# Equivalence principle

In relativity, the equivalence principle is applied to several related concepts dealing with the uniformity of physical measurements in different frames of reference. They are related to the Copernican idea that the laws of physics should be the same everywhere in the universe, but also to Albert Einstein's assertion 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. This principle is:

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.

This is a rule for determining if one is in an accelerated frame of reference. It played a crucial role in the development of general relativity.

 Contents

## History

The origins of the equivalence principle begin with Galileo demonstrating in the late 16th century that all objects are accelerated towards the center of the Earth at the same rate. This was codified by Newton with his gravitational theory in which it was postulated that inertial and gravitational masses are one and the same.

In Newtonian mechanics, gravity is assumed to be a force. This force draws objects towards the center of a massive body. At the Earth's surface, the force of gravity is counter-balanced by the mechanical resistance of the Earth's surface. So a person at rest on the surface of a (non-rotating) massive object is in an inertial frame of reference. The force of gravity is counter-balanced by the upward force of the surface on that person, and the net force is zero. While this picture works very well for most calculations, it remains a mystery why the inertial mass in Newton's second law, [itex]F=ma[itex], is equal to the gravitational mass in Newton's law of universal gravitation.

The equivalence principle proper was introduced by Albert Einstein in 1907. An that time, he made the observation that the acceleration of bodies towards the center of the Earth with acceleration 1g (g=9.81 m/s2 is the acceleration of gravity at the Earth's surface) is equivalent to the acceleration of inertially moving bodies that one would observe if one was 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, remaining at rest in a uniform gravitational field is physically equivalent to experiencing an acceleration (e.g. being at rest with respect to the Earth, while under the influence of its gravitational field, is an accelerated state of motion). From this principle, Einstein deduced that free-fall is actually inertial motion. The idea was precisely formulated by Einstein in 1911, referring to two frames of reference K and K'. The frame K is in a uniform gravitational field, whereas K' has no gravitational field but is uniformly accelerated such that objects in 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 led to the development of 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)

He used this principle, together with special relativity, to predict that clocks run at different rates in a gravitational potential and the bending of light-rays in a gravitational field, even before he developed the concept of curved-space time.

Thus, in general relativity the situation is quite different than in Newtonian mechanics. Since inertial mass is the same as gravitational mass in the gravitational fields of massive bodies, the equivalence principle indicates that free-fall is actually inertial motion. In that case, there is only one force acting on a person standing on the surface of a massive object, and that is the upward force of the surface on that person. Although the equivalence principle helped to guide the development of general relativity, the equivalence principle, rather than being a founding principle, is a simple consequence of the geometrical nature of the theory. There is no place in the theory to set the inertial and gravitational masses equal, because there is no force law for gravity. The inertial reference frames are defined as those which are freely falling.

Interest in the modern extensions of the equivalence principle was catalyzed in 1937 when Paul Dirac formulated his large numbers hypothesis which asserts that large, dimensionless numbers should not arise as fundamental quantities in physics: there should only be one fundamental energy scale in physics. He supported this by pointing out a coincidence: the dimensionless ratio of electric to gravitational forces in a hydrogen atom is about the same as the age of the universe, measured by the time it takes light to cross the hydrogen atom. Both are about 1040. To explain this surprising coincidence, Dirac postulated that Newton's constant varied as the inverse of the age of the universe, and the feebleness of gravity was due to the great age of the universe. While he turned out to be wrong, he led people to consider that the laws of physics may be different at different points in space and time, and the values of the physical constants, rather than being fundamental, may be set dynamically. These ideas, together with Mach's principle – roughly, the idea that inertia of a mass should be induced by the other masses in the universe – led physicists to develop scalar-tensor theories, in particular Brans-Dicke theory, in which the value of the gravitational constant is determined dynamically.

## Modern usage

A number of different forms of the equivalence principle are used in research today. The weak equivalence principle, also known as the universality of free fall:

The trajectory of a falling test body depends only on its initial position and velocity, and is independent of its composition.

The principle does not apply to large bodies, which might experience tidal forces, or heavy bodies, whose presence will substantially change the gravitational field around them. This form of the equivalence principle is closest to Einstein's original statement: in fact, his statements imply this one.

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 differ only in whether they apply to gravitational experiments or not.

The Einstein equivalence principle states that the result of a local non-gravitational experiment in an inertial frame of reference is independent of the velocity or location in the universe of the experiment. This is a kind of Copernican extension of Einstein's original formulation, which requires that suitable frames of reference all over the universe behave identically. It is an extension of the postulates of special relativity in that it requires that dimensionless physical values such as the fine-structure constant and electron-to-proton mass ratio be constant. Many physicists believe that any Lorentz invariant theory that satisfies the weak equivalence principle also satisfies the Einstein equivalence principle.

The strong equivalence principle states that the results of any local experiment, gravitational or not, in an inertial frame of reference are independent of where and when in the universe it is conducted. 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. General relativity is the only known theory of gravity compatible with this form of the equivalence principle.

### Tests of the weak equivalence principle

Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. These experiments demonstrate that all objects fall at the same rate when the effect of air resistance is either eliminated or negligible. The simplest way to test the weak equivalence principle is to drop two objects of different masses or compositions in a vacuum, and see if they hit the ground at the same time. More sophisticated tests use a torsion balance of a type invented by Roland Eötvös.

 Researcher Year Method Result Galileo Galilei ~1610 Dropping metal balls of different mass from the Tower of Pisa 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 Roland 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 a billion Roll, Krotkov and Dicke 1964 Torsion balance experiment, dropping aluminum and gold test masses difference is less than one part in one hundred billion David Scott 1971 Dropped an eagle feather and a hammer at the same time on the Moon no detectable difference (Not a very good experiment, but it was the first lunar one.) Branginsky and Panov 1971 Torsion balance, aluminum and platinum test masses, measuring acceleration towards the sun difference is less than 1 part in a trillion (most accurate to date) Eöt-Wash 1987– Torsion balance, measuring acceleration of different masses towards the earth, sun and galactic center, using several different kinds of masses difference is less than a few parts in a trillion

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 – 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 it is by far the most elegant theory of gravity known, and is perfectly compatible with all observations to date. 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

The outcome of any local non-gravitational experiment in a laboratory moving in an inertial frame of reference is independent of the velocity of the laboratory, or 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 moving inertially.

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 any of the fundamental physical parameters – other than masses and Newton's gravitational constant – must not depend on where in space or time we measure them. In practice, these are dimensionless numbers, such as the ratio of two masses, or coupling constants such as the fine-structure constant.

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.

#### 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 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. Other researchers dispute these findings. Other tests of the Einstein equivalence principle are gravitational redshift experiments, 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.

and

The outcome of any local experiment, whether gravitational or not, in a laboratory moving in an inertial frame of reference is independent of velocity of the laboratory, or its location in spacetime.

The first part is a version of the weak equivalence principle that it 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, 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. This idealized requirement has been misunderstood. This form of the equivalence principle does not imply that the effects of a gravitational field cannot be measured by observers in free-fall. For example, an observer in free-fall into a black hole will experience strong tidal forces: he will notice a more powerful force on his feet than his head.

The strong equivalence suggests that gravity is an entirely geometrical by nature (that is metric alone determines the effect of gravity) and does not have an 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 (Sattelite 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, using very long baseline interferometry, in particular the Lunar Laser Ranging Experiment. These measurements have put tight limits on Brans-Dicke theory.

## References

• 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, ISBN 0-486-60081-5. The two Einstein papers are discussed online at The Genesis of General Relativity (http://www1.kcn.ne.jp/~h-uchii/gen.GR.html).
• C. Brans, "The roots of scalar-tensor theory: an approximate history", arXiv:gr-qc/0506063 (http://www.arxiv.org/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).  (http://www.arxiv.org/abs/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, The Confrontation between General Relativity and Experiment, (http://relativity.livingreviews.org/Articles/lrr-2001-4/index.html) Living Reviews in Relativity (2001). An online, technical review, covering much of the material in Theory and experiment in gravitational physics. The Einstein and strong variants of the equivalence principles are discussed in sections 2.1 (http://relativity.livingreviews.org/Articles/lrr-2001-4/node3.html) and 3.1 (http://relativity.livingreviews.org/Articles/lrr-2001-4/node7.html), respectively.

## Experiments

• University of Washington Eöt-Wash group (http://www.npl.washington.edu/eotwash/)
• Lunar Laser Ranging  (http://funphysics.jpl.nasa.gov/technical/grp/lunar-laser.html)
• Galileo-Galilei satellite experiment  (http://eotvos.dm.unipi.it/nobili/)
• Satellite Test of the Equivalence Principle (STEP)  (http://einstein.stanford.edu/STEP/)
• MICROSCOPE  (http://smsc.cnes.fr/MICROSCOPE/index.htm)
• Satellite Energy Exchange (SEE)  (http://www.phys.utk.edu/see/)
• 16 November 2004, physicsweb: Equivalence principle passes atomic test (http://physicsweb.org/articles/news/8/11/8/1) Quote: "...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..."da:Ækvivalensprincip

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