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Strong equivalence principle

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Strong equivalence principle

In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

Einstein's statement of the equivalence principle

Albert Einstein


See momentum and velocity.

Development of gravitation theory

Something like the equivalence principle emerged in the late 16th and early 17th centuries, when Galileo expressed experimentally that the acceleration of a test mass due to gravitation is independent of the amount of mass being accelerated. These findings led to gravitational theory, in which the inertial and gravitational masses are identical.

The equivalence principle was properly introduced by Albert Einstein in 1907, when he observed that the acceleration of bodies towards the center of the Earth at a rate of 1g (g = 9.81 m/s2 being a standard reference of gravitational acceleration at the Earth's surface) is equivalent to the acceleration of an inertially moving body that would be observed on a rocket in free space being accelerated at a rate of 1g. Einstein stated it thus:

we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.
—Einstein, 1907

That is, being at rest on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines. From this principle, Einstein deduced that free-fall is actually inertial motion. Objects in free-fall do not really accelerate. In an inertial frame of reference bodies (and light) obey Newton's first law, moving at constant velocity in straight lines. Analogously, in a curved spacetime the worldline of an inertial particle or pulse of light is as straight as possible (in space and time).[2] Such a worldline is called a geodesic. Viewed across time from the viewpoint of an observer "stationary" on the surface of a gravitating body, the geodesics appear to curve towards the body. This is why an accelerometer in free-fall doesn't register any acceleration; there isn't any. By contrast, in Newtonian mechanics, gravity is assumed to be a force. This force draws objects having mass towards the center of any massive body. At the Earth's surface, the force of gravity is counteracted by the mechanical (physical) resistance of the Earth's surface. So in Newtonian physics, a person at rest on the surface of a (non-rotating) massive object is in an inertial frame of reference. These considerations suggest the following corollary to the equivalence principle, which Einstein formulated precisely in 1911:

Whenever an observer detects the local presence of a force that acts on all objects in direct proportion to the inertial mass of each object, that observer is in an accelerated frame of reference.

Einstein also referred to two reference frames, K and K'. K is a uniform gravitational field, whereas K' has no gravitational field but is uniformly accelerated such that objects in the two frames experience identical forces:

We arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K and K' are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitational fields, if we then regard K as uniformly accelerated. 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

This observation was the start of a process that culminated in general relativity. Einstein suggested that it should be elevated to the status of a general principle when constructing his theory of relativity:

As long as we restrict ourselves to purely mechanical processes in the realm where Newton's mechanics holds sway, we are certain of the equivalence of the systems K and K'. But this view of ours will not have any deeper significance unless the systems K and K' are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K'. By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field.
—Einstein, 1911

Einstein combined (postulated) the equivalence principle with special relativity to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field, even before he developed the concept of curved spacetime.

So the original equivalence principle, as described by Einstein, concluded that free-fall and inertial motion were physically equivalent. This form of the equivalence principle can be stated as follows. 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. This is not strictly true, because massive bodies give rise to tidal effects (caused by variations in the strength and direction of the gravitational field) which are absent from an accelerating spaceship in deep space.

Although the equivalence principle guided the development of general relativity, it is not a founding principle of relativity but rather a simple consequence of the geometrical nature of the theory. In general relativity, objects in free-fall follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics of spacetime, because the mechanical resistance of matter prevents us from doing so.

Since Einstein developed general relativity, there was a need to develop a framework to test the theory against other possible theories of gravity compatible with special relativity. This was developed by Robert Dicke as part of his program to test general relativity. Two new principles were suggested, the so-called Einstein equivalence principle and the strong equivalence principle, each of which assumes the weak equivalence principle as a starting point. They only differ in whether or not they apply to gravitational experiments.

Modern usage

Three forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.

The weak equivalence principle

The weak equivalence principle, also known as the universality of free fall or the Galilean equivalence principle can be stated in many ways. The strong EP includes (astronomic) bodies with gravitational binding energy[3] (e.g., 1.74 solar-mass pulsar PSR J1903+0327, 15.3% of whose separated mass is absent as gravitational binding energy[4]). The weak EP assumes falling bodies are bound by non-gravitational forces only. Either way,

The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.
All test particles at the alike spacetime point in a given gravitational field will undergo the same acceleration, independent of their properties, including their rest mass.[5]
All local centers of mass vacuum free fall along identical (parallel-displaced, same speed) minimum action trajectories independent of all observable properties.
The vacuum world line of a body immersed in a gravitational field is independent of all observable properties.
The local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat space, without exception.
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).

Locality eliminates measurable tidal forces originating from a radial divergent gravitational field (e.g., the Earth) upon finite sized physical bodies. The "falling" equivalence principle embraces Galileo's, Newton's, and Einstein's conceptualization. The equivalence principle does not deny the existence of measurable effects caused by a rotating gravitating mass (frame dragging), or bear on the measurements of light deflection and gravitational time delay made by non-local observers.

Active, passive, and inertial masses

By definition of active and passive gravitational mass, the force on M_1 due to the gravitational field of M_0 is:

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 = \frac{M_0^\mathrm{act} M_2^\mathrm{pass}}{r^2}

By definition of inertial mass:

F = m^\mathrm{inert} a

If m_1 and m_2 are the same distance r from m_0 then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same)

a_1 = \frac{F_1}{m_1^\mathrm{inert}} = a_2 = \frac{F_2}{m_2^\mathrm{inert}}


\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}}


\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 all objects.

Furthermore by Newton's third law of motion:

F_1 = \frac{M_0^\mathrm{act} M_1^\mathrm{pass}}{r^2}

must be equal and opposite to

F_0 = \frac{M_1^\mathrm{act} M_0^\mathrm{pass}}{r^2}

It follows that:

\frac{M_0^\mathrm{act}}{M_0^\mathrm{pass}} = \frac{M_1^\mathrm{act}}{M_1^\mathrm{pass}}

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

The dimensionless Eötvös-parameter \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."

\eta(A,B)=2\frac{ \left(\frac{m_g}{m_i}\right)_A-\left(\frac{m_g}{m_i}\right)_B }{\left(\frac{m_g}{m_i}\right)_A+\left(\frac{m_g}{m_i}\right)_B}

Tests of the weak equivalence principle

Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. An obvious test is dropping two contrasted objects in hard vacuum, e.g., inside Fallturm Bremen.

Researcher Year Method Result
John Philoponus 6th century Described correctly the effect of dropping balls of different masses no detectable difference
Simon Stevin[6] ~1586 Dropped lead balls of different masses off the Delft churchtower no detectable difference
Galileo Galilei ~1610 Rolling balls down inclined planes no detectable difference
Isaac Newton ~1680 measure the period of pendulums of different mass but identical length no measurable difference
Friedrich Wilhelm Bessel 1832 measure the period of pendulums of different mass but identical length no measurable difference
Loránd Eötvös 1908 measure the torsion on a wire, suspending a balance beam, between two nearly identical masses under the acceleration of gravity and the rotation of the Earth difference is less than 1 part in 109
Roll, Krotkov and Dicke 1964 Torsion balance experiment, dropping aluminum and gold test masses |\eta(\mathrm{Al},\mathrm{Au})|=(1.3\pm1.0)\times10^{-11}[7]
David Scott 1971 Dropped a falcon feather and a hammer at the same time on the Moon no detectable difference (not a rigorous experiment, but very dramatic being the first lunar one[8])
Braginsky and Panov 1971 Torsion balance, aluminum and platinum test masses, measuring acceleration towards the sun difference is less than 1 part in 1012
Eöt-Wash group 1987– Torsion balance, measuring acceleration of different masses towards the earth, sun and galactic center, using several different kinds of masses \eta(\mathrm{Earth},\mathrm{Be-Ti})=(0.3 \pm 1.8)\times 10^{-13}[9][10]


Year Investigator Sensitivity Method
500? Philoponus [12] "small" Drop Tower
1585 Stevin [13] 5×10-2 Drop Tower
1590? Galileo [14] 2×10-2 Pendulum, Drop Tower
1686 Newton [15] 10-3 Pendulum
1832 Bessel [16] 2×10-5 Pendulum
1910 Southerns [17] 5×10-6 Pendulum
1918 Zeeman [18] 3×10-8 Torsion Balance
1922 Eötvös [19] 5×10-9 Torsion Balance
1923 Potter [20] 3×10-6 Pendulum
1935 Renner [21] 2×10-9 Torsion Balance
1964 Dicke,Roll,Krotkov [22] 3x10-11 Torsion Balance
1972 Braginsky,Panov [23] 10-12 Torsion Balance
1976 Shapiro, et al.[24] 10-12 Lunar Laser Ranging
1981 Keiser,Faller [25] 4×10-11 Fluid Support
1987 Niebauer, et al.[26] 10-10 Drop Tower
1989 Heckel, et al.[27] 10-11 Torsion Balance
1990 Adelberger, et al.[28] 10-12 Torsion Balance
1999 Baessler, et al.[29] 5x10-14 Torsion Balance
cancelled? MiniSTEP 10-17 Earth Orbit
2015? MICROSCOPE 10-16 Earth Orbit
2015? 2×10-17 vacuum free fall

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. Future satellite experiments[31]STEP (Satellite Test of the Equivalence Principle), Galileo Galilei, and MICROSCOPE (MICROSatellite pour l'Observation de Principe d'Equivalence) – will test the weak equivalence principle in space, to much higher accuracy.

The need to continue testing Einstein's theory of gravity may seem superfluous, as the theory is elegant and is compatible with observations. However, no quantum theory of gravity is known, and most suggestions violate one of the equivalence principles at some level. String theory, supergravity and even quintessence, for example, seem to violate the weak equivalence principle because they contain many light scalar fields with long Compton wavelengths. These fields should generate fifth forces and variation of the fundamental constants. There are a number of mechanisms that have been suggested by physicists to reduce these violations of the equivalence principle to below observable levels.

The Einstein equivalence principle

The Einstein equivalence principle states that the weak equivalence principle holds, and that:[32]

The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

Here "local" has a very special meaning: not only must the experiment not look outside the laboratory, but it must also be small compared to variations in the gravitational field, tidal forces, so that the entire laboratory is freely falling. It also implies the absence of interactions with "external" fields other than the gravitational field.

The principle of relativity implies that the outcome of local experiments must be independent of the velocity of the apparatus, so the most important consequence of this principle is the Copernican idea that dimensionless physical values such as the fine-structure constant and electron-to-proton mass ratio must not depend on where in space or time we measure them. Many physicists believe that any Lorentz invariant theory that satisfies the weak equivalence principle also satisfies the Einstein equivalence principle.

Schiff's conjecture suggests that the weak equivalence principle actually implies the Einstein equivalence principle, but it has not been proven. 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[33] and Durand[34]).

Tests of the Einstein equivalence principle

In addition to the tests of the weak equivalence principle, the Einstein equivalence principle can be tested by searching for variation of dimensionless constants and mass ratios. The present best limits on the variation of the fundamental constants have mainly been set by studying the naturally occurring Oklo natural nuclear fission reactor, where nuclear reactions similar to ones we observe today have been shown to have occurred underground approximately two billion years ago. These reactions are extremely sensitive to the values of the fundamental constants.

Constant Year Method Limit on fractional change
fine structure constant 1976 Oklo 10−7
weak interaction constant 1976 Oklo 10−2
electron-proton mass ratio 2002 quasars 10−4
proton gyromagnetic factor 1976 astrophysical 10−1

There have been a number of controversial attempts to constrain the variation of the strong interaction constant. There have been several suggestions that "constants" do vary on cosmological scales. The best known is the reported detection of variation (at the 10−5 level) of the fine-structure constant from measurements of distant quasars, see Webb et al.[35] Other researchers dispute these findings. Other tests of the Einstein equivalence principle are gravitational redshift experiments, such as the Pound-Rebka experiment which test the position independence of experiments.

The strong equivalence principle

The strong equivalence principle suggests the laws of gravitation are independent of velocity and location. In particular,

The gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitution.


The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

The first part is a version of the weak equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments. The second part is the Einstein equivalence principle (with the same definition of "local"), restated to allow gravitational experiments and self-gravitating bodies. The freely-falling object or laboratory, however, must still be small, so that tidal forces may be neglected (hence "local experiment").

This is the only form of the equivalence principle that applies to self-gravitating objects (such as stars), which have substantial internal gravitational interactions. It requires that the gravitational constant be the same everywhere in the universe and is incompatible with a fifth force. It is much more restrictive than the Einstein equivalence principle.

The strong equivalence principle suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity) and does not have any extra fields associated with it. 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, satisfy only the Einstein equivalence principle.

Tests of the strong equivalence principle

The strong equivalence principle can be tested by searching for a variation of Newton's gravitational constant G over the life of the universe, or equivalently, variation in the masses of the fundamental particles. A number of independent constraints, from orbits in the solar system and studies of big bang nucleosynthesis have shown that G cannot have varied by more than 10%.

Thus, the strong equivalence principle can be tested by searching for fifth forces (deviations from the gravitational force-law predicted by general relativity). These experiments typically look for failures of the inverse-square law (specifically Yukawa forces or failures of Birkhoff's theorem) behavior of gravity in the laboratory. The most accurate tests over short distances have been performed by the Eöt-Wash group. A future satellite experiment, SEE (Satellite Energy Exchange), will search for fifth forces in space and should be able to further constrain violations of the strong equivalence principle. Other limits, looking for much longer-range forces, have been placed by searching for the Nordtvedt effect, a "polarization" of solar system orbits that would be caused by gravitational self-energy accelerating at a different rate from normal matter. This effect has been sensitively tested by the Lunar Laser Ranging Experiment. Other tests include studying the deflection of radiation from distant radio sources by the sun, which can be accurately measured by very long baseline interferometry. Another sensitive test comes from measurements of the frequency shift of signals to and from the Cassini spacecraft. Together, these measurements have put tight limits on Brans-Dicke theory and other alternative theories of gravity.

Challenges to the equivalence principle

One challenge to the equivalence principle is the Brans-Dicke theory. Self-creation cosmology is a modification of the Brans-Dicke theory. The Fredkin Finite Nature Hypothesis is an even more radical challenge to the equivalence principle and has even fewer supporters.

In August 2010, researchers from the School of Physics, University of New South Wales, Australia; the Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Australia; and the Institute of Astronomy, Cambridge, United Kingdom; published the paper "Evidence for spatial variation of the fine structure constant", whose tentative conclusion is that, "qualitatively, [the] results suggest a violation of the Einstein Equivalence Principle, and could infer a very large or infinite universe, within which our `local' Hubble volume represents a tiny fraction."[36]

Explanations of the equivalence principle

Dutch physicist and string theorist Erik Verlinde has generated a self-contained, logical derivation of the equivalence principle based on the starting assumption of a holographic universe. Given this situation, gravity would not be a true fundamental force as is currently thought but instead an "emergent property" related to entropy. Verlinde's approach to explaining gravity apparently leads naturally to the correct observed strength of dark energy; previous failures to explain its incredibly small magnitude have been called by such people as cosmologist Michael Turner (who is credited as having coined the term "dark energy") as "the greatest embarrassment in the history of theoretical physics".[37] However, it should be noted that these ideas are far from settled and still very controversial.


See also



  • R. H. Dicke, "New Research on Old Gravitation," Science 129, 3349 (1959). This paper is the first to make the distinction between the strong and weak equivalence principles.
  • R. H. Dicke, "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.
  • Albert Einstein, "Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen," Jahrbuch der Radioaktivitaet und Elektronik 4 (1907); translated "On the relativity principle and the conclusions drawn from it," in The collected papers of Albert Einstein. Vol. 2 : The Swiss years: writings, 1900–1909 (Princeton University Press, Princeton, NJ, 1989), Anna Beck translator. This is Einstein's first statement of the equivalence principle.
  • Albert Einstein, "Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes," Annalen der Physik 35 (1911); translated "On the Influence of Gravitation on the Propagation of Light" in The collected papers of Albert Einstein. Vol. 3 : The Swiss years: writings, 1909–1911 (Princeton University Press, Princeton, NJ, 1994), Anna Beck translator, and in The Principle of Relativity, (Dover, 1924), pp 99–108, W. Perrett and G. B. Jeffery translators, The Genesis of General Relativity.
  • C. Brans, "The roots of scalar-tensor theory: an approximate history", gr-qc/0506063. Discusses the history of attempts to construct gravity theories with a scalar field and the relation to the equivalence principle and Mach's principle.
  • C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company, New York (1973), Chapter 16 discusses the equivalence principle.
  • Hans Ohanian and Remo Ruffini Gravitation and Spacetime 2nd edition, Norton, New York (1994). ISBN 0-393-96501-5 Chapter 1 discusses the equivalence principle, but incorrectly, according to modern usage, states that the strong equivalence principle is wrong.
  • J. P. Uzan, "The fundamental constants and their variation: Observational status and theoretical motivations," Rev. Mod. Phys. 75, 403 (2003). hep-ph/0205340 This technical article reviews the best constraints on the variation of the fundamental constants.
  • C. M. Will, Theory and experiment in gravitational physics, Cambridge University Press, Cambridge (1993). This is the standard technical reference for tests of general relativity.
  • C. M. Will, Was Einstein Right?: Putting General Relativity to the Test, Basic Books (1993). This is a popular account of tests of general relativity.
  • C. M. Will, 3.1, respectively.
  • Michael Friedman, Foundations of Space-Time Theories, Princeton University Press, Princeton (1983). Chapter V discusses the equivalence principle.

External links

  • Equivalence Principle at NASA, including tests
  • Introducing The Einstein Principle of Equivalence from Syracuse University
  • The Equivalence Principle at MathPages
  • The Einstein Equivalence Principle at Living Reviews on General Relativity

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