFT description of gravity is that there is an exchange of virtual gravitons.[30][31]

This description reproduces general relativity in the classical limit. However, this approach fails at shortdistances of the order of the Planck length,[29] where a more complete theory of quantum gravity (or a newapproach to quantum mechanics) is required.

Tests

Gravity and quantum mechanics

An initially-stationary objectthat is allowedto fall freelyunder gravitydrops a distancethat isproportional tothe square ofthe elapsedtime. Thisimage spanshalf a secondand wascaptured at 20flashes persecond.

If an object withcomparable mass to thatof the Earth were to falltowards it, then thecorrespondingacceleration of the Earthwould be observable.

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 onall objects. Assuming a spherically symmetrical planet, the strength of this field at anygiven point above the surface is proportional to the planetary body's mass and inverselyproportional to the square of the distance from the center of the body.

The strength of the gravitational field is numerically equalto the acceleration of objects under its influence.[32] Therate of acceleration of falling objects near the Earth's surfacevaries very slightly depending on latitude, surface featuressuch as mountains and ridges, and perhaps unusually highor low sub-surface densities.[33] For purposes of weightsand measures, a standard gravity value is defined by theInternational Bureau of Weights and Measures, under theInternational System of Units (SI).

That value, denoted g, is g = 9.80665 m/s2

(32.1740 ft/s2).[34][35]

The standard value of 9.80665 m/s2 is the one originally adopted by the InternationalCommittee on Weights and Measures in 1901 for 45° latitude, even though it has beenshown to be too high by about five parts in ten thousand.[36] This value has persisted inmeteorology and in some standard atmospheres as the value for 45° latitude even thoughit applies more precisely to latitude of 45°32'33".[37]

Assuming the standardized value for g and ignoring air resistance, this means that anobject falling freely near the Earth's surface increases its velocity by 9.80665 m/s(32.1740 ft/s or 22 mph) for each second of its descent. Thus, an object starting from restwill attain a velocity of 9.80665 m/s (32.1740 ft/s) after one second, approximately19.62 m/s (64.4 ft/s) after two seconds, and so on, adding 9.80665 m/s (32.1740 ft/s) toeach resulting velocity. Also, again ignoring air resistance, any and all objects, whendropped from the same height, will hit the ground at the same time.

According to Newton's 3rd Law, the Earth itself experiences a force equal in magnitudeand opposite in direction to that which it exerts on a falling object. This means that theEarth also accelerates towards the object until they collide. Because the mass of the Earthis huge, however, the acceleration imparted to the Earth by this opposite force is negligible in comparison tothe object's. If the object does not bounce after it has collided with the Earth, each of them then exerts arepulsive contact force on the other which effectively balances the attractive force of gravity and preventsfurther acceleration.

The force of gravity on Earth is the resultant (vector sum) of two forces:[38] (a) The gravitational attraction inaccordance with Newton's universal law of gravitation, and (b) the centrifugal force, which results from thechoice of an earthbound, rotating frame of reference. The force of gravity is weakest at the equator because ofthe centrifugal force caused by the Earth's rotation and because points on the equator are furthest from thecenter of the Earth. The force of gravity varies with latitude and increases from about 9.780 m/s2 at theEquator to about 9.832 m/s2 at the poles.

Specifics

Earth's gravity

Gravity acts on stars that form theMilky Way.[39]

Under an assumption of constant gravitational attraction, Newton's law of universal gravitation simplifies to F= mg, where m is the mass of the body and g is a constant vector with an average magnitude of 9.81 m/s2 onEarth. This resulting force is the object's weight. The acceleration due to gravity is equal to this g. An initiallystationary object which is allowed to fall freely under gravity drops a distance which is proportional to thesquare of the elapsed time. The image on the right, spanning half a second, was captured with a stroboscopicflash at 20 flashes per second. During the first 1⁄20 of a second the ball drops one unit of distance (here, a unitis about 12 mm); by 2⁄20 it has dropped at total of 4 units; by 3⁄20, 9 units and so on.

Under the same constant gravity assumptions, the potential energy, Ep, of a body at height h is given by Ep =mgh (or Ep = Wh, with W meaning weight). This expression is valid only over small distances h from the

surface of the Earth. Similarly, the expression for the maximum height reached by a vertically

projected body with initial velocity v is useful for small heights and small initial velocities only.

The application of Newton's law of gravity has enabled theacquisition of much of the detailed information we have about theplanets in the Solar System, the mass of the Sun, and details ofquasars; even the existence of dark matter is inferred using Newton'slaw of gravity. Although we have not traveled to all the planets nor tothe Sun, we know their masses. These masses are obtained byapplying the laws of gravity to the measured characteristics of theorbit. In space an object maintains its orbit because of the force ofgravity acting upon it. Planets orbit stars, stars orbit galactic centers,galaxies orbit a center of mass in clusters, and clusters orbit insuperclusters. The force of gravity exerted on one object by another isdirectly proportional to the product of those objects' masses andinversely proportional to the square of the distance between them.

The earliest gravity (possibly in the form of quantum gravity, supergravity or a gravitational singularity), alongwith ordinary space and time, developed during the Planck epoch (up to 10−43 seconds after the birth of theUniverse), possibly from a primeval state (such as a false vacuum, quantum vacuum or virtual particle), in acurrently unknown manner.[5]

General relativity predicts that energy can be transported out of a system through gravitational radiation. Anyaccelerating matter can create curvatures in the space-time metric, which is how the gravitational radiation istransported away from the system. Co-orbiting objects can generate curvatures in space-time such as the Earth-Sun system, pairs of neutron stars, and pairs of black holes. Another astrophysical system predicted to loseenergy in the form of gravitational radiation are exploding supernovae.

The first indirect evidence for gravitational radiation was through measurements of the Hulse–Taylor binary in1973. This system consists of a pulsar and neutron star in orbit around one another. Its orbital period hasdecreased since its initial discovery due to a loss of energy, which is consistent for the amount of energy lossdue to gravitational radiation. This research was awarded the Nobel Prize in Physics in 1993.

Equations for a falling body near the surface of the Earth

Gravity and astronomy

Gravitational radiation

The LIGO Hanford Observatorylocated in Washington, US wheregravitational waves were firstobserved in September 2015.

Rotation curve of a typical spiral galaxy:predicted (A) and observed (B). Thediscrepancy between the curves isattributed to dark matter.

The first direct evidence for gravitational radiation was measured on14 September 2015 by the LIGO detectors. The gravitational wavesemitted during the collision of two black holes 1.3 billion-light yearsfrom Earth were measured.[40][41] This observation confirms thetheoretical predictions of Einstein and others that such waves exist. Italso opens the way for practical observation and understanding of thenature of gravity and events in the Universe including the BigBang.[42] Neutron star and black hole formation also create detectableamounts of gravitational radiation.[43] This research was awarded theNobel Prize in physics in 2017.[44]

As of 2020, the gravitational radiation emitted by the Solar System isfar too small to measure with current technology.

In December 2012, a research team in China announced that it had produced measurements of the phase lag ofEarth tides during full and new moons which seem to prove that the speed of gravity is equal to the speed oflight.[45] This means that if the Sun suddenly disappeared, the Earth would keep orbiting it normally for 8minutes, which is the time light takes to travel that distance. The team's findings were released in the ChineseScience Bulletin in February 2013.[46]

In October 2017, the LIGO and Virgo detectors received gravitational wave signals within 2 seconds ofgamma ray satellites and optical telescopes seeing signals from the same direction. This confirmed that thespeed of gravitational waves was the same as the speed of light.[47]

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

Extra-fast stars: Stars in galaxies follow a distributionof velocities where stars on the outskirts are movingfaster than they should according to the observeddistributions of normal matter. Galaxies within galaxyclusters show a similar pattern. Dark matter, whichwould interact through gravitation but notelectromagnetically, would account for thediscrepancy. Various modifications to Newtoniandynamics have also been proposed.Flyby anomaly: Various spacecraft have experiencedgreater acceleration than expected during gravityassist maneuvers.Accelerating expansion: The metric expansion ofspace seems to be speeding up. Dark energy hasbeen proposed to explain this. A recent alternativeexplanation is that the geometry of space is not homogeneous (due to clusters of galaxies) andthat when the data are reinterpreted to take this into account, the expansion is not speeding upafter all,[48] however this conclusion is disputed.[49]

Anomalous increase of the astronomical unit: Recent measurements indicate that planetaryorbits are widening faster than if this were solely through the Sun losing mass by radiating

Speed of gravity

Anomalies and discrepancies

energy.Extra energetic photons: Photons travelling through galaxy clusters should gain energy andthen lose it again on the way out. The accelerating expansion of the Universe should stop thephotons returning all the energy, but even taking this into account photons from the cosmicmicrowave background radiation gain twice as much energy as expected. This may indicatethat gravity falls off faster than inverse-squared at certain distance scales.[50]

Extra massive hydrogen clouds: The spectral lines of the Lyman-alpha forest suggest thathydrogen clouds are more clumped together at certain scales than expected and, like dark flow,may indicate that gravity falls off slower than inverse-squared at certain distance scales.[50]

Aristotelian theory of gravityLe Sage's theory of gravitation (1784) also called LeSage gravity, proposed by Georges-LouisLe Sage, based on a fluid-based explanation where a light gas fills the entire Universe.Ritz's theory of gravitation, Ann. Chem. Phys. 13, 145, (1908) pp. 267–271, Weber-Gausselectrodynamics applied to gravitation. Classical advancement of perihelia.Nordström's theory of gravitation (1912, 1913), an early competitor of general relativity.Kaluza Klein theory (1921)Whitehead's theory of gravitation (1922), another early competitor of general relativity.

Brans–Dicke theory of gravity (1961)[51]

Induced gravity (1967), a proposal by Andrei Sakharov according to which general relativitymight arise from quantum field theories of matterString theory (late 1960s)ƒ(R) gravity (1970)Horndeski theory (1974)[52]

Supergravity (1976)In the modified Newtonian dynamics (MOND) (1981), Mordehai Milgrom proposes amodification of Newton's second law of motion for small accelerations[53]

The self-creation cosmology theory of gravity (1982) by G.A. Barber in which the Brans-Dicketheory is modified to allow mass creationLoop quantum gravity (1988) by Carlo Rovelli, Lee Smolin, and Abhay AshtekarNonsymmetric gravitational theory (NGT) (1994) by John MoffatTensor–vector–scalar gravity (TeVeS) (2004), a relativistic modification of MOND by JacobBekensteinChameleon theory (2004) by Justin Khoury and Amanda Weltman.Pressuron theory (2013) by Olivier Minazzoli and Aurélien Hees.Conformal gravity[54]

Gravity as an entropic force, gravity arising as an emergent phenomenon from thethermodynamic concept of entropy.In the superfluid vacuum theory the gravity and curved space-time arise as a collectiveexcitation mode of non-relativistic background superfluid.

Alternative theories

Historical alternative theories

Modern alternative theories

Massive gravity, a theory where gravitons and gravitational waves have a non-zero mass

Anti-gravity, the idea of neutralizing or repelling gravityArtificial gravityGauss's law for gravityGravitational potentialGravitational waveKepler's third law of planetary motionMicro-g environment, also called microgravityNewton's laws of motionStandard gravitational parameterWeightlessness

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6. Reviel Neitz; William Noel (13 October 2011). The Archimedes Codex: Revealing The Secretsof the World's Greatest Palimpsest (https://books.google.com/books?id=ZC1MOaAkKnsC&pg=PT125). Hachette UK. p. 125. ISBN 978-1-78022-198-4.

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8. Vitruvius, Marcus Pollio (1914). "7" (http://www.gutenberg.org/files/20239/20239-h/29239-h.htm#Page_215). In Alfred A. Howard (ed.). De Architectura libri decem [Ten Books onArchitecture]. VII. Herbert Langford Warren, Nelson Robinson (illus), Morris Hicky Morgan.Harvard University, Cambridge: Harvard University Press. p. 215.

9. Pickover, Clifford (16 April 2008). Archimedes to Hawking: Laws of Science and the GreatMinds Behind Them (https://books.google.com/books?id=SQXcpvjcJBUC&pg=PA105). OxfordUniversity Press. ISBN 9780199792689.

10. *Sen, Amartya (2005). The Argumentative Indian. Allen Lane. p. 29. ISBN 978-0-7139-9687-6.11. Ball, Phil (June 2005). "Tall Tales". Nature News. doi:10.1038/news050613-10 (https://doi.org/1

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12. Galileo (1638), Two New Sciences, First Day Salviati speaks: "If this were what Aristotle meantyou would burden him with another error which would amount to a falsehood; because, sincethere is no such sheer height available on earth, it is clear that Aristotle could not have madethe experiment; yet he wishes to give us the impression of his having performed it when hespeaks of such an effect as one which we see."

13. Bongaarts, Peter (2014). Quantum Theory: A Mathematical Approach (https://books.google.com/books?id=Cc6lBQAAQBAJ&pg=PA11) (illustrated ed.). Springer. p. 11. ISBN 978-3-319-09561-5.

14. *Chandrasekhar, Subrahmanyan (2003). Newton's Principia for the common reader. Oxford:Oxford University Press. (pp. 1–2). The quotation comes from a memorandum thought to havebeen written about 1714. As early as 1645 Ismaël Bullialdus had argued that any force exertedby the Sun on distant objects would have to follow an inverse-square law. However, he alsodismissed the idea that any such force did exist. See, for example, Linton, Christopher M.(2004). From Eudoxus to Einstein – A History of Mathematical Astronomy (https://archive.org/details/fromeudoxustoein00lint_648). Cambridge: Cambridge University Press. p. 225 (https://archive.org/details/fromeudoxustoein00lint_648/page/n237). ISBN 978-0-521-82750-8.

15. Nobil, Anna M. (March 1986). "The real value of Mercury's perihelion advance". Nature. 320(6057): 39–41. Bibcode:1986Natur.320...39N (https://ui.adsabs.harvard.edu/abs/1986Natur.320...39N). doi:10.1038/320039a0 (https://doi.org/10.1038%2F320039a0). S2CID 4325839 (https://api.semanticscholar.org/CorpusID:4325839).

16. M.C.W.Sandford (2008). "STEP: Satellite Test of the Equivalence Principle" (https://web.archive.org/web/20110928041608/http://www.sstd.rl.ac.uk/fundphys/step/). Rutherford AppletonLaboratory. Archived from the original (http://www.sstd.rl.ac.uk/fundphys/step/) on 28September 2011. Retrieved 14 October 2011.

17. Paul S Wesson (2006). Five-dimensional Physics (https://archive.org/details/fivedimensionalp0000wess). World Scientific. p. 82 (https://archive.org/details/fivedimensionalp0000wess/page/82). ISBN 978-981-256-661-4.

18. Haugen, Mark P.; C. Lämmerzahl (2001), "Principles of Equivalence: Their Role in GravitationPhysics and Experiments that Test Them", Gyros, Lecture Notes in Physics, 562 (562, Gyros,Clocks, and Interferometers...: Testing Relativistic Gravity in Space): 195–212, arXiv:gr-qc/0103067 (https://arxiv.org/abs/gr-qc/0103067), Bibcode:2001LNP...562..195H (https://ui.adsabs.harvard.edu/abs/2001LNP...562..195H), doi:10.1007/3-540-40988-2_10 (https://doi.org/10.1007%2F3-540-40988-2_10), S2CID 15430387 (https://api.semanticscholar.org/CorpusID:15430387)

19. "Gravity and Warped Spacetime" (https://web.archive.org/web/20110621005940/http://www.black-holes.org/relativity6.html). black-holes.org. Archived from the original (https://www.black-holes.org/relativity6.html) on 21 June 2011. Retrieved 16 October 2010.

20. Dmitri Pogosyan. "Lecture 20: Black Holes – The Einstein Equivalence Principle" (https://www.ualberta.ca/~pogosyan/teaching/ASTRO_122/lect20/lecture20.html). University of Alberta.Retrieved 14 October 2011.

21. Pauli, Wolfgang Ernst (1958). "Part IV. General Theory of Relativity". Theory of Relativity.Courier Dover Publications. ISBN 978-0-486-64152-2.

22. Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a tabledocumenting the observed and calculated values for the precession of the perihelion ofMercury, Venus, and Earth.)

23. Dyson, F.W.; Eddington, A.S.; Davidson, C.R. (1920). "A Determination of the Deflection ofLight by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29,1919" (https://zenodo.org/record/1432106). Phil. Trans. Roy. Soc. A. 220 (571–581): 291–333.Bibcode:1920RSPTA.220..291D (https://ui.adsabs.harvard.edu/abs/1920RSPTA.220..291D).doi:10.1098/rsta.1920.0009 (https://doi.org/10.1098%2Frsta.1920.0009).. Quote, p. 332: "Thusthe results of the expeditions to Sobral and Principe can leave little doubt that a deflection oflight takes place in the neighbourhood of the sun and that it is of the amount demanded byEinstein's generalised theory of relativity, as attributable to the sun's gravitational field."

24. Weinberg, Steven (1972). Gravitation and cosmology (https://archive.org/details/gravitationcosmo00stev_0). John Wiley & Sons.. Quote, p. 192: "About a dozen stars in all were studied, andyielded values 1.98 ± 0.11" and 1.61 ± 0.31", in substantial agreement with Einstein'sprediction θ☉ = 1.75"."

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29. Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions (https://archive.org/details/warpedpassagesun00rand_1). Ecco. ISBN 978-0-06-053108-9.

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54. Haugan, Mark P; Lämmerzahl, C (2011). "Einstein gravity from conformal gravity".arXiv:1105.5632 (https://arxiv.org/abs/1105.5632) [hep-th (https://arxiv.org/archive/hep-th)].

References

Further reading

External links