In

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, the

**sticky bead argument** is a simple

thought experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

designed to show that gravitational radiation is indeed predicted by

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, and can have physical effects. These claims were not widely accepted prior to about 1955, but after the introduction of the bead

argumentIn philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

, any remaining doubts soon disappeared from the research literature.

The argument is often credited to

Hermann BondiSir Hermann Bondi, KCB, FRS was an Anglo-Austrian mathematician and cosmologist. He is best known for developing the steady-state theory of the universe with Fred Hoyle and Thomas Gold as an alternative to the Big Bang theory, but his most lasting legacy will probably be his important...

, who popularized it, but it was apparently originally proposed anonymously by

Richard FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

.

## Description of the thought experiment

The thought experiment was first described by Feynman (under the pseudonym "Mr. Smith") in 1957, at a conference at

Chapel HillChapel Hill is a town in Orange County, North Carolina, United States and the home of the University of North Carolina at Chapel Hill and UNC Health Care...

,

North CarolinaNorth Carolina is a state located in the southeastern United States. The state borders South Carolina and Georgia to the south, Tennessee to the west and Virginia to the north. North Carolina contains 100 counties. Its capital is Raleigh, and its largest city is Charlotte...

. His insight was that a passing gravitational wave should in principle cause a bead on a stick (with the stick parallel to the wave velocity) to slide back and forth, thus heating the bead and the stick by

frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

. A gravitational wave pulse will stretch spacetime behind the bead, pushing the bead forward; after the wave passes through the bead the stretching will occur in front of the bead, accelerating the bead in the opposite direction. This heating, said Feynman, showed that the wave did indeed impart energy to the bead and stick system, so it must indeed transport energy.

### Einstein's double reversal

The creator of the theory of general relativity,

Albert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

, argued in 1916 that gravitational radiation should be produced, according to his theory, by any mass-energy configuration which has a time-varying quadrupole moment (or higher multipole moment). Using a linearized field equation (appropriate for the study of

*weak* gravitational fields), he derived the famous quadrupole radiation formula quantifying the rate at which such radiation should carry away energy. Examples of systems with time varying quadrupole moments include vibrating strings, bars rotating about an axis perpendicular to the symmetry axis of the bar, and binary star systems, but not rotating disks.

In 1922,

Arthur Stanley EddingtonSir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...

wrote a paper expressing (apparently for the first time) the view that gravitational waves are in essence ripples in coordinates, and have no physical meaning. He did not appreciate Einstein's arguments that the waves are real.

In 1936, together with

Nathan RosenNathan Rosen was an American-Israeli physicist noted for his study on the structure of the hydrogen molecule and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox.-Background:Nathan Rosen was born into a Jewish family in Brooklyn, New York...

, Einstein rediscovered the Beck vacuums, a family of exact gravitational wave solutions with cylindrical symmetry (sometimes also called

*Einstein-Rosen waves*). While investigating the motion of test particles in these solutions, Einstein and Rosen became convinced that gravitational waves were unstable to collapse. Einstein reversed himself and declared that gravitational radiation was

*not* after all a prediction of his theory. Einstein wrote to his friend

Max BornMax Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...

Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist, though they had been assumed a certainty to the first approximation. This shows that the nonlinear field equations can show us more, or rather limit us more, than we have believed up till now.

In other words, Einstein believed that he and Rosen had established that their new argument showed that the prediction of gravitational radiation was a

*mathematical artifact* of the linear approximation he had employed in 1916. Einstein believed these plane waves would gravitationally collapse into points; he had long hoped something like this would explain quantum mechanical wave-particle duality.

Einstein and Rosen accordingly submitted a paper entitled

*Do gravitational waves exist?* to a leading physics journal, the

Physical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...

, in which they described their wave solutions and concluded that the "radiation" which seemed to appear in general relativity was not genuine radiation capable of transporting energy or having (in principle) measurable physical effects. The anonymous referee, who—as the current editor of the Physical Review recently confirmed, all parties now being deceased—was the combative cosmologist,

Howard Percy RobertsonHoward Percy Robertson was an American mathematician and physicist known for contributions related to physical cosmology and the uncertainty principle...

, pointed out the error described below, and the manuscript was returned to the authors with a note from the editor asking them to revise the paper to address these concerns. Quite uncharacteristically, Einstein took this criticism very badly, angrily replying "I see no reason to address the, in any case erroneous, opinion expressed by your referee." He vowed never again to submit a paper to the Physical Review. Instead, Einstein and Rosen resubmitted the paper without change to another and much less well known journal, the Journal of the Franklin Institute. He kept his vow regarding the Physical Review.

Leopold InfeldLeopold Infeld was a Polish physicist who worked mainly in Poland and Canada . He was a Rockefeller fellow at Cambridge University and a member of the Polish Academy of Sciences....

, who arrived at

Princeton UniversityPrinceton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

at this time, later remembered his utter astonishment on hearing of this development, since radiation is such an essential element for any

classical field theoryA classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....

worthy of the name. Infeld expressed his doubts to a leading expert on general relativity: H. P. Robertson, who had just returned from a visit to Caltech. Going over the argument as Infeld remembered it (apparently from a conversation with Einstein), Robertson was able to show Infeld the mistake: locally, the Einstein-Rosen waves are

gravitational plane waveIn general relativity, a gravitational plane wave is a special class of a vacuum pp-wave spacetime, and may be defined in terms of Brinkmann coordinates byds^2=[a+2bxy]du^2+2dudv+dx^2+dy^2...

s (which had been studied earlier by O. R. Baldwin and

George Barker JefferyGeorge Barker Jeffery was a leading mathematical physicist in the early twentieth century. He is probably best known to the scientifically literate public as the translator of papers by Einstein, Lorentz, and other fathers of relativity theory .-Career:Jeffery was born in 1891 and educated at...

, and even earlier by Hans W. Brinkmann). Einstein and Rosen had correctly shown that a cloud of test particles would, in sinusoidal plane waves, form

causticIn differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the optical concept of caustics...

s, but changing to another chart (essentially the

Brinkmann coordinatesBrinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor can be written asds^2 \, = H du^2 + 2 du dv + dx^2 + dy^2...

) shows that the formation of the caustic is

*not a contradiction at all*, but in fact just what one would expect in this situation. Infeld then approached Einstein, who concurred with this analysis (still not knowing it was him who reviewed the Physical Review submission).

Since Rosen had recently departed for the Soviet Union, Einstein acted alone in promptly and thoroughly revising their joint paper. This third version was retitled

*On gravitational waves*, and, following Robertson's suggestion of a transformation to cylindrical coordinates, presented what are now called Einstein-Rosen cylindrical waves (these are locally isometric to plane waves). This is the version which eventually appeared. However, Rosen was unhappy with this revision and eventually published his own version, which retained the erroneous "disproof" of the prediction of gravitational radiation.

In a letter to the editor of the Physical Review, Robertson wryly reported that in the end, Einstein had fully accepted the objections which had initially so upset him.

### The Bern and Chapel Hill conferences

In 1955, an important conference honoring the semi-centennial of

special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

was held in Bern, the Swiss town where Einstein was working in the famous patent office during the

Annus mirabilisAnnus mirabilis is a Latin phrase meaning "wonderful year" or "year of wonders" . It was used originally to refer to the year 1666, but is today also used to refer to different years with events of major importance...

. Rosen attended and gave a talk in which he computed the

*Einstein pseudotensor* and

*Landau-Lifschitz pseudotensor* (two alternative, non-covariant, descriptions of the energy carried by a

*gravitational* field, a notion which is notoriously difficult to pin down in general relativity). These turn out to be zero for the Einstein-Rosen waves, and Rosen argued that this reaffirmed the negative conclusion he had reached with Einstein in 1936.

However, by this time a few physicists, such as Felix A. E. Pirani and Ivor Robinson, had come to appreciate the role played by curvature in producing tidal accelerations, and were able to convince many peers that gravitational radiation would indeed be produced, at least in cases such as a vibrating spring where different pieces of the system were clearly not in

inertiaInertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

l motion. Nonetheless, some physicists continued to doubt whether radiation would be produced by a binary star system, where the

world lineIn physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...

s of the

centers of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

of the two stars should, according to EIH approximation (dating from 1938 and due to Einstein, Infeld, and

Banesh HoffmannBanesh Hoffmann was a British mathematician and physicist known for his association with Albert Einstein.-Life:Banesh Hoffmann was born in Richmond, England, on 6 September 1906...

), follow timelike geodesics.

Inspired by conversations by Felix Pirani, Hermann Bondi took up the study of gravitational radiation, in particular the question of quantifying the energy and momentum carried off 'to infinity' by a radiating system. During the next few years, Bondi developed the Bondi radiating chart and the notion of Bondi energy to rigorously study this question in maximal generality.

In 1957, at a conference at

Chapel HillChapel Hill is a town in Orange County, North Carolina, United States and the home of the University of North Carolina at Chapel Hill and UNC Health Care...

,

North CarolinaNorth Carolina is a state located in the southeastern United States. The state borders South Carolina and Georgia to the south, Tennessee to the west and Virginia to the north. North Carolina contains 100 counties. Its capital is Raleigh, and its largest city is Charlotte...

, appealing to various mathematical tools developed by

John Lighton SyngeJohn Lighton Synge was an Irish mathematician and physicist.-Background:Synge was born 1897 in Dublin, Ireland, in a Protestant family and educated at St. Andrew's College, Dublin. He entered Trinity College, Dublin in 1915...

, A. Z. Petrov and

André LichnerowiczAndré Lichnerowicz was a noted French differential geometer and mathematical physicist of Polish descent.-Biography:...

, Pirani explained more clearly than had previously been possible the central role played by the Riemann tensor and in particular the

tidal tensorIn general relativity, the tidal tensor or gravitoelectric tensor is one of the pieces in the Bel decomposition of the Riemann tensor. It is physically interpreted as giving the tidal stresses on small bits of a material object , or the tidal accelerations of a small cloud of test particles in a...

in general relativity. He gave the first correct description of the relative (tidal) acceleration of initially mutually static test particles which encounter a sinusoidal gravitational plane wave.

### Feynman's argument

Later in the Chapel Hill conference,

Richard FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

— who had insisted on registering under a pseudonym to express his disdain for the contemporary state of gravitational physics — used Pirani's description to point out that a passing gravitational wave should in principle cause a bead on a stick (oriented parallel to the direction of propagation of the wave) to slide back and forth, thus heating the bead and the stick by

frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

. This heating, said Feynman, showed that the wave did indeed impart energy to the bead and stick system, so it must indeed transport energy, contrary to the view expressed in 1955 by Rosen.

In two 1957 papers, Bondi and (separately)

Joseph WeberJoseph Weber was an American physicist. He gave the earliest public lecture on the principles behind the laser and the maser and developed the first gravitational wave detectors .-Early education:...

and

John Archibald WheelerJohn Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...

used this bead argument to present detailed refutations of Rosen's argument.

### Rosen's final views

Nathan Rosen continued to argue as late as the 1970s, on the basis of a supposed paradox involving the radiation reaction, that gravitational radiation is not in fact predicted by general relativity. His arguments were generally regarded as invalid, but in any case the sticky bead argument had by then long since convinced other physicists of the reality of the prediction of gravitational radiation.

## See also

- monochromatic electromagnetic plane wave
In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell's theory...

and monochromatic gravitational plane wave, for a modern account of two exact solutions which should clarify the point which confused Einstein and Rosen in 1936,
- pp-wave spacetime
In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. These solutions model radiation moving at the speed of light...

, for the Brinkmann gravitational wave solutions,
- Gravitational plane wave
In general relativity, a gravitational plane wave is a special class of a vacuum pp-wave spacetime, and may be defined in terms of Brinkmann coordinates byds^2=[a+2bxy]du^2+2dudv+dx^2+dy^2...

, for the Baldwin-Jeffery gravitational plane wave solutions,
- Brinkmann coordinates
Brinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor can be written asds^2 \, = H du^2 + 2 du dv + dx^2 + dy^2...

and Rosen coordinates for the two coordinate charts,
- Beck vacuums, for the Beck or Einstein-Rosen family of vacuum solutions,
- Cooperstock's Energy Localization Hypothesis
In physics, the Cooperstock's energy-localization hypothesis is a hypothesis proposed by Fred Cooperstock that in general relativity, energy only exists in regions of non-vanishing energy–momentum tensor....

, for a conflicting hypothesis (potentially, therefore, a test).