Editing Equivalence principle (section)

== Definitions ==
[[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.]]Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.<ref name=CliftonFerreiraPadilla>{{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 }}</ref>{{rp|6}} Some proposals also suggest finer divisions or minor alterations.<ref>{{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),}}</ref><ref>{{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 }}</ref>

=== <span class="anchor" id="Weak"></span> Weak equivalence principle ===
<!--[[Universality of free fall]], [[weak equivalence principle]], and [[Galilean equivalence principle]] redirect here-->
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">{{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 }}</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/>{{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."<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">{{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 }}</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 principle ===
<!-- This section is linked from [[Gravitational field]] -->
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.''<ref name="Lāmmerzahl">{{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}}</ref>}}
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/>{{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.<ref name=Will2014/>{{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<ref>{{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 }}</ref> and Durand<ref>{{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 }}</ref>).

=== 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.<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 experiment]]s. It requires that the [[gravitational constant]] be the same everywhere in the universe<ref name=Will2014/>{{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 tensor (general relativity)|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 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:<ref name=Will2014>{{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 }}</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 mass<sub>2</sub> due to the gravitational field of mass<sub>0</sub> 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 experiment|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/>{{rp|10}}

A similar parameter can be used to compare passive and active mass.
By [[Newton's laws of motion#Third law|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>{{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}}</ref>
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