Editing Equivalence principle (section)

=== 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>
<!-- I have created [[User talk:Lemmiwinks2#Redundant edits (active, passive, and inertial masses)|a list of known places where the above material appears]]. -->