History of gravitational theory

Classical antiquity Heraclitus, Anaxagoras, Empedocles and Leucippus The pre-Socratic Greek philosopher Heraclitus () of the Ionian School used the word logos ('word') to describe a kind of law which keeps the cosmos in harmony, moving all objects, including the stars, winds, and waves. Anaxagoras (), another Ionian philosopher, introduced the concept of nous ('cosmic mind') as an ordering force. In the cosmogony of the Greek philosopher Empedocles (), there were two opposing fundamental cosmic forces of "attraction" and "repulsion", which Empedocles personified as "Love" and "Strife" (Philotes and Neikos). The ancient atomist Leucippus (5th century BC) proposed the cosmos was created when a large group of atoms came together and swirled as a vortex. The smaller atoms became the celestial bodies of the cosmos. The larger atoms in the center came together as a membrane from which the Earth was formed. Aristotle In the 4th century BC, Greek philosopher Aristotle taught that there is no effect or motion without a cause. The cause of the downward natural motion of heavy bodies, such as the classical elements of earth and water, was related to their nature (gravity), which caused them to move downward toward the center of the (geocentric) universe. For this reason Aristotle supported a spherical 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". On the other hand, light bodies such as the element fire and air, were moved by their nature (levity) upward toward the celestial sphere of the Moon (see sublunary sphere). Astronomical objects near the fixed stars are composed of aether, whose natural motion is circular. Beyond them is the prime mover, the final cause of all motion in the cosmos. In his Physics, Aristotle correctly asserted that objects immersed in a medium tend to fall at speeds proportional to their weight and inversely proportional to the density of the medium. Epicurus (270 BC) viewed weight as an inherent property of atoms which influences their movement. These atoms move downward in constant free fall within an infinite vacuum without friction at equal speed, regardless of their mass. On the other hand, upward motion is due to atomic collisions. Epicureans deviated from older atomist theories like that of Democritus () by proposing the idea that atoms may randomly deviate from their expected course. Greek astronomer Aristarchus of Samos () theorized Earth's rotation around its own axis, as well as Earth's orbit around the Sun in a heliocentric cosmology. Seleucus of Seleucia () supported his cosmology Archimedes The 3rd-century BC Greek physicist Archimedes () discovered the centre of mass of a triangle. He also postulated that if the centres of gravity of two equal weights was not the same, it would be located in the middle of the line that joins them, a result he used to prove the law of the lever and to extend his equilibrium analysis to floating bodies. In On Floating Bodies, Archimedes claimed that for any object submerged in a fluid there is an equivalent upward buoyant force to the weight of the fluid displaced by the object's volume. 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". Hipparchus of Nicaea, Lucretius and Vitruvius Greek astronomer Hipparchus of Nicaea () also rejected Aristotelian physics and followed Strato in adopting some form of theory of impetus to explain motion. The poem ' by Lucretius () asserts that more massive bodies fall faster in a medium because the latter resists less, but in a vacuum fall with equal speed. Roman engineer and architect Vitruvius () contends in his ' that gravity is not dependent on a substance's weight but rather on its 'nature' (specific gravity): If the quicksilver 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. (translated from the original Latin by W. Newton) Plutarch, Pliny the Elder, and Claudius Ptolemy Greek philosopher Plutarch () attested the existence of Roman astronomers who rejected Aristotelian physics, "even contemplating theories of inertia and universal gravitation". In his work De facie in orbe lunae, he suggested that gravitational attraction was not unique to the Earth, but applied to other bodies such as the Sun and the Moon, which were held to attract the parts of which they are made. His conception of gravity as a tendency of parts to unite with their whole coincides with Nicolaus Copernicus’s account of gravity. and Claudius Ptolemy () in his Tetrabiblos. Byzantine era John Philoponus In the 6th century AD, 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 which diminishes over time. In his commentary on Aristotle's Physics 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". == Indian subcontinent ==

Brahmagupta Brahmagupta () was the first Indian scholar to describe gravity as an attractive force: 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 only low thing, and seeds always return to it, in whatever direction you may throw them away, and never rise upwards from the earth. Bhāskara II Bhāskara II (), another Indian mathematician and astronomer, describes gravity as an inherent attractive property of Earth in the section "" ("On Spherics") of his treatise : 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]. == Islamic world ==

Abu Ma'shar Ancient Greeks like Posidonius 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. Ibn Sina In the 11th century, Persian polymath Ibn Sina (Avicenna) agreed with Philoponus' theory that "the moved object acquires an inclination from the mover" as an explanation for projectile motion. Ibn Sina then published his own theory of impetus in The Book of Healing (). 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 as air resistance to dissipate it. Ibn Sina made distinction between force and inclination (), 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. The Iraqi polymath Ibn 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. Al-Biruni Another 11th-century Persian polymath, Al-Biruni, proposed that heavenly bodies have mass, 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. The 12th-century scholar Al-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 or specific weight of objects, based the theory of balances and weighing. Abu'l-Barakāt al-Baghdādī In the 12th century, Ibn Malka al-Baghdadi adopted and modified Ibn Sina's theory on projectile motion. In his ''Kitab al-Mu'tabar'', Abu'l-Barakat stated that the mover imparts a violent inclination () on the moved and that this diminishes as the moving object distances itself from the mover. == European Renaissance ==

14th century .jpg|thumb|A 14th century illustration from Gautier de Metz's ''L'Image du monde'' showing the gravitational attraction of the Earth at its antipodes. Jean Buridan, the Oxford Calculators, Albert of Saxony In the 14th century, both the French philosopher Jean Buridan and the Oxford Calculators (the Merton School) of the Merton College of Oxford rejected the Aristotelian concept of gravity. They attributed the motion of objects to an impetus (akin to momentum), which varies according to velocity and mass; Buridan was influenced in this by Ibn Sina's Book of Healing. Influenced by Buridan, Albert developed a law of proportion regarding the relationship between the speed of an object in free fall and the time elapsed. He also theorized that mountains and valleys are caused by erosion—displacing the Earth's centre of gravity. Uniform and difform motion The roots of Domingo de Soto's expression motion [uniformly accelerated motion] lies in the Oxford Calculators terms "uniform" and "difform" motion: "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. 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. Mean speed theorem Also in the 14th century, the Merton School developed the mean speed theorem; a uniformly accelerated body starting from rest travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body. The mean speed theorem was proved by Nicole Oresme (1382) and would be influential in later gravitational equations. Written as a modern equation: \ s=\frac{1}{2}v_ft 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. 15th–16th centuries Leonardo da Vinci Leonardo da Vinci (1452–1519) made drawings recording the acceleration of falling objects. He wrote that the "mother and origin of gravity" is energy. He describes two pairs of physical powers which stem from a metaphysical 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. In Codex Arundel, Leonardo recorded that if a water-pouring vase moves transversally (sideways), simulating the trajectory of a vertically falling object, it produces a right triangle with equal leg length, composed of falling material that forms the hypotenuse and the vase trajectory forming one of the legs. On the hypotenuse, Leonardo noted the equivalence of the two orthogonal motions, one effected by gravity and the other proposed by the experimenter. In 1533, German humanist Petrus Apianus described the exertion of gravity: 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 a positional gravity [i.e. ] Francesco Beato and Luca Ghini By 1544, according to Benedetto Varchi, the experiments of at least two Italians, Francesco Beato, a Dominican philosopher at Pisa, and Luca Ghini, a physician and botanist from Bologna, had dispelled the Aristotelian claim that objects fall at speeds proportional to their weight. Domingo de Soto In 1551, Domingo de Soto theorized that objects in free fall accelerate uniformly in his book Physicorum Aristotelis quaestiones. This idea was subsequently explored in more detail by Galileo Galilei, who derived his kinematics from the 14th-century Merton College and Jean Buridan, and possibly De Soto as well. == Scientific Revolution ==

Simon Stevin In 1585, Flemish polymath Simon Stevin performed a demonstration for Jan Cornets de Groot, a local politician in the Dutch city of Delft. Stevin dropped two lead balls from the Nieuwe 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. Galileo Galilei Galileo successfully applied mathematics to the acceleration of falling objects, correctly hypothesizing in a 1604 letter to Paolo Sarpi that the distance of a falling object is proportional to the square of the time elapsed. Written with modern symbols: The result was published in Two 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. The relation of the distance of objects in free fall to the square of the time taken was confirmed by Italian Jesuits Grimaldi and Riccioli between 1640 and 1650. They also made a calculation of the gravity of Earth by recording the oscillations of a pendulum. Johannes Kepler In his Astronomia 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.... 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. Giovanni Borelli 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. Mechanical explanations In 1644, René Descartes proposed that no empty space can exist and that a continuum of matter causes every motion to be curvilinear. Thus, centrifugal force thrusts relatively light matter away from the central vortices of celestial bodies, lowering density locally and thereby creating centripetal pressure. 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. In 1671, Robert Hooke speculated that gravitation is the result of bodies emitting waves in the aether. Nicolas Fatio de Duillier (1690) and Georges-Louis Le Sage (1748) proposed a 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 expanded Daniel Bernoulli's theory of corpuscular pressure to the universe as a whole. A similar model was later created by Hendrik Lorentz (1853–1928), who used electromagnetic radiation instead of corpuscles. English mathematician Isaac Newton used Descartes' argument that curvilinear motion constrains inertia, and in 1675, argued that aether streams attract all bodies to one another. Newton (1717) and Leonhard Euler (1760) proposed a model in which the aether loses density near mass, leading to a net force acting on bodies. Further mechanical explanations of gravitation (including Le 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 the energy conservation law and are incompatible with modern thermodynamics. 'Weight' before Newton Before Newton, 'weight' had the double meaning 'amount' and 'heaviness'. Mass as distinct from weight (1689) In 1686, Newton gave the concept of mass its name. In the first paragraph of Principia, Newton defined quantity of matter as "density and bulk conjunctly", and mass as quantity of matter. Newton's law of universal gravitation In 1679, Robert Hooke wrote to Isaac Newton of his hypothesis concerning orbital motion as a combination of tangential inertial motion and a central force. He also asked for the precise trajectory implied an inverse-square force. Newton was almost certainly influenced by this correspondence to do his subsequent work on gravitation, In January 1684 Hooke told Edmond Halley and Christopher Wren that he had proven the inverse-square law of planetary motion but he refused to produce his proof. That summer, Halley visited Newton and asked if Newton knew what trajectory an inverse square force would produce. Newton said an ellipse and by November 1684 he sent Halley De motu corporum in gyrum ('On the motion of bodies in an orbit'), in which he mathematically derives Kepler's laws of planetary motion. In Newton's theory (rewritten using more modern mathematics) the density of mass \rho\, generates a scalar field, the gravitational potential \varphi\, in joules per kilogram, by :{\partial^2 \varphi \over \partial x^j \, \partial x^j} = 4 \pi G \rho \,. Using the Nabla operator \nabla for the gradient and divergence (partial derivatives), this can be conveniently written as: :\nabla^2 \varphi = 4 \pi G \rho \,. This scalar field governs the motion of a free-falling particle by: :{ d^2x^j\over dt^2} = -{\partial\varphi\over\partial x^j\,}. At distance r from an isolated mass M, the scalar field is :\varphi = -\frac{GM} r \,. The Principia sold out quickly, inspiring Newton to publish a second edition in 1713. However the theory of gravity itself was not accepted quickly. The theory of gravity faced two barriers. First scientists like Gottfried Wilhelm Leibniz complained that it relied on action at a distance, that the mechanism of gravity was "invisible, intangible, and not mechanical". The French philosopher Voltaire countered these concerns, ultimately writing his own book to explain aspects of it to French readers in 1738, which helped to popularize Newton's theory. Second, detailed comparisons with astronomical data were not initially favorable. Among the most conspicuous issue was the so-called great 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 by Leonhard Euler in 1748, then Joseph-Louis Lagrange in 1763, by Pierre-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. Successes such the solution to the great inequality of Jupiter and Saturn mystery accumulated. In 1755, Prussian philosopher Immanuel Kant published a cosmological manuscript based on Newtonian principles, in which he develops an early version of the nebular hypothesis. 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. Newton's theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted by the actions of the other planets. Calculations by John Couch Adams and Urbain Le Verrier both predicted the general position of the planet. In 1846, Le Verrier sent his position to Johann Gottfried Galle, asking him to verify it. The same night, Galle spotted Neptune near the position Le Verrier had predicted. Not every comparison was successful. By the end of the 19th century, Le Verrier showed that the orbit of Mercury 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. Even so, Newton's theory is thought to be exceptionally accurate in the limit of weak gravitational fields and low speeds. At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics (like those of Wilhelm Eduard Weber, Carl Friedrich Gauss, and Bernhard Riemann) to explain the anomalous perihelion precession of Mercury. In 1890, Maurice Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed 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 of James Clerk Maxwell. == Modern era ==

In 1900, Hendrik Lorentz tried to explain gravity on the basis of his ether theory and Maxwell's equations. He assumed, like Ottaviano Fabrizio Mossotti and Johann 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. In the late 19th century, Lord Kelvin pondered the possibility of a theory of everything. He proposed that every body pulsates, which might be an explanation of gravitation and electric charges. His ideas were largely mechanistic and required the existence of the aether, which the Michelson–Morley experiment failed to detect in 1887. This, combined with Mach's principle, led to gravitational models which feature action at a distance. Albert Einstein developed his revolutionary theory of relativity in papers published in 1905 and 1915; these account for the perihelion precession of Mercury. Thereafter, German mathematician Theodor Kaluza promoted the idea of general relativity with a fifth dimension, which in 1921 Swedish physicist Oskar Klein gave a physical interpretation of in a prototypical string theory, a possible model of quantum gravity and potential theory of everything. Einstein's field equations include a cosmological constant to account for the alleged staticity 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. Alan Guth and Alexei Starobinsky proposed in 1980 that cosmic inflation in the very early universe could have been driven by a negative pressure field, a concept later coined 'dark energy'—found in 2013 to have composed around 68.3% of the early universe. In 1922, Jacobus Kapteyn proposed the existence of dark 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. Early theories of gravity attempted to explain planetary orbits (Newton) and more complicated orbits (e.g. Lagrange). Then came unsuccessful attempts to combine gravity and either wave or corpuscular theories of gravity. The whole landscape of physics was changed with the discovery of Lorentz 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, the gravitational deflection of light, the Eötvös experiment. These considerations led to and past the development of general relativity. Einstein (1905–1912) In 1905, Albert Einstein published a series of papers in which he established the special theory of relativity and the fact that mass 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 (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 the principle of least action: :\delta \int d\tau = 0\, :{d\tau}^2 = - \eta_{\mu \nu} \, dx^\mu \, dx^\nu \, where \eta_{\mu \nu} \, is the Minkowski metric, and there is a summation from 1 to 4 over indices \mu \, and \nu \,. Einstein and Grossmann includes Riemannian geometry and tensor calculus. :\delta \int d\tau = 0 \, :{d\tau}^2 = - g_{\mu \nu} \, dx^\mu \, dx^\nu \, The equations of electrodynamics exactly match those of general relativity. The equation :T^{\mu \nu} = \rho {dx^\mu \over d\tau} {dx^\nu \over d\tau} \, is not in general relativity. It expresses the stress–energy tensor as a function of the matter density. Lorentz-invariant models (1905–1910) Based on the principle of relativity, Henri Poincaré (1905, 1906), Hermann Minkowski (1908), and Arnold Sommerfeld (1910) tried to modify Newton's theory and to establish a Lorentz 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. Abraham (1912) 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. Nordström (1912) The first approach of Nordström (1912) was to retain the Minkowski metric and a constant value of c\, but to let mass depend on the gravitational field strength \varphi\,. Allowing this field strength to satisfy :\Box \varphi = \rho \, where \rho \, is rest mass energy and \Box \, is the d'Alembertian, :m = m_0 \exp\left( \frac \varphi {c^2} \right) \, where m_0 is the mass when gravitational potential vanishes and, :- {\partial \varphi \over \partial x^\mu} = \dot{u}_\mu + {u_\mu \over c^2 \dot{\varphi}} \, where u \, is the four-velocity and the dot is a differential with respect to time. The second approach of Nordström (1913) is remembered as the first logically consistent relativistic field theory of gravitation ever formulated. (notation from Pais not Nordström): :\delta \int \psi \, d\tau = 0 \, :{d\tau}^2 = - \eta_{\mu \nu} \, dx^\mu \, dx^\nu \, where \psi \, is a scalar field, :- {\partial T^{\mu \nu} \over \partial x^\nu} = T {1 \over \psi} {\partial \psi \over \partial x_\mu} \, This theory is Lorentz invariant, satisfies the conservation laws, correctly reduces to the Newtonian limit and satisfies the weak equivalence principle. Einstein and Fokker (1914) This theory is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing: :\delta \int ds = 0 \, :{ds}^2 = g_{\mu \nu} \, dx^\mu \, dx^\nu \, :g_{\mu \nu} = \psi^2 \eta_{\mu \nu} \, they relate Einstein–Grossmann It is believed that neutron star mergers (since detected in 2017) and black hole formation may also create detectable amounts of gravitational radiation. Quantum gravity Several decades after the discovery of general relativity, it was realized that it cannot be the complete theory of gravity because it is incompatible with quantum mechanics. Later it was understood that it is possible to describe gravity in the framework of quantum field theory like the other fundamental forces. In this framework, the attractive force of gravity arises due to exchange of virtual gravitons, in the same way as the electromagnetic force arises from exchange of virtual photons. This reproduces general relativity in the classical limit, but only at the linearized level and postulating that the conditions for the applicability of Ehrenfest theorem holds, which is not always the case. Moreover, this approach fails at short distances of the order of the Planck length. == See also ==