Equivalence principle

<h2 id="concept">Concept</h2>
In classical mechanics, <a href="/facts/Newton%2527s_law_of_universal_gravitation/F6qH2XC3">Newton's equation of motion in a gravitational field</a>, written out in full, is:

<a href="/facts/Inertial_mass/KXtSKomS">inertial mass</a> × <a href="/facts/Acceleration/UVDmjYvR">acceleration</a> = <a href="/facts/Gravitational_mass/KXtSKomS">gravitational mass</a> × <a href="/facts/Gravitational_field/K1uTmJFy">gravitational acceleration</a>
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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>: 32 
The equivalence principle can be considered an extension of the principle of relativity, the principle that the laws of physics are <a href="/facts/Invariant_(physics)/UfHcehYs">invariant</a> 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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>: 33 

<h2 id="history">History</h2>
See also: <a href="/facts/History_of_gravitational_theory/RWuCUDHj">History of gravitational theory</a>
By <a href="/facts/Experiment/esltM54f">experimenting</a> with the <a href="/facts/Acceleration/UVDmjYvR">acceleration</a> of different materials, <a href="/facts/Galileo_Galilei/98VT2z3v">Galileo</a> determined that <a href="/facts/Gravitation/bqAN7DdQ">gravitation</a> is independent of the amount of <a href="/facts/Mass/HHdc8XmF">mass</a> being accelerated.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
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 class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
A version of the equivalence principle consistent with special relativity was introduced by <a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a> 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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: 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 <a href="/facts/Accelerated_reference_frame/UGfKCASo">acceleration of the reference system</a>.— Einstein, 1907<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></blockquote>
In 1911 Einstein demonstrated the power of the equivalence principle by using it to predict that clocks run at different rates in a <a href="/facts/Gravitational_potential/x4ITT4gY">gravitational potential</a>, and light rays <a href="/facts/Bending_of_starlight/yDgwtNrw">bend</a> in a gravitational field.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>: 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, 1911<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></blockquote>
Soon after completing work on his theory of gravity (known as <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>)<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>: 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, 1922<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></blockquote>
Einstein's development of general relativity necessitated some means of empirically discriminating the theory from other theories of gravity compatible with <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>. Accordingly, <a href="/facts/Robert_H._Dicke/6LM6oN79">Robert Dicke</a> developed a test program incorporating two new principles – the § Einstein equivalence principle, and the § Strong equivalence principle – each of which assumes the weak equivalence principle as a starting point.

<h2 id="definitions">Definitions</h2>
Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>: 6  Some proposals also suggest finer divisions or minor alterations.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
<h3> Weak equivalence principle</h3>
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.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> Instead, the weak equivalence principle assumes falling bodies are self-bound by non-gravitational forces only (e.g. a stone). Either way:

<ul><li>"All uncharged, freely falling test particles follow the same trajectories, once an initial position and velocity have been prescribed".<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>: 6 </li>
<li>"... 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."<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li>
<li>"... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass."<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></li>
<li>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 <a href="/facts/Philosophi%25C3%25A6_Naturalis_Principia_Mathematica/DRdvI3yx">Philosophiæ Naturalis Principia Mathematica</a>, 1687).</li></ul>
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.

<h3>Einstein equivalence principle</h3>
What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that:

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.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
Here local means that experimental setup must be small compared to variations in the gravitational field, called <a href="/facts/Tidal_forces/GJgxo7gp">tidal forces</a>. 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 <a href="/facts/Lorentz_invariance/K5AEuGKg">Lorentz invariance</a>) and (2) independence of "where" (known as local positional invariance) − have far reaching consequences. With these constraints alone Einstein was able to predict the <a href="/facts/Gravitational_redshift/cEFgB3Ns">gravitational redshift</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> Theories of gravity that obey the Einstein equivalence principle must be "metric theories", meaning that trajectories of freely falling bodies are <a href="/facts/Geodesics/fbWR6l80">geodesics</a> of symmetric metric.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>: 9 
Around 1960 <a href="/facts/Leonard_I._Schiff/gz9hHnJe">Leonard I. Schiff</a> 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.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>: 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<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> and Durand<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>).

<h3>Strong equivalence principle</h3>
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.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>
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 <a href="/facts/Cavendish_experiment/18j8qa8d">Cavendish experiments</a>. It requires that the <a href="/facts/Gravitational_constant/GdQ8tcBD">gravitational constant</a> be the same everywhere in the universe<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>: 49  and is incompatible with a <a href="/facts/Fifth_force/SsPYPtTw">fifth force</a>. 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 <a href="/facts/Metric_tensor_(general_relativity)/lJoCMdve">metric</a> 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 <a href="/facts/Cosmological_constant/bjMksSA2">cosmological constant</a>) is thought to be the only theory of gravity that satisfies the strong equivalence principle. A number of alternative theories, such as <a href="/facts/Brans%25E2%2580%2593Dicke_theory/dcCnWqka">Brans–Dicke theory</a> and the <a href="/facts/Einstein-aether_theory/3poAhc0o">Einstein-aether theory</a> add additional fields.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>

<h3>Active, passive, and inertial masses</h3>
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:<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> 

<ol><li>Inertial mass intrinsic to an object, the sum of all of its mass–energy.</li>
<li>Passive mass, the response to gravity, the object's weight.</li>
<li>Active mass, the mass that determines the objects gravitational effect.</li></ol>
By definition of active and passive gravitational mass, the force on 
 
 
 
 
 M
 
 1
 
 
 
 
 {\displaystyle M_{1}}
 
 due to the gravitational field of 
 
 
 
 
 M
 
 0
 
 
 
 
 {\displaystyle M_{0}}
 
 is:

F
          
            1
          
        
        =
        
          
            
              
                M
                
                  0
                
                
                  
                    a
                    c
                    t
                  
                
              
              
                M
                
                  1
                
                
                  
                    p
                    a
                    s
                    s
                  
                
              
            
            
              r
              
                2
              
            
          
        
      
    
    {\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}}

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

F
          
            2
          
        
        =
        
          
            
              
                M
                
                  0
                
                
                  
                    a
                    c
                    t
                  
                
              
              
                M
                
                  2
                
                
                  
                    p
                    a
                    s
                    s
                  
                
              
            
            
              r
              
                2
              
            
          
        
      
    
    {\displaystyle F_{2}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}}}}

By definition of inertial mass:
 
 
 
 F
 =
 
 m
 
 
 i
 n
 e
 r
 t
 
 
 
 a
 
 
 {\displaystyle F=m^{\mathrm {inert} }a}
 
if 
 
 
 
 
 m
 
 1
 
 
 
 
 {\displaystyle m_{1}}
 
 and 
 
 
 
 
 m
 
 2
 
 
 
 
 {\displaystyle m_{2}}
 
 are the same distance 
 
 
 
 r
 
 
 {\displaystyle r}
 
 from 
 
 
 
 
 m
 
 0
 
 
 
 
 {\displaystyle m_{0}}
 
 then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same).

a
          
            1
          
        
        =
        
          
            
              F
              
                1
              
            
            
              m
              
                1
              
              
                
                  i
                  n
                  e
                  r
                  t
                
              
            
          
        
        =
        
          a
          
            2
          
        
        =
        
          
            
              F
              
                2
              
            
            
              m
              
                2
              
              
                
                  i
                  n
                  e
                  r
                  t
                
              
            
          
        
      
    
    {\displaystyle a_{1}={\frac {F_{1}}{m_{1}^{\mathrm {inert} }}}=a_{2}={\frac {F_{2}}{m_{2}^{\mathrm {inert} }}}}

Hence:

M
                
                  0
                
                
                  
                    a
                    c
                    t
                  
                
              
              
                M
                
                  1
                
                
                  
                    p
                    a
                    s
                    s
                  
                
              
            
            
              
                r
                
                  2
                
              
              
                m
                
                  1
                
                
                  
                    i
                    n
                    e
                    r
                    t
                  
                
              
            
          
        
        =
        
          
            
              
                M
                
                  0
                
                
                  
                    a
                    c
                    t
                  
                
              
              
                M
                
                  2
                
                
                  
                    p
                    a
                    s
                    s
                  
                
              
            
            
              
                r
                
                  2
                
              
              
                m
                
                  2
                
                
                  
                    i
                    n
                    e
                    r
                    t
                  
                
              
            
          
        
      
    
    {\displaystyle {\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} }}}}

Therefore:

M
              
                1
              
              
                
                  p
                  a
                  s
                  s
                
              
            
            
              m
              
                1
              
              
                
                  i
                  n
                  e
                  r
                  t
                
              
            
          
        
        =
        
          
            
              M
              
                2
              
              
                
                  p
                  a
                  s
                  s
                
              
            
            
              m
              
                2
              
              
                
                  i
                  n
                  e
                  r
                  t
                
              
            
          
        
      
    
    {\displaystyle {\frac {M_{1}^{\mathrm {pass} }}{m_{1}^{\mathrm {inert} }}}={\frac {M_{2}^{\mathrm {pass} }}{m_{2}^{\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 <a href="/facts/E%25C3%25B6tv%25C3%25B6s_experiment/0VADaCqR">Eötvös</a>-parameter or Eötvös ratio 
 
 
 
 η
 (
 A
 ,
 B
 )
 
 
 {\displaystyle \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".

η
        (
        A
        ,
        B
        )
        =
        2
        
          
            
              
                
                  (
                  
                    
                      
                        m
                        
                          
                            
                              p
                            
                          
                          a
                          s
                          s
                        
                      
                      
                        m
                        
                          
                            
                              i
                            
                          
                          n
                          e
                          r
                          t
                        
                      
                    
                  
                  )
                
                
                  A
                
              
              −
              
                
                  (
                  
                    
                      
                        m
                        
                          
                            
                              p
                            
                          
                          a
                          s
                          s
                        
                      
                      
                        m
                        
                          
                            
                              i
                            
                          
                          n
                          e
                          r
                          t
                        
                      
                    
                  
                  )
                
                
                  B
                
              
            
            
              
                
                  (
                  
                    
                      
                        m
                        
                          
                            
                              p
                            
                          
                          a
                          s
                          s
                        
                      
                      
                        m
                        
                          
                            
                              i
                            
                          
                          n
                          e
                          r
                          t
                        
                      
                    
                  
                  )
                
                
                  A
                
              
              +
              
                
                  (
                  
                    
                      
                        m
                        
                          
                            
                              p
                            
                          
                          a
                          s
                          s
                        
                      
                      
                        m
                        
                          
                            
                              i
                            
                          
                          n
                          e
                          r
                          t
                        
                      
                    
                  
                  )
                
                
                  B
                
              
            
          
        
        .
      
    
    {\displaystyle \eta (A,B)=2{\frac {\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{A}-\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{B}}{\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{A}+\left({\frac {m_{{\textrm {p}}ass}}{m_{{\textrm {i}}nert}}}\right)_{B}}}.}

Values of this parameter are used to compare tests of the equivalence principle.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>: 10 
A similar parameter can be used to compare passive and active mass.
By <a href="/facts/Newton%2527s_laws_of_motion/XKzGTXK1">Newton's third law of motion</a>:

must be equal and opposite to

F
          
            0
          
        
        =
        
          
            
              
                M
                
                  1
                
                
                  
                    a
                    c
                    t
                  
                
              
              
                M
                
                  0
                
                
                  
                    p
                    a
                    s
                    s
                  
                
              
            
            
              r
              
                2
              
            
          
        
      
    
    {\displaystyle F_{0}={\frac {M_{1}^{\mathrm {act} }M_{0}^{\mathrm {pass} }}{r^{2}}}}

It follows that:

M
              
                0
              
              
                
                  a
                  c
                  t
                
              
            
            
              M
              
                0
              
              
                
                  p
                  a
                  s
                  s
                
              
            
          
        
        =
        
          
            
              M
              
                1
              
              
                
                  a
                  c
                  t
                
              
            
            
              M
              
                1
              
              
                
                  p
                  a
                  s
                  s
                
              
            
          
        
      
    
    {\displaystyle {\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}={\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}}

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

S
          
            0
            ,
            1
          
        
        =
        
          
            
              M
              
                0
              
              
                
                  a
                  c
                  t
                
              
            
            
              M
              
                0
              
              
                
                  p
                  a
                  s
                  s
                
              
            
          
        
        −
        
          
            
              M
              
                1
              
              
                
                  a
                  c
                  t
                
              
            
            
              M
              
                1
              
              
                
                  p
                  a
                  s
                  s
                
              
            
          
        
      
    
    {\displaystyle S_{0,1}={\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}-{\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}}

is used to quantify differences between passive and active mass.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>

<h2 id="experimental-tests">Experimental tests</h2>
<h3>Tests of the weak equivalence principle</h3>
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 <a href="/facts/Galileo%2527s_Leaning_Tower_of_Pisa_experiment/7CPanLBu">Galileo's Leaning Tower of Pisa experiment</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>: 19–21  but instead earlier by <a href="/facts/Simon_Stevin/Nb7y9QvD">Simon Stevin</a>,<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> who dropped lead balls of different masses off the <a href="/facts/Nieuwe_Kerk_(Delft)/uMhXV0QN">Delft churchtower</a> and listened for the sound of them hitting a wooden plank.
<a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a> measured the <a href="/facts/Frequency/QUSbPaW8">period</a> of pendulums made with different materials as an alternative test giving the first precision measurements.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> <a href="/facts/Lor%25C3%25A1nd_E%25C3%25B6tv%25C3%25B6s/CP0YgoTC">Loránd Eötvös</a>'s approach in 1908 used a very sensitive <a href="/facts/Torsion_balance/qMrn3XGM">torsion balance</a> 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 <a href="/facts/David_Scott/PKu1nUlj">David Scott</a> in 1971. He dropped a falcon feather and a hammer at the same time, showing on video<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> that they landed at the same time.

Chronology of weak equivalence principles tests<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a><table><tbody><tr><th>Year</th><th>Investigator</th><th>Sensitivity</th><th>Method</th></tr><tr><td>500?</td><td><a href="/facts/John_Philoponus/rGRnd16b">John Philoponus</a><a class="footnote-ref" id="fnref:35" href="#fn:35">35</a></td><td>"small"</td><td>Drop tower</td></tr><tr><td>1585</td><td><a href="/facts/Simon_Stevin/Nb7y9QvD">Simon Stevin</a><a class="footnote-ref" id="fnref:36" href="#fn:36">36</a><a class="footnote-ref" id="fnref:37" href="#fn:37">37</a></td><td>5×10−2</td><td>Drop tower</td></tr><tr><td>1590?</td><td><a href="/facts/Galileo_Galilei/98VT2z3v">Galileo Galilei</a><a class="footnote-ref" id="fnref:38" href="#fn:38">38</a><a class="footnote-ref" id="fnref:39" href="#fn:39">39</a>: 91 </td><td>2×10−3</td><td>Pendulum, drop tower</td></tr><tr><td>1686</td><td><a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a><a class="footnote-ref" id="fnref:41" href="#fn:41">41</a>: 91 </td><td>10−3</td><td>Pendulum</td></tr><tr><td>1832</td><td><a href="/facts/Friedrich_Wilhelm_Bessel/82K1lJ7t">Friedrich Wilhelm Bessel</a><a class="footnote-ref" id="fnref:42" href="#fn:42">42</a><a class="footnote-ref" id="fnref:43" href="#fn:43">43</a>: 91 </td><td>2×10−5</td><td>Pendulum</td></tr><tr><td>1908 (1922)</td><td><a href="/facts/Lor%25C3%25A1nd_E%25C3%25B6tv%25C3%25B6s/CP0YgoTC">Loránd Eötvös</a><a class="footnote-ref" id="fnref:44" href="#fn:44">44</a><a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>: 92 </td><td>2×10−9</td><td>Torsion balance</td></tr><tr><td>1910</td><td>Southerns<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a><a class="footnote-ref" id="fnref:47" href="#fn:47">47</a>: 91 </td><td>5×10−6</td><td>Pendulum</td></tr><tr><td>1918</td><td>Zeeman<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a><a class="footnote-ref" id="fnref:49" href="#fn:49">49</a>: 91 </td><td>3×10−8</td><td>Torsion balance</td></tr><tr><td>1923</td><td>Potter<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a><a class="footnote-ref" id="fnref:51" href="#fn:51">51</a>: 91 </td><td>3×10−6</td><td>Pendulum</td></tr><tr><td>1935</td><td>Renner<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a><a class="footnote-ref" id="fnref:53" href="#fn:53">53</a>: 92 </td><td>2×10−9</td><td>Torsion balance</td></tr><tr><td>1964</td><td>Roll, Krotkov, <a href="/facts/Robert_H._Dicke/6LM6oN79">Dicke</a><a class="footnote-ref" id="fnref:54" href="#fn:54">54</a></td><td>3×10−11</td><td>Torsion balance</td></tr><tr><td>1972</td><td><a href="/facts/Vladimir_Braginsky/ls2AWrXC">Braginsky</a>, Panov<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a><a class="footnote-ref" id="fnref:56" href="#fn:56">56</a>: 92 </td><td>10−12</td><td>Torsion balance</td></tr><tr><td>1976</td><td>Shapiro, et al.<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a><a class="footnote-ref" id="fnref:58" href="#fn:58">58</a>: 92 </td><td>10−12</td><td>Lunar laser ranging</td></tr><tr><td>1979</td><td>Keiser, Faller<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a><a class="footnote-ref" id="fnref:60" href="#fn:60">60</a>: 93 </td><td>4×10−11</td><td>Fluid support</td></tr><tr><td>1987</td><td>Niebauer, et al.<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a><a class="footnote-ref" id="fnref:62" href="#fn:62">62</a>: 95 </td><td>10−10</td><td>Drop tower</td></tr><tr><td>1989</td><td>Stubbs, et al.<a class="footnote-ref" id="fnref:63" href="#fn:63">63</a><a class="footnote-ref" id="fnref:64" href="#fn:64">64</a>: 93 </td><td>10−11</td><td>Torsion balance</td></tr><tr><td>1990</td><td>Adelberger, Eric G.; et al.<a class="footnote-ref" id="fnref:65" href="#fn:65">65</a><a class="footnote-ref" id="fnref:66" href="#fn:66">66</a>: 95 </td><td>10−12</td><td>Torsion balance</td></tr><tr><td>1999</td><td>Baessler, et al.<a class="footnote-ref" id="fnref:67" href="#fn:67">67</a><a class="footnote-ref" id="fnref:68" href="#fn:68">68</a></td><td>5×10−14</td><td>Torsion balance</td></tr><tr><td>2008</td><td>Schlamminger, et al.<a class="footnote-ref" id="fnref:69" href="#fn:69">69</a></td><td>10−13</td><td>Torsion balance</td></tr><tr><td>2017</td><td><a href="/facts/MICROSCOPE/vfFoQ1gI">MICROSCOPE</a><a class="footnote-ref" id="fnref:70" href="#fn:70">70</a><a class="footnote-ref" id="fnref:71" href="#fn:71">71</a></td><td>10−15</td><td>Earth orbit</td></tr></tbody></table>
Experiments are still being performed at the <a href="/facts/University_of_Washington/36PG6qkG">University of Washington</a> which have placed limits on the differential acceleration of objects towards the Earth, the Sun and towards <a href="/facts/Dark_matter/y5HgX55N">dark matter</a> in the <a href="/facts/Galactic_Center/TI4MwWwM">Galactic Center</a>.<a class="footnote-ref" id="fnref:72" href="#fn:72">72</a> Future satellite experiments<a class="footnote-ref" id="fnref:73" href="#fn:73">73</a> – <a href="/facts/STEP_(satellite)/orpFTBY3">Satellite Test of the Equivalence Principle</a><a class="footnote-ref" id="fnref:74" href="#fn:74">74</a> and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy.<a class="footnote-ref" id="fnref:75" href="#fn:75">75</a>
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.<a class="footnote-ref" id="fnref:76" href="#fn:76">76</a>
Proposals that may lead to a <a href="/facts/Quantum_gravity/t7whBMNE">quantum theory of gravity</a> such as <a href="/facts/String_theory/HUJFIPYH">string theory</a> and <a href="/facts/Loop_quantum_gravity/6Xueeoux">loop quantum gravity</a> predict violations of the weak equivalence principle because they contain many light <a href="/facts/Scalar_field/8MtPTGob">scalar fields</a> with long <a href="/facts/Compton_wavelength/og2h1J2q">Compton wavelengths</a>, which should generate <a href="/facts/Fifth_force/SsPYPtTw">fifth forces</a> 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.<a class="footnote-ref" id="fnref:77" href="#fn:77">77</a>
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.<a class="footnote-ref" id="fnref:78" href="#fn:78">78</a>

<h3>Tests of the Einstein equivalence principle</h3>
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.<a class="footnote-ref" id="fnref:79" href="#fn:79">79</a>: 12  Nevertheless, attempts to look for quantum gravity require even more precise tests. The modern tests include looking for directional variations in the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> (called "clock anisotropy tests") and new forms of the <a href="/facts/Michelson%25E2%2580%2593Morley_experiment/yyrNEgA4">Michelson–Morley experiment</a>. The anisotropy measures less than one part in 10−20.<a class="footnote-ref" id="fnref:80" href="#fn:80">80</a>: 14 
Testing local positional invariance divides in to tests in space and in time.<a class="footnote-ref" id="fnref:81" href="#fn:81">81</a>: 17  Space-based tests use measurements of the <a href="/facts/Gravitational_redshift/cEFgB3Ns">gravitational redshift</a>, the classic is the <a href="/facts/Pound%25E2%2580%2593Rebka_experiment/3sALWlWJ">Pound–Rebka experiment</a> 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 <a href="/facts/Global_Positioning_System/DKVLnzF3">Global Positioning System</a> requires compensation for this redshift to give accurate position values.
Time-based tests search for variation of <a href="/facts/Dimensionless/TINrpQIq">dimensionless</a> <a href="/facts/Fundamental_physical_constants/79vy9fTj">constants and mass ratios</a>.<a class="footnote-ref" id="fnref:82" href="#fn:82">82</a> For example, Webb et al.<a class="footnote-ref" id="fnref:83" href="#fn:83">83</a> reported detection of variation (at the 10−5 level) of the fine-structure constant from measurements of distant <a href="/facts/Quasar/HS96328l">quasars</a>. Other researchers dispute these findings.<a class="footnote-ref" id="fnref:84" href="#fn:84">84</a>
The present best limits on the variation of the fundamental constants have mainly been set by studying the naturally occurring <a href="/facts/Oklo/p2jg2BuS">Oklo</a> <a href="/facts/Natural_nuclear_fission_reactor/G7FIGsEV">natural nuclear fission reactor</a>, 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<a class="footnote-ref" id="fnref:85" href="#fn:85">85</a>: 19 <table><tbody><tr><th>Constant</th><th>Year</th><th>Method</th><th>Limit on fractional change per year</th></tr><tr><td><a href="/facts/Weak_interaction/f6upmKMs">weak interaction</a> constant</td><td>1976</td><td>Oklo</td><td>10−11</td></tr><tr><td><a href="/facts/Fine-structure_constant/zyzpmTJV">fine-structure constant</a></td><td>1976</td><td>Oklo</td><td>10−16</td></tr><tr><td><a href="/facts/Electron/L0KBo5cW">electron</a>–<a href="/facts/Proton/clCjepXP">proton</a> mass ratio</td><td>2002</td><td>quasars</td><td>10−15</td></tr></tbody></table>
<h3>Tests of the strong equivalence principle</h3>
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<a class="footnote-ref" id="fnref:86" href="#fn:86">86</a>: 47 
Orbital variations due to gravitational self-energy should cause a "polarization" of solar system orbits called the <a href="/facts/Nordtvedt_effect/04rxnXBg">Nordtvedt effect</a>. This effect has been sensitively tested by <a href="/facts/Lunar_Laser_Ranging_experiments/YIAbdxGP">Lunar Laser Ranging experiments</a>.<a class="footnote-ref" id="fnref:87" href="#fn:87">87</a><a class="footnote-ref" id="fnref:88" href="#fn:88">88</a> 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 <a href="/facts/Pulsar/f8c7S5LQ">pulsar</a> timing data.<a class="footnote-ref" id="fnref:89" href="#fn:89">89</a>: 49  In 2014, astronomers discovered a stellar triple system containing a millisecond pulsar <a href="/facts/PSR_J0337%252B1715/y8sm6ooD">PSR J0337+1715</a> and two <a href="/facts/White_dwarf/gIYlTtrC">white dwarfs</a> orbiting it. The system provided them a chance to test the strong equivalence principle in a strong gravitational field with high accuracy.<a class="footnote-ref" id="fnref:90" href="#fn:90">90</a><a class="footnote-ref" id="fnref:91" href="#fn:91">91</a><a class="footnote-ref" id="fnref:92" href="#fn:92">92</a> If there is any departure from the strong equivalence principle, it is no more than two <a href="/facts/Parts_per_million/PozRuoLM">parts per million</a>.<a class="footnote-ref" id="fnref:93" href="#fn:93">93</a>
Most alternative theories of gravity predict a change in the gravity constant over time. Studies of <a href="/facts/Big_Bang_nucleosynthesis/IvM4gbKH">Big Bang nucleosynthesis</a>, 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 <a href="/facts/Ephemeris/vjQCBG29">ephemeris</a> of Mars, based on three successive NASA missions, <a href="/facts/Mars_Global_Surveyor/MGXbPxdx">Mars Global Surveyor</a>, <a href="/facts/Mars_Odyssey/KapXFdK4">Mars Odyssey</a>, and <a href="/facts/Mars_Reconnaissance_Orbiter/7odg7YfX">Mars Reconnaissance Orbiter</a>.<a class="footnote-ref" id="fnref:94" href="#fn:94">94</a>: 50 

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Classical_mechanics/ykG8upsN">Classical mechanics</a></li>
<li><a href="/facts/E%25C3%25B6tv%25C3%25B6s_experiment/0VADaCqR">Eötvös experiment</a></li>
<li><a href="/facts/Einstein%2527s_thought_experiments/ptuoP6uG">Einstein's thought experiments</a></li>
<li><a href="/facts/Gauge_gravitation_theory/wwRu94cD">Gauge gravitation theory</a></li>
<li><a href="/facts/General_covariance/KzTG2t9g">General covariance</a></li>
<li><a href="/facts/Mach%2527s_principle/J5EPg7dz">Mach's principle</a></li>
<li><a href="/facts/Tests_of_general_relativity/yDgwtNrw">Tests of general relativity</a></li>
<li><a href="/facts/Unsolved_problems_in_astronomy/CUEN4dsg">Unsolved problems in astronomy</a></li>
<li><a href="/facts/Unsolved_problems_in_physics/vJ3d1A83">Unsolved problems in physics</a></li></ul>

<h2 id="further-reading">Further reading</h2>

<ul><li>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.</li>
<li>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.</li>
<li>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.</li>
<li>Ohanian, Hans; and Ruffini, Remo; Gravitation and Spacetime 2nd edition, New York: Norton, 1994, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-393-96501-5 Chapter 1 discusses the equivalence principle, but incorrectly, according to modern usage, states that the strong equivalence principle is wrong.</li>
<li>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.</li>
<li>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.</li>
<li>Friedman, Michael; Foundations of Space-Time Theories, Princeton, New Jersey: Princeton University Press, 1983. Chapter V discusses the equivalence principle.</li></ul>

<h2 id="external-links">External links</h2>

Wikiquote has quotations related to Equivalence principle.

<ul><li><a href="https://feynmanlectures.caltech.edu/II_42.html#Ch42-S5">Gravity and the principle of equivalence – The Feynman Lectures on Physics</a></li>
<li><a href="https://web.archive.org/web/20080515172036/http://www.phy.syr.edu/courses/modules/LIGHTCONE/equivalence.html">Introducing The Einstein Principle of Equivalence</a> from Syracuse University</li>
<li><a href="http://www.mathpages.com/rr/s5-06/5-06.htm">The Equivalence Principle</a> at MathPages</li>
<li><a href="https://web.archive.org/web/20090329031728/http://emis.math.ecnu.edu.cn/journals/LRG/Articles/lrr-2001-4/node3.html">The Einstein Equivalence Principle</a> at Living Reviews on General Relativity</li>
<li><a href="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..."</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Einstein, Albert (2003). The Meaning of Relativity. Routledge. p. 59. ISBN 9781134449798. <a href="9781134449798" target="_blank">9781134449798</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Einstein, Albert (2003). The Meaning of Relativity. Routledge. p. 59. ISBN 9781134449798. <a href="9781134449798" target="_blank">9781134449798</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Everitt, C.W.F.; Damour, T.; Nordtvedt, K.; Reinhard, R. (October 2003). "Historical perspective on testing the Equivalence Principle". Advances in Space Research. 32 (7): 1297–1300. Bibcode:2003AdSpR..32.1297E. doi:10.1016/S0273-1177(03)90335-8. <a href="https://linkinghub.elsevier.com/retrieve/pii/S0273117703903358" target="_blank">https://linkinghub.elsevier.com/retrieve/pii/S0273117703903358</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Everitt, C.W.F.; Damour, T.; Nordtvedt, K.; Reinhard, R. (October 2003). "Historical perspective on testing the Equivalence Principle". Advances in Space Research. 32 (7): 1297–1300. Bibcode:2003AdSpR..32.1297E. doi:10.1016/S0273-1177(03)90335-8. <a href="https://linkinghub.elsevier.com/retrieve/pii/S0273117703903358" target="_blank">https://linkinghub.elsevier.com/retrieve/pii/S0273117703903358</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Whittaker, Sir Edmund (1 January 1989). A History of the Theories of Aether and Electricity. Vol. 2. Courier Dover Publications. ISBN 0-486-26126-3. <a href="0-486-26126-3" target="_blank">0-486-26126-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Einstein, Albert. "On the relativity principle and the conclusions drawn from it." Jahrb Radioaktivitat Elektronik 4 (1907): 411–462. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Whittaker, Sir Edmund (1 January 1989). A History of the Theories of Aether and Electricity. Vol. 2. Courier Dover Publications. ISBN 0-486-26126-3. <a href="0-486-26126-3" target="_blank">0-486-26126-3</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Einstein, Albert. "On the Influence of Gravitation on the Propagation of Light." Annalen der Physik 35.898–908 (1911): 906. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">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. <a href="https://books.google.com/books?id=oy_Z3o7p6zwC" target="_blank">https://books.google.com/books?id=oy_Z3o7p6zwC</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">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. <a href="https://web.archive.org/web/20151222085312/http://inpac.ucsd.edu/students/courses/winter2012/physics2d/einsteinonrelativity.pdf" target="_blank">https://web.archive.org/web/20151222085312/http://inpac.ucsd.edu/students/courses/winter2012/physics2d/einsteinonrelativity.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Clifton, Timothy; Ferreira, Pedro G.; Padilla, Antonio; Skordis, Constantinos (March 2012). "Modified gravity and cosmology". Physics Reports. 513 (1–3): 1–189. arXiv:1106.2476. Bibcode:2012PhR...513....1C. doi:10.1016/j.physrep.2012.01.001. <a href="https://linkinghub.elsevier.com/retrieve/pii/S0370157312000105" target="_blank">https://linkinghub.elsevier.com/retrieve/pii/S0370157312000105</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Di Casola, Eolo; Liberati, Stefano; Sonego, Sebastiano (1 January 2015). "Nonequivalence of equivalence principles". American Journal of Physics. 83 (1): 39–46. arXiv:1310.7426. Bibcode:2015AmJPh..83...39D. doi:10.1119/1.4895342. ISSN 0002-9505. 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), <a href="https://pubs.aip.org/ajp/article/83/1/39/1042100/Nonequivalence-of-equivalence-principles" target="_blank">https://pubs.aip.org/ajp/article/83/1/39/1042100/Nonequivalence-of-equivalence-principles</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Ghins, Michel; Budden, Tim (March 2001). "The Principle of Equivalence". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 32 (1): 33–51. Bibcode:2001SHPMP..32...33G. doi:10.1016/S1355-2198(00)00038-1. <a href="https://linkinghub.elsevier.com/retrieve/pii/S1355219800000381" target="_blank">https://linkinghub.elsevier.com/retrieve/pii/S1355219800000381</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
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</ol>

Equivalence principle open-in-new

Equivalence principle