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general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the sticky bead argument is a simple thought experiment designed to show that gravitational radiation is indeed predicted by
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and 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 argument, any remaining doubts soon disappeared from the research literature. The argument is often credited to Hermann Bondi, who popularized it, but it was originally proposed anonymously by Richard Feynman.Preskill, John and Kip S. Thorne. Foreword to ''Feynman Lectures On Gravitation''. Feynman et al. (Westview Press; 1st ed. (June 20, 2002) p. xxv–xxv
Link PDF (page 17-18)
/ref>DeWitt, Cecile M. (1957). Conference on the Role of Gravitation in Physics at the University of North Carolina, Chapel Hill, March 1957; WADC Technical Report 57-216 (Wright Air Development Center, Air Research and Development Command, United States Air Force, Wright Patterson Air Force Base, Ohio


Description

The thought experiment was first described by Feynman (under the pseudonym "Mr. Smith") in 1957 at a conference at
Chapel Hill Chapel Hill or Chapelhill may refer to: Places Antarctica * Chapel Hill (Antarctica) Australia *Chapel Hill, Queensland, a suburb of Brisbane *Chapel Hill, South Australia, in the Mount Barker council area Canada *Chapel Hill, Ottawa, a neighbou ...
,
North Carolina North Carolina () is a U.S. state, state in the Southeastern United States, Southeastern region of the United States. The state is the List of U.S. states and territories by area, 28th largest and List of states and territories of the United ...
, and later addressed in his private letter: As the gravitational waves are mainly transverse, the rod has to be oriented perpendicular to the propagation direction of the wave.


History of arguments on the properties of gravitational waves


Einstein's double reversal

The creator of the theory of general relativity,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, argued in 1916 that gravitational radiation should be produced, according to his theory, by any mass-energy configuration that has a time-varying
quadrupole moment A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure ref ...
(or higher multipole moment). Using a linearized field equation (appropriate for the study of ''weak'' gravitational fields), he derived the famous
quadrupole formula In general relativity, the quadrupole formula describes the rate at which gravitational waves are emitted from a system of masses based on the change of the (mass) quadrupole moment. The formula reads : \bar_(t,r) = \frac \ddot_(t-r/c), where \bar ...
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 Eddington 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 Rosen Nathan Rosen (Hebrew: נתן רוזן; March 22, 1909 – December 18, 1995) was an American-Israeli physicist noted for his study on the structure of the hydrogen atom and his work with Albert Einstein and Boris Podolsky on entangled wave functio ...
, Einstein rediscovered the
Beck vacuums In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non ...
, 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 Born
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, ''
Physical Review ''Physical Review'' is a peer-reviewed scientific journal established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the American Physical ...
'', in which they described their wave solutions and concluded that the "radiation" that 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 ''Physical Review'' recently confirmed, all parties now being deceased—was the combative cosmologist,
Howard Percy Robertson Howard Percy "Bob" Robertson (January 27, 1903 – August 26, 1961) was an American mathematician and physicist known for contributions related to physical cosmology and the uncertainty principle. He was Professor of Mathematical Physics at the C ...
, 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 ''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 ''Physical Review''.
Leopold Infeld Leopold Infeld (20 August 1898 – 15 January 1968) was a Polish physicist who worked mainly in Poland and Canada (1938–1950). He was a Rockefeller fellow at Cambridge University (1933–1934) and a member of the Polish Academy of Sciences. E ...
, who arrived at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
at this time, later remembered his utter astonishment on hearing of this development, since radiation is such an essential element for any
classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
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 The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
. Going over the argument as Infeld remembered it, Robertson was able to show Infeld the mistake: locally, the Einstein–Rosen waves are
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 by ds^2= (u)(x^2-y^2)+2b(u)xyu^2+2dudv+dx^2+dy^2 Here, a(u), b(u) can be any smooth functions ...
s. Einstein and Rosen had correctly shown that a cloud of test particles would, in sinusoidal plane waves, form
caustic Caustic most commonly refers to: * Causticity, a property of various corrosive substances ** Sodium hydroxide, sometimes called ''caustic soda'' ** Potassium hydroxide, sometimes called ''caustic potash'' ** Calcium oxide, sometimes called ''caust ...
s, but changing to another chart (essentially the
Brinkmann coordinates Brinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. They are named for Hans Brinkmann. In terms of these coordinates, the metric tensor can be written as :ds^2 = H(u,x,y) du^2 + 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 Robertson's analysis (still not knowing it was he 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 that 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 ''Physical Review'', Robertson wryly reported that in the end, Einstein had fully accepted the objections that had initially so upset him.


Bern and Chapel Hill conferences

In 1955, an important conference honoring the semi-centennial of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
was held in
Bern german: Berner(in)french: Bernois(e) it, bernese , neighboring_municipalities = Bremgarten bei Bern, Frauenkappelen, Ittigen, Kirchlindach, Köniz, Mühleberg, Muri bei Bern, Neuenegg, Ostermundigen, Wohlen bei Bern, Zollikofen , website ...
, the Swiss capital city where Einstein was working in the famous patent office during the Annus mirabilis. Rosen attended and gave a talk in which he computed the '' Einstein pseudotensor'' and '' Landau–Lifshitz pseudotensor'' (two alternative, non-covariant, descriptions of the energy carried by a ''gravitational'' field, a notion that 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 Pirani __NOTOC__ Felix Arnold Edward Pirani (2 February 1928 – 31 December 2015) was a British theoretical physicist, and professor at King's College London, specialising in gravitational physics and general relativity. Pirani and Hermann Bondi wr ...
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 inertial motion. Nonetheless, some physicists continued to doubt whether radiation would be produced by a binary star system, where the
world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
s of the
centers of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the two stars should, according to the EIH approximation (dating from 1938 and due to Einstein, Infeld, and
Banesh Hoffmann Banesh Hoffmann (6 September 1906 – 5 August 1986) was a British mathematician and physicist known for his association with Albert Einstein. Life Banesh Hoffmann was born in Richmond, Surrey, on 6 September 1906. He studied mathematics and ...
), follow timelike geodesics. Inspired by conversations by
Felix Pirani __NOTOC__ Felix Arnold Edward Pirani (2 February 1928 – 31 December 2015) was a British theoretical physicist, and professor at King's College London, specialising in gravitational physics and general relativity. Pirani and Hermann Bondi wr ...
, 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 Hill Chapel Hill or Chapelhill may refer to: Places Antarctica * Chapel Hill (Antarctica) Australia *Chapel Hill, Queensland, a suburb of Brisbane *Chapel Hill, South Australia, in the Mount Barker council area Canada *Chapel Hill, Ottawa, a neighbou ...
,
North Carolina North Carolina () is a U.S. state, state in the Southeastern United States, Southeastern region of the United States. The state is the List of U.S. states and territories by area, 28th largest and List of states and territories of the United ...
, appealing to various mathematical tools developed by
John Lighton Synge John Lighton Synge (; 23 March 1897 – 30 March 1995) was an Irish mathematician and physicist, whose seven-decade career included significant periods in Ireland, Canada, and the USA. He was a prolific author and influential mentor, and is cre ...
, A. Z. Petrov and André Lichnerowicz, Pirani explained more clearly than had previously been possible the central role played by the
Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
and in particular the
tidal tensor Tidal is the adjectival form of tide. Tidal may also refer to: * ''Tidal'' (album), a 1996 album by Fiona Apple * Tidal (king), a king involved in the Battle of the Vale of Siddim * TidalCycles, a live coding environment for music * Tidal (se ...
in general relativity. He gave the first correct description of the relative (tidal) acceleration of initially mutually static test particles that encounter a sinusoidal gravitational plane wave.


Feynman's argument

Later in the Chapel Hill conference, Richard Feynman used Pirani's description to point out that a passing gravitational wave should in principle cause a bead on a stick (oriented transversely to the direction of propagation of the wave) to slide back and forth, thus heating the bead and the stick by
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
. 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 Weber and John Archibald Wheeler 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

* Dashpot, of which the sticky-bead device is a variant. * Monochromatic electromagnetic plane wave and monochromatic gravitational plane wave, for a modern account of two exact solutions, which should clarify the point that 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. The term ''pp'' stands for ''plane-fronted waves with parallel propagation'', and was introduced in 19 ...
, 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 by ds^2= (u)(x^2-y^2)+2b(u)xyu^2+2dudv+dx^2+dy^2 Here, a(u), b(u) can be any smooth functions ...
, 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. They are named for Hans Brinkmann. In terms of these coordinates, the metric tensor can be written as :ds^2 = H(u,x,y) du^2 + 2 ...
and Rosen coordinates for the two coordinate charts. *
Beck vacuums In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non ...
, for the Beck or Einstein–Rosen family of vacuum solutions.


Notes


References

* See also th
on-line version
*Kennefick, Daniel
Controversies in the History of the Radiation Reaction problem in General Relativity
* * * * * * * {{DEFAULTSORT:Sticky Bead Argument Thought experiments in physics Gravitational waves History of physics Richard Feynman