Equivalence Principle

Type of physical science: Relativity

Field of study: General relativity

Experiments done in an accelerated frame of reference yield results that are equivalent to observations made in a gravitational field. This empirical fact, elevated to the status of a postulate, is one of the foundations of the general theory of relativity, Albert Einstein's theory of gravity.

Overview

Albert Einstein's equivalence principle states that it is impossible to distinguish by means of any physical measurement between a uniform gravitational field and a uniform acceleration. This principle explains the identity of gravitational and inertial mass and is one of the foundations of Einstein's theory of gravitation, general relativity theory. Understanding the logical distinction between these masses requires an acquaintance with Sir Isaac Newton's law of gravitation and his laws of motion. The numerical identity of these two masses for any particular object is established by experiment.

Newtonian theory describes gravitation in terms of a force that acts on any particle in a gravitational field. This force is the product of a property of the particle, its gravitational mass, and the field at the particle's location. (The field is produced by the gravitational masses of all other particles and depends on their locations.) Newton's second law of motion asserts that the acceleration of any particle that is subject to a resultant force is inversely proportional to its inertial mass. Numerous experiments have verified that if the only forces acting on an object are gravitational, its acceleration is independent of its composition. This requires that the gravitational mass of each particle must be numerically equal (or proportional through a universal constant) to its inertial mass. In the Newtonian description of gravitation and motion, there is no explanation of this mysterious coincidence.

What is the experimental evidence that inertial masses and gravitational masses are the same? The identity of inertial and gravitational mass implies that the times of fall from rest through a given height of two different weights will be equal. In spite of the authoritative statement of Aristotle to the contrary, precisely such an equality was proposed by Giambattista Benedetti in 1553, tested experimentally by Simon Stevin in 1586, and explored in detail through the study of balls rolling down inclined planes by Galileo, who was a professor at the University of Pisa from 1589 to 1592. Newton devoted the first paragraph of his PHILOSOPHIAE NATURALIS PRINCIPIA MATHEMATICA (1687; NEWTON'S PRINCIPIA: THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, 1846) to a careful statement of this result as a fundamental principle of mechanics. He also tested the result to a precision of 1 part in 1,000 in experiments involving periods of equal length and pendulums of different compositions. By the early twentieth century, the precision had been refined by a factor of 100, but this approach had reached the limits of its accuracy, which were imposed by factors such as the effects of air currents and difficulties in timing slowly moving objects.

In the late nineteenth century, Roland Eotvos first achieved substantial improvements in accuracy by conducting a new type of experiment based on a torsion balance, similar to that which had been used to determine the Newtonian gravitational constant. The experiments searched for and failed to detect a difference between the mass (inertial) on which centrifugal torques depend and the mass (gravitational) on which torques caused by the gravitational influence of the earth depend. In two series of experiments, which were performed in Budapest in 1889 and 1908, Eotvos and his colleagues confirmed the identification of these masses for numerous test bodies of various composition to the sensitivity of their apparatus, about 3 parts per billion.

To explain the identity of inertial and gravitational mass, Albert Einstein adopted the postulate that no experiment can inform an observer whether local phenomena are caused by the "fictitious forces" (for example, centrifugal and Coriolis) present in an accelerated frame of reference or by the gravitational field as a result of distant objects. This principle of equivalence is fundamental to the identification of the gravitational field with a distortion of the geometry of space-time, so that bodies subject only to gravitation follow local geodesics (extremal paths) in space-time but no longer appear to move along straight lines over extended regions of space.

When Einstein first proposed that the equivalence principle was important to the foundation of gravitational theory, in 1907, he was apparently unaware of the experimental work of Eotvos and the more precise support it gave this principle. Although the equivalence principle by itself dictates only half of gravitational theory (curved space-time tells matter how to move), in Einstein's thought it was a natural leap to a unique formulation of the other half (matter tells space-time how to curve), the general theory of relativity, which he published in 1916. In the meantime, he had become acquainted, apparently in 1912, with the experiments done in Budapest, and referred to them in his later writings.

Ever since, experimentalists have sought ways to improve the accuracy of equivalence principle tests, or new means of testing. Major improvements were achieved by Robert H. Dicke and his colleagues (1 part in 100 billion) at Princeton University in 1962 and Vladimir Braginsky and his colleagues (1 part in a trillion) at Moscow State University in 1970. They replaced the earth's gravitational field and the centrifugal effect of its rotation on its axis with the gravitational field of the sun and the centrifugal effect of the earth's revolution in orbit around the sun. This allowed the elimination of sources of noise and error associated with the earlier need to rotate bodily the apparatus in order to exchange the roles of two test objects with different compositions. This was now accomplished twice daily by the rotation of the earth on its axis.

Thus, both groups were able to exploit advances in torsion fiber technology, improved vacuum systems that minimized effects of air currents, automated temperature control of the experimental environment, and sophisticated electrical and optical monitoring techniques.

Applications

Beyond its role as an explanation of the equality of inertial and gravitational masses, the equivalence principle predicts the outcome of other experiments that depend only upon the response of bodies to a given gravitational field, not how the field is generated by its sources. Of primary importance among these, for reasons of both history and precision, are experiments involving the gravitational redshift of light in both astrophysical and laboratory contexts.

Similarly interesting is the predicted rate change of a clock that experiences varying gravitational potentials, perhaps before being returned for comparison of total aging with its twin, which has remained at rest in a laboratory.

In his 1916 paper, Einstein proposed that observations of spectral lines from a region of very great gravitational potential, such as a white dwarf star, should show detectable shifts toward longer wavelengths. This has been roughly verified for numerous examples, beginning with observations reported in 1926 of the spectrum of the white dwarf companion to the bright star Sirius, the Dog Star, and ultimately extending to observations of ordinary stars such as the sun using observational technology that became available in the early 1960's. Unfortunately, the precise quantitative shift is also influenced by complicated astrophysical processes, so the contribution of the equivalence principle, though theoretically calculable, is in practice impossible to untangle from other contributions. This prediction of the equivalence principle is so well supported by the laboratory experiments, however, that the gravitational redshift from astronomical sources is an accepted working tool of the theoretical astrophysicist. An interesting application may be seen in studies made in the late 1960's that showed that it is not possible to explain the dominant part of the observed redshifts of quasars in terms of a gravitational redshift from sources whose structures could be stable over the periods of time for which some quasars of large redshift had been under continuous observation. This development helped to convince most astronomers that the dominant part of the redshift of quasars had to be interpreted as being caused by huge Doppler effects, implying cosmological recession velocities larger than those of any other discrete objects that have been observed.

If astronomical phenomena mix gravitational redshift effects with other ingredients in messy recipes, there is one type of laboratory application that can be used for a nearly clinically pure measurement of this manifestation of the equivalence principle. This is the precision measurement of shifts in spectral lines emitted and absorbed at different heights in the earth's gravitational field that are made possible by the phenomenon of recoilless emission of γ rays, an effect named for its discoverer, Rudolf Mossbauer. Photons emitted at the top of a tower are blueshifted to shorter wavelengths at the base of the tower, and photons emitted at the base are redshifted in the same proportion at the top. Just such an experiment was first reported by Robert V. Pound and Glenn Rebka in 1960, who used the 23-meter tower of the physics building at Harvard University as a location in which to observe the 0.86-angstrom spectral line produced in the decay of the unstable isotope of iron known as iron 57. This first measurement yielded results within 10 percent of what was expected, and an improved experiment by Pound and Joseph Snider in 1963 yielded results within 1 percent.

If wavelengths of light are changed by vertical motion in a gravitational field, frequencies must change inversely so that the speed of light is invariant. The frequencies of radiation associated with transitions in electronic structure, however, are basic to the operation of atomic clocks, the devices used to define laboratory standards of time. Thus, the phenomenon of gravitational clock rate change is also an application of the equivalence principle. The analysis is complicated by the fact that at the speeds and altitudes of the commercial airliners that were used in the experiment performed by J. C. Hafele and Richard Keating in October of 1971, the effects predicted by the special theory of relativity as a result of relative velocity are of a magnitude comparable to that of the effects predicted by the equivalence principle as a result of variations in the gravitational potential between laboratory and flying clocks.

The special theory of relativity is so well established by numerous other experiments, however, that there is essentially no doubt of its correctness. Indeed, since November of 1983, international length and time standards have been related to each other so that the speed of light is defined as a constant (exactly 299,792,458 meters per second). Thus the effects of special relativity can be calculated with confidence and removed from measured "jet-lag" to yield effects of the equivalence principle. Comparisons of clocks flown eastward and westward around the earth with an identical clock left at rest in the laboratory showed agreement between observations of total aging differences as large as 273 nanoseconds and theoretical expectations to within the experimental error of 20 nanoseconds attributed to inaccuracies in logs of flight data and variations within and among the cesium beam atomic clocks.

Comparison of twin hydrogen maser clocks, one carried to an altitude of more than 10,000 kilometers by a Scout D rocket launched from Wallops Island, Virginia, in the morning of June 18, 1976, and the other at rest on Merritt Island, Florida, yielded a continuous and much more precise confirmation of the equivalence principle's effect on clock rates. Clever use of a transponder within the payload allowed complete removal of the Doppler effect contribution in a continuous comparison of clock rates observed through radio signals between ground and rocket, leaving only the effects of special relativistic time dilation and the equivalence principle. After more than two years of analysis of data from this two-hour flight, Robert Vessot and his colleagues demonstrated that the agreement between theory and observation was within 70 parts per million throughout the parts of the trajectory for which reliable tracking had been possible.

Context

Einstein first stated the equivalence of uniform acceleration and a uniform gravitational field in 1907, but gravitational fields in nature are often of varying strength and direction. The presentation in the "Overview" assumed that the portion of the frame of reference in which an experiment verifies the equivalence principle is a sufficiently small region of space-time that the effects of gravitation may be "transformed away" throughout that portion by changing to a freely falling frame of reference. In a uniform freely falling frame, a man, a feather, and a block of iron all have the same weight: no weight at all. In a gravitational field of varying strength and direction, however, no such global freely falling frame is possible over larger regions of space-time. The relationship of various locally freely falling frames will depend upon the ways in which the sources of the gravitational field produce large-scale space-time curvature. Einstein's general theory of relativity, first published in 1916, is a unique theory of gravitation that extends the earlier "weak" form of the equivalence principle to a "strong equivalence principle." This demands that even extended objects with significant internal gravitational potential energy must respond in exactly the same way to external gravitational fields. In Newton's theory of gravity or any other theory consistent with the original form of the equivalence principle, balls of platinum and aluminum must fall in the same way in the earth's gravitational field. Only in Einstein's general theory of relativity, however, must the earth and the Moon fall in the same way in the sun's gravitational field.

Analysis of possible deviations from the strong equivalence principle in a large set of alternatives to Einstein's theory of general relativity was first carried out by Kenneth Nordvedt in 1967. For example, the predicted Nordvedt effect amounted to an elongation in the Moon's orbit around the earth of as much as 1.3 meters, in the then viable Brans-Dicke theory of gravitation, along the direction toward the sun. Beginning with the American Apollo 11 landing on July 21, 1969, and continuing with the Apollo 14 and 15 missions and the unmanned Soviet Luna 17 and 21 missions, retroreflectors were placed on the Moon. Subsequently, it became possible, through precise laser ranging between the prime foci of observatory telescopes and these reflectors, to measure the distances between well-defined points on the earth and the Moon to a precision on the order of 0.15 meter. The orbit of the Moon is subject to a very large number of perturbing influences, caused, for example, by other planets, which are well understood in principle.

Fortunately, the locations of the planets at this time could be determined sufficiently well from radar ranging and spacecraft data that their effects on the Moon's orbit could be removed from the laser ranging data to within the combined experimental uncertainties of 0.3 meter, leaving no evidence for the Nordvedt effect. This means that the earth and the Moon fall in the same way in the gravitational field of the sun to a precision of 1 part in 100 billion. Thus, the strong equivalence principle is supported experimentally with a precision comparable to the support given the weak equivalence principle by the best laboratory tests of identity of test body gravitational and inertial masses.

Principal terms

ACCELERATION: the rate of change of velocity associated with a change in speed or direction of motion or both

FRAME OF REFERENCE: a system of space-time coordinates used by an observer to measure the physical world

GRAVITATIONAL FIELD: the region in which an object (a test particle)

experiences acceleration caused by the masses and locations of other objects (sources of the field)

GRAVITATIONAL MASS: the mass of a body defined as the quantity that determines its coupling to a gravitational field, either as a test particle or as a source

INERTIAL FRAME: a frame of reference in which a body that has no force acting on it has no acceleration

INERTIAL MASS: the mass of a body defined as the measure of its opposition to acceleration by external forces

Bibliography

Dicke, Robert H. THE THEORETICAL SIGNIFICANCE OF EXPERIMENTAL RELATIVITY. New York: Academic, 1964. This volume includes a masterly and readable presentation of the fundamental concepts and technical details involved in the refined Eotvos experiments.

Einstein, Albert. "The Foundation of the General Theory of Relativity." In THE PRINCIPLE OF RELATIVITY. Reprint. New York: Dover, 1952. This mathematically technical article includes descriptions of the basic principles that are accessible to the general reader and remain unsurpassed in insight and clarity.

Misner, Charles W., Kip S. Thorne, and John A. Wheeler. GRAVITATION. San Francisco: W. H. Freeman, 1973. This comprehensive textbook on general relativity is written in three clearly identified "tracks" that are suitable for a variety of readers, from the diligent layperson with some background in physics to the advanced graduate student of physics. Includes a clear exposition of basic principles and a discussion of experimental verification.

Schwinger, Julian. EINSTEIN'S LEGACY: THE UNITY OF SPACE AND TIME. New York: Scientific American Books, 1986. A lively and well-illustrated account of the special and general theories of relativity, this volume includes entertaining and illuminating thought experiments illustrating the basic principles and actual experimental verifications.

Will, Clifford M. WAS EINSTEIN RIGHT? New York: Basic Books, 1986. This engaging narrative presents the best popular description of the experimental basis for the equivalence principle and the general theory of relativity. The author was a principal developer of mathematical formulas for analyzing the predictions of alternative theories of gravitation.

Centrifugal/Centripetal and Coriolis Accelerations