History of gravitational theory

The Ionian Greek philosopher Heraclitus (c. 535 – c. 475 BC) used the word logos ('I say') to describe a kind of law which keeps the cosmos in harmony, moving all objects, including the stars, winds, and waves.[1]

In the 4th century BC, the Greek philosopher Aristotle taught that there is no effect or motion without a cause. The cause of the downward motion of heavy bodies, such as the element earth, was related to their nature, which caused them to move downward toward the centre of the universe, which was their natural place. Conversely, light bodies such as the element fire, move by their nature upward toward the inner surface of the sphere of the Moon. Thus in Aristotle's system heavy bodies are not attracted to the Earth by an external force, but tend toward the centre of the universe because of an inner gravitas or heaviness.[2][3]

The 3rd-century-BC Greek physicist Archimedes discovered the centre of mass of a triangle.[4] 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.[5] Two centuries later, the Roman engineer and architect Vitruvius contended in his De architectura that gravity is not dependent on a substance's weight but rather on its 'nature' (cf. 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.[6]

In the 6th century CE, 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.

The Indian mathematician/astronomer Brahmagupta (c. 598–c. 668) first described gravity as an attractive force, using the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" to describe it within a heliocentric view of the Solar System as had been defined by Aryabhata:[7][8][9][10]

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.[11][12][lower-alpha 1]

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.[41][42] Utilizing 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.[43] In 1671, Robert Hooke speculated that gravitation is the result of bodies emitting waves in the aether.[44][lower-alpha 10] 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.[45] A similar model was later created by Hendrik Lorentz (1853–1928), who used electromagnetic radiation instead of corpuscles.

English mathematician Isaac Newton utilized Descartes' argument that curvilinear motion constrains inertia,[46] and in 1675, argued that aether streams attract all bodies to one another.[lower-alpha 11] 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.[47]

Portrait of Isaac Newton (1642–1727) by Godfrey Kneller (1689)

In 1679, Robert Hooke wrote to Isaac Newton of his hypothesis concerning orbital motion which partly depends on an inverse-square force.[48] As a result, Newton was able to mathematically derive Kepler's laws of planetary motion, including the elliptical orbits for the six then-known planets and the Moon. In 1687, Newton published Philosophiæ Naturalis Principia Mathematica, which hypothesizes the inverse-square law of universal gravitation. 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:

${\displaystyle {\rm {Force\,of\,gravity}}\propto {\frac {\rm {mass\,of\,object\,1\,\times \,mass\,of\,object\,2}}{\rm {distance\,from\,centers^{2}}}}}$

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 (the gravitational constant). Newton would need an accurate measure of this constant to prove his inverse-square law. This was first performed by Henry Cavendish in 1797.[lower-alpha 12]

In Newton's theory[52] (rewritten using more modern mathematics) the density of mass ${\displaystyle \rho \,}$ generates a scalar field, the gravitational potential ${\displaystyle \varphi \,}$ in joules per kilogram, by

${\displaystyle {\partial ^{2}\varphi \over \partial x^{j}\,\partial x^{j}}=4\pi G\rho \,.}$

Using the Nabla operator ${\displaystyle \nabla }$ for the gradient and divergence (partial derivatives), this can be conveniently written as:

${\displaystyle \nabla ^{2}\varphi =4\pi G\rho \,.}$

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

${\displaystyle {d^{2}x^{j} \over dt^{2}}=-{\partial \varphi \over \partial x^{j}\,}.}$

At distance r from an isolated mass M, the scalar field is

${\displaystyle \varphi =-{\frac {GM}{r}}\,.}$

Failing to define a medium in which it interacted, Newton's theory seemed to require action at a distance.[53] His theory, and Joseph-Louis Lagrange's improvement on the calculation (applying the variational principle), do not take into account relativistic effects, which were unknown at that time. Even so, Newton's theory is thought to be exceptionally accurate in the limit of weak gravitational fields and low speeds.

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. Le Verrier sent his position to Johann Gottfried Galle, asking him to verify. The same night, Galle spotted Neptune near the position Le Verrier had predicted.[54]

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.[55]

At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Weber, Carl Friedrich Gauss, Bernhard Riemann and James Clerk Maxwell. Those models were used to explain the perihelion precession of Mercury. In 1890, 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 his theory. And in another attempt, Paul Gerber (1898) even succeeded in deriving the correct formula for the Perihelion shift (which was identical to that formula later used by Einstein). However, because the basic laws of Weber and others were wrong (for example, Weber's law was superseded by Maxwell's theory), those hypotheses were rejected.[59] In 1900, Hendrik Lorentz tried to explain gravity on the basis of his Lorentz ether theory and the Maxwell 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. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low.[60]

In the late 19th century, Lord Kelvin pondered the possibility of a theory of everything.[61] He proposed that every body pulsates, which might be an explanation of gravitation and electric charges. However, 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.

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.[62]

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[63][64] (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:

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

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

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

Einstein and Grossmann[65] includes Riemannian geometry and tensor calculus.

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

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

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

${\displaystyle T^{\mu \nu }=\rho {dx^{\mu } \over d\tau }{dx^{\nu } \over d\tau }\,}$

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

While this was going on, Abraham was developing 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 of Nordström (1912)[66] was to retain the Minkowski metric and a constant value of ${\displaystyle c\,}$ but to let mass depend on the gravitational field strength ${\displaystyle \varphi \,}$. Allowing this field strength to satisfy

${\displaystyle \Box \varphi =\rho \,}$

where ${\displaystyle \rho \,}$ is rest mass energy and ${\displaystyle \Box \,}$ is the d'Alembertian,

${\displaystyle m=m_{0}\exp \left({\frac {\varphi }{c^{2}}}\right)\,}$

and

${\displaystyle -{\partial \varphi \over \partial x^{\mu }}={\dot {u}}_{\mu }+{u_{\mu } \over c^{2}{\dot {\varphi }}}\,}$

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

The second approach of Nordström (1913)[67] is remembered as the first logically consistent relativistic field theory of gravitation ever formulated. (notation from Pais[68] not Nordström):

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

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

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

${\displaystyle -{\partial T^{\mu \nu } \over \partial x^{\nu }}=T{1 \over \psi }{\partial \psi \over \partial x_{\mu }}\,}$

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

This theory[69] is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing:

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

${\displaystyle {ds}^{2}=g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }\,}$

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

they relate Einstein–Grossmann[65] to Nordström.[67] 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 the equivalence principle, into his general theory of relativity, which fuses the three dimensions of space and the one dimension of time into the four-dimensional fabric of spacetime. However, it does not unify gravity with quanta—individual particles of energy, which Einstein himself had postulated the existence of in 1905.

Illustration explaining the relevance of the total solar eclipse of 29 May 1919, from the 22 November 1919 edition of The Illustrated London News

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 along locally straight paths in curved spacetime. More specifically, Einstein and David Hilbert discovered the field equations of general relativity, which relate the presence of matter and the curvature of spacetime. These field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor 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:

• The Schwarzschild solution, which describes spacetime surrounding a spherically symmetrical non-rotating uncharged massive object. For objects with radii smaller than the Schwarzschild radius, this solution generates a black hole with a central singularity.
• The Reissner–Nordström solution, in which the central object has an electrical charge. For charges with a geometrized length less than the geometrized length of the mass of the object, this solution produces black holes with an event horizon surrounding a Cauchy horizon.
• The Kerr solution for rotating massive objects. This solution also produces black holes with multiple horizons.
• The cosmological Robertson–Walker solution, which predicts the expansion of the universe.

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.[55]
• Gravitational lensing was first confirmed in 1919, and has more recently been strongly confirmed through the use of a quasar which passes behind the Sun as seen from the Earth.
• The expansion of the universe (predicted by the Robertson–Walker metric) was confirmed by Edwin Hubble in 1929.
• The prediction that time runs slower at lower potentials has been confirmed by the Pound–Rebka experiment, the Hafele–Keating experiment, and the GPS.
• The time delay of light passing close to a massive object was first identified by Irwin Shapiro in 1964 in interplanetary spacecraft signals.
• Gravitational radiation has been indirectly confirmed through studies of binary pulsars such as PSR 1913+16.
  • In 2015, the LIGO experiments directly detected gravitational radiation from two colliding black holes, making this the first direct observation of both gravitational waves and black holes.[70]

It is believed that neutron star mergers (since detected in 2017)[71] 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 with quantum mechanics.[72] 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.[73][74] 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.[72]

Theoretical models such as string theory and loop quantum gravity are current candidates for a possible 'theory of everything'.

• Gillispie, Charles Coulston (1960). The Edge of Objectivity: An Essay in the History of Scientific Ideas. Princeton University Press. ISBN 0-691-02350-6.