Equivalence principle

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In the physics of general relativity, the equivalence principle refers to several related concepts dealing with the equivalence of gravitational and inertial mass, and 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.

Contents

Einstein's statement of the equivalence principle

A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:

(Inertial mass) \cdot (Acceleration) = (Intensity of the gravitational field) \cdot (Gravitational mass).
It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.

– Albert Einstein, [1]

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 proper was 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. 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 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 discovery of weak equivalence principle

Loránd Eötvös published on surface tension between 1876 and 1886. The Torsion or Eötvös balance, designed by Hungarian Baron Loránd Eötvös, is a sensitive instrument for measuring the density of underlying rock strata. The device measures not only the direction of force of gravity, but the change in the force of gravity's extent in horizontal plane. It determines the distribution of masses in the Earth's crust. The Eötvös torsion balance, an important instrument of geodesy and geophysics throughout the whole world, studies the Earth's physical properties. It is used for mine exploration, and also in the search for minerals, such as oil, coal and ores. Eötvös' law of capillarity (weak equivalence principle) served as a basis for Einstein's theory of relativity. (Capillarity: the property or exertion of capillary attraction or repulsion, a force that is the resultant of adhesion, cohesion, and surface tension in liquids which are in contact with solids, causing the liquid surface to rise - or be depressed...)[2][3] These experiments demonstrate that all objects fall at the same rate with negligible friction (including air resistance). 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 Loránd Eötvös. Satellite experiments are planned for more accurate experiments in space.[4] They verify the weak principle.

The weak equivalence principle, also known as the universality of free fall, or the Galilean equivalence principle

The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition.

or

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]

The principle does not apply to physical bodies, which experience tidal forces, or heavy point masses, whose presence changes 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.

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}}

Hence:

\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:

\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.

Tests of the weak equivalence principle

Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass.

Researcher Year Method Result
John Philoponus 6th century Described correctly the effect of dropping balls of different masses no detectable difference
Simon Stevin ~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}[6]
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[7])
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})=(0.8 \pm 1.3)\times 10^{-13}[8][9]

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}

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 the theory is elegant and is compatible with almost all observations to date (except for instance the Pioneer anomaly). 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.

Laboratory equivalence principle composition and spin tests are supported by observation of binary pulsar PSR J0737-3039 (arXiv, Matters of Gravity). A neutron star core might be strange matter, pion condensate, lambda hyperon (Λ0), delta isobar, or free quark matter. Extreme bound (gravitational binding energy ~30% of disassembled rest mass), spinning (~20% of lightspeed at equator), magnetic (~108 tesla), dense (4-9x1014 g/cm3), superconducting neutronium obeys general relativity orbital predictions within 0.05% or better.

The equivalence principle is untested against opposite geometric parity (chirality in all directions) mass distributions. A parity Eötvös experiment contrasting solid single crystal spheres of identical composition α-quartz in enantiomorphic space groups P3121 (right-handed screw axis) versus P3221 (left-handed screw axis) is appropriate. Equivalence principle parity violation validates a chiral vacuum background forbidden within general relativity but allowed within Einstein-Cartan theory; affine, teleparallel, and noncomutative gravitation theories.

The Einstein equivalence principle

The Einstein equivalence principle states that the weak equivalence principle holds, and that:[10]

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[11] and Durand[12]).

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.[13]. 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.

and

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.

Controversial challenges to the equivalence principle

The best known challenge to the equivalence principle is the Brans-Dicke theory; however, the theory represents the viewpoint of a tiny minority in the physics community. 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.

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 be not 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 "the greatest embarassment in the history of theoretical physics".

Experiments

See also

References

  1. ↑ A. Einstein. “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 talk in Japan on 14 December 1922.
  2. ↑ http://www.gap-system.org/~history/Biographies/Eotvos.html
  3. ↑ http://zelmanov.ptep-online.com/papers/zj-2008-b2.pdf
  4. ↑ Dittus, H; C. Lāmmerzahl (pdf). Experimental Tests of the Equivalence Principle and Newton’s Law in Space. http://www.zarm.uni-bremen.de/2forschung/gravi/publications/papers/2005DittusLaemmerzahl.pdf. 
  5. ↑ Paul S Wesson (2006). Five-dimensional Physics. World Scientific. p. 82. ISBN 9812566619. http://books.google.com/books?id=dSv8ksxHR0oC&printsec=frontcover&dq=intitle:Five+intitle:Dimensional+intitle:Physics&lr=&as_brr=0#PPA82,M1. 
  6. ↑ P. G. Roll, R. Krotkov, R. H. Dicke, The equivalence of inertial and passive gravitational mass, Annals of Physics, Volume 26, Issue 3, 20 February 1964, Pages 442-517
  7. ↑ http://www.youtube.com/watch?v=MJyUDpm9Kvk
  8. ↑ http://link.aps.org/doi/10.1103/PhysRevLett.100.041101
  9. ↑ http://arxiv.org/abs/0712.0607
  10. ↑ Haugen, Mark P.; C. Lämmerzahl (2001). Principles of Equivalence: Their Role in Gravitation Physics and Experiments that Test Them. Springer. ISBN 978-3-540-41236-6. http://arxiv.org/abs/gr-qc/0103067v1. 
  11. ↑ http://arxiv.org/abs/quant-ph/9706018
  12. ↑ http://stacks.iop.org/ob/4/S351
  13. ↑ http://arxiv.org/abs/astro-ph/0012539
  14. ↑ Eöt-Wash group
  15. ↑ http://funphysics.jpl.nasa.gov/technical/grp/lunar-laser.html
  16. ↑ http://eotvos.dm.unipi.it/nobili/
  17. ↑ http://einstein.stanford.edu/STEP/
  18. ↑ http://smsc.cnes.fr/MICROSCOPE/index.htm
  19. ↑ http://www.phys.utk.edu/see/
  20. ↑ 16 November 2004, physicsweb: Equivalence principle passes atomic test
  • 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, ISBN 0-486-60081-5. The two Einstein papers are discussed online at The Genesis of General Relativity.
  • C. Brans, "The roots of scalar-tensor theory: an approximate history", arXiv: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). [1] 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, 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.

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