Equivalence principle
General relativity |
---|
The equivalence principle is the hypothesis that the observed equivalence of
Concept
In classical mechanics, Newton's equation of motion in a gravitational field, written out in full, is:
- gravitational mass × intensity of the gravitational field
Very 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.[1]: 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.[1]: 33
History
Newton, just 50 years after Galileo, developed the idea that gravitational and inertial mass were different concepts and compared the periods of pendulums composed of different materials to verify that these masses are the same. This form of the equivalence principle became known as "weak equivalence".[2]
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.[3]: 152 Since the physical laws are the same, Einstein assumed the graviational field and the acceleration were "physically equivalent". Einstein stated this hypothesis as:
we ... assume the complete physical equivalence of a gravitational field and a corresponding
acceleration of the reference system.— Einstein, 1907[4]
In 1911 Einstein demonstrated the power of the equivalence principle by using it to predict that clocks run at different rates in a
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, 1911[5]
Immediately after completing his work[6]: 111 on a theory of gravity (known as general relativity) and in later years Einstein recalled the role of the equivalence principle:
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, 1922[7]
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. These are discussed below.
Definitions
Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.[8]: 6 Some studies also create finer divisions or slight alternative.[9][10]
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 equivalence principle, a generalization of the weak equivalence principle, includes astronomic bodies with gravitational self-binding energy.[11] 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".[8]: 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."[11]
- "... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass."[12]
- 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.
Einstein equivalence principle
What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that:
Here local means that experimental setup must be small compared to variations in the gravitational field, called
The two additional constraints added to the weak principle to get the Einstein form − (1) the independence of the outcome on relative velocity (local
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.[14]: 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[15] and Durand[16]).
Strong equivalence principle
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.[8] 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[14]: 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 is 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.[8]
Active, passive, and inertial masses
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:[14]
- Inertial mass intrinsic to an object, the sum of all of its mass–energy.
- Passive mass, the response to gravity, the object's weight.
- Active mass, the mass that determines the objects gravitational effect.
By definition of active and passive gravitational mass, the force on due to the gravitational field of is:
By definition of inertial mass:
Hence:
Therefore:
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 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".
A similar parameter can be used to compare passive and active mass. By
It follows that:
In words, passive gravitational mass must be proportional to active gravitational mass for all objects. The difference,
Experimental tests
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 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[18]: 19–21 but earlier by Simon Stevin[19] who dropped lead balls of different masses off the Delft churchtower and listened for the sound they made on a wooden plank.
Isaac Newton measured the period of pendulums made with different materials as an alternative test giving the first precision measurements.[2]
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[20] that they landed at the same time.
Year | Investigator | Sensitivity | Method |
---|---|---|---|
500? | John Philoponus[22] | "small" | Drop tower |
1585 | Simon Stevin[23][19] | 5×10−2 | Drop tower |
1590? | Galileo Galilei[24][21]: 91 | 2×10−3 | Pendulum, drop tower |
1686 | Isaac Newton[25][21]: 91 | 10−3 | Pendulum |
1832 | : 91 | 2×10−5 | Pendulum |
1908 (1922) | Loránd Eötvös[27][21]: 92 | 2×10−9 | Torsion balance |
1910 | Southerns[28][21]: 91 | 5×10−6 | Pendulum |
1918 | Zeeman[29][21]: 91 | 3×10−8 | Torsion balance |
1923 | Potter[30][21]: 91 | 3×10−6 | Pendulum |
1935 | Renner[31][21]: 92 | 2×10−9 | Torsion balance |
1964 | Roll, Krotkov, Dicke[32] | 3×10−11 | Torsion balance |
1972 | Braginsky, Panov[33][21]: 92 | 10−12 | Torsion balance |
1976 | Shapiro, et al.[34][21]: 92 | 10−12 | Lunar laser ranging |
1979 | Keiser, Faller[35][21]: 93 | 4×10−11 | Fluid support |
1987 | Niebauer, et al.[36][21]: 95 | 10−10 | Drop tower |
1989 | Stubbs, et al.[37][21]: 93 | 10−11 | Torsion balance |
1990 | Adelberger, Eric G.; et al.[38][21]: 95 | 10−12 | Torsion balance |
1999 | Baessler, et al.[39][40] | 5×10−14 | Torsion balance |
2008 | Schlamminger, et al.[41] | 10−13 | Torsion balance |
2017 | MICROSCOPE[42][43] | 10−15 | Earth 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.[44] Future satellite experiments[45] – Satellite Test of the Equivalence Principle[46] and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy.[47]
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.[48]
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.[49]
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.[49]
Tests of the Einstein equivalence principle
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.
Testing local positional invariance divides in to tests in space and in time.
Time-based tests search for variation of
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.
Constant | Year | Method | Limit on fractional change per year |
---|---|---|---|
weak interaction constant | 1976 | Oklo | 10−11 |
fine-structure constant | 1976 | Oklo | 10−16 |
electron–proton mass ratio | 2002 | quasars | 10−15 |
Tests of the strong equivalence principle
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[14]: 47
Orbital variations due to gravitational self-energy should cause a "polarization" of solar system orbits called the
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.[14]: 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.[55][56][57][58]
Most alternative theories of gravity predict a change in the gravity constant over time. Studies of
See also
- 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
References
- ^ ISBN 9781134449798.
- ^ .
- ^ ISBN 0-486-26126-3.
- ^ Einstein, Albert. "On the relativity principle and the conclusions drawn from it." Jahrb Radioaktivitat Elektronik 4 (1907): 411–462.
- ^ Einstein, Albert. "On the Influence of Gravitation on the Propagation of Light." Annalen der Physik 35.898–908 (1911): 906.
- ^ Lorentz, Hendrik Antoon, et al. The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. United Kingdom, Dover Publications, 1923.
- ^ Einstein, Albert, 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.
- ^ .
- S2CID 119110646.
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),
- .
- ^ S2CID 59141292.
- ISBN 978-981-256-661-4.
- ^ S2CID 15430387.
- ^ PMID 28179848.
- S2CID 15007947.
- .
- PMID 37505941.
- ISBN 9780486495422.
- ^ ISBN 9781845643911.
- ^ "Weak Equivalence Principle test on the moon". YouTube. Archived from the original on 21 December 2021.
- ^ a b c d e f g h i j k l m n o Ciufolini, Ignazio; Wheeler, John A.; Gravitation and Inertia, Princeton, New Jersey: Princeton University Press, 1995, pp. 117–119
- ^ Philoponus, John; "Corollaries on Place and Void", translated by David Furley, Ithaca, New York: Cornell University Press, 1987
- ^ Stevin, Simon; De Beghinselen der Weeghconst ["Principles of the Art of Weighing"], Leyden, 1586; Dijksterhuis, Eduard J.; "The Principal Works of Simon Stevin", Amsterdam, 1955
- ^ Galilei, 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, 1638
- ^ Newton, 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, 1999
- ^ Bessel, 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)
- .
- .
- ^ Zeeman, Pieter (1918) "Some experiments on gravitation: The ratio of mass to weight for crystals and radioactive substances", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Amsterdam 20(4) 542–553
- .
- ^ Renner, János (1935). "Kísérleti vizsgálatok a tömegvonzás és tehetetlenség arányosságáról". Mathematikai és Természettudományi Értesítő. 53: 569.
- ^ 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
- ^ Braginski, Vladimir Borisovich; Panov, Vladimir Ivanovich (1971). "Журнал Экспериментальной и Теоретической Физики". (Zhurnal Éksperimental'noĭ I Teoreticheskoĭ Fiziki, Journal of Experimental and Theoretical Physics). 61: 873.
- doi:10.1103/physrevlett.36.555. Archived from the originalon 22 January 2014.
- ^ Keiser, George M.; Faller, James E. (1979). "New approach to the Eötvös experiment". Bulletin of the American Physical Society. 24: 579.
- PMID 10035824.
- PMID 10040283.
- PMID 10012726.
- S2CID 250758089.
- .
- S2CID 18653407.
- ^
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; et al. (2017). "MICROSCOPE Mission: First Results of a Space Test of the Equivalence Principle". Physical Review Letters. 119 (23): 231101. S2CID 6211162.
- ^
Touboul, P., Métris, G., Rodrigues, M., Bergé, J., Robert, A., Baghi, Q., André, Y., Bedouet, J., Boulanger, D., Bremer, S. and Carle, P. (2022). "MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle". Physical Review Letters. 129 (12): 121102. S2CID 252468544.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - ^ "The Eöt-Wash Group | Laboratory Tests of Gravitational and sub-Gravitational Physics". www.npl.washington.edu.
- doi:10.1063/1.1900510. Archived from the original(PDF) on 17 December 2008.
- ^ "S T e P".
- ^ ""GALILEO GALILEI" GG Small Mission Project".
- hdl:2434/457743.
- ^ S2CID 8019480.
- S2CID 118684485.
- S2CID 40461557.
- S2CID 9280269.
- ^ "Fundamental Physics of Space – Technical Details". Archived from the original on 28 November 2016. Retrieved 7 May 2005.
- S2CID 119454879.
- S2CID 4468698.
- S2CID 49578025.
- ^ "Even Phenomenally Dense Neutron Stars Fall like a Feather – Einstein Gets It Right Again". Charles Blue, Paul Vosteen. NRAO. 4 July 2018.
- S2CID 218486794.
Further reading
- 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 0-393-96501-5Chapter 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.
External links
- Gravity and the principle of equivalence – The Feynman Lectures on Physics
- Introducing The Einstein Principle of Equivalence from Syracuse University
- The Equivalence Principle at MathPages
- The Einstein Equivalence Principle at Living Reviews on General Relativity
- "...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..."