Gravity


from Wikipedia, the free encyclopedia
A fountain beam, which is directed obliquely upwards, is deformed into a parabola on earth under the influence of gravity .
Two spiral galaxies deforming under the influence of each other's gravity
David Randolph Scott , Commander of the Apollo 15 lunar mission (1971), uses a spring and a hammer that he drops in a vacuum on the moon to demonstrate Galileo's law of fall that all bodies fall at the same speed, regardless of their mass.

The gravity (of Latin gravitas for "severity"), even mass attraction or gravitational force , is one of the four fundamental forces of physics . It manifests itself in the mutual attraction of crowds . It decreases with increasing distance from the masses, but has unlimited range. In contrast to electrical or magnetic forces , it cannot be shielded .

On earth , gravitation causes all bodies to go " downwards ", i. H. fall towards the center of the earth unless they are prevented from doing so by other forces. In the solar system , gravity determines the orbits of planets, moons, satellites and comets and in the cosmos the formation of stars and galaxies and their development on a large scale.

Gravity is often equated with gravity . However, the force on a body (the weight of the body) determined by the locally prevailing gravitational field includes not only the gravitational force , but also the inertial effects acting on the body (in particular due to the rotation of the reference system).

In the context of classical physics , gravitation is described using Newton's law of gravitation , i.e. H. as an instantaneous force at a distance that acts directly and without loss of time through empty space . A fundamentally different understanding of gravity results from the general theory of relativity according to Albert Einstein . Here, gravity does not act in the form of a force on the body, but corresponds to a curvature of four-dimensional space - time , whereby the paths of the bodies, on which no other forces act, follow a shortest line (in curved space), i.e. H. a geodesic .

historical overview

Antiquity

The Greek philosopher Aristotle described gravity in his cosmology as that property of the sublunar elements (earth, water, air, fire), which makes all bodies consisting of these elements strive towards the center of the world. This idea has long been the main physical argument for the geocentric worldview .

Orient

Ancient Indian authors attributed free fall to a force that is proportional to the mass of an object and acts in the direction of the center of the earth. In the 9th century, the Persian astronomer Muhammad ibn Musa explained the movements of the celestial bodies through an attraction. Al-Biruni translated the works of Indian authors into Arabic and Persian in the 11th century. His contemporary Alhazen formulated a theory of mass attraction. The Persian Al-Khazini suggested in the 12th century that the strength of the gravity depends on the distance to the center of the earth, and differentiated between mass, weight and force.

Late scholasticism

A major critic of Peripatetic (Aristotelian) physics and a pioneer of the Copernican worldview is the late scholastic Nikolaus von Oresme . In the 14th century he considered the earth's rotation to be probable and described the possibility of many worlds as well as many gravitational centers - in contrast to an earth at rest, lying in the center of the universe and attracting all gravity.

Copernicus

In 1543, in De revolutionibus orbium coelestium, Nicolaus Copernicus assumed that besides the earth all other celestial bodies also exert gravity:

“... At least I am of the opinion that gravity is nothing other than a natural striving implanted in the parts by the divine providence of the world master, by virtue of which they form their unity and wholeness by joining together to form a sphere. And it can be assumed that this tendency is also inherent in the sun , the moon and the other planets ... "

Kepler

Johannes Kepler published the following axioms in his Astronomia nova in 1609 :

  • Any corporeal substance, in so far as it is corporeal, is inherently inclined to rest in any place it is alone, beyond the power range of a related body .
  • The gravity consists in the mutual physical striving between related bodies for union or connection (of this order also the magnetic force is), so that the earth attracts the stone much more ; as the stone strives for the earth.
  • The heavy is [...] not driven to the center of the world as such, but as the center of a related round body ...
  • If the earth were not round, then the heaviness would not be driven in a straight line everywhere towards the center of the earth , but would be driven from different sides to different points.
  • If you were to move two stones anywhere in the world, close to each other outside the range of force of a third related body, then those stones would unite like two magnetic bodies in an intermediate place, with one approaching the other by a distance, which is proportional to the mass of the other .
  • The area of the attraction force of the moon extends to the earth.

17th century

Also at the beginning of the 17th century, Galileo Galilei described the free fall of a body as a uniformly accelerated movement that is independent of its mass or other properties.

In his work “Traité de mécanique des poids soutenus par des puissances sur des plans inclinés à l'horizontale”, published in 1636, Gilles Personne de Roberval developed the idea of ​​a gravitational force, years before the corresponding publications by Robert Hooke and Isaac Newton. René Descartes explained gravity as a result of his vortex theory . In 1644 he published the Principia Philosophiae , which had a great influence - also on the criticism by Isaac Newton - because the comets could obviously not be explained with Descartes' model. That comets penetrate or cross the spheres or the orbits of the planets has been the prevailing opinion since Tycho Brahe and the comet of 1577 .

The English scholar Robert Hooke explained the effect of gravity around 1670 with "gravity funnels". He explained that gravity is a property of all bodies with mass and the greater the closer two bodies are to each other. The theory that gravity is inversely proportional to the square of the distance from the center of mass first appeared in a letter from Hooke to his compatriot Newton in 1680.

Newton

Animation: Newton's idea of ​​gravity

Isaac Newton was the first to describe gravity using a mathematical formula in his Principia (1687). This law of gravitation , which he formulated, is one of the basic equations of classical mechanics , the first physical theory that could also be used in astronomy . According to this, gravity is a force between two bodies that accelerates them towards their common center of gravity , whereby its strength decreases proportionally to the square of the distance between the bodies. Newton's theory, completed around 1800 by Pierre-Simon Laplace , provides a fundamental understanding of the dynamics of the solar system with the possibility of precise predictions of the motion of planets, moons and comets. It confirms Kepler's laws of planetary motion for individual planets, but also allows the disruptive influence of other planets and moons to be determined. For a long time, the values ​​calculated according to this corresponded perfectly with the corresponding astronomical and terrestrial observations and experiments. The first discrepancy that cannot be explained in this way was discovered in the mid-19th century at the perihelion of Mercury's orbit .

Alternative theories in the 18th and 19th centuries

To explain gravity in the sense of a process, a number of mechanical or kinetic explanations were proposed until the development of the general theory of relativity in the early 20th century (see mechanical explanations of gravity ). One of the best known is the theory of Le Sage gravitation developed by Fatio and Le Sage . This argues that the gravitational attraction of two bodies is based on the shielding of the pressure from the direction of the other . In connection with this are the theories of an ether as a mediator of interactions (instead of an action at a distance ), electromagnetism also belongs to these interactions. One of the last of these theories was Lorentz's theory of ethers , which emerged around 1900 and was eventually superseded by the novel approach of Einstein's theory of relativity.

Einstein

"Newton's apple bends space-time," vividly. William Stukeley reports that Newton came to gravity from a falling apple, see Memoirs of Sir Isaac Newton's life
Video: gravity as the curvature of space

In the general theory of relativity (ART) established by Albert Einstein in 1916 , gravity is traced back to a geometric property of space-time . He assumes that spacetime is warped by the presence of mass and any form of energy. This makes it possible to interpret gravity fundamentally differently than the other forces, namely as inertial force . According to the equivalence principle , the effect of gravity cannot be differentiated from the effect of an acceleration of the reference system ; in particular, the effects of gravity and acceleration cancel each other out exactly in a freely falling frame of reference. It is said that gravity is "transformed away" by the transition to the new coordinates. However, this only applies to one place ( locally ), because every real gravitational field causes different accelerations for neighboring places. In the general theory of relativity, each point in space is assigned its own local inertial system in which there is no gravitation and where the special theory of relativity with its four-dimensional, flat spacetime applies. Analogous to the fact that, according to Galileo, force-free movements run in a straight line and uniform, in the general theory of relativity bodies move without non-gravitational forces on geodesics in a " curved " space with Riemannian geometry . Einstein's field equations are used to determine the curvature of spacetime at a point . They were formulated in such a way that in the limit of weak gravitation the results calculated according to them agree with those calculated according to Newton's equation. The general theory of relativity treats gravity as a force of inertia and puts it on the same level as centrifugal force , Coriolis force or the force that is felt in a vehicle when starting or braking.

Within the solar system, where weak fields or a slight curvature of space-time predominate, there are only minor deviations from the predictions of Newton's law of gravitation. The first successful application example of general relativity was the explanation of the small discrepancy between the observed perihelion of the orbit of Mercury and the value that is predicted according to Newton's theory due to the orbital disturbances by the other planets.

With strong curvature, as it is caused by the concentration of a large mass in a small space, completely new phenomena such. B. Black holes predicted.

In Newtonian mechanics, mass is the only source and point of application of gravity. Based on the originally imprecise concept of a given amount of matter, mass was given its first precise physical definition here. In the general theory of relativity, gravity is an expression of the curvature of space-time, which in turn is influenced not only by the presence of matter, but also by energy in every form, including gravitational energy itself, and also by mass and energy flows. All predictions of the general theory of relativity accessible to observation were confirmed by measurements.

In Newtonian gravitation, one assumed an instantaneous or instantaneous spread of the gravitational effect, that is, that the effect occurs immediately even over great distances. However, from the Einsteinian point of view, no effect, including gravitational effects, spreads faster than the speed of light . A rapid change in the position of masses, such as in the case of rapidly circling double stars or the collapse of a star , therefore generates gravitational waves that propagate at the speed of light.

Quantum gravity

Extremely high concentrations of mass or energy in a very small space are not experimentally accessible, for the description of which, in addition to gravitation, quantum effects would have to be taken into account. Attempts at a quantum field theory of gravity are just beginning. However, there is a lack of predictions that are both predictable and observable. The basic problem is that at such concentrations black holes quickly form, inside of which quantum effects take place that cannot be observed.

Modified Newtonian Dynamics (MOON)

The phenomenon “ dark matter ” stands for the difference between the observed masses and the masses to be expected according to the models of Kepler, Newton and Einstein in the rotation behavior of galaxies and galaxy clusters . Instead of additional, invisible mass, Mordehai Milgrom suggested in 1983 that a change in Newton's laws of motion could be the cause of the observed rotation curves. According to the “ MOON hypothesis”, the change only has a relevant influence on the movements in the case of very small accelerations , as they occur on an astronomical scale.

Proponents of "Modified Newtonian Dynamics" claim that Newton's theory of gravity from 1686 had already undergone three modifications. In the case of very small distances, physicists only use quantum mechanics , in the case of very high speeds Einstein's special theory of relativity and near very large masses his general theory of relativity .

Gravitation in classical mechanics

Newton's law of gravitation

In classical mechanics , gravitation or general attraction of mass is a property of all matter that only depends on its mass, but not on its type or movement. Gravitation expresses itself in the gravitational force or the gravitational field, which is generated by every mass, attracts every other mass and (in classical mechanics) has infinite propagation speed and range. Newton's law of gravity indicates the momentary force with which two bodies, imaginary point-like, attract each other with their masses and at a distance ( is the universal gravitational constant ):

This force is the same for both bodies, but directed in opposite directions. If there are no other forces acting, each of the two bodies experiences an acceleration towards the other. This instantaneous acceleration can be calculated using Newton's second law . For example, it results for body 1:

The momentary acceleration of body 1 does not depend on its mass , but on the mass of the other body. The body 2 thus gives every other body, regardless of its mass, the same acceleration at a certain distance. Conversely, the same applies to the acceleration that body 1 gives to every other body at a distance :

The accelerations are therefore inversely proportional to the accelerated masses . If one takes the earth for body 2 and any object of daily life for body 1, this means that the earth experiences only an immeasurably small acceleration from body 1 due to its much larger mass. It can therefore be assumed to be dormant. Body 1 experiences an acceleration from it that depends on the distance from the center of the earth, but not on the mass . This explains the fact that Galileo Galilei first stated that (in empty space, i.e. unhindered by other forces or resistances) all bodies experience the same acceleration of gravity regardless of their mass . The equality of the gravitational acceleration is also referred to as the principle of the equivalence of inert and heavy mass (in its weak formulation).

If the masses of the two bodies are not so different from one another as in the previous example, then both bodies perform accelerated movements, whereby the total center of gravity between the two masses can be chosen as a reference point at rest (see the principle of centroid ). If both bodies start out of rest, they crash straight towards each other until they meet. (In the abstraction as point masses, this would happen in the overall center of gravity.)

If, however, they each have an initial speed in the center of gravity system , then they carry out movements whose trajectories lie in a common plane; this is required by the law of conservation of angular momentum . The shape of these trajectories depends on the speeds of the two bodies (see two-body problem ). One possible solution are elliptical trajectories, the center of gravity forming a focal point of the two ellipses. An example of this is the earth-moon system, in which the earth's mass is so great that the common center of gravity is even inside the earth.

Systems that consist of three or more bodies that attract each other often behave chaotically and cannot be calculated using analytical methods (see three-body problem ). However, there are helpful approximations. In the solar system, for example, the mass of the planets is very small compared to the solar mass. If one assumes that the sun is therefore not influenced by the planets and that the planets do not interact with one another, then calculations with Newton's law of gravity result in Kepler's orbital ellipses of the planets.

The classic description of gravity is therefore sufficiently precise for many applications. However, deviations occur in connection with the most precise measurements, e.g. B. in the perihelion of Mercury , and the classic description fails completely under extreme conditions, e.g. B. exist in black holes .

The weight of a body on the earth's surface is largely determined by gravity. Carry on inertial forces at the force of gravity; z. B. the centrifugal force , which results from the earth's rotation , counteracts gravity a little. Gravitation and inertial forces together form the gravitational field .

Gravitational constant

The gravitational constant is a fundamental constant in physics. Their exact determination is very difficult, because between two bodies, the mass of which can be determined by direct weighing, the gravitational force is extremely small. Its value is therefore only known to four decimal places, in contrast to the at least eight decimal places of other fundamental constants.

The first determination was made in 1798 by Henry Cavendish . The experiment ( gravitational balance ) carried out in his laboratory and devised by John Michell has historical significance for the development of the experimental and theoretical foundations of gravity.

general theory of relativity

The light from a distant galaxy is a very massive body are deflected so by gravity, that it on the earth as Einstein-Ring appears

In the general theory of relativity (GR) gravity is not treated like a force in the sense of classical physics. In contrast to the usual classical field theories , in which the coordinates for place and time are given in a fixed structure, the ART considers this structure itself to be changeable. As a basis, it uses the space-time known from the special theory of relativity , in which position and time coordinates are combined in a four-dimensional pseudo-Riemannian manifold . However, this spacetime is no longer “flat” in the GR as in Euclidean geometry, but is “curved” by the occurrence of mass or energy . The curvature results at each point in spacetime from the metric that defines the four-dimensional distance between two points in spacetime, i.e. between two events. The time interval is evaluated with a positive sign, but the spatial interval with a negative sign (occasionally also with the opposite sign). A body, on which no other forces apart from gravity act, now moves between two events (e.g. departure and arrival) always along the connecting line that is the longest according to this spatiotemetric metric ( geodesic ), which is because of the aforementioned choice of sign means the spatially shortest distance. Where space-time is flat, the geodesic is the straight line connecting the two points. Converted into the usual coordinates for place and time, this corresponds to a uniform movement on the shortest spatial path, i.e. along the spatial connecting straight line, analogous to the law of inertia of classical mechanics for the completely force-free body. In the case of a curved spacetime, however, a geodesic generally corresponds to an accelerated movement along a spatially curved path. The curvature of space-time caused by the presence of mass or energy is determined by Einstein's field equations in such a way that the geodesic reproduces a movement that corresponds exactly to the movement of the otherwise force-free body in the prevailing gravitational field (i.e. free fall, trajectory parabola, planetary orbit, etc. ). Since the mass of the considered body is not included, the same geodesic applies to bodies with different masses, i.e. i.e., they move the same in a given gravitational field. This also explains the equivalence principle , which in classical physics establishes the equality of heavy and inert mass. Gravity does not appear, as in classical physics, as a certain force that acts on the body and causes acceleration, but as a property of space-time in which the body moves without force. In this way, gravity is interpreted as a purely geometric phenomenon.

In this sense, the general theory of relativity reduces the gravitational force to the status of a pseudo-force : When sitting on a chair you feel like you are being pulled towards the earth by a "gravitational force", the GTR interprets this in such a way that you are constantly prevented from doing so by the surface of the chair is to follow the geodesic through space-time curved by the earth's mass, which would be free fall. The force with which the chair surface acts on the seat of the observer is by no means an apparent force. Ultimately, it goes back to the electrostatic repulsion when the atoms of the observer touch the atoms of the chair surface. According to the general theory of relativity, the interpretation of events is shifting. While according to classical mechanics the earth represents an inertial system in which the downwardly directed gravity on the observer is balanced by the upwardly directed supporting force of the chair so that the observer can remain at rest, the inertial system, which is correct according to the general theory of relativity, falls with it Acceleration due to gravity down. But in this inertial system, the chair exerts a force on the observer that constantly accelerates him upwards.

However, vertically free falling bodies, but also satellites, planets, comets or parabolic flights follow a geodesic through space-time. In the general theory of relativity, their movements are regarded as (net) free of forces, since the earth's (or solar) mass, through the curvature of space-time, influences the definition of what means “straight ahead” in the sense of the inertia of bodies. The space-time curvature occurs more directly (ie rather in accordance with the usual term of curvature). B. in appearance in astronomical observations in which it could be demonstrated (see Fig.) That large masses cause the curvature of light rays.

Due to the relativity principle and the resulting invariance towards Lorentz transformations , not only mass, but also every form of energy contributes to the curvature of spacetime. This applies including the energy associated with gravity itself. Hence, Einstein's field equations are non-linear. In the area of ​​weak curvature, however, they can be approximated by linear equations in which Newton's law of gravitation can be found. Compared to the phenomena calculated according to Newton's law, however, there are small corrections, which could all be confirmed by precise observations (see tests of general relativity ). Completely new phenomena, however, arise with strong curvature of space-time, here in particular the black holes .

Gravitation and quantum theory

Quantum Gravity, Quantum Field Theory

Within the framework of a quantum field theory , gravity is described in a linear approximation by the exchange of a massless particle called a graviton , which has spin 2. In addition, the formulation of a quantum theory of gravity leads to fundamental problems that have not yet been solved. Even the supersymmetric expansion has not yet led to a consistent theory. String theory and loop quantum gravity are currently considered to be the most promising candidates . An essential goal is to combine gravity with the other interactions to form a “Great Unified Theory” (GUT) in order to formulate a theory that can describe all natural forces at once. This means that gravitation, which does not take into account the effects of quantum field theory, would be expanded to include these. In the context of the unified superstring theories , the M-theory , the universe is described as an eleven-dimensional manifold. The part of the universe in which we exist represents a higher-dimensional membrane ( p-brane ), which is itself embedded in an even higher-dimensional manifold, in which other branes could vibrate and thus represent parallel spacetime within the same universe. In M-theory, the gravitons are represented as closed strings that are not tied to the boundaries of a brane. Therefore, they are able to expand through all additional spatial dimensions and also get into other branes. In this way the strength of gravity is weakened sufficiently so that it appears as the weakest of the four interactions in the context of our four-dimensional world of experience.

Fundamental interactions and their descriptions
Strong interaction Electromagnetic interaction Weak interaction Gravity
classic Electrostatics & magnetostatics ,
electrodynamics
Newton's law of gravitation ,
general relativity
quantum
theory
Quantum
( standard model )
Quantum electrodynamics Fermi theory Quantum gravity  ?
Electroweak Interaction
( Standard Model )
Big Unified Theory  ?
World formula ("theory of everything")?
Theories at an early stage of development are grayed out.

Quantum physical effects of the gravitational field

The effect of the gravitational potential on the quantum mechanical phase of the wave function was proven in 1975 by an interference experiment on free neutrons . The wave function and energy of neutrons, which occupy a state bound in the gravitational field, could be measured in 2012. In both cases, the measurement results confirm the predictions calculated on the basis of quantum mechanics.

Gravitation on earth

Gravitation (more precisely: acceleration due to gravity ) in the interior of the earth according to the seismic PREM earth model, as well as approximations through constant and linearly increasing rock density .

The earth has a mass of 5.974 · 10 24  kg. Its radius is 6356 km at the poles and, due to the flattening of the earth , 6378 km at the equator. With the help of Newton's law of gravitation, it follows that the gravitational acceleration is between (at the equator) and (at the poles). The actually effective gravitational acceleration differs from the value calculated in this way, which is why it is also referred to as the location factor . This location dependency, which also affects the direction of the gravitational acceleration, is related to the centrifugal effect caused by the earth's rotation, the altitude of the location and local gravity anomalies . Accordingly, the weight force in the earth's gravity field is not just a pure gravitational force in the sense of the law of gravitation.

weightlessness

When "weightlessness" is spoken of, it is often not the absence of gravity that is meant, but only the absence of a holding force that is contrary to the force of weight. A body on which only the force of gravity acts is in a state of free fall. In this sense, a space station is also in free fall in earth orbit , whereby the flight path does not end at the surface of the earth, but leads around the earth due to the sufficiently high horizontal orbit speed . In a freely falling frame of reference, no gravitational effects are noticeable. Consequently, this state is called weightlessness. This applies under the condition that the gravitational field is at least locally approximately homogeneous. Slight deviations from this lead to phenomena of microgravity .

Lagrange points

In a system of two celestial bodies circling each other (e.g. earth and sun) there are constantly rotating points at which other celestial bodies could have a path on which all forces cancel each other - the so-called Lagrange points . There the gravitational forces of the heavenly bodies and the centrifugal force of the orbital movement cancel each other out. A corresponding trajectory following a Lagrange point can be stable or unstable - a slight deviation from the Lagrange point can bring about a corrective force back to the point (stable) or lead to a breakout (unstable). The Planck space telescope was stationed at a Lagrange point.

Gravisphere

Near masses have more influence on gravitational acceleration than distant masses. Therefore satellite orbits are possible around relatively small bodies in the gravitational field of large bodies. The space in which this is the case is the gravisphere of the respective celestial body. For the same reason, the gravitational acceleration of an irregularly shaped body is not aligned with its barycentre at all points in space .

Shielding from gravity and anti-gravity

In the field of science fiction and frontier science there are numerous concepts of gravitational shielding or antigravity . Experiments by Quirino Majorana , who claims to have found a shielding effect through heavy elements around 1920 (invalidated by Henry Norris Russell , among others ), and by Jewgeni Podkletnow , who in 1995 claimed a decrease in the weight of rotating superconductors , but this was also not confirmed, are relatively well known could be.

literature

Web links

Commons : Gravitation  - collection of images, videos and audio files
Wiktionary: Gravitation  - explanations of meanings, word origins, synonyms, translations
Wiktionary: gravity  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. PONS German - Latin, first meanings of gravitas
  2. ^ Edward Grant : The Nature of Natural Philosophy in the Late Middle Ages. Washington 2010, p. 63 ; Planets, Stars, and Orbs: The Medieval Cosmos, 1200–1687. 1994/96, p. 165 ; A Source Book in Medieval Science, Volume 1. Compiled by Edward Grant, 1974, p. 551 ; Paul S. Agutter, Denys N. Wheatley: Thinking about Life: The history and philosophy of biology and other sciences. 2008, p. 59
  3. quoted from: Nicolaus Coppernicus from Thorn on the circular motions of the cosmic bodies. (German translation by CL Menzzer, 1879.), p. 23 @wikisource; Original: “Equidem existimo, gravitatem non aliud esse, quam appetentiam quandam naturalem partibus inditam a divina providentia opificis universorum, ut in unitatem integritatemque suam sese conferant in formam globi coeuntes. Quam affectionem credibile est etiam Soli, Lunae, caeterisque errantium fulgoribus inesse… “ Lib. I, Cap. IX @wikisource; s. a. Johann Samuel Traugott Gehler : Physical dictionary. Volume 2, Leipzig 1789, p. 519
  4. Astronomia nova. New causally established astronomy (translation by Max Caspar , 1929 and Fritz Krafft , Wiesbaden 2005.), pp. 28–29; Original: Scan of the printed copy from the ETH Library Zurich (accessed March 24, 2014); s. a. Johann Samuel Traugott Gehler : Physical dictionary. Volume 2, Leipzig 1789, p. 519 and Florian Freistetter: Johannes Kepler: Astronomia Nova - The Introduction (3) @ scienceblogs.de / astrodicticum-simplex (accessed March 24, 2014)
  5. Heinz Klaus Strick: Gilles Personne Roberval (1602–1675): Discoverer of gravity. Spektrum.de article
  6. Harry Nussbaumer: Revolution in the sky: how the Copernican turn changed astronomy. Zurich 2011, p. 237 ; Eberhard Knobloch : The worldview in the sciences - history of a conception. in: Christoph Markschies, Johannes Zachhuber (Hrsg.): The world as image: Interdisciplinary contributions to the visuality of world views. Berlin 2008, pp. 227-246, p. 242
  7. Albert Einstein: The foundations of the general theory of relativity. In: Annals of Physics. 4, 49. PDF
  8. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 26, 2015 . Relative uncertainty 4.7 · 10 −5
  9. H. Colella, A. W. Overhauser, S. A. Werner: Observation of Gravitationally Induced Quantum Interference , Phys. Rev. Lett. 34 (1975) p. 1472
  10. Hartmut Abele, Helmut Leeb: Gravitation and quantum interference experiments with neutrons , New Journal of Physics 14 (2012) 055010, doi: 10.1088 / 1367-2630 / 14/5/055010 .
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  12. ^ Q. Majorana: On gravitation. Theoretical and experimental researches. In: Philosophical Magazine. Volume 39, 1920. pp. 488-504.
  13. H. N. Russell: On Majorana's theory of gravitation. In: Astrophysical Journal. Volume 54, 1921. pp. 334-346. bibcode : 1921ApJ .... 54..334R .
  14. arxiv : physics / 0108005 Podkletnov's original publication
  15. N. Li, D. Noever, T. Robertson, R. Koczor, W. Brantley: Static Test for a Gravitational Force Coupled to Type II YBCO Superconductors ; In: Physica C , Vol. 281, 1997, pp. 260-267.
  16. C. Woods, S. Cooke, J. Helme, C. Caldwell: Gravity Modification by High Temperature Superconductors ; In: Joint Propulsion Conference, AIAA 2001-3363 , 2001.
  17. Hathaway, Cleveland, Bao: Gravity modification experiment using a rotating superconducting disk and radio frequency fields ; In: Physica C , Volume 385, 2003, pp. 488-500.