equivalence principle

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equivalence principle
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{{General relativity sidebar |fundamentals}}In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

Einstein's statement of the equality of inertial and gravitational mass

{{Quotation|A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:
(Inertial mass) cdot (Acceleration) = (Intensity of the gravitational field) cdot (Gravitational mass).
It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.Einstein, Albert, weblink" title="">How I Constructed the Theory of Relativity, translated by Masahiro Morikawa from the text recorded in Japanese by Jun Ishiwara, Association of Asia Pacific Physical Societies (AAPPS) Bulletin, Vol. 15, No. 2, pp. 17–19, April 2005. Einstein recalls events of 1907 in a talk in Japan on 14 December 1922.BOOK, Einstein, Albert, The Meaning of Relativity, 2003, Routledge, 9781134449798, 59, }}

Development of gravitational theory

File:Apollo 15 feather and hammer drop.ogv|thumb|During the Apollo 15 mission in 1971, astronaut (David Scott]] showed that Galileo was right: acceleration is the same for all bodies subject to gravity on the Moon, even for a hammer and a feather.)Something like the equivalence principle emerged in the early 17th century, when Galileo expressed experimentally that the acceleration of a test mass due to gravitation is independent of the amount of mass being accelerated.Kepler, using Galileo's discoveries, showed knowledge of the equivalence principle by accurately describing what would occur if the moon were stopped in its orbit and dropped towards Earth. This can be deduced without knowing if or in what manner gravity decreases with distance, but requires assuming the equivalency between gravity and inertia.The 1/54 ratio is Kepler's estimate of the Moon–Earth mass ratio, based on their diameters. The accuracy of his statement can be deduced by using Newton's inertia law F=ma and Galileo's gravitational observation that distance D = (1/2)at^2. Setting these accelerations equal for a mass is the equivalence principle. Noting the time to collision for each mass is the same gives Kepler's statement that Dmoon/DEarth=MEarth/Mmoon, without knowing the time to collision or how or if the acceleration force from gravity is a function of distance.Newton's gravitational theory simplified and formalized Galileo's and Kepler's ideas by recognizing Kepler's "animal force or some other equivalent" beyond gravity and inertia were not needed, deducing from Kepler's planetary laws how gravity reduces with distance.The equivalence principle was properly introduced by Albert Einstein in 1907, when he observed that the acceleration of bodies towards the center of the Earth at a rate of 1g (g = 9.81 m/s2 being a standard reference of gravitational acceleration at the Earth's surface) is equivalent to the acceleration of an inertially moving body that would be observed on a rocket in free space being accelerated at a rate of 1g. Einstein stated it thus:That is, being on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines. The direction or vector of acceleration equivalence on the surface of the earth is "up" or directly opposite the center of the planet while the vector of acceleration in a spaceship is directly opposite from the mass ejected by its thrusters. From this principle, Einstein deduced that free-fall is inertial motion. Objects in free-fall do not experience being accelerated downward (e.g. toward the earth or other massive body) but rather weightlessness and no acceleration. In an inertial frame of reference bodies (and photons, or light) obey Newton's first law, moving at constant velocity in straight lines. Analogously, in a curved spacetime the world line of an inertial particle or pulse of light is as straight as possible (in space and time).WEB,weblink 32, General Relativity in a Nutshell, Alan, Macdonald, September 15, 2012, February 8, 2013, Luther College (Iowa), Luther College, Such a world line is called a geodesic and from the point of view of the inertial frame is a straight line. This is why an accelerometer in free-fall doesn't register any acceleration; there isn't any.As an example: an inertial body moving along a geodesic through space can be trapped into an orbit around a large gravitational mass without ever experiencing acceleration. This is possible because spacetime is radically curved in close vicinity to a large gravitational mass. In such a situation the geodesic lines bend inward around the center of the mass and a free-floating (weightless) inertial body will simply follow those curved geodesics into an elliptical orbit. An accelerometer on-board would never record any acceleration.By contrast, in Newtonian mechanics, gravity is assumed to be a force. This force draws objects having mass towards the center of any massive body. At the Earth's surface, the force of gravity is counteracted by the mechanical (physical) resistance of the Earth's surface. So in Newtonian physics, a person at rest on the surface of a (non-rotating) massive object is in an inertial frame of reference. These considerations suggest the following corollary to the equivalence principle, which Einstein formulated precisely in 1911:Einstein also referred to two reference frames, K and K'. K is a uniform gravitational field, whereas K' has no gravitational field but is uniformly accelerated such that objects in the two frames experience identical forces:This observation was the start of a process that culminated in general relativity. Einstein suggested that it should be elevated to the status of a general principle, which he called the "principle of equivalence" when constructing his theory of relativity:Einstein combined (postulated) the equivalence principle with special relativity to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field, even before he developed the concept of curved spacetime.So the original equivalence principle, as described by Einstein, concluded that free-fall and inertial motion were physically equivalent. This form of the equivalence principle can be stated as follows. An observer in a windowless room cannot distinguish between being on the surface of the Earth, and being in a spaceship in deep space accelerating at 1g. This is not strictly true, because massive bodies give rise to tidal effects (caused by variations in the strength and direction of the gravitational field) which are absent from an accelerating spaceship in deep space. The room, therefore, should be small enough that tidal effects can be neglected.Although the equivalence principle guided the development of general relativity, it is not a founding principle of relativity but rather a simple consequence of the geometrical nature of the theory. In general relativity, objects in free-fall follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics of spacetime, because the mechanical resistance of matter prevents us from doing so.Since Einstein developed general relativity, there was a need to develop a framework to test the theory against other possible theories of gravity compatible with special relativity. This was developed by Robert Dicke as part of his program to test general relativity. Two new principles were suggested, the so-called Einstein equivalence principle and the strong equivalence principle, each of which assumes the weak equivalence principle as a starting point. They only differ in whether or not they apply to gravitational experiments.Another clarification needed is that the equivalence principle assumes a constant acceleration of 1g without considering the mechanics of generating 1g. If we do consider the mechanics of it, then we must assume the aforementioned windowless room has a fixed mass. Accelerating it at 1g means there is a constant force being applied, which = m*g where m is the mass of the windowless room along with its contents (including the observer). Now, if the observer jumps inside the room, an object lying freely on the floor will decrease in weight momentarily because the acceleration is going to decrease momentarily due to the observer pushing back against the floor in order to jump. The object will then gain weight while the observer is in the air and the resulting decreased mass of the windowless room allows greater acceleration; it will lose weight again when the observer lands and pushes once more against the floor; and it will finally return to its initial weight afterwards. To make all these effects equal those we would measure on a planet producing 1g, the windowless room must be assumed to have the same mass as that planet. Additionally, the windowless room must not cause its own gravity, otherwise the scenario changes even further. These are technicalities, clearly, but practical ones if we wish the experiment to demonstrate more or less precisely the equivalence of 1g gravity and 1g acceleration.

Modern usage

Three forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.

The weak equivalence principle

The weak equivalence principle, also known as the universality of free fall or the Galilean equivalence principle can be stated in many ways. The strong EP includes (astronomic) bodies with gravitational binding energyWagner, Todd A.; Schlamminger, Stephan; Gundlach, Jens H.; Adelberger, Eric G.; "Torsion-balance tests of the weak equivalence principle", Classical and Quantum Gravity 29, 184002 (2012);weblink (e.g., 1.74 solar-mass pulsar PSR J1903+0327, 15.3% of whose separated mass is absent as gravitational binding energyChampion, David J.; Ransom, Scott M.; Lazarus, Patrick; Camilo, Fernando; et al.; Science 320(5881), 1309 (2008),weblink{{failed verification|date=June 2018}}). The weak EP assumes falling bodies are bound by non-gravitational forces only. Either way:
The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.
All test particles at the alike spacetime point, in a given gravitational field, will undergo the same acceleration, independent of their properties, including their rest mass.BOOK, Five-dimensional Physics, Paul S., Wesson, 82,weblink 978-981-256-661-4, World Scientific, 2006,
All local centers of mass free-fall (in vacuum) along identical (parallel-displaced, same speed) minimum action trajectories independent of all observable properties.
The vacuum world-line of a body immersed in a gravitational field is independent of all observable properties.
The local effects of motion in a curved spacetime (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime, without exception.
Mass (measured with a balance) and weight (measured with a scale) are locally in identical ratio for all bodies (the opening page to Newton's Philosophiæ Naturalis Principia Mathematica, 1687).
Locality eliminates measurable tidal forces originating from a radial divergent gravitational field (e.g., the Earth) upon finite sized physical bodies. The "falling" equivalence principle embraces Galileo's, Newton's, and Einstein's conceptualization. The equivalence principle does not deny the existence of measurable effects caused by a rotating gravitating mass (frame dragging), or bear on the measurements of light deflection and gravitational time delay made by non-local observers.

Active, passive, and inertial masses

By definition of active and passive gravitational mass, the force on M_1 due to the gravitational field of M_0 is:
F_1 = frac{M_0^mathrm{act} M_1^mathrm{pass}}{r^2}
Likewise the force on a second object of arbitrary mass2 due to the gravitational field of mass0 is:
F_2 = frac{M_0^mathrm{act} M_2^mathrm{pass}}{r^2}
By definition of inertial mass:
F = m^mathrm{inert} a
If m_1 and m_2 are the same distance r from m_0 then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same)
a_1 = frac{F_1}{m_1^mathrm{inert}} = a_2 = frac{F_2}{m_2^mathrm{inert}}
frac{M_0^mathrm{act} M_1^mathrm{pass}}{r^2 m_1^mathrm{inert}} = frac{M_0^mathrm{act} M_2^mathrm{pass}}{r^2 m_2^mathrm{inert}}
frac{M_1^mathrm{pass}}{m_1^mathrm{inert}} = frac{M_2^mathrm{pass}}{m_2^mathrm{inert}}
In other words, passive gravitational mass must be proportional to inertial mass for all objects.Furthermore, by Newton's third law of motion:
F_1 = frac{M_0^mathrm{act} M_1^mathrm{pass}}{r^2}
must be equal and opposite to
F_0 = frac{M_1^mathrm{act} M_0^mathrm{pass}}{r^2}
It follows that:
frac{M_0^mathrm{act}}{M_0^mathrm{pass}} = frac{M_1^mathrm{act}}{M_1^mathrm{pass}}
In other words, passive gravitational mass must be proportional to active gravitational mass for all objects.The dimensionless Eötvös-parameter eta(A,B) is the difference of the ratios of gravitational and inertial masses divided by their average for the two sets of test masses "A" and "B."
eta(A,B)=2frac{ left(frac{m_g}{m_i}right)_A-left(frac{m_g}{m_i}right)_B }{left(frac{m_g}{m_i}right)_A+left(frac{m_g}{m_i}right)_B}

Tests of the weak equivalence principle

Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. An obvious test is dropping different objects, ideally in a vacuum environment, e.g., inside the Fallturm Bremen drop tower.{| class="wikitable"
John Philoponus6th centurySaid that by observation, two balls of very different weights will fall at nearly the same speedno detectable difference
Simon StevinDEVREESE >FIRST1=JOZEF T. LAST2=VANDEN BERGHE YEAR=2008 URL=HTTPS://BOOKS.GOOGLE.COM/BOOKS?ISBN=1845643917 ISBN=9781845643911, ~1586Dropped lead balls of different masses off the Delft churchtowerno detectable difference
Galileo Galilei~1610Rolling balls of varying weight down inclined planes to slow the speed so that it was measurableno detectable difference
Isaac Newton~1680Measure the period of pendulums of different mass but identical lengthdifference is less than 1 part in 103
Friedrich Wilhelm Bessel1832Measure the period of pendulums of different mass but identical lengthno measurable difference
Loránd Eötvös1908Measure the torsion on a wire, suspending a balance beam, between two nearly identical masses under the acceleration of gravity and the rotation of the Earthdifference is 10±2 part in 109 (H2O/Cu)Eötvös, Loránd; Annalen der Physik (Leipzig) 68 11 (1922)
P. G. Roll, Robert Krotkov>Krotkov and Dicke1964Torsion balance experiment, dropping aluminum and gold test masses=(1.3pm1.0)times10^{-11}Roll, Peter G.; Krotkov, Robert; Dicke, Robert H.; The equivalence of inertial and passive gravitational mass, Annals of Physics, Volume 26, Issue 3, 20 February 1964, pp. 442–517
David Scott1971Dropped a falcon feather and a hammer at the same time on the Moonno detectable difference (not a rigorous experiment, but very dramatic being the first lunar oneHTTPS://WWW.YOUTUBE.COM/WATCH?V=MJYUDPM9KVK > TITLE=WEAK EQUIVALENCE PRINCIPLE TEST ON THE MOON, )
Vladimir Borisovich Braginski and Vladimir Ivanovich Panov>Panov1971Torsion balance, aluminum and platinum test masses, measuring acceleration towards the Sundifference is less than 1 part in 1012
Eöt-Wash group1987–Torsion balance, measuring acceleration of different masses towards the Earth, Sun and galactic center, using several different kinds of masseseta(text{Earth},text{Be-Ti})=(0.3 pm 1.8)times 10^{-13}SCHLAMMINGER >FIRST1=STEPHAN FIRST2=KI-YOUNG FIRST3=TODD A. FIRST4=JENS H. FIRST5=ERIC G. JOURNAL=PHYSICAL REVIEW LETTERS ISSUE=4 DOI=10.1103/PHYSREVLETT.100.041101 ARXIV=0712.0607 PAGE=041101,
See:Ciufolini, Ignazio; Wheeler, John A.; "Gravitation and Inertia", Princeton, New Jersey: Princeton University Press, 1995, pp. 117–119{| class="wikitable"
500?PhiloponusPhiloponus, John; "Corollaries on Place and Void", translated by David Furley, Ithaca, New York: Cornell University Press, 1987"small"Drop tower
1585StevinStevin, Simon; De Beghinselen der Weeghconst ["Principles of the Art of Weighing"], Leyden, 1586; Dijksterhuis, Eduard J.; "The Principal Works of Simon Stevin", Amsterdam, 19555×10−2Drop tower
1590?GalileoGalilei, Galileo; "Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze", Leida: Appresso gli Elsevirii, 1638; "Discourses and Mathematical Demonstrations Concerning Two New Sciences", Leiden: Elsevier Press, 16382×10−2Pendulum, drop tower
1686NewtonNewton, Isaac; "Philosophiae Naturalis Principia Mathematica" [Mathematical Principles of Natural Philosophy and his System of the World], translated by Andrew Motte, revised by Florian Cajori, Berkeley, California: University of California Press, 1934; Newton, Isaac; "The Principia: Mathematical Principles of Natural Philosophy", translated by I. Bernard Cohen and Anne Whitman, with the assistance of Julia Budenz, Berkeley, California: University of California Press, 199910−3Pendulum
1832BesselBessel, Friedrich W.; "Versuche Uber die Kraft, mit welcher die Erde Körper von verschiedner Beschaffenhelt anzieht", Annalen der Physik und Chemie, Berlin: J. Ch. Poggendorff, 25 401–408 (1832)2×10−5Pendulum
1908 (1922)EötvösEötvös, Loránd; Mathematische and naturnissenschaftliche Berichte aus Ungarn 8 65 (1889); Annalen der Physik (Leipzig) 68 11 (1922); Physical Review D 61(2) 022001 (1999)2×10−9Torsion balance
1910SouthernsSoutherns, Leonard; "A Determination of the Ratio of Mass to Weight for a Radioactive Substance", Proceedings of the Royal Society of London, 84 325–344 (1910), {{DOI|10.1098/rspa.1910.0078}}5×10−6Pendulum
1918ZeemanZeeman, Pieter; "Some experiments on gravitation: The ratio of mass to weight for crystals and radioactivesubstances", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Amsterdam 20(4) 542–553 (1918)3×10−8Torsion balance
1935RennerRenner, János; "Kísérleti vizsgálatok a tömegvonzás és tehetetlenség arányosságáról", Mathematikai és Természettudományi Értesítő 53 569 (1935), Budapest2×10−9Torsion balance
1964Dicke, Roll, Krotkov3x10−11Torsion balance
1972Braginsky, PanovBraginski, Vladimir Borisovich; Panov, Vladimir Ivanovich; Журнал Экспериментальной и Теоретической Физики (Zhurnal Éksperimental’noĭ i Teoreticheskoĭ Fiziki, Journal of Experimental and Theoretical Physics) 61 873 (1971)10−12Torsion balance
1981Keiser, FallerKeiser, George M.; Faller, James E.; Bulletin of the American Physical Society 24 579 (1979)4×10−11Fluid support
1999Baessler, et al.Baeßler, Stefan; et al.; Classical and Quantum Gravity 18(13) 2393 (2001); Baeßler, Stefan; Heckel, Blayne R.; Adelberger, Eric G.; Gundlach, Jens H.; Schmidt, Ulrich; Swanson, H. Erik; "Improved Test of the Equivalence Principle for Gravitational Self-Energy", Physical Review Letters 83(18) 3585 (1999)5x10−14Torsion balance
2017MICROSCOPE (satellite)2017PHRVL.119W1101T, MICROSCOPE Mission: First Results of a Space Test of the Equivalence Principle, Touboul, Pierre, Métris, Gilles, Rodrigues, Manuel, André, Yves, Baghi, Quentin, Bergé, Joël, Boulanger, Damien, Bremer, Stefanie, Carle, Patrice, Chhun, Ratana, Christophe, Bruno, Cipolla, Valerio, Damour, Thibault, Danto, Pascale, Dittus, Hansjoerg, Fayet, Pierre, Foulon, Bernard, Gageant, Claude, Guidotti, Pierre-Yves, Hagedorn, Daniel, Hardy, Emilie, Huynh, Phuong-Anh, Inchauspe, Henri, Kayser, Patrick, Lala, Stéphanie, Lämmerzahl, Claus, Lebat, Vincent, Leseur, Pierre, Liorzou, Françoise, List, Meike, 29, Physical Review Letters, 119, 23, 2017, 231101, 1712.01176, 10.1103/PhysRevLett.119.231101, 10−15Earth orbit
Experiments are still being performed at the University of Washington which have placed limits on the differential acceleration of objects towards the Earth, the Sun and towards dark matter in the galactic center. Future satellite experimentsJOURNAL, Dittus, Hansjörg, Claus, Lāmmerzahl, Experimental Tests of the Equivalence Principle and Newton's Law in Space, 95, 758, 10.1063/1.1900510, Gravitation and Cosmology: 2nd Mexican Meeting on Mathematical and Experimental Physics, AIP Conference Proceedings,weblink 2005AIPC..758...95D, dead,weblink" title="">weblink 17 December 2008, dmy, 2005, – STEP (Satellite Test of the Equivalence Principle), Galileo Galilei, and MICROSCOPE (MICROSatellite à traînée Compensée pour l'Observation du Principe d'Équivalence) – will test the weak equivalence principle in space, to much higher accuracy.With the first successful production of antimatter, in particular anti-hydrogen, a new approach to test the weak equivalence principle has been proposed. Experiments to compare the gravitational behavior of matter and antimatter are currently being developed.JOURNAL, Testing the Weak Equivalence Principle with an antimatter beam at CERN, Journal of Physics: Conference Series, 2015, 631, 1, 012047,weblink 2015JPhCS.631a2047K, 10.1088/1742-6596/631/1/012047, Kimura, M., Aghion, S., Amsler, C., Ariga, A., Ariga, T., Belov, A., Bonomi, G., Bräunig, P., Bremer, J., Brusa, R. S., Cabaret, L., Caccia, M., Caravita, R., Castelli, F., Cerchiari, G., Chlouba, K., Cialdi, S., Comparat, D., Consolati, G., Demetrio, A., Derking, H., Di Noto, L., Doser, M., Dudarev, A., Ereditato, A., Ferragut, R., Fontana, A., Gerber, S., Giammarchi, M., Gligorova, A., 29, Proposals that may lead to a quantum theory of gravity such as string theory and loop quantum gravity predict violations of the weak equivalence principle because they contain many light scalar fields with long Compton wavelengths, which should generate fifth forces and variation of the fundamental constants. Heuristic arguments suggest that the magnitude of these equivalence principle violations could be in the 10−13 to 10−18 range. Currently envisioned tests of the weak equivalence principle are approaching a degree of sensitivity such that non-discovery of a violation would be just as profound a result as discovery of a violation. Non-discovery of equivalence principle violation in this range would suggest that gravity is so fundamentally different from other forces as to require a major reevaluation of current attempts to unify gravity with the other forces of nature. A positive detection, on the other hand, would provide a major guidepost towards unification.JOURNAL, Overduin, James, Everitt, Francis, Mester, John, Worden, Paul, The Science Case for STEP, 10.1016/j.asr.2009.02.012, Advances in Space Research, 43, 10, 1532–1537, 2009, 0902.2247, 2009AdSpR..43.1532O,

The Einstein equivalence principle

What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that:BOOK, Haugen, Mark P., Claus, Lämmerzahl, Principles of Equivalence: Their Role in Gravitation Physics and Experiments that Test Them, Gyros, 562, 195–212, 2001, 978-3-540-41236-6, gr-qc/0103067, 2001LNP...562..195H, 10.1007/3-540-40988-2_10,
The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
Here "local" has a very special meaning: not only must the experiment not look outside the laboratory, but it must also be small compared to variations in the gravitational field, tidal forces, so that the entire laboratory is freely falling. It also implies the absence of interactions with "external" fields other than the gravitational field.{{Citation needed|reason = Definition of "local" in this context. In particular its relation to external fields as stated is dubious. If there are external fields than the equivalence principle doesn't trigger due to the "in a freely falling"-part in the statement. If *no external fields* would be part of the definition of "local", then "free falling" would be redundant.|date=August 2011}}The principle of relativity implies that the outcome of local experiments must be independent of the velocity of the apparatus, so the most important consequence of this principle is the Copernican idea that dimensionless physical values such as the fine-structure constant and electron-to-proton mass ratio must not depend on where in space or time we measure them. Many physicists believe that any Lorentz invariant theory that satisfies the weak equivalence principle also satisfies the Einstein equivalence principle.Schiff's conjecture suggests that the weak equivalence principle implies the Einstein equivalence principle, but it has not been proven. Nonetheless, the two principles are tested with very different kinds of experiments. The Einstein equivalence principle has been criticized as imprecise, because there is no universally accepted way to distinguish gravitational from non-gravitational experiments (see for instance HadleyJOURNAL, Mark J., Hadley, 10.1007/BF02764119, Foundations of Physics Letters, 10, 1, The Logic of Quantum Mechanics Derived from Classical General Relativity, 43–60, 1997, quant-ph/9706018, 1997FoPhL..10...43H,, and DurandJOURNAL, Durand, Stéphane, 2002, An amusing analogy: modelling quantum-type behaviours with wormhole-based time travel,weblink Journal of Optics B: Quantum and Semiclassical Optics, 4, 4, S351–S357, 10.1088/1464-4266/4/4/319, 2002JOptB...4S.351D, ).

Tests of the Einstein equivalence principle

In addition to the tests of the weak equivalence principle, the Einstein equivalence principle can be tested by searching for variation of dimensionless constants and mass ratios. The present best limits on the variation of the fundamental constants have mainly been set by studying the naturally occurring Oklo natural nuclear fission reactor, where nuclear reactions similar to ones we observe today have been shown to have occurred underground approximately two billion years ago. These reactions are extremely sensitive to the values of the fundamental constants.{| class="wikitable"
ConstantYearMethodLimit on fractional change
proton gyromagnetic factor1976astrophysical10−1
weak interaction constant1976Oklo10−2
fine structure constant1976Oklo10−7
electronproton mass ratio2002quasars10−4
There have been a number of controversial attempts to constrain the variation of the strong interaction constant. There have been several suggestions that "constants" do vary on cosmological scales. The best known is the reported detection of variation (at the 10−5 level) of the fine-structure constant from measurements of distant quasars, see Webb et al.JOURNAL, John K., Webb, Michael T., Murphy, Victor V., Flambaum, Vladimir A., Dzuba, John D., Barrow, Chris W., Churchill, Jason X., Prochaska, Arthur M., Wolfe, 10.1103/PhysRevLett.87.091301, Physical Review Letters, Further Evidence for Cosmological Evolution of the Fine Structure Constant, 87, 9, 2001, astro-ph/0012539, 11531558, 2001PhRvL..87i1301W, 091301, Other researchers{{who|date=November 2018}} dispute these findings. Other tests of the Einstein equivalence principle are gravitational redshift experiments, such as the Pound–Rebka experiment which test the position independence of experiments.

The strong equivalence principle

The strong equivalence principle suggests the laws of gravitation are independent of velocity and location. In particular,
The gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitution.
The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
The first part is a version of the weak equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments. The second part is the Einstein equivalence principle (with the same definition of "local"), restated to allow gravitational experiments and self-gravitating bodies. The freely-falling object or laboratory, however, must still be small, so that tidal forces may be neglected (hence "local experiment").This is the only form of the equivalence principle that applies to self-gravitating objects (such as stars), which have substantial internal gravitational interactions. It requires that the gravitational constant be the same everywhere in the universe and is incompatible with a fifth force. It is much more restrictive than the Einstein equivalence principle.The strong equivalence principle suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity) and does not have any extra fields associated with it. If an observer measures a patch of space to be flat, then the strong equivalence principle suggests that it is absolutely equivalent to any other patch of flat space elsewhere in the universe. Einstein's theory of general relativity (including the cosmological constant) is thought to be the only theory of gravity that satisfies the strong equivalence principle. A number of alternative theories, such as Brans–Dicke theory, satisfy only the Einstein equivalence principle.

Tests of the strong equivalence principle

The strong equivalence principle can be tested by searching for a variation of Newton's gravitational constant G over the life of the universe, or equivalently, variation in the masses of the fundamental particles. A number of independent constraints, from orbits in the solar system and studies of Big Bang nucleosynthesis have shown that G cannot have varied by more than 10%.Thus, the strong equivalence principle can be tested by searching for fifth forces (deviations from the gravitational force-law predicted by general relativity). These experiments typically look for failures of the inverse-square law (specifically Yukawa forces or failures of Birkhoff's theorem) behavior of gravity in the laboratory. The most accurate tests over short distances have been performed by the Eöt–Wash group. A future satellite experiment, SEE (Satellite Energy Exchange), will search for fifth forces in space and should be able to further constrain violations of the strong equivalence principle. Other limits, looking for much longer-range forces, have been placed by searching for the Nordtvedt effect, a "polarization" of solar system orbits that would be caused by gravitational self-energy accelerating at a different rate from normal matter. This effect has been sensitively tested by the Lunar Laser Ranging Experiment. Other tests include studying the deflection of radiation from distant radio sources by the sun, which can be accurately measured by very long baseline interferometry. Another sensitive test comes from measurements of the frequency shift of signals to and from the Cassini spacecraft. Together, these measurements have put tight limits on Brans–Dicke theory and other alternative theories of gravity.In 2014, astronomers discovered a stellar triple system including a millisecond pulsar PSR J0337+1715 and two white dwarfs orbiting it. The system provided them a chance to test the strong equivalence principle in a strong gravitational field with high accuracy.JOURNAL,weblink A millisecond pulsar in a stellar triple system, Scott M., Ransom, etal, Nature, 2014, 8 January 2014, 10.1038/nature12917, 1401.0535, 2014Natur.505..520R, 505, 7484, 520–524, 24390352, JOURNAL, Universality of free fall from the orbital motion of a pulsar in a stellar triple system, Anne M. Archibald, etal, Nature, 559, 7712, 73–76, 4 July 2018, 10.1038/s41586-018-0265-1, 29973733, 1807.02059, 2018Natur.559...73A, NEWS, Even Phenomenally Dense Neutron Stars Fall like a Feather – Einstein Gets It Right Again,weblink NRAO, 4 July 2018, 28 July 2018, Charles Blue, Paul Vosteen,


One challenge to the equivalence principle is the Brans–Dicke theory. Self-creation cosmology is a modification of the Brans–Dicke theory. The Fredkin Finite Nature Hypothesis is an even more radical challenge to the equivalence principle and has even fewer supporters.In August 2010, researchers from the University of New South Wales, Swinburne University of Technology, and Cambridge University published a paper titled "Evidence for spatial variation of the fine structure constant", whose tentative conclusion is that, "qualitatively, [the] results suggest a violation of the Einstein Equivalence Principle, and could infer a very large or infinite universe, within which our 'local' Hubble volume represents a tiny fraction."JOURNAL, 1008.3907, John K., Webb, Julian A., King, Michael T., Murphy, Victor V., Flambaum, Robert F., Carswell, Matthew B., Bainbridge, Evidence for spatial variation of the fine structure constant, 2010, 10.1103/PhysRevLett.107.191101, 22181590, 107, 19, 191101, Physical Review Letters, 2011PhRvL.107s1101W,


Dutch physicist and string theorist Erik Verlinde has generated a self-contained, logical derivation of the equivalence principle based on the starting assumption of a holographic universe. Given this situation, gravity would not be a true fundamental force as is currently thought but instead an "emergent property" related to entropy. Verlinde's entropic gravity theory apparently leads naturally to the correct observed strength of dark energy; previous failures to explain its incredibly small magnitude have been called by such people as cosmologist Michael Turner (who is credited as having coined the term "dark energy") as "the greatest embarrassment in the history of theoretical physics".WEB, Wright, Karen, Very Dark Energy,weblink Discover Magazine, 26 February 2013, 1 March 2001, These ideas are far from settled and still very controversial.


  • University of WashingtonEöt–Wash group
  • Lunar Laser RangingWEB,weblink Fundamental Physics of Space - Technical Details, JOURNAL, Viswanathan, V, Fienga, A, Minazzoli, O, Bernus, L, Laskar, J, Gastineau, M, The new lunar ephemeris INPOP17a and its application to fundamental physics, Monthly Notices of the Royal Astronomical Society, May 2018, 476, 2, 1877–1888, 10.1093/mnras/sty096, 1710.09167,
  • Galileo-Galilei satellite experimentWEB,weblink "GALILEO GALILEI" GG Small Mission Project,
  • Satellite Test of the Equivalence Principle (STEP)WEB,weblink S T e P,
  • MICROSCOPweblink
  • Satellite Energy Exchange (SEE)WEB,weblink Archived copy, 2005-05-07, dead,weblink" title="">weblink 7 May 2005, dmy-all,
  • "...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..."16 November 2004, physicsweb: Equivalence principle passes atomic test

See also

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  • Dicke, Robert H.; "New Research on Old Gravitation", Science 129, 3349 (1959). This paper is the first to make the distinction between the strong and weak equivalence principles.
  • Dicke, Robert H.; "Mach's Principle and Equivalence", in Evidence for gravitational theories: proceedings of course 20 of the International School of Physics "Enrico Fermi", ed. C. Møller (Academic Press, New York, 1962). This article outlines the approach to precisely testing general relativity advocated by Dicke and pursued from 1959 onwards.
  • Einstein, Albert; "Ãœber das Relativitätsprinzip und die aus demselben gezogene Folgerungen", Jahrbuch der Radioaktivitaet und Elektronik 4 (1907); translated "On the relativity principle and the conclusions drawn from it", in The collected papers of Albert Einstein. Vol. 2 : The Swiss years: writings, 1900–1909 (Princeton University Press, Princeton, New Jersey, 1989), Anna Beck translator. This is Einstein's first statement of the equivalence principle.
  • Einstein, Albert; "Ãœber den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes", Annalen der Physik 35 (1911); translated "On the Influence of Gravitation on the Propagation of Light" in The collected papers of Albert Einstein. Vol. 3 : The Swiss years: writings, 1909–1911 (Princeton University Press, Princeton, New Jersey, 1994), Anna Beck translator, and in The Principle of Relativity, (Dover, 1924), pp 99–108, W. Perrett and G. B. Jeffery translators, {{ISBN|0-486-60081-5}}. The two Einstein papers are discussed online at The Genesis of General Relativity.
  • Brans, Carl H.; "The roots of scalar-tensor theory: an approximate history", {{arxiv|gr-qc/0506063}}. Discusses the history of attempts to construct gravity theories with a scalar field and the relation to the equivalence principle and Mach's principle.
  • Misner, Charles W.; Thorne, Kip S.; and Wheeler, John A.; Gravitation, New York: W. H. Freeman and Company, 1973, Chapter 16 discusses the equivalence principle.
  • Ohanian, Hans; and Ruffini, Remo; Gravitation and Spacetime 2nd edition, New York: Norton, 1994, {{ISBN|0-393-96501-5}} Chapter 1 discusses the equivalence principle, but incorrectly, according to modern usage, states that the strong equivalence principle is wrong.
  • Uzan, Jean-Philippe; "The fundamental constants and their variation: Observational status and theoretical motivations", Reviews of Modern Physics 75, 403 (2003). {{arxiv|hep-ph/0205340}} This technical article reviews the best constraints on the variation of the fundamental constants.
  • Will, Clifford M.; Theory and experiment in gravitational physics, Cambridge, UK: Cambridge University Press, 1993. This is the standard technical reference for tests of general relativity.
  • Will, Clifford M.; Was Einstein Right?: Putting General Relativity to the Test, Basic Books (1993). This is a popular account of tests of general relativity.
  • Will, Clifford M.; The Confrontation between General Relativity and Experiment, Living Reviews in Relativity (2006). An online, technical review, covering much of the material in Theory and experiment in gravitational physics. The Einstein and strong variants of the equivalence principles are discussed in sections 2.1 and weblink" title="">3.1, respectively.
  • Friedman, Michael; Foundations of Space-Time Theories, Princeton, New Jersey: Princeton University Press, 1983. Chapter V discusses the equivalence principle.
  • {{citation | first = Michel | last = Ghins| first2 = Tim |last2 = Budden |title= The Principle of Equivalence| journal = Stud. Hist. Phil. Mod. Phys.| volume = 32 | year = 2001 |issue = 1| pages = 33–51 | doi=10.1016/S1355-2198(00)00038-1}}
  • {{citation|first = Hans C.|last = Ohanian|title= What is the Principle of Equivalence?|journal = American Journal of Physics| volume = 45 |year =1977|issue=10|pages=903–909|doi=10.1119/1.10744|bibcode=1977AmJPh..45..903O}}
  • {{citation|first = E.|last = Di Casola|first2=S.|last2=Liberati|first3=S.|last3=Sonego|title= Nonequivalence of equivalence principles|journal = American Journal of Physics| volume =83 |year=2015|issue=1|pages=39|doi=10.1119/1.4895342|arxiv=1310.7426|bibcode=2015AmJPh..83...39D}}

External links

  • weblink" title="">Equivalence Principle at NASA, including tests
  • weblink" title="">Introducing The Einstein Principle of Equivalence from Syracuse University
  • The Equivalence Principle at MathPages
  • weblink" title="">The Einstein Equivalence Principle at Living Reviews on General Relativity
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