What does the statement “waves of the coordinate system” mean? It is a tenet of general relativity that you can label points in space and time in essentially any way you like and measurements made of the physical phenomena using devices such as clocks or telescopes or laser beams will be the same. Let us describe a simple example, not in spacetime but on a two-dimensional surface in a suitable city like Manhattan in New York City, where a very nice Starbucks store is located. There are a number of ways to tell your friend where the Starbucks is. The city itself has a grid of numbered avenues, say First, Second, Third Avenues pointing in one direction, and of numbered streets, say 65th, 66th, 67th Streets, pointing in the perpendicular direction. Many US cities are laid out in this regular way. This grid provides a “coordinate system” for the city, and let’s say the nice Starbucks is located at the corner of Third Avenue and East 66th Street. However, you could use a very different coordinate system to direct your friend. You could give her the GPS coordinates of the store, and she could get there using her smartphone. The GPS coordinates are based on a grid of lines of longitude and latitude, which are not in general aligned with the avenue–street grid. Starbucks is there with your latte waiting no matter what coordinate system you use.
Figure 7.2 Coordinate systems for finding Starbucks in upper east side Manhattan. Left: System based on the grid of streets. Middle: System based on GPS coordinates. Right: System based on grid of numbered ropes being jiggled crazily by giants.
Now consider a third coordinate system, defined by a grid of parallel ropes held taut by giants standing at the edge of the neighborhood where Starbucks is located. There are two sets of parallel ropes, one set perpendicular to the other, with the ropes in each direction labeled E1, E2, E3, etc., and N15, N16, N17, etc. In this system, Starbucks might be located at E3–N17, i.e. at the intersection of rope #E3 in one direction and rope #N17 in the other direction. Your friend could just as easily get to Starbucks using this system. Now suppose that the giants start jiggling the ropes so that they all oscillate in some complicated manner in the horizontal plane. In this coordinate system, the Starbucks store seems to be going crazy, now at E3–N17, now at E2.9–N16.8, then back to E3–N17, then E3.1–N17.2, and so on. Using this coordinate system to get your friend to Starbucks would be complicated (in fact the values of the coordinates would depend on her time of arrival), but it could be done. Yet the customers at Starbucks feel nothing, since after all, the store is not actually moving. They have no idea that the location of the store is sloshing back and forth when described using the giants’ wavy coordinates.
This is the bottom line: coordinates are merely convenient labels of points and have no physical significance. In general relativity there are four coordinates, three for space and one for time, but the basic idea is the same. We can label points in spacetime using any convenient coordinates, but they will have no consequences for the physics that goes on, including the experiments we do using physical apparatus to measure the physical phenomena. Today, students in our general relativity courses learn this up front and eventually become comfortable with it.
But Einstein was not totally comfortable with it, even though he used the concept as the central guidepost for developing his theory. After all, he called it the “general” theory (as opposed to his 1905 “special” theory of relativity) because the theory was valid in “general” coordinate systems. Even Eddington, who understood general relativity at least as well as Einstein did, and probably better than any one else at that time, seemed slightly unsure what to make of these wavy coordinates. Today we think of their ruminations on the meaning of these modes as rather quaint, but we have the benefit of hindsight based on a century of research and teaching in the subject.
Still, Eddington’s comment about some gravitational waves traveling at the “speed of thought” served to make the whole topic seem rather dubious, and it would be almost thirty-five years before the issue was fully settled.
The case for gravitational waves was not helped when, in 1936, Einstein tried to claim that they do not exist.
In 1933 Einstein moved to the United States to escape Nazi Germany, taking a job at the Institute for Advanced Studies, near Princeton University in New Jersey. This institute had been founded three years earlier to serve as a center for knowledge and discovery, and also became a refuge for intellectuals escaping Nazi oppression. In 1934 Einstein hired Nathan Rosen as his assistant, a physicist from Brooklyn, New York who had studied physics at MIT. An “assistant” in those days played the role of what today we would call a “postdoc,” someone typically with a physics Ph.D. who acts as a research assistant to a senior scientist in a hands-on apprenticeship for a few years.
Einstein and Rosen set out to revisit Einstein’s 1918 calculation to determine whether gravitational waves were truly real. Einstein’s 1918 paper had used an approximate version of his theory to predict the existence of such waves, but they now wanted to determine whether the exact theory led to the same prediction. To their surprise, the exact theory seemed to predict the opposite! The solution they found was singular, or infinite, in certain parts of spacetime, just as Schwarzschild’s “Massenpunkt” solution was singular at the Schwarzschild radius. Reasoning that singularities are unphysical, they concluded the entire solution also had to be unphysical. In 1936, Einstein and Rosen submitted their disproof of gravitational waves for publication in the Physical Review scientific journal.
Following a policy that had only recently been instituted, the editor of Physical Review, John Tate, sent the Einstein–Rosen paper to an external scientist to be reviewed. Given that one of the authors was Einstein, Tate had some misgivings about this, but saw no reason to make an exception, especially considering the surprising claim made in the paper. The anonymous referee recommended that the paper not be accepted for publication without major corrections.
Today, peer review is the norm for publication in serious scientific journals, and it is an essential tool to ensure the validity of new results. Typically, a journal’s editor will send a new submission to one or more experts for comments and critiques, and it is only once these experts reach a consensus for publication that the submission is accepted. But in 1936 this practice was not common at all and was almost unheard of in Europe, where Einstein was used to publishing papers. So when Einstein received Tate’s reply with the anonymous referee’s criticism, he was offended and withdrew his submission. He wrote to Tate that he had “sent the manuscript for publication,” not to be disclosed to an anonymous expert. Einstein never submitted another paper to Physical Review.
If Einstein had actually read the report of the referee, our story would be quite different, since the referee had spotted a serious mistake in their paper. Instead, Einstein submitted the original paper, without any modifications, to the Journal of the Franklin Institute, a small journal published in Philadelphia, which accepted it without refereeing in 1937. By this point, Rosen had moved to the University of Kiev in present-day Ukraine, so Einstein had hired a new assistant, Leopold Infeld, a Polish physicist. Infeld arrived at the Institute for Advanced Studies around the time that the Einstein–Rosen paper was being accepted by the Franklin Institute journal, and Einstein was excited to talk with him about his new paper and the discovery that gravitational waves did not exist after all.
Infeld was initially skeptical. It was hard to believe that Einstein’s theory, which resembled Maxwell’s theory of electricity and magnetism in so many respects, did not have gravitational waves similar to Maxwell’s waves. But Einstein was an eminence in physics, and he soon convinced Infeld that his argument was correct. Around this time, however, Howard Percy Robertson, a Princeton professor who had done work that would lay the foundations of general relativistic cosmology, was returning from a sabbatical at Caltech. When Infeld met Robertson, he told him about the Einstein–Rosen result, but Robertson dismissed it and, a few days later, showed Infeld exactly what the problem was. Once again it was the coordinates!
Robertson explained to Infeld that if one were to transform the solution that Einstein and Rosen had found to coordinates adapted to a cylinder, then the infinities that so troubled them would all be pushed to the axis of the cylinder, where the source of the gravitational waves should reside, and where their solution would not be applicable. Infeld was impressed that Robertson appeared to have solved the problem just based on their brief discussion.
The singularities found by Einstein and Rosen are called “coordinate singularities” and are today understood to be artifacts of the choice of coordinates and therefore have no effect on the physics. Even as mundane a place as the South Pole on the surface of the Earth is the location of a coordinate singularity. There, the latitude is − 90 degrees, but there are an infinite number of possibilities for the longitude, since all the lines of longitude converge there. But standing at the scientific research station at the South Pole you would never know this, and in fact you could lay down a nice grid of streets on the ice and find Starbucks (if there were one there) just as easily as you could in Manhattan. As we discussed before, physics cannot depend on the coordinate choices one makes. Infeld rushed to tell Einstein about this, only to be told by Einstein that he had just reached the same conclusion on his own.
With input and advice from Infeld and Robertson, Einstein heavily revised the paper, already in galley proofs from the journal. He changed the title, added a new section on cylindrical gravitational waves, and altered the main conclusion: his exact theory of general relativity did predict the existence of (at least cylindrical) gravitational waves. In 2005, through some inventive detective work, our colleague Daniel Kennefick was given permission by the current editors of Physical Review to examine Tate’s log book of submitted papers and confirmed that the anonymous referee of the Einstein–Rosen paper had been none other than Robertson himself!
Once again, sorting out what was real about gravitational waves proved to be a problem, and would not really be resolved for another twenty years. Recall that this period was the “low water mark” for the field of general relativity, when few scientists were interested in it or worked on it. But in the mid 1950s the beginnings of a rebirth in the subject began to take hold, leading to the great renaissance for general relativity during the 1960s described in Chapter 1. Science historians who study our field point to two conferences on general relativity as being particularly influential. The first was in Bern, Switzerland in 1955. The meeting was convened to celebrate the fiftieth anniversary of Einstein’s “miracle year” when, while working as a patent clerk in that city, he developed special relativity and made groundbreaking discoveries in quantum mechanics and atomic physics. It also turned out to memorialize Einstein, who had died three months before the meeting. The second conference was in Chapel Hill, North Carolina in 1957.
In fact, these two meetings became so legendary that when the International Committee on General Relativity and Gravitation was formed a few years later and decided to organize a big “GR” conference every three years, they retroactively renamed these two meetings “GR0” and “GR1.” In 2019, GR22 was held in Valencia, Spain.
Although gravitational radiation was not a major topic in either meeting, it was discussed. In Bern, Nathan Rosen reviewed cylindrical gravitational waves. In Chapel Hill, John Wheeler and Joseph Weber argued that, despite the rather unrealistic physical setup, cylindrical waves were physically measurable. And a 27-year-old British graduate student named Felix Pirani discussed his recent paper showing precisely how the waves would affect material particles and how you would measure the effects.
Imagine a ring of eight disks (hockey pucks, for example) arrayed in a circle on a table, free to slide across the table without any kind of friction (see the snapshot on the far left of the top panel of Figure 7.3). A gravitational wave passes vertically through the table (into the page of the figure). The strength or amplitude of the wave is a sine wave, starting at zero, growing to a maximum, returning to zero, then to a minimum and then returning to zero. According to Einstein’s theory, the disks on opposite sides of the center of the circle are pushed apart in one direction while being pushed together in the perpendicular directions, as in the second snapshot in the top panel of Figure 7.3. As the wave amplitude passes through zero, the disks then return to the circular shape (third snapshot). As the amplitude goes negative, the disks are pushed together and apart in the opposite sense (fourth snapshot). Finally, after one complete cycle of the wave, the disks return to the circular shape (fifth snapshot). As we have already discussed, a general gravitational wave can contain two modes of polarization. The action of the second mode is shown in the bottom panel of Figure 7.3. It is the same as the first mode, except that the pushing together and apart is along the diagonals, at 45 degrees compared to the first mode. The two modes are conventionally called the “plus (+)” and “cross (×)” modes, because if you overlay the five snapshots in each case, the pattern reminds you of a plus sign or of a multiplication or cross-out sign.
Figure 7.3 Two modes of polarization of a gravitational wave propagating perpendicularly into the page. Snapshots show a set of eight disks on a frictionless surface. From left to right, the snapshots show the disks every quarter of a cycle. The top row displays the plus polarization; the bottom row shows the cross polarization.
Now, we have mentioned many times already that gravity is really due to the warping or distortion of spacetime, and you may have read many newspaper articles or internet stories describing gravitational waves as stretching and squeezing spacetime in order to produce the patterns shown in Figure 7.3. So it is fair to ask, doesn’t the wave also stretch and squeeze the table so that the disks don’t actually slide? The answer is that the response of an object to warping spacetime depends on what other forces are acting on it. Our disks have no forces acting on them in the horizontal direction (assuming zero friction) and so they respond “fully” to the spacetime distortions induced by the wave. On the other hand, the atoms that make up the table are being acted upon by the interatomic electric forces produced by all the surrounding atoms, and these forces are enormous compared to the force of gravity. So the distortion of the table is much, much, much smaller than the displacement of the disks. Thus, the disks will actually slide.
In fact, this was the basis of an ingenious argument made at the Chapel Hill conference by a “Mr. Smith” to underscore the reality of gravitational waves. Mr. Smith was actually the famous American physicist Richard Feynman, who had registered for the meeting under the pseudonym in part to indicate his low opinion about the status of general relativity research at the time. Drawing on Pirani’s description of how objects would move, he pointed out that if you introduced a bit of friction onto the surface of the table, then as the disks slide back and forth, the table would heat up a bit. Therefore some energy would have been transferred from the gravitational waves to the tabletop, a clear and unambiguous sign of a physical effect. He actually used a pair of beads sliding on a rod in his discussion, and so his argument has gone down in physics history as Feynman’s “sticky beads” proof of the reality of gravitational waves.
Pirani received his Ph.D. in 1958 under Hermann Bondi at King’s College, London, and over the next few years he, Bondi and others would show in clear and unambiguous terms that gravitational waves are real, can be measured and carry energy away from their sources. The long period of uncertainty inspired by Eddington’s jibe was over.
So we have answered one question: gravitational waves are physically real and measurable.
But an equally important question is: are there any waves to measure? Any decent physicist would say, let’s generate the waves and then detect them. In 1887, Hertz had created electromagnetic waves using an electrical discharge, and then detected the effects of the waves at the other end of his lab. Could one take one of the dumbbells that Einstein imagined in his 1916 calculations and generate gravitational waves to be detected somewhere else? Unfortunately, it was not too hard to show that the
waves produced by such a setup would be hopelessly weak, utterly undetectable by even the most far-fetched scheme. However, gravity depends on how much mass you have, and so perhaps one needs astronomical bodies to generate gravitational waves. But in the late 1950s, this did not seem very promising either. According to the astronomers, the universe was a very quiet place, where almost nothing happened. Planets revolved sedately around the Sun, stars hardly ever changed, and galaxies just sat there, moving slowly away from each other as the universe gradually expanded. To be fair, there were supernovae, stellar explosions seen, for example, in 1054, 1572 and 1604, but these were rare events, and it was not known how much if any gravitational radiation they emitted.
A third question is: if there were gravitational waves passing through the Earth, could one build a practical detector sensitive enough to sense them? The person who took on this challenge was Joseph P. Weber. To his many detractors Weber was a tragic figure, whose work was flawed and whose claims were roundly refuted. To others, he was the father of gravitational wave detection whose insights established many of the principles that enabled the successful detections using laser interferometers. His story illustrates the sometimes fitful and complex ways in which science advances, and is also a case study in how science works.
Weber was the son of Lithuanian Jews who settled in New Jersey and New York in the early 1900s. Born in 1919, he graduated from the US Naval Academy in 1940, served in World War II, and after the war led the Navy’s electronic countermeasures section, retiring from the Navy as lieutenant commander.
In 1948, the University of Maryland appointed him as an engineering professor under the stipulation that he quickly earn a Ph.D. degree. Weber asked George Gamow, a physics professor at George Washington University in Washington DC, famous for his explanation of radioactivity through quantum physics, if he would be his Ph.D. advisor. Ironically, this was the same year that Gamow and his student Ralph Alpher had theoretically predicted the existence of the first light after the big bang, today called the cosmic microwave background (CMB) radiation. But instead of suggesting to Weber that he should work on the experimental detection of this first light, Gamow turned Weber away. In 1965 this radiation was detected, almost by accident, by Arno Penzias and Robert Wilson, two Bell Telephone Laboratory scientists.
Is Einstein Still Right? Page 22
