Equivalence principle

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A falling object behaves exactly the same on a planet or in an equivalent accelerating frame of reference.

The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same trajectories and landing at identical times. The extended form by Albert Einstein requires special relativity to also hold in free fall and requires the weak equivalence to be valid everywhere. This form was a critical input for the development of the theory of general relativity. The strong form requires Einstein's form to work for stellar objects. Highly precise experimental tests of the principle limit possible deviations from equivalence to be very small.

ConceptEdit

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.<ref name="EinsteinMeaning">Template:Cite book</ref>Template:Rp

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.<ref name="EinsteinMeaning" />Template:Rp

HistoryEdit

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By experimenting with the acceleration of different materials, Galileo Galilei determined that gravitation is independent of the amount of mass being accelerated.<ref name=Everitt>Template:Cite journal</ref>

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".<ref name=Everitt/>

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.<ref name="WhittakerII"/>Template:Rp 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:

...assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.{{#if:Einstein, 1907<ref>Einstein, Albert. "On the relativity principle and the conclusions drawn from it." Jahrb Radioaktivitat Elektronik 4 (1907): 411–462.</ref>|{{#if:|}}
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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.<ref name="WhittakerII">Template:Cite book</ref>Template:Rp He connected the equivalence principle to his earlier principle of special relativity:

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.{{#if:Einstein, 1911<ref>Einstein, Albert. "On the Influence of Gravitation on the Propagation of Light." Annalen der Physik 35.898–908 (1911): 906.</ref>|{{#if:|}}
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Soon after completing work on his theory of gravity (known as general relativity)<ref>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.</ref>Template:Rp and then also in later years, Einstein recalled the importance of the equivalence principle to his work:

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.{{#if: Einstein, 1922<ref>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.</ref>|{{#if:|}}

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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 Template:Section link, and the Template:Section link – each of which assumes the weak equivalence principle as a starting point.

DefinitionsEdit

File:Apollo 15 feather and hammer drop.ogv

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.

Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.<ref name=CliftonFerreiraPadilla>Template:Cite journal</ref>Template:Rp Some proposals also suggest finer divisions or minor alterations.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Weak equivalence principleEdit

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.<ref name="WagnerSchalmminger">Template:Cite journal</ref> 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".<ref name=CliftonFerreiraPadilla/>Template:Rp
"... 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."<ref name=WagnerSchalmminger/>
"... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass."<ref name="Wesson">Template:Cite book</ref>
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 principleEdit

What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that: Template:Block indent 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.<ref name="Lāmmerzahl" /> 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.<ref name=Will2014/>Template:Rp

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.<ref name=Will2014/>Template:Rp 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<ref>Template:Cite journal</ref> and Durand<ref>Template:Cite journal</ref>).

Strong equivalence principleEdit

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.<ref name=CliftonFerreiraPadilla/> 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<ref name=Will2014/>Template:Rp 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.<ref name=CliftonFerreiraPadilla/>

Active, passive, and inertial massesEdit

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:<ref name=Will2014>Template:Cite journal</ref>

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 <math>M_1</math> due to the gravitational field of <math>M_0</math> is: <math display="block">F_1 = \frac{M_0^\mathrm{act} M_1^\mathrm{pass}}{r^2}</math> Likewise the force on a second object of arbitrary mass₂ due to the gravitational field of mass₀ is: <math display="block">F_2 = \frac{M_0^\mathrm{act} M_2^\mathrm{pass}}{r^2}</math>

By definition of inertial mass:<math display="block">F = m^\mathrm{inert} a</math>if <math>m_1</math> and <math>m_2</math> are the same distance <math>r</math> from <math>m_0</math> then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same). <math display="block">a_1 = \frac{F_1}{m_1^\mathrm{inert}} = a_2 = \frac{F_2}{m_2^\mathrm{inert}}</math>

Hence: <math display="block">\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}}</math>

Therefore: <math display="block">\frac{M_1^\mathrm{pass}}{m_1^\mathrm{inert}} = \frac{M_2^\mathrm{pass}}{m_2^\mathrm{inert}}</math>

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 <math>\eta(A,B)</math> 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". <math display="block">\eta(A,B)=2\frac{ \left(\frac{m_{\textrm pass}}{m_{\textrm inert}}\right)_A-\left(\frac{m_{\textrm pass}}{m_{\textrm inert}}\right)_B }{\left(\frac{m_{\textrm pass}}{m_{\textrm inert}}\right)_A+\left(\frac{m_{\textrm pass}}{m_{\textrm inert}}\right)_B}.</math> Values of this parameter are used to compare tests of the equivalence principle.<ref name=Will2014/>Template:Rp

A similar parameter can be used to compare passive and active mass. By Newton's third law of motion: <math display="block">F_1 = \frac{M_0^\mathrm{act} M_1^\mathrm{pass}}{r^2}</math> must be equal and opposite to <math display="block">F_0 = \frac{M_1^\mathrm{act} M_0^\mathrm{pass}}{r^2}</math>

It follows that: <math display="block">\frac{M_0^\mathrm{act}}{M_0^\mathrm{pass}} = \frac{M_1^\mathrm{act}}{M_1^\mathrm{pass}}</math>

In words, passive gravitational mass must be proportional to active gravitational mass for all objects. The difference, <math display="block">S_{0,1} = \frac{M_0^\mathrm{act}}{M_0^\mathrm{pass}} - \frac{M_1^\mathrm{act}}{M_1^\mathrm{pass}}</math> is used to quantify differences between passive and active mass.<ref>Template:Cite journal</ref>

Experimental testsEdit

Tests of the weak equivalence principleEdit

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<ref name=Drake>Template:Cite book</ref>Template:Rp but instead earlier by Simon Stevin,<ref name=Devreese>Template:Cite book</ref> 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.<ref name=Everitt/> 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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref> that they landed at the same time.

Chronology of weak equivalence principles tests<ref name=CiufoliniWheeler>Ciufolini, Ignazio; Wheeler, John A.; *Gravitation and Inertia*, Princeton, New Jersey: Princeton University Press, 1995, pp. 117–119</ref>
Year	Investigator	Sensitivity	Method
500?	John Philoponus<ref>Philoponus, John; "Corollaries on Place and Void", translated by David Furley, Ithaca, New York: Cornell University Press, 1987</ref>	"small"	Drop tower
1585	Simon Stevin<ref>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</ref><ref name="Devreese"/>	Template:Val	Drop tower
1590?	Galileo Galilei<ref>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</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Pendulum, drop tower
1686	Isaac Newton<ref>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</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Pendulum
1832	Friedrich Wilhelm Bessel<ref>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)</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Pendulum
1908 (1922)	Loránd Eötvös<ref>R. v. Eötvös 1890 Mathematische und Naturwissenschaftliche Berichte aus Ungarn, 8, 65; Annalen der Physik (Leipzig) 68 11 (1922); Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Torsion balance
1910	Southerns<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Pendulum
1918	Zeeman<ref>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</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Torsion balance
1923	Potter<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Pendulum
1935	Renner<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Torsion balance
1964	Roll, Krotkov, Dicke<ref name="RollKrotkovDicke">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</ref>	Template:Val	Torsion balance
1972	Braginsky, Panov<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Torsion balance
1976	Shapiro, et al.<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Lunar laser ranging
1979	Keiser, Faller<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Fluid support
1987	Niebauer, et al.<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Drop tower
1989	Stubbs, et al.<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Torsion balance
1990	Adelberger, Eric G.; et al.<ref>Template:Cite journal</ref><ref name=CiufoliniWheeler/>Template:Rp	Template:Val	Torsion balance
1999	Baessler, et al.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>	Template:Val	Torsion balance
2008	Schlamminger, et al.<ref>Template:Cite journal</ref>	Template:Val	Torsion balance
2017	MICROSCOPE<ref> Template:Cite journal</ref><ref> Template:Cite journal</ref>	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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Future satellite experiments<ref name="Dittus">Template:Cite conference</ref> – Satellite Test of the Equivalence Principle<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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.<ref>Template:Cite journal</ref>

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.<ref name="Overduin2009" />

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.<ref name="Overduin2009">Template:Cite journal</ref>

Tests of the Einstein equivalence principleEdit

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.<ref name=Will2014/>Template:Rp 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.<ref name=Will2014/>Template:Rp

Testing local positional invariance divides in to tests in space and in time.<ref name=Will2014/>Template:Rp 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.<ref name="Uzan">Template:Cite journal</ref> For example, Webb et al.<ref>Template:Cite journal</ref> reported detection of variation (at the 10⁻⁵ level) of the fine-structure constant from measurements of distant quasars. Other researchers dispute these findings.<ref>Template:Cite journal</ref>

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.

Tests of changes in fundamental constants<ref name=Will2014/>Template:Rp
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 principleEdit

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<ref name=Will2014/>Template:Rp

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal</ref> 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.<ref name=Will2014/>Template:Rp 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.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite news</ref> If there is any departure from the strong equivalence principle, it is no more than two parts per million.<ref>Template:Cite journal</ref>

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.<ref name=Will2014/>Template:Rp