Is Einstein Still Right?

by Clifford M. Will

  Now consider the same laboratory in distant space being accelerated by a rocket attached to it (the bottom panel of Figure 3.3). The observer can now stand on the floor of the lab because of the rocket’s thrust. Let us assume that, when the light ray enters from the left, the speed of the laboratory is the same as it was in the top panel. The sequence of events now is different: (d) As the ray enters, it seems to be traveling once again horizontally as it initially enters the laboratory, because of aberration. (e) By the time the ray is half way across, the lab has traveled a bit farther than before because its speed is now a bit higher, so the ray is a bit below the lab’s midpoint. (f) By the time the ray exits the lab, the lab has moved even more than before, and so the exit point is well below the midpoint. According to external observers, the light has traveled on a perfectly straight line (the dashed line), but according to the observer in the accelerating lab, the light ray (the dotted line) appears to have bent slightly toward the floor as it crossed the lab.

  But, according to Einstein’s principle of equivalence, the accelerating lab in the bottom panels of Figure 3.3 is equivalent to a lab at rest in a gravitational field. Therefore light should be bent by gravity! By considering a sequence of such laboratories all along the trajectory of a light ray passing by the Sun, and adding up all the tiny deflections, Einstein could conclude that the net deflection of a ray that just grazes the Sun would be 0.875 arcseconds. Therefore, whether we use the Newtonian theory of gravity combined with the corpuscular theory of light, as Cavendish and von Soldner did, or the principle of equivalence, as in this derivation, we predict the same deflection of light.

  Yet, in November 1915 Einstein doubled the prediction. By that time, he had completed the full general theory of relativity, and found that, in a first approximation to the equations of the theory, the deflection had to be 1.75 arcseconds, not 0.875 arcseconds.

  Was this doubling completely arbitrary? Were the previous calculations wrong? Not at all. They are correct as far as they go. They simply did not, indeed could not, take into account an important circumstance that only the complete general theory of relativity could cope with: the curvature of space. As we saw in Chapter 2, the principle of equivalence tells us that time must be warped, but it says nothing about space. It is natural to assume that space might also be warped, since relativity involves uniting space and time into a unified spacetime. But it turns out that to determine by how much space is curved, we need the complete equations of general relativity, not just the principle of equivalence.

  Indeed, general relativity predicts that space is curved near gravitating bodies, the curvature being greater the closer one gets to the body, and negligibly small at large distances. This is all embodied in the complicated mathematics of a four-dimensional spacetime continuum, and is very difficult to describe in words. However, if we strip away some unnecessary details, it is possible to present a qualitative picture that might give you a sense of some of the effects of curved space.

  Let’s imagine what curved space might look like around an object such as the Sun. Since we imagine the Sun is unchanging in time, we can strip away the time dimension and focus just on space. Also, since the Sun is spherical to a good approximation, like a soccer ball or basketball, any radial direction in space from its center is as good as any other, so we can pick one and then focus on the two-dimensional plane perpendicular to that direction. If we now ask what does that curved two-dimensional plane look like, it turns out that a good analogy is to imagine a rubber sheet stretched taut across a room on Earth, with a heavy bowling ball, representing the Sun, in the center (Figure 3.4). Because of the weight of the ball, it sinks and stretches the sheet. At the edges of the sheet, far from the ball, the sheet is approximately flat, with a geometry that obeys the usual rules of Euclid, but close to the ball it is warped.

  Figure 3.4 Rubber sheet analogy for curved space. A massive ball sits in the center of a rubber sheet stretched taut across a room. A small ball rolls around the banked surface, metaphorically representing a body in orbit around the large ball. Two fast bodies, metaphorically representing light rays, cross the sheet. The ray passing close to the ball is deflected because of the banking or warping of the surface.

  As a result, the distance measured along the rubber sheet from the center of the indentation to the edge of the sheet is longer than the distance measured “as the crow flies,” because of the stretching and warping of the sheet. The circumference of a circle drawn on the sheet around the ball would be smaller than 2π times the distance from the center measured along the sheet. In Euclidean geometry, the circumference of a circle is exactly 2π times the radius (with π a constant that is approximately 3.14159265 …). A word of caution, however: this rubber sheet picture is only an analogy, and an imperfect one at that. Spacetime is not a rubber sheet; it is four-dimensional, not two, and it includes time as well as space.

  Einstein’s principle of equivalence says that, when viewed from a local freely falling frame, bodies move on straight lines, as if there were no gravity. But this is only a local statement, confined to a small region near the observer. Einstein argued that in a more general situation, in the presence of gravity, a body would move on the “straightest” line possible within the curved spacetime in which it found itself.

  Such “straightest” lines are familiar to international air travelers. On the surface of the Earth, which is a true curved two-dimensional surface, such “straightest” lines are called geodesics. The equator is a geodesic, as are all the lines of longitude. The curve produced by the intersection of the Earth’s surface with any plane passing through the Earth’s center is also a geodesic. It is the path you would take if you traveled along the Earth’s surface veering neither left nor right, but keeping the same forward heading. Your path is not straight in the Euclidean sense, because you are going around the Earth, ultimately returning to your starting point, but it is as straight as it can be. On Earth, such geodesic paths also happen to provide the shortest distance between two points, which is why your air trip from Los Angeles to Paris goes way north over Hudson’s Bay and Greenland, and not over the mid-Atlantic. With the equivalence principle, Einstein extended this idea, asserting that freely moving particles travel on geodesics of a curved spacetime.

  Let’s now try to imagine how this idea about geodesics affects the path particles follow. In Figure 3.4, the small ball is orbiting the large ball by merely following the steeply banked rubber surface in as straight a manner as possible. This is just as in a roller coaster that turns to the right because the tracks are steeply banked in that direction, and not because it actually veers off the tracks (hopefully!). In a similar manner, a very fast particle, such as light, passes across the sheet near the edge with very little deflection (the nearly straight black line in Figure 3.4). But if such a particle passes close to the ball, it must dip down into the sheet, and because of the steep banking it will be deflected more.

  The orbits and paths depicted in Figure 3.4 should actually be drawn using three dimensions, with time plotted somehow in addition, but that would severely challenge our artistic capabilities. Nevertheless, we hope that the rubber sheet metaphor, however imperfect, helps readers to visualize some of the consequences of curved spacetime.

  The curvature of space explains Einstein’s doubling. The previous calculations, such as the one using the accelerating laboratory or the one using Newtonian gravity, gave the deflection of light relative to space. If we thought that space was flat, that would be it. However, general relativity predicts that space is warped or “banked” near the Sun relative to space far from the Sun, and this adds an additional 0.875 arcseconds for a ray that grazes the Sun. Thus, the total deflection must be the sum of these two effects, or 1.75 arcseconds.

  This space curvature effect is the important difference in the predictions of different theories of gravity. Any theory of gravity that is compatible with the equivalence principle (and almost all current theories are) predicts the first 0.875 a
rcseconds part. The second part comes from the curvature of space. Newtonian theory is a flat-space theory, so there is no further effect; the prediction remains at 0.875 arcseconds. General relativity, purely by coincidence, predicts an amount of space curvature that just doubles the deflection. Some theories predict slightly less curvature than general relativity, resulting in a slightly smaller value for the second part and a slightly smaller total deflection. Other theories predict more curvature, and thus a larger deflection angle.

  Einstein’s doubling of the predicted deflection had important consequences, for it meant that the effect was now a bit easier to observe. But the fact that a successful observation came as early as 1919, only four years after the publication of the general theory, must be credited to the pivotal role played by Arthur Stanley Eddington. We already encountered Eddington in Chapters 1 and 2. By the time of the outbreak of World War I he was one of the foremost observational astronomers of the day, and had recently been elected a Fellow of the Royal Society and appointed the Plumian Professor at Cambridge University. The war had effectively cut off direct communication between British and German scientists, but the Dutch scientist Willem de Sitter managed to forward to Eddington Einstein’s latest paper together with several of his own on the general theory of relativity. Eddington recognized the deep implications of this new theory, and he immediately set out to learn the mathematics required to master it. In 1917 he prepared a detailed report on the general theory for the Physical Society of London. This helped spread the word.

  Eddington and Astronomer Royal Frank Dyson also began to contemplate an eclipse expedition to measure the predicted deflection of light. As an astronomer at the Royal Greenwich Observatory from 1906 to 1913, Eddington had made an eclipse expedition in 1912 to study features of the Sun, such as the solar corona, and was familiar with the techniques and problems involved. Dyson had pointed out that the eclipse of 29 May 1919 would be an excellent opportunity because of the large number of bright stars expected to form the field around the Sun. A grant of 1,000 pounds sterling (around 70,000 US dollars today) was obtained from the government, and planning began in earnest. The outcome of the war was still in doubt at this time, and a danger arose that Eddington would be drafted. As a devout Quaker and ardent pacifist he had pleaded exemption from military service as a conscientious objector, but, in its desperate need for more manpower, the Ministry of National Service appealed the exemption. Finally, after three hearings and a last-minute appeal from Dyson attesting to Eddington’s importance to the eclipse expedition, the exemption from service was upheld on 11 July 1918. This was just one week before the second Battle of the Marne, a pivotal event in that war. Eddington also firmly believed, perhaps naively, that the example of British scientists verifying the theory of a German physicist would demonstrate how science could lead the world toward peace.

  On 8 March 1919, just four months after the end of hostilities, two expeditions set sail from England. After a brief stop at the island of Madeira, the teams split up. Eddington, accompanied by Edwin Cottingham, headed for the island of Principe, off the coast of present-day Equatorial Guinea; Charles Davidson and Andrew Crommelin headed for the city of Sobral, in northern Brazil. The principle of the experiment is deceptively simple. During a total solar eclipse, the Moon hides the Sun completely, revealing the field of stars around it. Using a telescope and photographic plates, the astronomers take pictures of the obscured Sun and the surrounding star field. These pictures are then compared with pictures of the same star field taken when the Sun is not present. The comparison pictures are taken at night, weeks or months before or after the eclipse, when the Sun is nowhere near that part of the sky and the stars are in their true, undeflected positions. In the eclipse pictures, the stars whose light is deflected would appear to be displaced away from the Sun relative to their actual positions (see Figure 3.2).

  One property of the predicted deflection is important: Although a star whose image is at the edge of the Sun is deflected by 1.75 arcseconds, a star whose image is twice as far from the center of the Sun is deflected by half as much, and a star ten times as far is deflected by one tenth; in other words, the deflection varies inversely as the angular distance of the star from the Sun (see Figure 3.2). Now, because the eclipse pictures and the comparison pictures are taken at different times, under different conditions (and sometimes using different telescopes), their overall magnifications may not be the same. Therefore, the stars in the photographs that are farthest from the Sun, undeflected on the comparison plate, deflected only negligibly on the eclipse plate, can be used to determine an overall magnification correction. Then, the true deflection of the stars closest to the Sun can be measured.

  In practice, of course, nothing is ever this simple. One important complication is a phenomenon that astronomers call “seeing.” Because of turbulence in the Earth’s atmosphere, starlight passing through it can be refracted or bent by the warmer and colder pockets of moving air and can suffer deflections of as much as a few arcseconds (this is part of what makes stars twinkle to the naked eye). These deflections are comparable to the effect being measured. But because they are random in nature (as likely to be toward the Sun as away from it), they can be averaged away if one has many images. The larger the number of star images, the more accurately this effect can be removed. Therefore, it is absolutely crucial to obtain as many photographs with as many star images as possible. To this end, of course, it helps to have a clear sky.

  We can therefore imagine Eddington’s emotional state when, on the day of the eclipse, “a tremendous rainstorm came on.” As the morning wore on, he began to lose all hope. Before the expedition, Dyson had joked about the possible outcomes: no deflection would show that light was not affected by gravity, a half deflection would confirm Newton, and a full deflection would confirm Einstein. Eddington’s companion on Principe had asked Dyson before the departure what would happen if they found double the deflection. Dyson had answered, “Then, my dear Cottingham, Eddington will go mad, and you will have to come home alone.” Now Eddington had to consider the possibility of getting no results at all. But at the last moment, the weather began to change for the better: “The rain stopped about noon, and about 1:30, when the partial phase [of the eclipse] was well advanced, we began to get a glimpse of the Sun.” Of the sixteen photographs taken through the remaining cloud cover, only two had reliable images, totaling only about five stars. Nevertheless, comparison of the two eclipse plates with a comparison plate taken at the Oxford University telescope before the expedition yielded results in agreement with general relativity, corresponding to a deflection for a grazing ray of 1.60 ± 0.31 arcseconds, or 0.91 ± 0.18 times the Einsteinian prediction. The Sobral expedition, blessed with better weather, managed to obtain eight usable plates showing at least seven stars each. The nineteen plates taken on a second telescope turned out to be worthless because the telescope apparently changed its focal length just before totality of the eclipse, possibly as a result of heating by the Sun. Analysis of the good plates yielded a grazing deflection of 1.98 ± 0.12 arcseconds, or 1.13 ± 0.07 times the Einsteinian value.

  Eddington made the announcement of the measurements at a joint meeting of the Royal Society of London and the Royal Astronomical Society on 6 November 1919. He may be the first scientist to fully appreciate the power of the media of his day, and engineered some adroit advance publicity. The mathematician Alfred North Whitehead described the scene: “The whole atmosphere … was exactly that of a Greek drama … in the background the picture of Newton to remind us that the greatest of scientific generalizations was now, after more than two centuries, to receive its first modification.” Before this, Einstein had been an obscure Swiss/German scientist, well known and respected within the small European community of physicists, but largely unknown to the outside world. With newspaper headlines spreading worldwide during the following days, everything changed, and Einstein and his theory became immediate sensations. The Einstein aura has not abated sinc
e.

  On the other hand, Einstein’s fame did engender a backlash, especially in Germany. The rise of nationalism and anti-Semitism in Germany between the world wars had its counterpart in scientific circles. In 1920, Paul Weyland organized a public forum in which Einstein and his theories were denounced. One of the leading exponents of this view was Philipp Lenard, a Nobel Laureate in Physics (1905) for his work on cathode rays (electron beams in modern parlance). An avowed sympathizer of the nascent Nazi movement, Lenard spent much of his time between the wars attempting to cleanse German science of the “Jewish taint.” Relativity represented the epitome of “Jewish science,” and much effort was expended by Lenard and others in attempts to discredit it. In early 1921, while preparing an article against general relativity, Lenard learned of the existence of Georg von Soldner’s 1804 paper. This discovery delighted him, because it showed the precedence of von Soldner’s “Aryan” work over Einstein’s “Jewish” theory. The fact that the eclipse results appeared to favor Einstein over von Soldner did not appear to faze him. Lenard prepared a lengthy introductory essay, incorporated the first two pages of von Soldner’s paper verbatim and summarized the rest, and had the whole thing published under von Soldner’s name in the 27 September 1921 issue of the journal Annalen der Physik.

  The vast majority of non-Jewish German physicists did not share these views, however, and despite the Nazi takeover in Germany and the subsequent dismissal and emigration of many Jewish physicists (including Einstein), the anti-relativity program became little more than a footnote in the history of science.