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# Equivalence principle

In relativity the equivalence principle is applied to several related concepts dealing with frames of reference. In particular, the principles describes the unformity of physical measurements in inertial frames of reference.

The original equivalence principle was introduced by Einstein in 1907, and is now known as the weak equivalence principle. It may be summarized by the following statement:

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

The weak equivalence principle is therefore a rule for determining if one is in an accelerated frame of reference. It is equivalent to the universality of free fall which states that

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

Another form is the Einstein equivalence principle which states that the result 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 an extension of the postulates of special relativity which requires that dimensionless physical values such as the fine-structure constant and electron-to-proton mass ratio be constant. Many workers believe that any Lorentz invariant theory that satisfies the weak equivalence principle also satisfies the Einstein equivalence principle.

Finally, there is the strong equivalence principle, which states that the results of any local experiment, gravitational or not, in an inertial frame of reference is 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. Itrequires 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.

 Contents

## Historical Development

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, F = ma, 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 at 1g (g=9.81 m/s2) 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. From this principle, Einstein deduced that free-fall is actually inertial motion, while being at rest with respect to the Earth (while under the influence of its gravitational field) is really an accelerated state of motion. This observation is now known as the weak equivalence principle. This observation was the start of a process that eventually led to the development of general relativity.

Therefore, 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.

Modern investigation into the equivalence principle was catalyzed in 1937 when 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. Dirac postulated that Newton's constant varied as the inverse of the age of the universe. While he turned out to be wrong, he led people to consider that the fundamental constants may evolve in space and time, and their present values, rather than being fundamental, may be set dynamically.

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. Two new principles were suggested, the so-called Einstein equivalence principle and the strong equivalence principle. They differ only in whether they apply to gravitational experiments or not.

## Tests of the weak equivalence principle

Tests of the validity of the equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. This is evidenced by all objects falling at the same rate when the effect of air resistance is either eliminated or negligible. Notable test are: 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 (MICROSattelite 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 changes in the gravitational field, tidal forces, so that the entire laboratory is moving inertially.

The most important consequence of this principle is 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, we measure dimensionless numbers, such as the ratio of two masses or coupling constants such as the fine-structure constant.

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 occuring Oklo fission reactor, where nuclear reactions similar to ones we observe today have been shown to have occured 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 is the only known theory that satisfies the strong equivalence principle. A number of alternative theories, such as Brans-Dicke theory, satisfy only the Einstein 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 measurments have put tight limits on Brans-Dicke theory.

## References

• Albert Einstein On the influence of gravitation on the propagation of light, Annalen der Physik, 35 (1911), translated in The Principle of Relativity, Dover (1924), pp 99-108. ISBN 0-486-60081-5
• 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). [1] This technical article reviews the best constraints on the variation of the fundamenal 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, 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 and 3.1, respectively.