# Gravity

In physics, gravity (from Latin gravitas 'weight'[1]) is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is by far the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong interaction, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weak interaction. As a result, it has no significant influence at the level of subatomic particles.[2] However, gravity is the most significant interaction between objects at the macroscopic scale, and it determines the motion of planets, stars, galaxies, and even light.

Artist's conception of gravity pulling mass away from a star and into a black hole

On Earth, gravity gives weight to physical objects, and the Moon's gravity causes tides in the oceans. Gravity also has many important biological functions, helping to guide the growth of plants through the process of gravitropism and influencing the circulation of fluids in multicellular organisms. Investigation into the effects of weightlessness has shown that gravity may play a role in immune system function and cell differentiation within the human body.

The gravitational attraction between the original gaseous matter in the Universe allowed it to coalesce and form stars which eventually condensed into galaxies, so gravity is responsible for many of the large-scale structures in the Universe. Gravity has an infinite range, although its effects become weaker as objects get farther away.

Gravity is most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915), which describes gravity not as a force, but as the curvature of spacetime, caused by the uneven distribution of mass, and causing masses to move along geodesic lines. The most extreme example of this curvature of 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 well approximated by Newton's law of universal gravitation, which describes gravity as a force causing any two bodies to be attracted toward each other, with magnitude proportional to the product of their masses and inversely proportional to the square of the distance between them.

Current models of particle physics imply that the earliest instance of gravity in the Universe, possibly in the form 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 a primeval state, such as a false vacuum, quantum vacuum or virtual particle, in a currently unknown manner.[4] Scientists are currently working to develop a theory of gravity consistent with quantum mechanics, a quantum gravity theory, which would allow gravity to be united in a common mathematical framework (a theory of everything) with the other three fundamental interactions of physics.

## History

### Ancient world

The nature and mechanism of gravity was explored by a wide range of ancient scholars. In Greece, Aristotle believed that objects fell towards the Earth because the Earth was the center of the Universe and attracted all of the mass in the Universe towards it. He also thought that the speed of a falling object should increase with its weight, a conclusion which was later shown to be false.[5] While Aristotle's view was widely accepted throughout Ancient Greece, there were other thinkers such as Plutarch who correctly predicted that the attraction of gravity was not unique to the Earth.[6]

Although he didn't understand gravity as a force, the ancient Greek philosopher Archimedes discovered the center of gravity of a triangle.[7] He also postulated that if two equal weights did not have the same center of gravity, the center of gravity of the two weights together would be in the middle of the line that joins their centers of gravity.[8]

In India, the mathematician-astronomer Aryabhata first identified gravity to explain why objects are not driven away from the Earth by the centrifugal force of the planet's rotation. Later, in the seventh century CE, Brahmagupta proposed the idea that gravity is an attractive force which draws objects to the Earth and used the term gurutvākarṣaṇ to describe it.[9][10][11]

In the ancient Middle East, gravity was a topic of fierce debate. The Persian intellectual Al-Biruni believed that the force of gravity was not unique to the Earth, and he correctly assumed that other heavenly bodies should exert a gravitational attraction as well.[12] In contrast, Al-Khazini held the same position as Aristotle that all matter in the Universe is attracted to the center of the Earth.[13]

The Leaning Tower of Pisa, where Galileo performed one of the most famous experiments about the speed of falling objects

### Scientific revolution

In the mid-16th century, various European scientists experimentally disproved the Aristotelian notion that heavier objects fall at a faster rate.[14] In particular, the Spanish Dominican priest Domingo de Soto wrote in 1551 that bodies in free fall uniformly accelerate.[14] De Soto may have been influenced by earlier experiments conducted by other Dominican priests in Italy, including those by Benedetto Varchi, Francesco Beato, Luca Ghini, and Giovan Bellaso which contradicted Aristotle's teachings on the fall of bodies.[14] The mid-16th century Italian physicist Giambattista Benedetti published papers claiming that, due to specific gravity, objects made of the same material but with different masses would fall at the same speed.[15] With the 1586 Delft tower experiment, the Flemish physicist Simon Stevin observed that two cannonballs of differing sizes and weights fell at the same rate when dropped from a tower.[16] Finally, in the late 16th century, Galileo Galilei performed his famous Leaning Tower of Pisa experiment in order to show once again that balls of different weights would fall at the same speed.[17] Combining this knowledge with careful measurements of balls rolling down inclines, Galileo firmly established that gravitational acceleration is the same for all objects.[18] Galileo postulated that air resistance is the reason that objects with a low density and high surface area fall more slowly in an atmosphere.

In 1604, Galileo correctly hypothesized that the distance of a falling object is proportional to the square of the time elapsed.[19] This was later confirmed by Italian scientists Jesuits Grimaldi and Riccioli between 1640 and 1650. They also calculated the magnitude of the Earth's gravity by measuring the oscillations of a pendulum.[20]

### Newton's theory of gravitation

English physicist and mathematician, Sir Isaac Newton (1642–1727)

In 1684, Newton sent a manuscript to Edmond Halley titled De motu corporum in gyrum ('On the motion of bodies in an orbit'), which provided a physical justification for Kepler's laws of planetary motion.[21] Halley was impressed by the manuscript and urged Newton to expand on it, and a few years later Newton published a groundbreaking book called Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). In this book, Newton described gravitation as a universal force, and claimed that "the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve." This statement was later condensed into the following inverse-square law:

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}$

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 the gravitational constant.

Newton's Principia was well-received by the scientific community, and his law of gravitation quickly spread across the European world.[22] More than a century later, in 1821, his theory of gravitation rose to even greater prominence when it was used to predict the existence of Neptune. In that year, the French astronomer Alexis Bouvard used this theory to create a table modeling the orbit of Uranus, which was shown to differ significantly from the planet's actual trajectory. In order to explain this discrepancy, many astronomers speculated that there might be a large object beyond the orbit of Uranus which was disrupting its orbit. In 1846, the astronomers John Couch Adams and Urbain Le Verrier independently used Newton's law to predict Neptune's location in the night sky, and the planet was discovered there within a day.[23]

### General relativity

Eventually, astronomers noticed an eccentricity in the orbit of the planet Mercury which could not be explained by Newton's theory: the perihelion of the orbit was increasing by about 42.98 arcseconds per century. The most obvious explanation for this discrepancy was an as-yet-undiscovered celestial body (such as a planet orbiting the Sun even closer than Mercury), but all efforts to find such a body turned out to be fruitless. Finally, in 1915, Albert Einstein developed a theory of general relativity which was able to accurately model Mercury's orbit.[24]

In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of a force. Einstein began to toy with this idea in the form of the equivalence principle, a discovery which he later described as "the happiest thought of my life."[25] In this theory, free fall is considered to be equivalent to inertial motion, meaning that free-falling inertial objects are accelerated relative to non-inertial observers on the ground.[26][27] In contrast to Newtonian physics, Einstein believed that it was possible for this acceleration to occur without any force being applied to the object.

Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime. These straight paths are called geodesics. As in Newton's first law of motion, Einstein believed that a force applied to an object would cause it to deviate from a geodesic. For instance, people standing on the surface of the Earth are prevented from following a geodesic path because the mechanical resistance of the Earth exerts an upward force on them. This explains why moving along the geodesics in spacetime is considered inertial.

Einstein's description of gravity was quickly accepted by the majority of physicists, as it was able to explain a wide variety of previously baffling experimental results.[28] In the coming years, a wide range of experiments provided additional support for the idea of general relativity.[29][30][31][32] Today, Einstein's theory of relativity is used for all gravitational calculations where absolute precision is desired, although Newton's inverse-square law continues to be a useful and fairly accurate approximation.[33]

## Modern research

In modern physics, general relativity remains the framework for the understanding of gravity.[34] Physicists continue to work to find solutions to the Einstein field equations that form the basis of general relativity, while some scientists have speculated that general relativity may not be applicable at all in certain scenarios.[33]

### Einstein field equations

The Einstein field equations are a system of 10 partial differential equations which describe how matter effects the curvature of spacetime. The system is often expressed in the following simplified form:

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu },}$

where Gμν is the Einstein tensor, gμν is the metric tensor, Tμν is the stress–energy tensor, Λ is the cosmological constant, ${\displaystyle G}$  is Newton's gravitational constant and ${\displaystyle c}$  is the speed of light.[35] The term ${\displaystyle {\frac {8\pi G}{c^{4}}}}$  is sometimes referred to as the Einstein gravitational constant ${\displaystyle \kappa }$ .[36]

An illustration of the Schwarzchild metric, which describes spacetime around a spherical, uncharged, and nonrotating object with mass

A major area of research is the discovery of exact solutions to the Einstein field equations. Solving these equations amounts to calculating a precise value for the metric tensor (which defines the curvature and geometry of spacetime) under certain physical conditions. There is no formal definition for what constitutes such solutions, but most scientists agree that they should be expressable using elementary functions or linear differential equations.[37] Some of the most notable solutions of the equations include:

• The Schwarzschild solution, which describes spacetime surrounding a spherically symmetric non-rotating uncharged massive object. For compact enough objects, this solution generated a black hole with a central singularity.[38] At points far away from the central mass, the accelerations predicted by the Schwarzschild solution are practically identical to those predicted by Newton's theory of gravity.[39]
• The Reissner-Nordström solution, which analyzes a non-rotating spherically symmetric object with charge and was independently discovered by several different researchers between 1916 and 1921.[40] In some cases, this solution can predict the existence of black holes with double event horizons.[41]
• The Kerr solution, which generalizes the Schwarzchild solution to rotating massive objects. Because of the difficulty of factoring in the effects of rotation into Einstein's field equations, this solution was not discovered until 1963.[42]
• The Kerr-Newman solution for charged, rotating massive objects. This solution was derived in 1964, using the same technique of complex coordinate transformation that was used for the Kerr solution.[43]
• The cosmological Friedmann-Lemaître-Robertson-Walker solution, discovered in 1922 by Alexander Friedmann and then confirmed in 1927 by Georges Lemaître. This solution was revolutionary for predicting the expansion of the Universe, which was confirmed seven years later after a series of measurements by Edwin Hubble.[44] It even showed that general relativity was incompatible with a static universe, and Einstein later conceded that he had been wrong to design his field equations to account for a Universe that was not expanding.[45]

Today, there remain many important situations in which the Einstein field equations have not been solved. Chief among these is the two-body problem, which concerns the geometry of spacetime around two mutually interacting massive objects (such as the Sun and the Earth, or the two stars in a binary star system). The situation gets even more complicated when considering the interactions of three or more massive bodies (the "n-body problem"), and some scientists suspect that the Einstein field equations will never be solved in this context.[46] However, it is still possible to construct an approximate solution to the field equations in the n-body problem by using the technique of post-Newtonian expansion.[47] In general, the extreme nonlinearity of Einstein's field equations makes it difficult to solve them in all but the most specific cases.[48]

### Gravity and quantum mechanics

Despite its success in predicting the effects of gravity at large scales, general relativity is ultimately a classical theory which is incompatible with quantum mechanics. In most cases, this doesn't matter, because gravity is usually only relevant at large scales while quantum physics is only relevant at very small ones. However, in order to understand environments such as the singularity of a black hole (where an extremely large amount of mass is compressed into a small space), many physicists have begun to search for a theory that could unite both gravity and quantum mechanics under a more general framework.[49]

One path is to describe gravity in the framework of quantum field theory, which has been successful to accurately describe the other fundamental interactions. The electromagnetic force arises from an exchange of virtual photons, where the QFT description of gravity is that there is an exchange of virtual gravitons.[50][51] This description reproduces general relativity in the classical limit. However, this approach fails at short distances of the order of the Planck length,[52] where a more complete theory of quantum gravity (or a new approach to quantum mechanics) is required.

### Tests of general relativity

It is difficult to test the theory of general relativity on Earth, because for small masses and velocities its predictions are almost identical to that of Newtonian gravity.[53] Still, since its development, an ongoing series of experimental results have provided support for the theory:[54]

The 1919 total solar eclipse provided one of the first opportunities to test the predictions of general relativity
• In 1919, the British astrophysicist Arthur Eddington was able to confirm the predicted gravitational lensing of light during that year's solar eclipse.[55][56] Eddington measured starlight deflections twice those predicted by Newtonian corpuscular theory, in accordance with the predictions of general relativity. Although Eddington's analysis was later disputed, this experiment made Einstein famous almost overnight and caused general relativity to become widely accepted in the scientific community.[57]
• In 1959, American physicists Robert Pound and Glen Rebka performed an experiment in which they used gamma rays to confirm the prediction of gravitational time dilation. By sending the rays down a 74-foot tower and measuring their frequency at the bottom, the scientists confirmed that light is redshifted as it moves towards a source of gravity. The observed redshift also supported the idea that time runs more slowly in the presence of a gravitational field.[58]
• The time delay of light passing close to a massive object was first identified by Irwin I. Shapiro in 1964 in interplanetary spacecraft signals.[59]
• In 1971, scientists discovered the first-ever black hole in the galaxy Cygnus. The black hole was detected because it was emitting bursts of x-rays as it consumed a smaller star, and it came to be known as Cygnus X-1.[60] This discovery confirmed yet another prediction of general relativity, because Einstein's equations implied that light could not escape from a sufficiently large and compact object.[61]
• General relativity states that gravity acts on light and matter equally, meaning that a sufficiently massive object could warp light around it and create a gravitational lens. This phenomenon was observed for the first time in 1979, when the Hubble telescope saw two mirror images of the same quasar whose light had been bent around the galaxy YGKOW G1.[62]
• Frame dragging, the idea that a rotating massive object should twist spacetime around it, was confirmed by Gravity Probe B results in 2011.[63][64]
• In 2015, the LIGO observatory detected faint gravitational waves, the existence of which had been predicted by general relativity. Scientists believe that the waves emanated from a black hole merger that occurred 1.5 billion light-years away.[65]

## Specifics

### Earth's gravity

An initially-stationary object that is allowed to fall freely under gravity drops a distance that is proportional to the square of the elapsed time. This image spans half a second and was captured at 20 flashes per second.

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

If an object with comparable mass to that of the Earth were to fall towards it, then the corresponding acceleration of the Earth would be observable.

The strength of the gravitational field is numerically equal to the acceleration of objects under its influence.[66] The rate of acceleration of falling objects near the Earth's surface varies very slightly depending on latitude, surface features such as mountains and ridges, and perhaps unusually high or low sub-surface densities.[67] For purposes of weights and measures, a standard gravity value is defined by the International Bureau of Weights and Measures, under the International System of Units (SI).

The force of gravity on Earth is the resultant (vector sum) of two forces:[68] (a) The gravitational attraction in accordance with Newton's universal law of gravitation, and (b) the centrifugal force, which results from the choice of an earthbound, rotating frame of reference. The force of gravity is weakest at the equator because of the centrifugal force caused by the Earth's rotation and because points on the equator are furthest from the center of the Earth. The force of gravity varies with latitude and increases from about 9.780 m/s2 at the Equator to about 9.832 m/s2 at the poles.[citation needed]

### Origin

The earliest gravity (possibly in the form 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 a primeval state (such as a false vacuum, quantum vacuum or virtual particle), in a currently unknown manner.[4]

The LIGO Hanford Observatory located in Washington, United States, where gravitational waves were first observed in September 2015.

General relativity predicts that energy can be transported out of a system through gravitational radiation. The first indirect evidence for gravitational radiation was through measurements of the Hulse–Taylor binary in 1973. This system consists of a pulsar and neutron star in orbit around one another. Its orbital period has decreased since its initial discovery due to a loss of energy, which is consistent for the amount of energy loss due to gravitational radiation. This research was awarded the Nobel Prize in Physics in 1993.[citation needed]

The first direct evidence for gravitational radiation was measured on 14 September 2015 by the LIGO detectors. The gravitational waves emitted during the collision of two black holes 1.3 billion-light years from Earth were measured.[69][70] This observation confirms the theoretical predictions of Einstein and others that such waves exist. It also opens the way for practical observation and understanding of the nature of gravity and events in the Universe including the Big Bang.[71] Neutron star and black hole formation also create detectable amounts of gravitational radiation.[72] This research was awarded the Nobel Prize in physics in 2017.[73]

### Speed of gravity

In December 2012, a research team in China announced that it had produced measurements of the phase lag of Earth tides during full and new moons which seem to prove that the speed of gravity is equal to the speed of light.[74] This means that if the Sun suddenly disappeared, the Earth would keep orbiting the vacant point normally for 8 minutes, which is the time light takes to travel that distance. The team's findings were released in the Chinese Science Bulletin in February 2013.[75]

In October 2017, the LIGO and Virgo detectors received gravitational wave signals within 2 seconds of gamma ray satellites and optical telescopes seeing signals from the same direction. This confirmed that the speed of gravitational waves was the same as the speed of light.[76]

## Anomalies and discrepancies

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

Rotation curve of a typical spiral galaxy: predicted (A) and observed (B). The discrepancy between the curves is attributed to dark matter.
• Extra-fast stars: Stars in galaxies follow a distribution of velocities where stars on the outskirts are moving faster than they should according to the observed distributions of normal matter. Galaxies within galaxy clusters show a similar pattern. Dark matter, which would interact through gravitation but not electromagnetically, would account for the discrepancy. Various modifications to Newtonian dynamics have also been proposed.
• Flyby anomaly: Various spacecraft have experienced greater acceleration than expected during gravity assist maneuvers.
• Accelerating expansion: The metric expansion of space seems to be speeding up. Dark energy has been proposed to explain this. A recent alternative explanation is that the geometry of space is not homogeneous (due to clusters of galaxies) and that when the data are reinterpreted to take this into account, the expansion is not speeding up after all,[77] however this conclusion is disputed.[78]
• Anomalous increase of the astronomical unit: Recent measurements indicate that planetary orbits are widening faster than if this were solely through the Sun losing mass by radiating energy.
• Extra energetic photons: Photons travelling through galaxy clusters should gain energy and then lose it again on the way out. The accelerating expansion of the Universe should stop the photons returning all the energy, but even taking this into account photons from the cosmic microwave background radiation gain twice as much energy as expected. This may indicate that gravity falls off faster than inverse-squared at certain distance scales.[79]
• Extra massive hydrogen clouds: The spectral lines of the Lyman-alpha forest suggest that hydrogen 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.[79]

## Footnotes

1. ^ "dict.cc dictionary :: gravitas :: English-Latin translation". Archived from the original on 13 August 2021. Retrieved 11 September 2018.
2. ^ Krebs, Robert E. (1999). Scientific Development and Misconceptions Through the Ages: A Reference Guide (illustrated ed.). Greenwood Publishing Group. p. 133. ISBN 978-0-313-30226-8.
3. ^ "HubbleSite: Black Holes: Gravity's Relentless Pull". hubblesite.org. Archived from the original on 26 December 2018. Retrieved 7 October 2016.
4. ^ a b Staff. "Birth of the Universe". University of Oregon. Archived from the original on 28 November 2018. Retrieved 24 September 2016. – discusses "Planck time" and "Planck era" at the very beginning of the Universe
5. ^ Cappi, Alberto. "The concept of gravity before Newton" (PDF). Culture and Cosmos.{{cite web}}: CS1 maint: url-status (link)
6. ^ Bakker, Frederik; Palmerino, Carla Rita (1 June 2020). "Motion to the Center or Motion to the Whole? Plutarch's Views on Gravity and Their Influence on Galileo". Isis. 111 (2): 217–238. doi:10.1086/709138. ISSN 0021-1753. S2CID 219925047.
7. ^ Reviel Neitz; William Noel (13 October 2011). The Archimedes Codex: Revealing The Secrets of the World's Greatest Palimpsest. Hachette UK. p. 125. ISBN 978-1-78022-198-4. Archived from the original on 7 January 2020. Retrieved 10 April 2019.{{cite book}}: CS1 maint: multiple names: authors list (link)
8. ^ CJ Tuplin, Lewis Wolpert (2002). Science and Mathematics in Ancient Greek Culture. Hachette UK. p. xi. ISBN 978-0-19-815248-4. Archived from the original on 17 January 2020. Retrieved 10 April 2019.
9. ^ Pickover, Clifford (16 April 2008). Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 9780199792689. Archived from the original on 18 January 2017. Retrieved 29 August 2017.
10. ^ Bose, Mainak Kumar (1988). Late classical India. A. Mukherjee & Co. Archived from the original on 13 August 2021. Retrieved 28 July 2021.
11. ^ * Sen, Amartya (2005). The Argumentative Indian. Allen Lane. p. 29. ISBN 978-0-7139-9687-6.
12. ^ Starr, S. Frederick (2015). Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane. Princeton University Press. p. 260. ISBN 9780691165851.
13. ^ Rozhanskaya, Mariam; Levinova, I. S. (1996). "Statics". In Rushdī, Rāshid (ed.). Encyclopedia of the History of Arabic Science. Vol. 2. Psychology Press. pp. 614–642. ISBN 9780415124119.
14. ^ a b c Wallace, William A. (2018) [2004]. Domingo de Soto and the Early Galileo: Essays on Intellectual History. Abingdon, UK: Routledge. pp. 119, 121–22. ISBN 978-1-351-15959-3. Archived from the original on 16 June 2021. Retrieved 4 August 2021.
15. ^ Drabkin, I. E. (1963). "Two Versions of G. B. Benedetti's Demonstratio Proportionum Motuum Localium". Isis. 54 (2): 259–262. doi:10.1086/349706. ISSN 0021-1753. JSTOR 228543. S2CID 144883728.
16. ^ Schilling, Govert (31 July 2017). Ripples in Spacetime: Einstein, Gravitational Waves, and the Future of Astronomy. Harvard University Press. p. 26. ISBN 9780674971660. Archived from the original on 16 December 2021. Retrieved 16 December 2021.
17. ^ Ball, Phil (June 2005). "Tall Tales". Nature News. doi:10.1038/news050613-10.
18. ^ Galileo (1638), Two New Sciences, First Day Salviati speaks: "If this were what Aristotle meant you would burden him with another error which would amount to a falsehood; because, since there is no such sheer height available on earth, it is clear that Aristotle could not have made the experiment; yet he wishes to give us the impression of his having performed it when he speaks of such an effect as one which we see."
19. ^ Gillispie, Charles Coulston (1960). The Edge of Objectivity: An Essay in the History of Scientific Ideas. Princeton University Press. pp. 3–6. ISBN 0-691-02350-6.
20. ^ J.L. Heilbron, Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics (Berkeley: University of California Press, 1979), 180.
21. ^ Sagan, Carl & Druyan, Ann (1997). Comet. New York: Random House. pp. 52–58. ISBN 978-0-3078-0105-0. Archived from the original on 15 June 2021. Retrieved 5 August 2021.
22. ^ "The Reception of Newton's Principia" (PDF). Retrieved 6 May 2022.
23. ^ "This Month in Physics History". www.aps.org. Retrieved 6 May 2022.
24. ^ Nobil, Anna M. (March 1986). "The real value of Mercury's perihelion advance". Nature. 320 (6057): 39–41. Bibcode:1986Natur.320...39N. doi:10.1038/320039a0. S2CID 4325839.
25. ^ Webb, Joh; Dougan, Darren (23 November 2015). "Without Einstein it would have taken decades longer to understand gravity". Retrieved 21 May 2022.{{cite web}}: CS1 maint: url-status (link)
26. ^ "Gravity and Warped Spacetime". black-holes.org. Archived from the original on 21 June 2011. Retrieved 16 October 2010.
27. ^ Dmitri Pogosyan. "Lecture 20: Black Holes – The Einstein Equivalence Principle". University of Alberta. Archived from the original on 8 September 2013. Retrieved 14 October 2011.
28. ^ Brush, S. G. (1 January 1999). "Why was Relativity Accepted?". Physics in Perspective. 1 (2): 184–214. Bibcode:1999PhP.....1..184B. doi:10.1007/s000160050015. ISSN 1422-6944. S2CID 51825180.
29. ^ Lindley, David (12 July 2005). "The Weight of Light". Physics. 16.
30. ^ "Hafele-Keating Experiment". hyperphysics.phy-astr.gsu.edu. Retrieved 22 May 2022.
31. ^ "How the 1919 Solar Eclipse Made Einstein the World's Most Famous Scientist". Discover Magazine. Retrieved 22 May 2022.
32. ^ "At Long Last, Gravity Probe B Satellite Proves Einstein Right". www.science.org. Retrieved 22 May 2022.
33. ^ a b "Einstein showed Newton was wrong about gravity. Now scientists are coming for Einstein". NBC News. Retrieved 22 May 2022.
34. ^ Stephani, Hans (2003). Exact Solutions to Einstein's Field Equations. p. 1. ISBN 978-0-521-46136-8.
35. ^ "Einstein Field Equations (General Relativity)". University of Warwick. Retrieved 24 May 2022.
36. ^ "How to understand Einstein's equation for general relativity". Big Think. Retrieved 24 May 2022.
37. ^ Ishak, Mustafa. "Exact Solutions to Einstein's Equations in Astrophysics" (PDF). University of Texas at Dallas. Retrieved 25 May 2022.{{cite web}}: CS1 maint: url-status (link)
38. ^ "The Schwarzchild Metric and Applications" (PDF). p. 36. Retrieved 26 May 2022.{{cite web}}: CS1 maint: url-status (link)
39. ^ Ehlers, Jurgen (1997). "Examples of Newtonian limits of relativistic spacetimes". Classical Quantum Gravity. 14 (1A): 122–123. Bibcode:1997CQGra..14A.119E. doi:10.1088/0264-9381/14/1A/010. hdl:11858/00-001M-0000-0013-5AC5-F.
40. ^ "Surprise: the Big Bang isn't the beginning of the universe anymore". Big Think. Retrieved 26 May 2022.
41. ^ Norebo, Jonatan (16 March 2016). "The Reissner-Nordström metric" (PDF).{{cite web}}: CS1 maint: url-status (link)
42. ^ Teukolsky, Saul (1 June 2015). "The Kerr metric" (PDF). Classical and Quantum Gravity. 32 (12): 124006. arXiv:1410.2130. Bibcode:2015CQGra..32l4006T. doi:10.1088/0264-9381/32/12/124006. S2CID 119219499.
43. ^ Newman, E. T.; Couch, E.; Chinnapared, K.; Exton, A.; Prakash, A.; Torrence, R. (June 1965). "Metric of a Rotating, Charged Mass". Journal of Mathematical Physics. 6 (6): 918–919. Bibcode:1965JMP.....6..918N. doi:10.1063/1.1704351. ISSN 0022-2488. S2CID 122962090.
44. ^ Pettini, M. "RELATIVISTIC COSMOLOGY" (PDF). Retrieved 27 May 2022.{{cite web}}: CS1 maint: url-status (link)
45. ^ O’Raifeartaigh, Cormac; O’Keeffe, Michael (2017). "Einstein's 1917 Static Model of the Universe: A Centennial Review". The European Physical Journal H. 42 (3): 41. arXiv:1701.07261. Bibcode:2017EPJH...42..431O. doi:10.1140/epjh/e2017-80002-5. S2CID 119461771.
46. ^ Siegel, Ethan. "This Is Why Scientists Will Never Exactly Solve General Relativity". Forbes. Retrieved 27 May 2022.
47. ^ Spyrou, N. (1 May 1975). "The N-body problem in general relativity". The Astrophysical Journal. 197: 725–743. Bibcode:1975ApJ...197..725S. doi:10.1086/153562. ISSN 0004-637X.
48. ^ Sleator, Daniel (6 June 1996). "Hermeneutics of Classical General Relativity". Retrieved 23 May 2022.{{cite web}}: CS1 maint: url-status (link)
49. ^ Huggett, Nick; Matsubara, Keizo; Wüthrich, Christian (2020). Beyond Spacetime: The Foundations of Quantum Gravity. Cambridge University Press. p. 6. ISBN 9781108655705.
50. ^ Feynman, R.P.; Morinigo, F.B.; Wagner, W.G.; Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 978-0-201-62734-3.
51. ^ Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press. ISBN 978-0-691-01019-9.
52. ^ Randall, Lisa (2005). Warped Passages: Unraveling the Universe's Hidden Dimensions. Ecco. ISBN 978-0-06-053108-9.
53. ^ "Testing General Relativity". NASA Blueshift. Retrieved 29 May 2022.
54. ^ "Testing General Relativity". NASA Blueshift. Retrieved 29 May 2022.
55. ^ Dyson, F.W.; Eddington, A.S.; Davidson, C.R. (1920). "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919". Phil. Trans. Roy. Soc. A. 220 (571–581): 291–333. Bibcode:1920RSPTA.220..291D. doi:10.1098/rsta.1920.0009. Archived from the original on 15 May 2020. Retrieved 1 July 2019.. Quote, p. 332: "Thus the results of the expeditions to Sobral and Principe can leave little doubt that a deflection of light takes place in the neighbourhood of the sun and that it is of the amount demanded by Einstein's generalised theory of relativity, as attributable to the sun's gravitational field."
56. ^ Weinberg, Steven (1972). Gravitation and cosmology. John Wiley & Sons. ISBN 9780471925675.. Quote, p. 192: "About a dozen stars in all were studied, and yielded values 1.98 ± 0.11" and 1.61 ± 0.31", in substantial agreement with Einstein's prediction θ = 1.75"."
57. ^ Gilmore, Gerard; Tausch-Pebody, Gudrun (20 March 2022). "The 1919 eclipse results that verified general relativity and their later detractors: a story re-told". Notes and Records: The Royal Society Journal of the History of Science. 76 (1): 155–180. doi:10.1098/rsnr.2020.0040. S2CID 225075861.
58. ^ "General Astronomy Addendum 10: Graviational Redshift and time dilation". homepage.physics.uiowa.edu. Retrieved 29 May 2022.
59. ^ Asada, Hideki (20 March 2008). "Gravitational time delay of light for various models of modified gravity". Physics Letters B. 661 (2–3): 78–81. arXiv:0710.0477. Bibcode:2008PhLB..661...78A. doi:10.1016/j.physletb.2008.02.006. S2CID 118365884.
60. ^ "The Fate of the First Black Hole". www.science.org. Retrieved 30 May 2022.
61. ^ "Black Holes Science Mission Directorate". webarchive.library.unt.edu. Retrieved 30 May 2022.
62. ^ "Hubble, Hubble, Seeing Double!". NASA. 24 January 2014. Retrieved 31 May 2022.{{cite web}}: CS1 maint: url-status (link)
63. ^ "NASA's Gravity Probe B Confirms Two Einstein Space-Time Theories". Nasa.gov. Archived from the original on 22 May 2013. Retrieved 23 July 2013.
64. ^ ""Frame-Dragging" in Local Spacetime" (PDF). Stanford University.{{cite web}}: CS1 maint: url-status (link)
65. ^ "Gravitational Waves Detected 100 Years After Einstein's Prediction". LIGO Lab Caltech. Retrieved 30 May 2022.
66. ^ Cantor, G.N.; Christie, J.R.R.; Hodge, M.J.S.; Olby, R.C. (2006). Companion to the History of Modern Science. Routledge. p. 448. ISBN 978-1-134-97751-2. Archived from the original on 17 January 2020. Retrieved 22 October 2017.
67. ^ Nemiroff, R.; Bonnell, J., eds. (15 December 2014). "The Potsdam Gravity Potato". Astronomy Picture of the Day. NASA.
68. ^ Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). Springer. ISBN 978-3-211-33544-4. § 2.1: "The total force acting on a body at rest on the earth's surface is the resultant of gravitational force and the centrifugal force of the earth's rotation and is called gravity".{{cite book}}: CS1 maint: postscript (link)
69. ^ Clark, Stuart (11 February 2016). "Gravitational waves: scientists announce 'we did it!' – live". the Guardian. Archived from the original on 22 June 2018. Retrieved 11 February 2016.
70. ^ Castelvecchi, Davide; Witze, Witze (11 February 2016). "Einstein's gravitational waves found at last". Nature News. doi:10.1038/nature.2016.19361. S2CID 182916902. Archived from the original on 12 February 2016. Retrieved 11 February 2016.
71. ^ "WHAT ARE GRAVITATIONAL WAVES AND WHY DO THEY MATTER?". popsci.com. 13 January 2016. Archived from the original on 3 February 2016. Retrieved 12 February 2016.
72. ^ Abbott, B. P.; et al. (LIGO Scientific Collaboration & Virgo Collaboration) (October 2017). "GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral" (PDF). Physical Review Letters. 119 (16): 161101. arXiv:1710.05832. Bibcode:2017PhRvL.119p1101A. doi:10.1103/PhysRevLett.119.161101. PMID 29099225. Archived (PDF) from the original on 8 August 2018. Retrieved 28 September 2019.
73. ^ Devlin, Hanna (3 October 2017). "Nobel prize in physics awarded for discovery of gravitational waves". the Guardian. Archived from the original on 3 October 2017. Retrieved 3 October 2017.
74. ^ Chinese scientists find evidence for speed of gravity Archived 8 January 2013 at the Wayback Machine, astrowatch.com, 12/28/12.
75. ^ TANG, Ke Yun; HUA ChangCai; WEN Wu; CHI ShunLiang; YOU QingYu; YU Dan (February 2013). "Observational evidences for the speed of the gravity based on the Earth tide". Chinese Science Bulletin. 58 (4–5): 474–477. Bibcode:2013ChSBu..58..474T. doi:10.1007/s11434-012-5603-3.
76. ^ "GW170817 Press Release". LIGO Lab – Caltech. Archived from the original on 17 October 2017. Retrieved 24 October 2017.
77. ^ Dark energy may just be a cosmic illusion Archived 13 August 2021 at the Wayback Machine, New Scientist, issue 2646, 7 March 2008.
78. ^ Swiss-cheese model of the cosmos is full of holes Archived 6 May 2015 at the Wayback Machine, New Scientist, issue 2678, 18 October 2008.
79. ^ a b Chown, Marcus (16 March 2009). "Gravity may venture where matter fears to tread". New Scientist. Archived from the original on 18 December 2012. Retrieved 4 August 2013.
80. ^ Brans, C.H. (March 2014). "Jordan-Brans-Dicke Theory". Scholarpedia. 9 (4): 31358. arXiv:gr-qc/0207039. Bibcode:2014Schpj...931358B. doi:10.4249/scholarpedia.31358.
81. ^ Horndeski, G.W. (September 1974). "Second-Order Scalar-Tensor Field Equations in a Four-Dimensional Space". International Journal of Theoretical Physics. 88 (10): 363–384. Bibcode:1974IJTP...10..363H. doi:10.1007/BF01807638. S2CID 122346086.
82. ^ Milgrom, M. (June 2014). "The MOND paradigm of modified dynamics". Scholarpedia. 9 (6): 31410. Bibcode:2014SchpJ...931410M. doi:10.4249/scholarpedia.31410.
83. ^ Haugan, Mark P; Lämmerzahl, C (2011). "Einstein gravity from conformal gravity". arXiv:1105.5632 [hep-th].