Theoreticians have formulated a set of fundamental criteria that any theory of gravity should satisfy, two purely theoretical and two that are based on experimental evidence. Thus, a theory must be:
- complete (capable of analyzing from the “first principles” the result of any experiment of interest).
- self-consistent (its prediction for the outcome of each experiment must be unique)
- relativistic (at the limit when gravity is neglected compared to other physical interactions, non-gravitational laws of physics must be reduced to special relativity laws)
- with the correct Newtonian limit (within the limits of weak gravitational fields and slow motions, they must reproduce Newton’s laws)
The main theories of gravity from 1686-1900, until the development by Lorentz of his own theory and then the elaboration the theories of relativity by Einstein, are
- Newton’s Law of Universal Gravity (1686): Newton’s theory is considered to be exactly within the limits of low gravity fields and low velocities, and all other theories of gravity must reproduce Newton’s theory within the appropriate limits.
- Mechanistic explanations (1650-1900): Bifurcated theories having as hard core the mechanistic theory; they failed because most led to an unacceptably high value of aether dragging, which is unconfirmed, violates the law on energy conservation and is incompatible with modern thermodynamics.
- René Descartes (1644) and Christiaan Huygens (1690) used vortexes to explain the mechanistic gravity. Newton opposed the theory arguing with the lack of deviations of the orbits due to the fluid-dynamic resistance, the sometimes-different direction of the natural satellites from the direction of the vortex, and of Huygens’s circular explanations.
- Electrostatic models (1870-1900): They tried to combine Newton’s laws with those of electrodynamics (Weber, Carl Friedrich Gauss, Bernhard Riemann, James Clerk Maxwell), trying to explain the perihelion precession of Mercury. There were partial successes, in 1890 Lévy and in 1898 Paul Gerber, but the models were rejected because were based on assumptions that later proved to be wrong.
- Robert Hooke (1671) and James Challis (1869) assumed that each body emits waves whose effect is the attraction between bodies. Maxwell argued that this theory requires constant production of waves, which must be accompanied by infinite energy consumption. Challis himself acknowledged that he did not reach a precise result due to the complexity of the processes.
- Including Isaac Newton (1675), and later Bernhard Riemann (1853) proposed a theory that aetheric flows move all bodies to one another. As with Le Sage’s theory, the theory violates the law of energy conservation. There are also problems related to the interaction of bodies with aether.
- Nicolas Fatio de Duillier (1690) and Georges-Louis Le Sage (1748) proposed a corpuscular model, using some sort of screening or shading mechanism – a bifurcation of Newton’s law that respects the law of inverse squares. It was re-invented, among others, by Lord Kelvin (1872) and Hendrik Lorentz (1900), and criticized by James Clerk Maxwell (1875) and Henri Poincaré (1908) in particular for thermodynamic anomalies. Le Sage’s theory was studied by Radzievskii and Kagalnikova (1960), Shneiderov (1961), Buonomano and Engels (1976), Adamut (1982), Jaakkola (1996), Tom Van Flandern (1999) and Edwards (2007). A variety of Le Sage models and related topics are discussed in Edwards, et al. 
- Newton proposed a second theory based on aether (1717) developed later by Leonhard Euler (1760), in which the aether loses its density near mass, leading to a net force directed toward bodies. James Clerk Maxwell pointed out that in this “hydrostatic” model “the state of stress… which we must suppose to exist in the invisible medium, is 3000 times greater than that which the strongest steel could support.”
- Later, a similar model was created by Hendrik Lorentz, who used electromagnetic radiation instead of corpuscles.
- Lord Kelvin (1871) and Carl Anton Bjerknes (1871) considered that each body pulsates, which could be an explanation of gravity and electrical charges. This hypothesis was also studied by George Gabriel Stokes and Woldemar Voigt. But the theory forces the assumption that all pulsations in the universe are in phase, which seems highly unlikely. And the aether should be incompressible. Maxwell argued that this process must be accompanied by new production and permanent destruction of aether.
Clifford M. Will explains, in Theory and experiment in gravitational physics, the motivations of some of these theories, including the elaboration of general relativity and quantum theory, which include bifurcations of Newton’s initial theory, or do not meet the current criteria of a gravitational theory, with the observation that it is possible that, in the case of the modification of the present forms, some of these theories may subsequently meet these criteria:
- Newtonian theory of gravity: it is not relativistic
- Milne’s kinematic relativity: it was initially designed to solve certain cosmological problems. It is incomplete – it does not predict gravitational redshift.
- The various vector theories of Kustaanheimo  contain a vector gravitational field in flat space-time. They are incomplete – they cannot be coupled with the other laws of non-gravitational physics (Maxwell’s equations), unless we impose a flat space-time. They are inconsistent – they give different results in the propagation of light for the corpuscular and undulatory aspects of light.
- Poincare’s theory (generalized by Whitrow and Morduch): the theory of action at a distance in flat space-time. It is incomplete or inconsistent in the same way as Kustaanheimo’s theories.
- Whitrow-Morduch vector theory (1965): contains a vector gravitational field in flat spacetime. It is incomplete or inconsistent in the same way as Kustaanheimo’s theories.
- Birkhoff’s Theory (1943): contains a tensor gravitational field used to construct a metric. It violates the Newtonian limit by the specific conditions imposed.
- Yilmaz’s Theory (1971, 1973): contains a tensor gravitational field used to construct a metric. It is mathematically inconsistent – the functional dependence of the metrics on the tensor field is not well defined.
Other alternative historical theories developed over time have been refuted by experimental checks or replaced by better corroborated theories:
- In 1690, Pierre Varignon assumed that all bodies are exposed to thrusts of aether particles from all directions, with a limitation to a certain distance from the Earth’s surface, under which bodies would experience greater attraction to Earth.
- In 1748, Mikhail Lomonosov assumed that the effect of aether is proportional to the complete surface of the elemental components of which matter is composed.
- In 1821, John Herapath tried to apply the co-developed model of kinetic gas theory to gravity. He assumed that the aether is heated by bodies and density decreases occur that push the bodies in that direction. Taylor showed that the low density due to thermal expansion is compensated by the increased velocity of the heated particles; therefore, no attractions appear.
- Ritz gravity theory, Weber-Gauss electrodynamics applied to gravity. Classical promotion of perihelions.
- Nordström’s theory of gravity (1912, 1913), an early competitor of general relativity.
- Kaluza Klein’s Theory (1921) 
- Whitehead’s theory of gravity (1922), another early competitor to general relativity.
Lorentz aether theory was developed from Hendrik Lorentz’s “electron theory”, between 1892 and 1895 considering it as a completely immobile aether. It introduced an ad-hoc hypothesis to cancel the failure of negative first order aether deviation experiments in v/c by introducing an auxiliary variable called “local time”. The negative result of the Michelson-Morley experiment resulted in the introduction of another ad-hoc hypothesis, length contraction, in 1892. But neither did the subsequent experiments confirm the theory, which became a degenerate theory according to Lakatos. Lorentz tried to revitalize it in 1899 and 1904 by introducing the Lorentz transformation. But neither the new theoretical models solved the problem of aether. Henri Poincaré corrected the errors in 1905 and incorporated the non-electromagnetic effects into the theory, calling it “New mechanics” and using for the first time the expression “the principle of relativity.”  He also criticized Lorentz for introducing too many helpful assumptions into his theory. Later, Minkowski (1908) and Arnold Sommerfeld (1910) also tried to develop a Lorentz invariant gravity law. Poincaré’s theory resisted a period due to his greater heuristic power, but he was defeated by the special relativity of Albert Einstein, who also took over some of the ideas of this theory. Lorentz acknowledged in 1914 that his theory was incompatible with the principle of relativity and rejected it. At present some physicists consider the Lorentz theory developed later by Poincaré as a special, “Lorentzian” or “neo-Lorenzian” interpretation of special relativity. Since both use Lorentz transformations and the same mathematical formalism, it is not possible to distinguish between the two theories by experiment. The difference between them is that Lorentz assumes the existence of an undetectable aether.
Modified Newtonian Dynamics (MOND) is a theory that proposes to modify Newton’s law of universal gravity with the intention of taking into account the observed properties of galaxies. MOND is trying to eliminate the controversial theory of dark matter. It was developed in 1982 and published in 1983 by Israeli physicist Mordehai Milgrom. Milgrom introduced the hypothesis that the gravitational force experienced by a star in the outer regions of a galaxy is proportional to the square of centripetal acceleration (as opposed to simple proportionality, from Newton’s second law) or, alternatively, that the gravitational force in these cases vary inversely proportional to radius (as opposed to the inverse square of radius in Newton’s law of gravity). In the MOND, the modification of Newton’s laws takes place only for the movement of galaxies, at extremely small accelerations.
MOND successfully predicted galactic phenomena unexplained by the theory of dark matter, but fails to confirm the properties of galaxy clusters, nor to develop a cosmological model that competes with the current ΛCDM model. Accurate measurement of the speed of gravitational waves in comparison to the speed of light in 2017 did not exclude MOND theories.
- Concrete relationship between the total baryonic mass of the galaxy and the asymptotic rotation speed according to the MOND prediction.
- MOND predicts a much better correlation between characteristics in the distribution of the nonbarionic mass and the rotation curve than the dark matter hypothesis, observed in several spiral galaxies.
- MOND predicts a specific relationship between the acceleration of stars at any distance from the center of a galaxy and the amount of dark matter in this radius that would be deduced in a Newtonian analysis, an observationally verified prediction.
- Confirms the stability of disk galaxies for galaxy regions within the deep MOND regime.
- For particularly massive galaxies, MOND predicts that the rotation curve should decrease by 1/r, according to Kepler’s law, confirmed by observations of elliptical galaxies with large masses.
From the initial MOND theory, several competing theories have been branched off that are based on the same hard core (negative heuristics) but with different development strategies (positive heuristics):
- AQUAL was developed in 1984 by Milgrom and Jacob Bekenstein, generating MOND behavior by modifying the gravitational term in the classical Lagrangian.
- QUMOND introduces a distinction between the MOND acceleration field and the Newtonian acceleration field.
- TeVeS starts from the behavior of the MOND but considers a relativistic framework. TeVeS has been successful in gravitational lens observations and structure formation but fails to explain other cosmological aspects.
There are other alternative relativistic generalizations of the MOND, such as BIMOND and the generalized Einstein-Aeter theories.
The external field effect implies a fundamental break of MOND by the principle of strong equivalence (but not necessarily by the principle of weak equivalence), this being recognized as a crucial element of the MOND paradigm.
Supporters of MOND theory have proposed several observational and experimental tests to help establish the best-corrected theory between MOND models and dark matter, such as: the existence of abnormal accelerations on Earth that could be detected in a precision experiment; testing in the solar system using the LISA Pathfinder mission by observing the tides predicted by the MOND and a Sun-Earth saddle point of Newtonian gravitational potential; measuring the MOND corrections to the precession of the perihelion of the planets in the Solar System; an astrophysical test to investigate the behavior of isolated galaxies, and non-Newtonian behavior in binary star systems; testing using the redshift dependence of radial acceleration.
The “Fifth Force” is a theory that changes Newton’s law of universal gravity. The initial experiments gave contradictory results: one claimed the existence of the fifth force, while the other contradicted this theory. After numerous repetitions of the experiment, the discord was resolved, and the consensus was reached that the Fifth Force does not exist.
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Sfetcu, Nicolae, “Proliferation of Post-Newtonian, Non-Relativistic Theories”, SetThings (August 29, 2019), URL = https://www.setthings.com/en/proliferation-of-post-newtonian-non-relativistic-theories/