Sticky bead argument: Encyclopedia II - Sticky bead argument - Einstein's double reversal

Sticky bead argument - Einstein's double reversal

The creator of general relativity, Albert Einstein, 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 orthogonal 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 spacetime curvature. Nonetheless, during the next several decades, few physicists appreciated either the role played by the Riemann tensor in general relativity, or the pitfalls of working with local coordinate charts.

Case in point: in 1936, together with Nathan Rosen, 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 so confused by certain features of the coordinate chart they were using that Einstein (incorrectly) 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. Specifically, they believed they had shown that sinusoidal type gravitational waves could not exist, because assuming their existence led to an apparent contradiction.

Einstein and Rosen accordingly submitted a paper entitled Do gravitational waves exist? to a leading physics journal, the Physical Review, 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) measureable 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 Robertson, pointed out the error, 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, and vowing never again to submit a paper to the Physical Review (he never did). Instead, Einstein and Rosen resubmitted the paper without change to another and much less well known journal, the Journal of the Franklin Institute.

Leon Infeld, who arrived at Princeton University 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 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 waves (which had been studied earlier by O. R. Baldwin and George Barker Jeffery, 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 caustics, but changing to another chart (essentially the Brinkmann coordinates) 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.

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.

Adapted from the Wikipedia article "Einstein's double reversal", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki