Pioneers of gravitational theory

Aristotle

Galileo Galilei

Isaac Newton

Albert Einstein

Inphysics, theories ofgravitation postulate mechanisms of interaction governing the movements of bodies with mass. There have been numerous theories of gravitation since ancient times. The first extant sources discussing such theories are found inancient Greek philosophy. This work was furthered through theMiddle Ages byIndian,Islamic, andEuropean scientists, before gaining great stridesduring the Renaissance andScientific Revolution—culminating in the formulation ofNewton's law of gravity. This was superseded byAlbert Einstein'stheory of relativity in the early 20th century.

Greek philosopherAristotle (fl. 4th century BC) found that objects immersed in a medium tend to fall at speeds proportional to their weight.Vitruvius (fl. 1st century BC) understood that objects fall based on theirspecific gravity. In the 6th century AD, ByzantineAlexandrian scholarJohn Philoponus modified theAristotelian concept of gravity with thetheory of impetus. In the 7th century, Indian astronomerBrahmagupta spoke of gravity as an attractive force. In the 14th century, European philosophersJean Buridan andAlbert of Saxony—who were influenced by Islamic scholarsIbn Sina andAbu'l-Barakat respectively—developed the theory of impetus and linked it to the acceleration and mass of objects. Albert also developed a law of proportion regarding the relationship between the speed of an object infree fall and the time elapsed.

Italians of the 16th century found that objects in free fall tend to accelerate equally. In 1632,Galileo Galilei put forththe basic principle of relativity. The existence of thegravitational constant was explored by various researchers from the mid-17th century, helpingIsaac Newton formulate his law of universal gravitation. Newton'sclassical mechanics were superseded in the early 20th century, when Einstein developed thespecial andgeneral theories of relativity. An elementalforce carrier of gravity is hypothesized inquantum gravity approaches such asstring theory, in a potentially unifiedtheory of everything.

Thepre-Socratic Greek philosopherHeraclitus (c. 535 – c. 475 BC) of theIonian School used the wordlogos ('word') to describe a kind of law which keeps thecosmos in harmony, moving all objects, including the stars, winds, and waves.^[1]Anaxagoras (c. 500 – c. 428 BC), another Ionian philosopher, introduced the concept ofnous ('cosmic mind') as an ordering force.^[2]

In the cosmogony of the Greek philosopherEmpedocles (c. 494 – c. 434/443 BC), there were two opposingfundamental cosmic forces of "attraction" and "repulsion", which Empedocles personified as "Love" and "Strife" (Philotes andNeikos).^[3][4]

The ancient atomistLeucippus (5th century BC) proposed the cosmos was created when a large group ofatoms came together and swirled as avortex. The smaller atoms became the celestial bodies of the cosmos. The larger atoms in the center came together as a membrane from which theEarth was formed.^[5][6]

In the 4th century BC, Greek philosopher Aristotle taught that there is noeffect ormotion without a cause. The cause of the downward natural motion of heavy bodies, such as theclassical elements ofearth andwater, was related to theirnature (gravity), which caused them to move downward toward the center of the (geocentric) universe. For this reason Aristotle supported aspherical Earth, since "every portion of earth has weight until it reaches the centre, and the jostling of parts greater and smaller would bring about not a waved surface, but rather compression and convergence of part and part until the centre is reached".^[10] On the other hand, light bodies such as the elementfire andair, were moved by their nature (levity) upward toward thecelestial sphere of theMoon (seesublunary sphere).Astronomical objects near thefixed stars are composed ofaether, whose natural motion is circular. Beyond them is theprime mover, thefinal cause of all motion in the cosmos.^[11][12] In hisPhysics, Aristotle correctly asserted that objects immersed in a medium tend to fall at speeds proportional to theirweight and inversely proportional to thedensity of the medium.^[7][9]

Greek philosopherStrato of Lampsacus (c. 335 – c. 269 BC) rejected the Aristotelian belief of "natural places" in exchange for amechanical view in which objects do not gainweight as they fall, instead arguing that the greater impact was due to an increase in speed.^[13][14]

Epicurus (c. 341 – 270 BC) viewed weight as an inherent property ofatoms which influences their movement.^[15] These atoms move downward in constantfree fall within an infinite vacuum withoutfriction at equal speed, regardless of their mass. On the other hand, upward motion is due toatomic collisions.^[16]Epicureans deviated from olderatomist theories like that ofDemocritus (c. 460 – c. 370 BC) by proposing the idea that atoms mayrandomly deviate from their expected course.^[17]

Greek astronomerAristarchus of Samos (c. 310 – c. 230 BC) theorizedEarth's rotation around its own axis, as well asEarth's orbit around theSun in aheliocentric cosmology.^[18]Seleucus of Seleucia (c. 190 – c. 150 BC) supported his cosmology^[18] and also describedgravitational effects of the Moon on thetidal range.^[19]

The 3rd-century BC Greek physicistArchimedes (c. 287 – c. 212 BC) discovered thecentre of mass of a triangle.^[20] He also postulated that if thecentres of gravity of two equal weights was not the same, it would be located in the middle of the line that joins them.^[21] InOn Floating Bodies, Archimedes claimed that for any object submerged in a fluid there is an equivalent upwardbuoyant force to the weight of the fluid displaced by the object's volume.^[22] The fluids described by Archimedes are not self-gravitating, since he assumes that "any fluid at rest is the surface of a sphere whose centre is the same as that of the Earth".^[23][24]

Greek astronomerHipparchus of Nicaea (c. 190 – c. 120 BC) also rejectedAristotelian physics and followed Strato in adopting some form oftheory of impetus to explain motion.^[25][26] The poemDe rerum natura byLucretius (c. 99 – c. 55 BC) asserts that more massive bodies fall faster in a medium because the latter resists less, but in avacuum fall with equal speed.^[27] Roman engineer and architectVitruvius (c. 85 – c. 15 BC) contends in hisDe architectura that gravity is not dependent on a substance's weight but rather on its 'nature' (cf.specific gravity):

If thequicksilver is poured into a vessel, and a stone weighing one hundred pounds is laid upon it, the stone swims on the surface, and cannot depress the liquid, nor break through, nor separate it. If we remove the hundred pound weight, and put on a scruple of gold, it will not swim, but will sink to the bottom of its own accord. Hence, it is undeniable that the gravity of a substance depends not on the amount of its weight, but on its nature.^[28][29] (translated from the original Latin by W. Newton)

Greek philosopherPlutarch (c. 46 – c. 120 AD) attested the existence of Roman astronomers who rejected Aristotelian physics, "even contemplating theories ofinertia anduniversal gravitation",^[30][31] and suggested that gravitational attraction was not unique to the Earth.^[32] The gravitational effects of the Moon on the tides were noticed byPliny the Elder (23–79 AD) in hisNaturalis Historia^[33] andClaudius Ptolemy (c. 100 – c. 170 AD) in hisTetrabiblos.^[34]

In the 6th century AD, the ByzantineAlexandrian scholarJohn 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 which diminishes over time. In hiscommentary on Aristotle'sPhysics that "if one lets fall simultaneously from the same height two bodies differing greatly in weight, one will find that the ratio of the times of their motion does not correspond to the ratios of their weights, but the difference in time is a very small one".^[35]

Brahmagupta (c. 598 – c. 668 AD) was the first Indian scholar to describe gravity as an attractive force:^{[36][37][failed verification][38][39][failed verification]}

The earth on all its sides is the same; all people on the earth stand upright, and all heavy things fall down to the earth by a law of nature, for it is the nature of the earth to attract and to keep things, as it is the nature of water to flow ... If a thing wants to go deeper down than the earth, let it try. The earth is the onlylow thing, and seeds always return to it, in whatever direction you may throw them away, and never rise upwards from the earth.^[40][41][a]

Bhāskara II (c. 1114 – c. 1185), another Indian mathematician and astronomer, describes gravity as an inherent attractive property of Earth in the section "Golādhyāyah" ("On Spherics") of his treatiseSiddhānta Shiromani:

The property of attraction is inherent in the Earth. By this property the Earth attracts any unsupported heavy thing towards it: The thing appears to be falling but it is in a state of being drawn to Earth. ... It is manifest from this that ... people situated at distances of a fourth part of the circumference [of earth] from us or in the opposite hemisphere, cannot by any means fall downwards [in space].^[42][43]

Ancient Greeks likePosidonius had associated the tides in the sea with to be influenced by moonlight. Around 850,Abu Ma'shar al-Balkhi recorded the tides and the moon position and noticed high-tides when the Moon was below the horizon. Abu Ma'shar considered an alternative explanation where the Moon and the sea had to share some astrological virtue that attracted each other. This work was translated into Latin and became one of the two main theories for tides for European scholars.^[44]

In the 11th century, Persian polymathIbn Sina (Avicenna) agreed with Philoponus' theory that "the moved object acquires an inclination from the mover" as an explanation forprojectile motion.^[45] Ibn Sina then publishedhis own theory of impetus inThe Book of Healing (c. 1020). Unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, Ibn Sina viewed it as a persistent, requiring external forces such asair resistance to dissipate it.^[46][47][48] Ibn Sina made distinction between force and inclination (mayl), and argued that an object gained inclination when the object is in opposition to its natural motion. He concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until the inclination is spent.^[49] The Iraqi polymathIbn al-Haytham describes gravity as a force in which heavier body moves towards the centre of the earth. He also describes the force of gravity will only move towards the direction of the centre of the earth not in different directions.^[50]

Another 11th-century Persian polymath,Al-Biruni, proposed thatheavenly bodies havemass, weight, and gravity, just like the Earth. He criticized both Aristotle and Ibn Sina for holding the view that only the Earth has these properties.^[51] The 12th-century scholarAl-Khazini suggested that the gravity an object contains varies depending on its distance from the centre of the universe (referring to the centre of the Earth). Al-Biruni and Al-Khazini studied the theory of the centre of gravity, and generalized and applied it to three-dimensional bodies. Fine experimental methods were also developed for determining the specific gravity orspecific weight of objects, based the theory ofbalances andweighing.^[52]

In the 12th century,Ibn Malka al-Baghdadi adopted and modified Ibn Sina's theory onprojectile motion. In hisKitab al-Mu'tabar, Abu'l-Barakat stated that the mover imparts a violent inclination (mayl qasri) on the moved and that this diminishes as the moving object distances itself from the mover.^[53] According toShlomo Pines, al-Baghdādī's theory of motion was "the oldest negation of Aristotle's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law ofclassical mechanics [namely, that a force applied continuously producesacceleration]."^[54]

In the 14th century, both the French philosopherJean Buridan and theOxford Calculators (the Merton School) of theMerton College ofOxford rejected theAristotelian concept of gravity.^[55][b] They attributed the motion of objects to an impetus (akin tomomentum), which varies according to velocity and mass;^[55] Buridan was influenced in this by Ibn Sina'sBook of Healing.^[48] Buridan and the philosopherAlbert of Saxony (c. 1320 – c. 1390) adopted Abu'l-Barakat's theory that the acceleration of a falling body is a result of its increasing impetus.^[53] Influenced by Buridan, Albert developed a law of proportion regarding the relationship between the speed of an object infree fall and the time elapsed.^[56] He also theorized that mountains and valleys are caused byerosion^[c]—displacing the Earth's centre of gravity.^[57][d]

The roots ofDomingo de Soto's expressionuniform difform motion [uniformly accelerated motion] lies in the Oxford Calculators terms "uniform" and "difform" motion:^[59] "uniform motion" was used differently then than it would be by later writers, and might have referred both to constant speed and to motion in which all parts of a body are moving at equal speed. The Calculators did not illustrate the different types of motion with real-world examples.^[59] John of Holland at the University of Prague, illustrated uniform motion with what would later be called uniform velocity, but also with a falling stone (all parts moving at the same speed), and with a sphere in uniform rotation. He did, however, make distinctions between different kinds of "uniform" motion. Difform motion was exemplified by walking at increasing speed.^[59]

Also in the 14th century, the Merton School developed themean speed theorem; a uniformly accelerated body starting from rest travels the same distance as a body withuniform speed whose speed is half the final velocity of the accelerated body. The mean speed theorem was proved byNicole Oresme (c. 1323 – 1382) and would be influential in latergravitational equations.^[55] Written as a modern equation:

${\displaystyle s=\frac{1}{2}{v}_{f}t}$

However, since small time intervals could not be measured, the relationship between time and distance was not so evident as the equation suggests. More generally; equations, which were not widely used until after Galileo's time, imply a clarity that was not there.

Leonardo da Vinci (1452–1519) made drawings recording the acceleration of falling objects.^[60] He wrote that the "mother and origin of gravity" isenergy. He describes two pairs of physical powers which stem from ametaphysical origin and have an effect on everything:abundance of force and motion, and gravity and resistance. He associates gravity with the 'cold'classical elements,water and earth, and calls its energy infinite.^[61][e] InCodex Arundel, Leonardo recorded that if a water-pouring vase moves transversally (sideways), simulating the trajectory of a vertically falling object, it produces aright triangle with equal leg length, composed of falling material that forms thehypotenuse and the vase trajectory forming one of the legs.^[63] On the hypotenuse, Leonardo noted the equivalence of the two orthogonal motions, one effected by gravity and the other proposed by the experimenter.^[63]

By 1514,Nicolaus Copernicus had writtenan outline ofhis heliocentric model, in which he stated that Earth's centre is the centre of bothits rotation and theorbit of the Moon.^[64][f] In 1533, German humanistPetrus Apianus described theexertion of gravity:^[g]

Since it is apparent that in the descent [along the arc] there is more impediment acquired, it is clear that gravity is diminished on this account. But because this comes about by reason of the position of heavy bodies, let it be called apositional gravity [i.e.gravitas secundum situm]^[67]

By 1544, according toBenedetto Varchi, the experiments of at least two Italians, Francesco Beato, a Dominican philosopher at Pisa, andLuca Ghini, a physician and botanist from Bologna, had dispelled the Aristotelian claim that objects fall at speeds proportional to their weight.^[68]

In 1551,Domingo de Soto theorized that objects in free fall accelerate uniformly in his bookPhysicorum Aristotelis quaestiones.^[69] This idea was subsequently explored in more detail by Galileo Galilei, who derived hiskinematics from the 14th-century Merton College and Jean Buridan,^[55] and possibly De Soto as well.^[69]

In 1585, Flemish polymathSimon Stevin performed a demonstration forJan Cornets de Groot, a local politician in the Dutch city ofDelft.^[70] Stevin dropped two lead balls from theNieuwe Kerk in that city. From the sound of the impacts, Stevin deduced that the balls had fallen at the same speed. The result was published in 1586.^[71][72]

Let us take (as ... Jan Cornets de Groot ... and I have done) two balls of lead, the one ten times larger and heavier than the other, and drop them together from a height of 30 feet on to a board or something on which they give a perceptible sound. Then it will be found that the lighter will not be ten times longer on its way than the heavier, but that they fall together on to the board so simultaneously that their two sounds seem to be one and the same. ... Therefore Aristotle ... is wrong.

This section is an excerpt fromGalileo's Leaning Tower of Pisa experiment.[edit]

Between 1589 and 1592,^[73] the Italian scientistGalileo Galilei (then professor of mathematics at theUniversity of Pisa) is said to have dropped "unequal weights of the same material" from theLeaning Tower of Pisa to demonstrate that their time of descent was independent of their mass, according to a biography by Galileo's pupilVincenzo Viviani, composed in 1654 and published in 1717.^{[74][75]: 19–21 [76][77]} The basic premise had already been demonstrated by Italian experimenters a few decades earlier.

According to the story, Galileo discovered through this experiment that the objects fell with the same acceleration, proving his prediction true, while at the same time disprovingAristotle's theory of gravity (which states that objects fall at speed proportional to their mass). Though Viviani wrote that Galileo conducted "repeated experiments made from the height of the Leaning Tower of Pisa in the presence of other professors and all the students,"^[74] most historians consider it to have been athought experiment rather than a physical test.^[78]

Galileo successfully applied mathematics to the acceleration of falling objects,^[79] correctly hypothesizing in a 1604 letter toPaolo Sarpi that the distance of a falling object is proportional to thesquare of the time elapsed.^[80][h]

I have arrived at a proposition, ... namely, that spaces traversed in natural motion are in the squared proportion of the times.

Written with modern symbols:s ∝t²

The result was published inTwo New Sciences in 1638. In the same book, Galileo suggested that the slight variance of speed of falling objects of different mass was due to air resistance, and that objects would fall completely uniformly in a vacuum.^[81] The relation of the distance of objects in free fall to the square of the time taken was confirmed by ItalianJesuitsGrimaldi andRiccioli between 1640 and 1650. They also made a calculation of thegravity of Earth by recording the oscillations of a pendulum.^[82]

In hisAstronomia nova (1609),Johannes Kepler proposed an attractive force of limited radius between any "kindred" bodies:

Gravity is a mutual corporeal disposition among kindred bodies to unite or join together; thus the earth attracts a stone much more than the stone seeks the earth. (The magnetic faculty is another example of this sort).... If two stones were set near one another in some place in the world outside the sphere of influence of a third kindred body, these stones, like two magnetic bodies, would come together in an intermediate place, each approaching the other by a space proportional to the bulk [moles] of the other....^[83]

Kepler claimed that if the Earth and Moon were not held apart by some force they would come together. He recognized that mechanical forces cause action, resulting in a more modern view of planetary motion, in his view a 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.^{[84][85]: 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.^[85]: 848

A disciple of Galileo,Evangelista Torricelli reiterated Aristotle's model involving a gravitational centre, adding his view that a system can only be in equilibrium when the common centre itself is unable to fall.^[66]

The relation of the distance of objects in free fall to the square of the time taken was confirmed byFrancesco Maria Grimaldi andGiovanni Battista Riccioli between 1640 and 1650. They also made a calculation of thegravity of Earth constant by recording the oscillations of a pendulum.^[86]

In 1644,René Descartes proposed that no empty space can exist and that acontinuum of matter causes every motion to becurvilinear. Thus,centrifugal force thrusts relatively light matter away from the centralvortices of celestial bodies, lowering density locally and thereby creatingcentripetal pressure.^[87][88] Using aspects of this theory, between 1669 and 1690,Christiaan Huygens designed a mathematical vortex model. In one of his proofs, he shows that the distance elapsed by an object dropped from a spinning wheel will increase proportionally to the square of the wheel's rotation time.^[89] In 1671,Robert Hooke speculated that gravitation is the result of bodies emitting waves in theaether.^[90][i]Nicolas Fatio de Duillier (1690) andGeorges-Louis Le Sage (1748) proposeda corpuscular model using some sort of screening or shadowing mechanism. In 1784, Le Sage posited that gravity could be a result of the collision of atoms, and in the early 19th century, he expandedDaniel Bernoulli'stheory of corpuscular pressure to the universe as a whole.^[91] A similar model was later created byHendrik Lorentz (1853–1928), who usedelectromagnetic radiation instead of corpuscles.

English mathematician Isaac Newton used Descartes' argument that curvilinear motion constrains inertia,^[92] and in 1675, argued that aether streams attract all bodies to one another.^[j] Newton (1717) andLeonhard Euler (1760) proposed a model in which the aether loses density near mass, leading to a net force acting on bodies.^{[citation needed]} Further mechanical explanations of gravitation (includingLe Sage's theory) were created between 1650 and 1900 to explain Newton's theory, but mechanistic models eventually fell out of favor because most of them lead to an unacceptable amount of drag (air resistance), which was not observed. Others violate theenergy conservation law and are incompatible with modernthermodynamics.^[93]

Before Newton, 'weight' had the double meaning 'amount' and 'heaviness'.^[94]

What we now know as mass was until the time of Newton called "weight." ... A goldsmith believed that an ounce of gold was a quantity of gold. ... But the ancients believed that a beam balance also measured "heaviness" which they recognized through their muscular senses. ... Mass and its associated downward force were believed to be the same thing.Kepler formed a [distinct] concept of mass ("amount of matter" (copia materiae), but called it "weight" as did everyone at that time.

In 1686, Newton gave the concept of mass its name. In the first paragraph ofPrincipia, Newton defined quantity of matter as "density and bulk conjunctly", and mass as quantity of matter.^[95]

The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight.

In 1679, Robert Hooke wrote to Isaac Newton of his hypothesis concerning orbital motion, which partly depends on aninverse-square force.^[96][k] In 1684, both Hooke and Newton toldEdmond Halley that they had proven the inverse-square law of planetary motion, in January and August, respectively.^[98] While Hooke refused to produce his proofs, Newton was prompted to composeDe motu corporum in gyrum ('On the motion of bodies in an orbit'), in which he mathematically derivesKepler's laws of planetary motion.^[98] In 1687, with Halley's support (and toHooke's dismay), Newton publishedPhilosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which hypothesizes the inverse-squarelaw of universal gravitation.^[98] In his own words:

I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve; and thereby compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth; and found them to answer pretty nearly.

Newton's original formula was:

${\displaystyle \mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}\propto \frac{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}1\times \mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}2}{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}{\mathrm{s}}^2}}$

where the symbol${\displaystyle \propto }$ means "is proportional to". To make this into an equal-sided formula or equation, there needed to be a multiplying factor or constant that would give the correct force of gravity no matter the value of the masses or distance between them – thegravitational constant. Newton would need an accurate measure of this constant to prove his inverse-square law. Reasonably accurate measurements were not available in until theCavendish experiment byHenry Cavendish in 1797.^[99]

In Newton's theory^[100] (rewritten using more modern mathematics) the density of mass${\displaystyle \rho }$ generates a scalar field, the gravitational potential${\displaystyle \phi }$ in joules per kilogram, by

${\displaystyle \frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{x}^j\mathrm{\partial }{x}^j}=4\pi G\rho .}$

Using theNabla operator${\displaystyle \mathrm{\nabla }}$ for thegradient anddivergence (partial derivatives), this can be conveniently written as:

${\displaystyle {\mathrm{\nabla }}^2\phi =4\pi G\rho .}$

This scalar field governs the motion of afree-falling particle by:

${\displaystyle \frac{d^2{x}^j}{d{t}^2}=-\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}^j}.}$

At distancer from an isolated massM, the scalar field is

${\displaystyle \phi =-\frac{GM}{r}.}$

ThePrincipia sold out quickly, inspiring Newton to publish a second edition in 1713.^[101][102]However the theory of gravity itself was not accepted quickly.

The theory of gravity faced two barriers. First scientists likeGottfried Wilhelm Leibniz complained that it relied onaction at a distance, that the mechanism of gravity was "invisible, intangible, and not mechanical".^{[103]: 339 [104]: 144} The French philosopherVoltaire countered these concerns, ultimately writinghis own book to explain aspects of it to French readers in 1738, which helped to popularize Newton's theory.^[105]

Second, detailed comparisons with astronomical data were not initially favorable. Among the most conspicuous issue was the so-calledgreat inequality of Jupiter and Saturn. Comparisons of ancient astronomical observations to those of the early 1700s implied that the orbit of Saturn was increasing in diameter while that of Jupiter was decreasing. Ultimately this meant Saturn would exit the Solar System and Jupiter would collide with other planets or the Sun. The problem was tackled first byLeonhard Euler in 1748, thenJoseph-Louis Lagrange in 1763, byPierre-Simon Laplace in 1773. Each effort improved the mathematical treatment until the issue was resolved by Laplace in 1784 approximately 100 years after Newton's first publication on gravity. Laplace showed that the changes were periodic but with immensely long periods beyond any existing measurements.^[106]: 144

Successes such the solution to the great inequality of Jupiter and Saturn mystery accumulated. In 1755, Prussian philosopherImmanuel Kant publisheda cosmological manuscript based on Newtonian principles, in which he develops an early version of thenebular hypothesis.^[107]Edmond Halley proposed that similar looking objects appearing every 76 years was in fact a single comet. The appearance of the comet in 1759, now named after him, within a month of predictions based on Newton's gravity greatly improved scientific opinion of the theory.^[108] Newton's theory enjoyed its greatest success when it was used to predict the existence ofNeptune based on motions ofUranus that could not be accounted by the actions of the other planets. Calculations byJohn Couch Adams andUrbain Le Verrier both predicted the general position of the planet. In 1846, Le Verrier sent his position toJohann Gottfried Galle, asking him to verify it. The same night, Galle spotted Neptune near the position Le Verrier had predicted.^[109]

Not every comparison was successful. By the end of the 19th century, Le Verrier showed that the orbit ofMercury could not be accounted for entirely under Newtonian gravity, and all searches for another perturbing body (such as a planet orbiting the Sun even closer than Mercury) were fruitless.^[110] Even so, Newton's theory is thought to be exceptionally accurate in the limit of weakgravitational fields and low speeds.

At the end of the 19th century, many tried to combine Newton's force law with the established laws ofelectrodynamics (like those ofWilhelm Eduard Weber,Carl Friedrich Gauss, andBernhard Riemann) to explain the anomalousperihelion precession of Mercury. In 1890,Maurice Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby thespeed of gravity is equal to the speed of light. In another attempt,Paul Gerber (1898) succeeded in deriving the correct formula for the perihelion shift (which was identical to the formula later used by Albert Einstein). These hypotheses were rejected because of the outdated laws they were based on, being superseded by those ofJames Clerk Maxwell.^[93]

In 1900,Hendrik Lorentz tried to explain gravity on the basis ofhis ether theory andMaxwell's equations. He assumed, likeOttaviano Fabrizio Mossotti andJohann Karl Friedrich Zöllner, that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. Lorentz calculated that the value for the perihelion advance of Mercury was much too low.^[111]

In the late 19th century,Lord Kelvin pondered the possibility of atheory of everything.^[112] He proposed that every body pulsates, which might be an explanation of gravitation andelectric charges. His ideas were largely mechanistic and required the existence of the aether, which theMichelson–Morley experiment failed to detect in 1887. This, combined withMach's principle, led to gravitational models which featureaction at a distance.

Albert Einstein developed his revolutionarytheory of relativity in papers published in 1905 and 1915; these account for the perihelion precession of Mercury.^[110] In 1914,Gunnar Nordström attempted to unify gravity andelectromagnetism inhis theory offive-dimensional gravitation.^[l] General relativity was proven in 1919, whenArthur Eddington observedgravitational lensing around a solar eclipse, matching Einstein's equations. This resulted in Einstein's theory superseding Newtonian physics.^[113] Thereafter, German mathematicianTheodor Kaluza promoted the idea of general relativity with a fifth dimension, which in 1921 Swedish physicistOskar Klein gavea physical interpretation of in a prototypicalstring theory, a possible model ofquantum gravity and potential theory of everything.

Einstein's field equations include acosmological constant to account for the allegedstaticity of the universe. However,Edwin Hubble observed in 1929 that the universe appears to be expanding. By the 1930s,Paul Dirac developed the hypothesis that gravitation should slowly and steadily decrease over the course of the history of the universe.^[114]Alan Guth andAlexei Starobinsky proposed in 1980 thatcosmic inflation in the very early universe could have been driven by a negativepressure field, a concept later coined 'dark energy'—found in 2013 to have composed around 68.3% of the early universe.^[115]

In 1922,Jacobus Kapteyn proposed the existence ofdark matter, an unseen force that moves stars in galaxies at higher velocities than gravity alone accounts for. It was found in 2013 to have comprised 26.8% of the early universe.^[115] Along with dark energy, dark matter is an outlier in Einstein's relativity, and an explanation for its apparent effects is a requirement for a successful theory of everything.

In 1957,Hermann Bondi proposed thatnegative gravitational mass (combined with negative inertial mass) would comply with thestrong equivalence principle of general relativity andNewton's laws of motion. Bondi's proof yieldedsingularity-free solutions for the relativity equations.^[116]

Early theories of gravity attempted to explain planetary orbits (Newton) and more complicated orbits (e.g. Lagrange). Then came unsuccessful attempts tocombine gravity and either wave or corpuscular theories of gravity. The whole landscape of physics was changed with the discovery ofLorentz transformations, and this led to attempts to reconcile it with gravity. At the same time, experimental physicists started testing the foundations of gravity and relativity—Lorentz invariance, thegravitational deflection of light, theEötvös experiment. These considerations led to and past the development ofgeneral relativity.

In 1905, Albert Einstein published a series of papers in which he established thespecial theory of relativity and the fact thatmass and energy are equivalent. In 1907, in what he described as "the happiest thought of my life", Einstein realized that someone who is in free fall experiences no gravitational field. In other words, gravitation is exactly equivalent to acceleration.

Einstein's two-part publication in 1912^[117][118] (and before in 1908) is really only important for historical reasons. By then he knew of the gravitational redshift and the deflection of light. He had realized that Lorentz transformations are not generally applicable, but retained them. The theory states that the speed of light is constant in free space but varies in the presence of matter. The theory was only expected to hold when the source of the gravitational field is stationary. It includes theprinciple of least action:

${\displaystyle \delta \int d\tau =0}$

${\displaystyle {d\tau }^2=-{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$

where${\displaystyle {\eta }_{\mu \nu }}$ is theMinkowski metric, and there is a summation from 1 to 4 over indices${\displaystyle \mu }$ and${\displaystyle \nu }$.

Einstein and Grossmann^[119] includesRiemannian geometry andtensor calculus.

${\displaystyle \delta \int d\tau =0}$

${\displaystyle {d\tau }^2=-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$

The equations of electrodynamics exactly match those of general relativity. The equation

${\displaystyle T^{\mu \nu }=\rho \frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }}$

is not in general relativity. It expresses thestress–energy tensor as a function of the matter density.

Based on theprinciple of relativity,Henri Poincaré (1905, 1906),Hermann Minkowski (1908), andArnold Sommerfeld (1910) tried to modify Newton's theory and to establish aLorentz invariant gravitational law, in which the speed of gravity is that of light. As in Lorentz's model, the value for the perihelion advance of Mercury was much too low.^[120]

Meanwhile,Max Abraham developed an alternative model of gravity in which the speed of light depends on the gravitational field strength and so is variable almost everywhere. Abraham's 1914 review of gravitation models is said to be excellent, but his own model was poor.

The first approach ofNordström (1912)^[121] was to retain the Minkowski metric and a constant value of${\displaystyle c}$ but to let mass depend on the gravitational field strength${\displaystyle \phi }$. Allowing this field strength to satisfy

${\displaystyle ◻\phi =\rho }$

where${\displaystyle \rho }$ is rest mass energy and${\displaystyle ◻}$ is thed'Alembertian,

${\displaystyle m=m_0\exp \left(\frac{\phi }{c^2}\right)}$

where${\displaystyle m_0}$ is the mass when gravitational potential vanishes and,

${\displaystyle -\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}^{\mu }}={\dot{u}}_{\mu }+\frac{u_{\mu }}{c^2\dot{\phi }}}$

where${\displaystyle u}$ is the four-velocity and the dot is a differential with respect to time.

The second approach ofNordström (1913)^[122] is remembered as the firstlogically consistent relativistic field theory of gravitation ever formulated. (notation from Pais^[123] not Nordström):

${\displaystyle \delta \int \psi d\tau =0}$

${\displaystyle {d\tau }^2=-{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$

where${\displaystyle \psi }$ is a scalar field,

${\displaystyle -\frac{\mathrm{\partial }{T}^{\mu \nu }}{\mathrm{\partial }{x}^{\nu }}=T\frac{1}{\psi }\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{x}_{\mu }}}$

This theory is Lorentz invariant, satisfies the conservation laws, correctly reduces to the Newtonian limit and satisfies theweak equivalence principle.

This theory^[124] is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing:

${\displaystyle \delta \int ds=0}$

${\displaystyle {ds}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$

${\displaystyle g_{\mu \nu }={\psi }^2{\eta }_{\mu \nu }}$

they relate Einstein–Grossmann^[119] to Nordström.^[122] They also state:

${\displaystyle T\propto R.}$

That is, the trace of the stress energy tensor is proportional to the curvature of space.

Between 1911 and 1915, Einstein developed the idea that gravitation is equivalent to acceleration, initially stated as theequivalence principle, into his general theory of relativity, which fuses thethree dimensions of space and the one dimension oftime into thefour-dimensional fabric ofspacetime. However, it does not unify gravity withquanta—individual particles of energy, which Einstein himself had postulated the existence of in 1905.

In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of to a force. The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion. The issue that this creates is that free-falling objects can accelerate with respect to each other. To deal with this difficulty, Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving alonglocally straight paths in curved spacetime. More specifically, Einstein andDavid Hilbert discovered thefield equations of general relativity, which relate the presence of matter and the curvature of spacetime.These field equations are a set of 10simultaneous,non-linear,differential equations. The solutions of the field equations are the components of themetric tensor of spacetime, which describes its geometry. The geodesic paths of spacetime are calculated from the metric tensor.

Notable solutions of the Einstein field equations include:

• TheSchwarzschild solution, which describes spacetime surrounding aspherically symmetrical non-rotating uncharged massive object. For objects with radii smaller than theSchwarzschild radius, this solution generates ablack hole with a central singularity.
• TheReissner–Nordström solution, in which the central object has an electrical charge. For charges with ageometrized length less than the geometrized length of the mass of the object, this solution produces black holes with anevent horizon surrounding aCauchy horizon.
• TheKerr solution for rotating massive objects. This solution also produces black holes with multiple horizons.
• ThecosmologicalRobertson–Walker solution (from 1922 and 1924), which predicts the expansion of the universe.^{[citation needed]}

General relativity has enjoyed much success because its predictions (not called for by older theories of gravity) have been regularly confirmed. For example:

• General relativity accounts for the anomalous perihelion precession of Mercury.^[110]
• Gravitational lensing was first confirmed in 1919, and has more recently been strongly confirmed through the use of aquasar which passes behind the Sun as seen from the Earth.
• The expansion of the universe (predicted by theRobertson–Walker metric) was confirmed by Edwin Hubble in 1929.
• The prediction that time runs slower at lower potentials has been confirmed by thePound–Rebka experiment, theHafele–Keating experiment, and theGPS.
• Thetime delay of light passing close to a massive object was first identified byIrwin Shapiro in 1964 in interplanetary spacecraft signals.
• Gravitational radiation has been indirectly confirmed through studies of binarypulsars such asPSR 1913+16.
  • In 2015, theLIGO experiments directlydetected gravitational radiation fromtwo colliding black holes, making this the first direct observation of both gravitational waves and black holes.^[125]

It is believed thatneutron star mergers (since detected in 2017)^[126] and black hole formation may also create detectable amounts of gravitational radiation.

Several decades after the discovery of general relativity, it was realized that it cannot be the complete theory of gravity because it is incompatible withquantum mechanics.^[127] Later it was understood that it is possible to describe gravity in the framework ofquantum field theory like the otherfundamental forces. In this framework, the attractive force of gravity arises due to exchange ofvirtualgravitons, in the same way as the electromagnetic force arises from exchange of virtualphotons.^[128][129] This reproduces general relativity in theclassical limit, but only at the linearized level and postulating that the conditions for the applicability ofEhrenfest theorem holds, which is not always the case. Moreover, this approach fails at short distances of the order of thePlanck length.^[127]

• Anti-gravity
• History of physics

• Gillispie, Charles Coulston (1960).The Edge of Objectivity: An Essay in the History of Scientific Ideas. Princeton University Press.ISBN 0-691-02350-6.{{cite book}}:ISBN / Date incompatibility (help)
• Wallace, W. A. (2004a). "The enigma of Domingo de Soto: Uniformiter difformis and falling bodies in late medieval physics". In Wallace, W. A. (ed.).Domingo de Soto and the early Galileo: Essays on intellectual history. Routledge. (Reprinted from "The enigma of Domingo de Soto: Uniformiter difformis and falling bodies in late medieval physics". (1968). Isis, 59(4), 384–401).
• Wallace, W. A. (2004b). "Domingo de Soto and the Iberian roots of Galileo's science". In Wallace, W. A. (ed.).Domingo de Soto and the early Galileo: Essays on intellectual history. Routledge. (Reprinted from White, K. (Ed.). (1997). Hispanic philosophy in the age of discovery. Studies in Philosophy and the History of Philosophy 29. Catholic University of America Press).

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