Equivalence principle#The weak equivalence principle

In classical mechanics, Newton's equation of motion in a gravitational field, written out in full, is:

: inertial mass × acceleration = gravitational mass × gravitational acceleration

Careful experiments have shown that the inertial mass on the left side and gravitational mass on the right side are numerically equal and independent of the material composing the masses. The equivalence principle is the hypothesis that this numerical equality of inertial and gravitational mass is a consequence of their fundamental identity.{{cite book|last1=Einstein|first1=Albert|title=The Meaning of Relativity|url=https://archive.org/details/meaningrelativit00aein|url-access=limited|date=2003|publisher=Routledge|isbn=9781134449798|page=[https://archive.org/details/meaningrelativit00aein/page/n67 59]}}{{rp|32}}

The equivalence principle can be considered an extension of the principle of relativity, the principle that the laws of physics are invariant under uniform motion. 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 and the laws of physics are unable to distinguish these cases.{{rp|33}}

{{See also|History of gravitational theory}}

By experimenting with the acceleration of different materials, Galileo Galilei determined that gravitation is independent of the amount of mass being accelerated.{{Cite journal |last1=Everitt |first1=C.W.F. |last2=Damour |first2=T. |last3=Nordtvedt |first3=K. |last4=Reinhard |first4=R. |date=October 2003 |title=Historical perspective on testing the Equivalence Principle |url=https://linkinghub.elsevier.com/retrieve/pii/S0273117703903358 |journal=Advances in Space Research |language=en |volume=32 |issue=7 |pages=1297–1300 |doi=10.1016/S0273-1177(03)90335-8|bibcode=2003AdSpR..32.1297E }}

Isaac Newton, just 50 years after Galileo, investigated whether gravitational and inertial mass might be different concepts. He compared the periods of pendulums composed of different materials and found them to be identical. From this, he inferred that gravitational and inertial mass are the same thing. The form of this assertion, where the equivalence principle is taken to follow from empirical consistency, later became known as "weak equivalence".

A version of the equivalence principle consistent with special relativity was introduced by Albert Einstein in 1907, when he observed that identical physical laws are observed in two systems, one subject to a constant gravitational field causing acceleration and the other subject to constant acceleration, like a rocket far from any gravitational field.{{rp|152}} Since the physical laws are the same, Einstein assumed the gravitational field and the acceleration were "physically equivalent". Einstein stated this hypothesis by saying he would:

{{blockquote|...assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.|Einstein, 1907Einstein, Albert. "On the relativity principle and the conclusions drawn from it." Jahrb Radioaktivitat Elektronik 4 (1907): 411–462.}}

In 1911 Einstein demonstrated the power of the equivalence principle by using it to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field.{{cite book | last=Whittaker | first=Sir Edmund | title=A History of the Theories of Aether and Electricity | publisher=Courier Dover Publications | date=1989-01-01 | isbn=0-486-26126-3 | volume=2}}{{rp|153}} He connected the equivalence principle to his earlier principle of special relativity:

{{blockquote| This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course.|Einstein, 1911Einstein, Albert. "On the Influence of Gravitation on the Propagation of Light." Annalen der Physik 35.898–908 (1911): 906.}}

Soon after completing work on his theory of gravity (known as general relativity)Lorentz, Hendrik Antoon, et al. [https://books.google.com/books?id=oy_Z3o7p6zwC The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity]. United Kingdom, Dover Publications, 1923.{{rp|111}} and then also in later years, Einstein recalled the importance of the equivalence principle to his work:

{{blockquote| The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me: If a man falls

freely, he would not feel his weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity.| Einstein, 1922Einstein, Albert, [https://web.archive.org/web/20151222085312/http://inpac.ucsd.edu/students/courses/winter2012/physics2d/einsteinonrelativity.pdf 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.}}

Einstein's development of general relativity necessitated some means of empirically discriminating the theory from other theories of gravity compatible with special relativity. Accordingly, Robert Dicke developed a test program incorporating two new principles – the {{section link||Einstein equivalence principle}}, and the {{section link||Strong equivalence principle}} – each of which assumes the weak equivalence principle as a starting point.

File:Apollo 15 feather and hammer drop.ogv 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.]]Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.{{Cite journal |last1=Clifton |first1=Timothy |last2=Ferreira |first2=Pedro G. |last3=Padilla |first3=Antonio |last4=Skordis |first4=Constantinos |date=March 2012 |title=Modified gravity and cosmology |url=https://linkinghub.elsevier.com/retrieve/pii/S0370157312000105 |journal=Physics Reports |language=en |volume=513 |issue=1–3 |pages=1–189 |doi=10.1016/j.physrep.2012.01.001|arxiv=1106.2476 |bibcode=2012PhR...513....1C }}{{rp|6}} Some proposals also suggest finer divisions or minor alterations.{{Cite journal |last1=Di Casola |first1=Eolo |last2=Liberati |first2=Stefano |last3=Sonego |first3=Sebastiano |date=2015-01-01 |title=Nonequivalence of equivalence principles |url=https://pubs.aip.org/ajp/article/83/1/39/1042100/Nonequivalence-of-equivalence-principles |journal=American Journal of Physics |language=en |volume=83 |issue=1 |pages=39–46 |doi=10.1119/1.4895342 |arxiv=1310.7426 |bibcode=2015AmJPh..83...39D |s2cid=119110646 |issn=0002-9505 |quote=We have seen that the various formulations of the equivalence principle form hierarchy (or rather, a nested sequence of statements narrowing down the type of gravitational theory),}}{{Cite journal |last1=Ghins |first1=Michel |last2=Budden |first2=Tim |date=March 2001 |title=The Principle of Equivalence |url=https://linkinghub.elsevier.com/retrieve/pii/S1355219800000381 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |language=en |volume=32 |issue=1 |pages=33–51 |doi=10.1016/S1355-2198(00)00038-1|bibcode=2001SHPMP..32...33G }}

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 equivalence principle, a generalization of the weak equivalence principle, includes astronomic bodies with gravitational self-binding energy.{{cite journal | last1 = Wagner | first1 = Todd A. | last2 = Schlamminger | first2 = Stephan | last3 = Gundlach | first3 = Jens H. | last4 = Adelberger | first4 = Eric G. | year = 2012 | title = Torsion-balance tests of the weak equivalence principle | journal = Classical and Quantum Gravity | volume = 29 | issue = 18| page = 184002 | doi = 10.1088/0264-9381/29/18/184002 | arxiv = 1207.2442 | bibcode = 2012CQGra..29r4002W | s2cid = 59141292 }} Instead, the weak equivalence principle assumes falling bodies are self-bound by non-gravitational forces only (e.g. a stone). Either way:

• "All uncharged, freely falling test particles follow the same trajectories, once an initial position and velocity have been prescribed".{{rp|6}}
• "... in a uniform gravitational field all objects, regardless of their composition, fall with precisely the same acceleration." "The weak equivalence principle implicitly assumes that the falling objects are bound by non-gravitational forces."
• "... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass."{{cite book |title=Five-dimensional Physics |first=Paul S. |last=Wesson |page=82 |url=https://books.google.com/books?id=dSv8ksxHR0oC&q=intitle:Five+intitle:Dimensional+intitle:Physics |isbn=978-981-256-661-4 |publisher=World Scientific |year=2006 }}
• 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).

Uniformity of the gravitational field eliminates measurable tidal forces originating from a radial divergent gravitational field (e.g., the Earth) upon finite sized physical bodies.

What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that:

{{block indent|em=1.5|text=the outcome of any local, non-gravitational test experiment is independent of the experimental apparatus' velocity relative to the gravitational field and is independent of where and when in the gravitational field the experiment is performed.{{Cite book |last1=Haugen |first1=Mark P. |title=Gyros, Clocks, Interferometers...: Testing Relativistic Gravity in Space. Lecture Notes in Physics |last2=Lämmerzahl |first2=Claus |year=2001 |isbn=978-3-540-41236-6 |volume=562 |pages=195–212 |chapter=Principles of Equivalence: Their Role in Gravitation Physics and Experiments That Test Them. |journal=Gyros |bibcode=2001LNP...562..195H |doi=10.1007/3-540-40988-2_10 |arxiv=gr-qc/0103067 |s2cid=15430387}}}}

Here local means that experimental setup must be small compared to variations in the gravitational field, called tidal forces. The test experiment must be small enough so that its gravitational potential does not alter the result.

The two additional constraints added to the weak principle to get the Einstein form − (1) the independence of the outcome on relative velocity (local Lorentz invariance) and (2) independence of "where" (known as local positional invariance) − have far reaching consequences. With these constraints alone Einstein was able to predict the gravitational redshift. Theories of gravity that obey the Einstein equivalence principle must be "metric theories", meaning that trajectories of freely falling bodies are geodesics of symmetric metric.{{rp|9}}

Around 1960 Leonard I. Schiff conjectured that any complete and consistent theory of gravity that embodies the weak equivalence principle implies the Einstein equivalence principle; the conjecture can't be proven but has several plausibility arguments in its favor.{{rp|20}} 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 Hadley{{cite journal |first=Mark J. |last=Hadley |doi=10.1007/BF02764119 |journal=Foundations of Physics Letters |volume=10 |issue=1 |title=The Logic of Quantum Mechanics Derived from Classical General Relativity |pages=43–60 |year=1997 |arxiv=quant-ph/9706018|bibcode = 1997FoPhL..10...43H |citeseerx=10.1.1.252.6335 |s2cid=15007947 }} and Durand{{cite journal | last1 = Durand | first1 = Stéphane | year = 2002| title = An amusing analogy: modelling quantum-type behaviours with wormhole-based time travel | url = http://stacks.iop.org/ob/4/S351 | journal = Journal of Optics B: Quantum and Semiclassical Optics | volume = 4 | issue = 4 | pages = S351–S357| doi = 10.1088/1464-4266/4/4/319 | bibcode = 2002JOptB...4S.351D }}).

The strong equivalence principle applies the same constraints as the Einstein equivalence principle, but allows the freely falling bodies to be massive gravitating objects as well as test particles.

Thus this is a version of the equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments. It requires that the gravitational constant be the same everywhere in the universe{{rp|49}} and is incompatible with a fifth force. It is much more restrictive than the Einstein equivalence principle.

Like the Einstein equivalence principle, the strong equivalence principle requires gravity to be geometrical by nature, but in addition it forbids any extra fields, so the metric alone determines all of the effects of gravity. 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 and the Einstein-aether theory add additional fields.

Some of the tests of the equivalence principle use names for the different ways mass appears in physical formulae. In nonrelativistic physics three kinds of mass can be distinguished:{{Cite journal |last=Will |first=Clifford M. |date=Dec 2014 |title=The Confrontation between General Relativity and Experiment |journal=Living Reviews in Relativity |language=en |volume=17 |issue=1 |page=4 |doi=10.12942/lrr-2014-4 |doi-access=free |issn=2367-3613 |pmc=5255900 |pmid=28179848|arxiv=1403.7377 |bibcode=2014LRR....17....4W }}

1. Inertial mass intrinsic to an object, the sum of all of its mass–energy.
2. Passive mass, the response to gravity, the object's weight.
3. Active mass, the mass that determines the objects gravitational effect.

By definition of active and passive gravitational mass, the force on $M\_1$ due to the gravitational field of $M\_0$ is:

$$F\_1\; =\; \backslash frac\{M\_0^\backslash mathrm\{act\}\; M\_1^\backslash mathrm\{pass\}\}\{r^2\}$$

Likewise the force on a second object of arbitrary mass₂ due to the gravitational field of mass₀ is:

$$F\_2\; =\; \backslash frac\{M\_0^\backslash mathrm\{act\}\; M\_2^\backslash mathrm\{pass\}\}\{r^2\}$$

By definition of inertial mass:$$F\; =\; m^\backslash mathrm\{inert\}\; a$$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).

$$a\_1\; =\; \backslash frac\{F\_1\}\{m\_1^\backslash mathrm\{inert\}\}\; =\; a\_2\; =\; \backslash frac\{F\_2\}\{m\_2^\backslash mathrm\{inert\}\}$$

Hence:

$$\backslash frac\{M\_0^\backslash mathrm\{act\}\; M\_1^\backslash mathrm\{pass\}\}\{r^2\; m\_1^\backslash mathrm\{inert\}\}\; =\; \backslash frac\{M\_0^\backslash mathrm\{act\}\; M\_2^\backslash mathrm\{pass\}\}\{r^2\; m\_2^\backslash mathrm\{inert\}\}$$

Therefore:

$$\backslash frac\{M\_1^\backslash mathrm\{pass\}\}\{m\_1^\backslash mathrm\{inert\}\}\; =\; \backslash frac\{M\_2^\backslash mathrm\{pass\}\}\{m\_2^\backslash mathrm\{inert\}\}$$

In other words, passive gravitational mass must be proportional to inertial mass for objects, independent of their material composition if the weak equivalence principle is obeyed.

The dimensionless Eötvös-parameter or Eötvös ratio $\backslash eta(A,B)$ 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".

$$\backslash eta(A,B)=2\backslash frac\{\; \backslash left(\backslash frac\{m\_\{\backslash textrm\; pass\}\}\{m\_\{\backslash textrm\; inert\}\}\backslash right)\_A-\backslash left(\backslash frac\{m\_\{\backslash textrm\; pass\}\}\{m\_\{\backslash textrm\; inert\}\}\backslash right)\_B\; \}\{\backslash left(\backslash frac\{m\_\{\backslash textrm\; pass\}\}\{m\_\{\backslash textrm\; inert\}\}\backslash right)\_A+\backslash left(\backslash frac\{m\_\{\backslash textrm\; pass\}\}\{m\_\{\backslash textrm\; inert\}\}\backslash right)\_B\}.$$

Values of this parameter are used to compare tests of the equivalence principle.{{rp|10}}

A similar parameter can be used to compare passive and active mass.

By Newton's third law of motion:

$$F\_1\; =\; \backslash frac\{M\_0^\backslash mathrm\{act\}\; M\_1^\backslash mathrm\{pass\}\}\{r^2\}$$

must be equal and opposite to

$$F\_0\; =\; \backslash frac\{M\_1^\backslash mathrm\{act\}\; M\_0^\backslash mathrm\{pass\}\}\{r^2\}$$

It follows that:

$$\backslash frac\{M\_0^\backslash mathrm\{act\}\}\{M\_0^\backslash mathrm\{pass\}\}\; =\; \backslash frac\{M\_1^\backslash mathrm\{act\}\}\{M\_1^\backslash mathrm\{pass\}\}$$

In words, passive gravitational mass must be proportional to active gravitational mass for all objects. The difference,

$$S\_\{0,1\}\; =\; \backslash frac\{M\_0^\backslash mathrm\{act\}\}\{M\_0^\backslash mathrm\{pass\}\}\; -\; \backslash frac\{M\_1^\backslash mathrm\{act\}\}\{M\_1^\backslash mathrm\{pass\}\}$$

is used to quantify differences between passive and active mass.{{Cite journal |last1=Singh |first1=Vishwa Vijay |last2=Müller |first2=Jürgen |last3=Biskupek |first3=Liliane |last4=Hackmann |first4=Eva |last5=Lämmerzahl |first5=Claus |date=2023-07-13 |title=Equivalence of Active and Passive Gravitational Mass Tested with Lunar Laser Ranging |url=https://link.aps.org/doi/10.1103/PhysRevLett.131.021401 |journal=Physical Review Letters |language=en |volume=131 |issue=2 |page=021401 |doi=10.1103/PhysRevLett.131.021401 |pmid=37505941 |arxiv=2212.09407 |bibcode=2023PhRvL.131b1401S |issn=0031-9007}}

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 and verifying that they land at the same time. Historically this was the first approach – though probably not by Galileo's Leaning Tower of Pisa experiment{{cite book|last=Drake|first=Stillman|title=Galileo at Work: His Scientific Biography|date=2003|publisher=Dover publ.|location=Mineola (N.Y.)|isbn=9780486495422|edition=Facsim.}}{{rp|19–21}} but instead earlier by Simon Stevin,{{cite book |last1=Devreese |first1=Jozef T. |author-link1=Jozef T. Devreese |last2=Vanden Berghe |first2=Guido |year=2008 |title='Magic Is No Magic': The Wonderful World of Simon Stevin |url=https://books.google.com/books?isbn=1845643917 |page=154 |publisher=WIT Press |isbn=9781845643911 }} who dropped lead balls of different masses off the Delft churchtower and listened for the sound of them hitting a wooden plank.

Newton measured the period of pendulums made with different materials as an alternative test giving the first precision measurements. Loránd Eötvös's approach in 1908 used a very sensitive torsion balance to give precision approaching 1 in a billion. Modern experiments have improved this by another factor of a million.

A popular exposition of this measurement was done on the Moon by David Scott in 1971. He dropped a falcon feather and a hammer at the same time, showing on video{{cite web | url=https://www.youtube.com/watch?v=MJyUDpm9Kvk |archive-url=https://ghostarchive.org/varchive/youtube/20211221/MJyUDpm9Kvk |archive-date=2021-12-21 |url-status=live |title=Weak Equivalence Principle test on the moon |website=YouTube |date=18 May 2007 }}{{cbignore}} that they landed at the same time.

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.{{Cite web|url=http://www.npl.washington.edu/eotwash/|title=The Eöt-Wash Group | Laboratory Tests of Gravitational and sub-Gravitational Physics|website=www.npl.washington.edu}} Future satellite experiments{{cite conference|last1=Dittus |first1=Hansjörg |first2=Claus |last2=Lāmmerzahl |title=Experimental Tests of the Equivalence Principle and Newton's Law in Space |page=95 |volume=758 |doi=10.1063/1.1900510 |conference=Gravitation and Cosmology: 2nd Mexican Meeting on Mathematical and Experimental Physics, AIP Conference Proceedings |url=http://www.zarm.uni-bremen.de/2forschung/gravi/publications/papers/2005DittusLaemmerzahl.pdf |bibcode=2005AIPC..758...95D |url-status=dead |archive-url=https://web.archive.org/web/20081217040845/http://www.zarm.uni-bremen.de/2forschung/gravi/publications/papers/2005DittusLaemmerzahl.pdf |archive-date=17 December 2008|year=2005 }} – Satellite Test of the Equivalence Principle{{cite web | url=http://einstein.stanford.edu/STEP/ | title=S T e P}} and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy.{{cite web | url=http://eotvos.dm.unipi.it/nobili/ | title="GALILEO GALILEI" GG Small Mission Project}}

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.{{cite journal|title=Testing the Weak Equivalence Principle with an antimatter beam at CERN|journal=Journal of Physics: Conference Series|date=2015|volume=631|issue=1|page=012047|url=http://stacks.iop.org/1742-6596/631/i=1/a=012047|bibcode = 2015JPhCS.631a2047K |doi = 10.1088/1742-6596/631/1/012047 |last1=Kimura|first1=M.|last2=Aghion|first2=S.|last3=Amsler|first3=C. |last4=Ariga|first4=A.|last5=Ariga|first5=T.|last6=Belov|first6=A.|last7=Bonomi|first7=G. |last8=Bräunig|first8=P. |last9=Bremer|first9=J.|last10=Brusa|first10=R. S.|last11=Cabaret|first11=L.|last12=Caccia |first12=M. |last13=Caravita|first13=R. |last14=Castelli|first14=F.|last15=Cerchiari|first15=G. |last16=Chlouba|first16=K. |last17=Cialdi|first17=S. |last18=Comparat|first18=D. |last19=Consolati|first19=G.|last20=Demetrio|first20=A. |last21=Derking|first21=H. |last22=Di Noto|first22=L. |last23=Doser |first23=M.|last24=Dudarev|first24=A. |last25=Ereditato|first25=A. |last26=Ferragut|first26=R.|last27=Fontana|first27=A. |last28=Gerber|first28=S. |last29=Giammarchi|first29=M. |last30=Gligorova|first30=A.|display-authors=29|doi-access=free|hdl=2434/457743|hdl-access=free}}

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.{{Cite journal | last1 = Overduin | first1 = James | last2 = Everitt | first2 = Francis | last3 = Mester | first3 = John | last4 = Worden | first4 = Paul | title = The Science Case for STEP | doi = 10.1016/j.asr.2009.02.012 | journal = Advances in Space Research | volume = 43 | issue = 10 | pages = 1532–1537 | year = 2009 |arxiv = 0902.2247 |bibcode = 2009AdSpR..43.1532O | s2cid = 8019480 }}

In addition to the tests of the weak equivalence principle, the Einstein equivalence principle requires testing the local Lorentz invariance and local positional invariance conditions.

Testing local Lorentz invariance amounts to testing special relativity, a theory with vast number of existing tests.{{rp|12}} Nevertheless, attempts to look for quantum gravity require even more precise tests. The modern tests include looking for directional variations in the speed of light (called "clock anisotropy tests") and new forms of the Michelson–Morley experiment. The anisotropy measures less than one part in 10⁻²⁰.{{rp|14}}

Testing local positional invariance divides in to tests in space and in time.{{rp|17}} Space-based tests use measurements of the gravitational redshift, the classic is the Pound–Rebka experiment in the 1960s. The most precise measurement was done in 1976 by flying a hydrogen maser and comparing it to one on the ground. The Global Positioning System requires compensation for this redshift to give accurate position values.

Time-based tests search for variation of dimensionless constants and mass ratios.{{Cite journal |last=Uzan |first=Jean-Philippe |date=2003-04-07 |title=The fundamental constants and their variation: observational and theoretical status |url=https://link.aps.org/doi/10.1103/RevModPhys.75.403 |journal=Reviews of Modern Physics |language=en |volume=75 |issue=2 |pages=403–455 |doi=10.1103/RevModPhys.75.403 |arxiv=hep-ph/0205340 |bibcode=2003RvMP...75..403U |s2cid=118684485 |issn=0034-6861}} For example, Webb et al.{{cite journal |first1=John K. |last1=Webb |first2=Michael T. |last2=Murphy |first3=Victor V. |last3=Flambaum |first4=Vladimir A. |last4=Dzuba |first5=John D. |last5=Barrow |first6=Chris W. |last6=Churchill |first7=Jason X. |last7=Prochaska |first8=Arthur M. |last8=Wolfe |doi=10.1103/PhysRevLett.87.091301 |journal=Physical Review Letters |title=Further Evidence for Cosmological Evolution of the Fine Structure Constant |volume=87 |issue=9 |year=2001 |arxiv=astro-ph/0012539 |pmid=11531558 |bibcode=2001PhRvL..87i1301W |pages=091301|s2cid=40461557 }} reported detection of variation (at the 10⁻⁵ level) of the fine-structure constant from measurements of distant quasars. Other researchers dispute these findings.{{Cite journal |last1=Rocha |first1=G |last2=Trotta |first2=R |last3=Martins |first3=C.J.A.P |last4=Melchiorri |first4=A |last5=Avelino |first5=P.P |last6=Viana |first6=P.T.P |date=Nov 2003 |title=New constraints on varying α |url=https://linkinghub.elsevier.com/retrieve/pii/S1387647303001532 |journal=New Astronomy Reviews |language=en |volume=47 |issue=8–10 |pages=863–869 |doi=10.1016/j.newar.2003.07.018|arxiv=astro-ph/0309205 |bibcode=2003NewAR..47..863R |s2cid=9280269 }}

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.

The strong equivalence principle can be tested by 1) finding orbital variations in massive bodies (Sun-Earth-Moon), 2) variations in the gravitational constant (G) depending on nearby sources of gravity or on motion, or 3) searching for a variation of Newton's gravitational constant over the life of the universe{{rp|47}}

Orbital variations due to gravitational self-energy should cause a "polarization" of solar system orbits called the Nordtvedt effect. This effect has been sensitively tested by Lunar Laser Ranging experiments.{{cite web | url=http://funphysics.jpl.nasa.gov/technical/grp/lunar-laser.html | title=Fundamental Physics of Space – Technical Details | access-date=7 May 2005 | archive-url=https://web.archive.org/web/20161128185551/http://funphysics.jpl.nasa.gov/technical/grp/lunar-laser.html | archive-date=28 November 2016 | url-status=dead }}{{cite journal |last1=Viswanathan |first1=V |last2=Fienga |first2=A |last3=Minazzoli |first3=O |last4=Bernus |first4=L |last5=Laskar |first5=J |last6=Gastineau |first6=M |title=The new lunar ephemeris INPOP17a and its application to fundamental physics |journal=Monthly Notices of the Royal Astronomical Society |date=May 2018 |volume=476 |issue=2 |pages=1877–1888 |doi=10.1093/mnras/sty096|doi-access=free |arxiv=1710.09167 |bibcode=2018MNRAS.476.1877V |s2cid=119454879 }} Up to the limit of one part in 10¹³ there is no Nordtvedt effect.

A tight bound on the effect of nearby gravitational fields on the strong equivalence principle comes from modeling the orbits of binary stars and comparing the results to pulsar timing data.{{rp|49}} In 2014, astronomers discovered a stellar triple system containing 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.{{cite journal |url=http://www.nature.com/nature/journal/vaop/ncurrent/full/nature12917.html#ref7 |title=A millisecond pulsar in a stellar triple system |first1=Scott M. |last1=Ransom |display-authors=etal |journal=Nature |year=2014|doi=10.1038/nature12917 |arxiv = 1401.0535 |bibcode = 2014Natur.505..520R |volume=505 |issue=7484 |pages=520–524 |pmid=24390352|s2cid=4468698 }}{{cite journal|title=Universality of free fall from the orbital motion of a pulsar in a stellar triple system|author=Anne M. Archibald|author-link=Anne Archibald|display-authors=etal |journal=Nature|volume=559 |issue=7712|pages=73–76 |date=4 July 2018|doi=10.1038/s41586-018-0265-1|pmid=29973733 |arxiv=1807.02059 | bibcode=2018Natur.559...73A|s2cid=49578025}}{{cite news|title=Even Phenomenally Dense Neutron Stars Fall like a Feather – Einstein Gets It Right Again|url=https://public.nrao.edu/news/neutron-stars-fall/ |publisher=NRAO|date=4 July 2018 |work=Charles Blue, Paul Vosteen}} If there is any departure from the strong equivalence principle, it is no more than two parts per million.{{Cite journal|last1=Voisin|first1=G. |last2=Cognard|first2=I. |last3=Freire|first3=P. C. C.|last4=Wex|first4=N. |last5=Guillemot|first5=L. |last6=Desvignes|first6=G. |last7=Kramer|first7=M. |last8=Theureau|first8=G. |date=2020-06-01|title=An improved test of the strong equivalence principle with the pulsar in a triple star system |url=https://www.aanda.org/articles/aa/abs/2020/06/aa38104-20/aa38104-20.html |journal=Astronomy & Astrophysics|language=en |volume=638 |pages=A24 |arxiv=2005.01388|doi=10.1051/0004-6361/202038104|bibcode=2020A&A...638A..24V|s2cid=218486794 |issn=0004-6361}}

Most alternative theories of gravity predict a change in the gravity constant over time. Studies of Big Bang nucleosynthesis, analysis of pulsars, and the lunar laser ranging data have shown that G cannot have varied by more than 10% since the creation of the universe. The best data comes from studies of the ephemeris of Mars, based on three successive NASA missions, Mars Global Surveyor, Mars Odyssey, and Mars Reconnaissance Orbiter.{{rp|50}}

{{cols|colwidth=20em}}

• Classical mechanics
• Eötvös experiment
• Einstein's thought experiments
• Gauge gravitation theory
• General covariance
• Mach's principle
• Tests of general relativity
• Unsolved problems in astronomy
• Unsolved problems in physics

{{colend}}

{{reflist|30em}}

{{refbegin}}

• Dicke, Robert H.; "New Research on Old Gravitation", Science 129, 3349 (1959). Explains the value of research on gravitation and distinguishes between the strong (later renamed "Einstein") 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.
• 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.
• 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.
• Friedman, Michael; Foundations of Space-Time Theories, Princeton, New Jersey: Princeton University Press, 1983. Chapter V discusses the equivalence principle.

{{refend}}

{{wikiquote}}

• [https://feynmanlectures.caltech.edu/II_42.html#Ch42-S5 Gravity and the principle of equivalence – The Feynman Lectures on Physics]
• [https://web.archive.org/web/20080515172036/http://www.phy.syr.edu/courses/modules/LIGHTCONE/equivalence.html Introducing The Einstein Principle of Equivalence] from Syracuse University
• [http://www.mathpages.com/rr/s5-06/5-06.htm The Equivalence Principle] at MathPages
• [https://web.archive.org/web/20090329031728/http://emis.math.ecnu.edu.cn/journals/LRG/Articles/lrr-2001-4/node3.html The Einstein Equivalence Principle] at Living Reviews on General Relativity
• [https://physicsworld.com/a/equivalence-principle-passes-atomic-test/ "...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..."]

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