CHAPTER 3
How Light Sheds Light on Gravity
After over a year of painstaking preparation and numerous rehearsals, Don Bruns felt he was ready. The skies over Casper, Wyoming that morning were clear and blue, with only a few thin clouds. Winds were calm. His TeleVue Optics NP101is telescope and its attached CCD camera were ready. The computer programs he had written to send commands to the telescope during the crucial two and a half minutes had been tested and rehearsed. All that remained was to sit back and wait for what was being called the Great American Eclipse.
Bruns was one of an estimated 215 million people who viewed the eclipse of the Sun either in person or electronically that 21 August 2017. But Bruns, a retired physicist and amateur astronomer, was not in Wyoming to ooh and ahh over the sight of the eclipse, he was there hoping to repeat one of the most famous measurements of the twentieth century, an experiment that made Albert Einstein famous to the world at large.
That earlier experiment generated over-the-top headlines in the autumn of 1919. “Revolution in Science / New Theory of the Universe / Newtonian Ideas Overthrown,” proclaimed The Times of London on 7 November. “Lights all Askew in the Heavens / Men of Science More or Less Agog over Results of Eclipse Observations,” declared the New York Times three days later (notice only “men” of science, a typical attitude of that era). It heralded a brave new world in which the old values of absolute space and absolute time were lost forever. To a world emerging from the devastation of World War I, it meant the overthrow of all absolute standards, whether in morality or philosophy, music or art. In a 1983 survey of twentieth-century history, the British historian Paul Johnson argued that the “modern era” began not in 1900 or in August 1914, but with the event that spawned these headlines in 1919.
This event made Einstein a celebrity. Set aside for a moment his genius, the triumph of his theories, and the new scientific order he created almost singlehandedly. That alone might have been enough, but Einstein was also, in today’s terminology, a very “media-friendly” fellow. His absentmindedness, his wit, his willingness to expound upon politics, religion and philosophy in addition to science, his violin playing—all these characteristics sparked an intense curiosity on the part of the public. The press, tired of printing battle reports and casualty lists from the war, was only too eager to satisfy its readers’ curiosity.
The event that caused such a commotion was the successful measurement of the bending of starlight by the Sun. The amount of bending agreed with the prediction of Einstein’s general theory of relativity, but disagreed with the prediction of Newton’s gravitational theory. This was the experiment that Don Bruns planned to repeat, and if all went well, to beat in precision.
The story of how gravity affects the trajectory of light is one of the most fascinating in all science. It actually has its roots in the eighteenth century, yet the story continues to evolve to this day. It journeys from the heights of theoretical and experimental accomplishments to the depths of racist propaganda, from our solar system to the most distant galaxies.
It is believed that the first person to consider seriously the possible effect of gravity on light was a British theologian, geophysicist and astronomer, Reverend John Michell (1724–1793). Ever since the time of Newton, who had himself speculated vaguely that gravity might affect light, it had been assumed that light consisted of particles or “corpuscles.” In 1783, Michell reasoned that light would be attracted by gravity in the same way that ordinary matter is attracted. He noted that light emitted outward from the surface of a body such as the Earth or the Sun would slow down after traveling great distances (Michell, of course, did not know the theory of special relativity, which requires the speed of light to be the same from the viewpoint of any inertial observer). He then asked how large would a body of the same density (same number of grams per cubic centimeter) as the Sun have to be in order that light emitted from it would be stopped by gravity and pulled back before escaping? The answer he obtained was 500 times the diameter of the Sun. Light could never escape from such a body.
This remarkable idea describes what we now refer to as a black hole. In today’s language, Michell’s object would be 100 million times more massive than our Sun. Fifteen years later, the great French mathematician Pierre Simon Laplace performed a similar calculation. Although Michell and Laplace were wrong in the fundamental theory, their basic premise is right: gravity affects light.
But Michell didn’t stop there. He then asked, how would one ever detect such a body if light could not escape from it? His remarkable answer was that if such a dark body were to be in a double star or binary orbit with a normal star, one could infer its existence by measuring the wobble in the normal star’s position as the two bodies revolved around each other. What made this remarkable was that in 1783 there was no evidence that such binary star systems existed. It appears that Michell’s writings and speculations on such possibilities, along with his groundbreaking statistical analyses of close associations of stars on the sky, played a role in getting astronomers to start looking for evidence of binaries. The first solid discovery of two regular stars in a mutual orbit was announced by Wilhelm Herschel in 1803.
Michell’s friend and colleague, Henry Cavendish (1731–1810), shared his interest in gravity. Already famous for his discovery of hydrogen, Cavendish inherited from his recently deceased friend an instrument that Michell had built to measure gravity, and after some modifications of it, he used it to measure what we now call Newton’s constant of gravitation (called “big G” by physicists), which relates the gravitational force between two bodies to their masses and separation.
But around 1784, Cavendish also asked the question: If gravity affects light as Michell suggested, would it not also bend it? According to Newtonian gravity, the orbit of one body about another is a “conic section,” the figure formed by the intersection of a cone with a plane tilted at various angles: an ellipse or a circle if the orbit is bound, so that the body never escapes, or a hyperbola if it is unbound (Figure 3.1). If light is a corpuscle undergoing the same gravitational attraction as a material particle, then because its speed is so large, its orbit will be a hyperbola that is very close to being a straight line [the bottom panel of Figure 3.1(b)]. However, the deviation, while small, is calculable, and apparently Cavendish did the calculation.
Figure 3.1 Newtonian orbits. Orbits (a) are bound, either circular or elliptical (also called eccentric). Orbits (b) are unbound hyperbolae of ever greater speed, moving from top to bottom.
Why apparently? Cavendish was notorious for not bothering to publish his work, or even to discuss it with colleagues (the neurologist Oliver Sacks has speculated that Cavendish may have had Asperger syndrome). Around 1920, during a project to complete the publication of Cavendish’s work in physics (his work on chemistry having been compiled and published earlier), researchers discovered a scrap of paper among his documents which stated “To find the bending of a ray of light which passes near the surface of any body by the attraction of the body …,” followed by a formula. No calculational details, just the correct answer to the problem posed.
Some fifteen years after Michell and Cavendish, a similar story played out on the other side of the European continent, but with a somewhat different outcome. Prompted by Laplace’s speculations, a Bavarian astronomer named Johann Georg von Soldner (1776–1833) asked the same question: Would gravity bend light? Von Soldner was a largely self-taught man who became a highly respected astronomer. He made fundamental contributions to the field of precision astronomical measurements known as astrometry, and eventually rose to the position of director of the observatory of the Munich Academy of Sciences. But in 1801, he was still an assistant to the astronomer Johann Bode in the Berlin Observatory. Von Soldner calculated the bending (his and Cavendish’s answers agree), and determined that, for a path that skims the surface of the Sun, the bending would be 0.875 seconds of arc. An arcsecond is the angle subtended by a human finger at a distance of about 4 kilometers or 2.5 miles (3,600 arcseconds equals 1 degree of arc).
Von Soldner’s work was published in 1804 in one of the German astronomical journals. It was then immediately forgotten, partly because the effect was beyond the current limits of telescope precision, and partly because of the rise during most of the nineteenth century of the wave theory of light, according to which light moves as a wave through an imponderable “aether,” and presumably suffers no deflection. Einstein was certainly not aware of either von Soldner’s paper or Cavendish’s calculation. It was not until 1921 that von Soldner’s work was rediscovered and resurrected, but then it was for a different, more unsavory purpose.
Like Cavendish and von Soldner over a century before, Einstein in 1907 was also interested in the effect of gravity on light. He recognized that if the principle of equivalence led to an effect on the frequency of light, the gravitational redshift (Chapter 2), it should also result in an effect on its trajectory. In 1911 he determined that the deflection of a ray grazing the Sun should be 0.875 arcseconds. He proposed that astronomers should look for the effect during a total solar eclipse, when stars near the Sun would be visible and any bending of their rays could be detected through the displacement of the stars from their normal positions (Figure 3.2). Several teams, including one headed by Erwin Finlay-Freundlich of the Berlin Observatory, one headed by William Campbell of the Lick Observatory in the USA, and one headed by Charles Perrine of the National Argentinian Observatory, traveled to the Crimea to observe the eclipse of 21 August 1914. But World War I intervened, and Russia sent many of the astronomers home, interned others, and temporarily confiscated much of the equipment; in any case, the weather at the site on eclipse day would have been too bad to permit useful observations.
Figure 3.2 Deflection of light by the Sun. Top: Deflection of light causes the apparent position of a star to be displaced away from the Sun. Bottom left: A field of stars viewed at night. Bottom right: The same field with the Sun in the middle, obscured by the Moon. Stars whose light passes close to the Sun (the six stars in black) have their locations displaced more than stars farther from the Sun (the stars in gray). The amount of bending is greatly exaggerated, of course.
It is quite easy to see how Einstein’s equivalence principle leads to a deflection of light. Imagine a laboratory with glass sides containing an observer well versed in the equivalence principle (see Figure 3.3). The laboratory is moving with constant speed far from any star or galaxy; it is therefore an inertial reference frame in which special relativity is valid. Because there is no gravity, the observer inside floats freely. The following sequence of events is shown in the upper panel of Figure 3.3: (a) A light ray enters the laboratory from the left at a spot right at the middle of the lab. (b) As the ray crosses the lab in a straight line, the lab moves forward (upwards in the figure), so the ray is still at the midpoint. (c) The ray exits the lab also at the midpoint. The ray crosses the lab in a straight line.
Figure 3.3 Einstein’s principle of equivalence and the bending of light. Top panel: A light ray (dashed lines) enters a laboratory moving at constant speed in empty space. It crosses the laboratory in a straight line, with the angle changed because of aberration. Bottom panel: A light ray enters a laboratory accelerating via the thrust of a rocket. Because the laboratory is moving faster in each panel, the light ray leaves the laboratory slightly below the level where it entered, as if it were deflected downward.
But the angle of the ray as seen by the observer in the lab is different from the angle seen by external observers at rest! This is the well-known phenomenon of aberration, discovered by James Bradley in 1725. It manifests itself in an annual back and forth motion of stellar images by about 40 arcseconds as the Earth moves around the Sun. On a more mundane level, it is the same phenomenon that occurs when you carry an umbrella quickly through a vertical rainfall. You observe that the drops are tilted toward you and get your feet wet. In the top panel of Figure 3.3, the difference in angle between what we outside observers see for the light ray entering the lab and what the observer in the moving lab sees is substantial because we have made our lab move at around half the speed of light.
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 arcseconds 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 since.
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.
There were legitimate scientific questions about Eddington’s results, however. Given the poor quality of the data, did they really support Einstein or not? In 1980, some historians of science wondered whether Eddington’s enthusiasm for the theory of general relativity caused him to select or massage the data to get the desired result. Numerous reanalyses between 1923 and 1956 of the plates used by Eddington yielded the same results as he obtained within 10 percent. In 1979, on the occasion of the centenary of Einstein’s birth, astronomers at the Royal Greenwich Observatory near London reanalysed both sets of Sobral plates using a modern tool called the Zeiss Ascorecord and its data reduction software. The plates from the first Sobral telescope yielded virtually the same deflection as that obtained by Davidson and Crommelin, with the errors actually reduced by 40 percent. Despite the scale changes in the plates from the second Sobral telescope, the analysis still gave a result 1.55 ± 0.34 arcseconds for a grazing ray, consistent with general relativity, albeit with much larger errors, reflecting the problem with the telescope focal length. Looking back on the British astronomers’ treatment of the data, our colleague Daniel Kennefick has argued that there is no credible evidence of bias on their part.
But scientists are reluctant to adopt a world-changing theory on the basis of the measurements of a single team. Any new theory of nature must stand the test of many experimental checks by different groups with different methods and techniques. Strangely, one set of measurements made before the 1919 eclipse failed to confirm Einstein’s prediction. William Campbell and Heber Curtis of the Lick Observatory analyzed plates from a 1900 eclipse near Augusta, Georgia and a 1918 eclipse at Goldendale, Washington in the USA, hoping to beat the British to the punch. Unfortunately the quality of the images was poor, and they found no unambiguous evidence for the Einstein deflection; ironically, Campbell reported this negative result at the Royal Astronomical Society meeting on 11 July 1919 while Eddington was still at sea returning from Principe. At the meeting, Dyson reported that Eddington had telegraphed that his prelimary measurements indicated a positive result.
Following up on Eddington’s success, seven teams tried the measurement during a 1922 eclipse in Australia, although only three succeeded in getting usable data. Campbell and Robert Trumpler of the Lick team reported a result for the grazing deflection of 1.72 ± 0.11 arcseconds, while a Canadian team and an English/Australian team reported values between 1.2 and 2.3 arcseconds. Later eclipse measurements continued to support general relativity: one in 1929, two in 1936, one in 1947, one in 1952 and one in 1973. Surprisingly, there was very little improvement in accuracy, with different measurements giving values ranging as far as 30 percent away from the Einstein value. Still, there was little doubt that Einstein beat Newton.
The 1973 expedition is a case in point. Organized by the University of Texas and Princeton University, the observation took place in June at Chinguetti Oasis in Mauritania. The observers had the benefit of 1970s technology: Kodak photographic emulsions, a temperature-controlled building housing the telescope (the outside temperature at mid-eclipse was 97°F), sophisticated motor drives to control the direction of the telescope accurately, and computerized analysis of the photographs. Unfortunately they couldn’t control the weather any better than Eddington could. Eclipse morning brought high winds, drifting sand, and dust too thick to see the Sun. But as totality of the eclipse approached, the winds died down, the dust began to settle, and the astronomers took a sequence of photographs during what they have described as the shortest six minutes of their lives. They had hoped to gather over 1,000 star images, but the dust cut the visibility to less than 20 percent and only a disappointing 150 were obtained. After a follow-up expedition to the site in November to take comparison plates, the photographs were analyzed using a special automated device called the GALAXY Measuring Engine at the Royal Greenwich Observatory. The result agreed with the Einsteinian prediction within the measurement error of about 10 percent, still only a modest improvement over previous eclipse measurements.
This was the backdrop for Don Bruns’ attempt to do an improved eclipse measurement in 2017. Bruns had retired in 2014 after a career in the optics industry, working on lasers and advanced optics for both military and commercial applications. He knew astronomical instrumentation inside and out, and decided to employ twenty-first-century technology in his attempt to redo this historic measurement. Among his advantages were the CCD camera, promising greatly improved response to the incoming starlight and improved image stability over photographic emulsions or the glass plates used by Eddington. He also did not have to worry about taking comparison images of the star field before or after the experiment, because an orbiting telescope known as Gaia, launched in 2013, was providing undeflected positions of all the relevant stars with an accuracy far better than he could ever obtain himself. Finally, the telescope and camera could be completely controlled by a computer using software written, tested and rehearsed in advance. In fact, unlike the many teams before him, Bruns reported that he could actually sit back and enjoy the eclipse, because everything was pre-programmed. Excellent weather didn’t hurt. Nevertheless, he had to sweat many tedious details in his analysis of the data before he could report a value of 1.75 arcseconds for a grazing ray, with a probable uncertainty of 3 percent, in excellent agreement with general relativity and with about three times smaller uncertainty than Eddington had claimed.
Bruns’ measurement is of mainly historical and personal interest, because by the late 1960s testing of Einstein’s deflection during solar eclipses was already being superseded by a technique that was a marriage of two of the most important astronomical discoveries of the twentieth century: the radio telescope and the quasar.
Radio astronomy began in 1931, when Karl Jansky of the Bell Telephone Laboratories in New Jersey found that the noise in the radio antenna he was trying to improve for use in radio telecommunications was coming from the direction of the center of our galaxy (we will return to this in Chapter 6). The development of radar during World War II led to new receivers and techniques, and to the rapid development of radio telescopes as new astronomical tools. Among the sources of radio waves that were discovered were the Sun itself, interstellar gas clouds such as the Crab Nebula, clouds of hydrogen atoms and of complex molecules, and radio galaxies. Radio waves are the same as ordinary visible light, only of longer wavelength. Whereas visible light spans a wavelength range from 400 to 700 nanometers (a nanometer is a billionth of a meter), radio waves span the range from a tenth of a millimeter to several meters. General relativity predicts exactly the same deflection of radio waves as visible light; the effect is independent of wavelength.
To measure the deflection of radio waves, we need to be able to measure to high precision the direction from which they come. To this end, the radio interferometer is the ideal instrument. In its simplest form, a radio interferometer consists of two radio telescopes separated by some distance, called the baseline (Figure 3.5). As a radio wave from some external source approaches the pair, the wavefronts may arrive at one telescope before they arrive at the other, depending on the location of the source on the sky. The difference in the time of arrival of a given wave front at the two telescopes is measured by comparing the signals using a precise atomic clock. For a given wavelength of the radio waves, the longer the baseline, the larger the time delay for a given angle of approach, and thus the more accurately the angle can be measured. Radio interferometers range in baseline from the 1 kilometer instrument in Owens Valley, California to the 42 kilometer long “Y” containing twenty-seven linked antennae along its three legs at the Very Large Array in New Mexico, to the Event Horizon Telescope (EHT), which links antennae as far apart as Hawaii, Chile, Europe and the South Pole in a global interferometer (we will return to EHT in Chapter 6). When the telescopes are separated by transcontinental and intercontinental distances, the technique is known as Very Long Baseline Interferometry (VLBI). The resolution of some of these VLBI interferometers can be better than one ten-thousandth of an arcsecond, or 100 microarcseconds. That would be good enough to resolve this book from Earth if it were sitting on the surface of the Moon.
Figure 3.5 Radio interferometry. A radio wave approaches two radio telescopes. Each wave arrives first at one telescope, then at the other. The times are compared very precisely using an atomic clock linked to the two telescopes, leading to accurate determinations of the direction of the source.
We also need a very sharp source of radio waves. Most astronomical sources are unsuitable for this purpose because they are extended in space. For example, most galaxies that emit radio waves (or radio galaxies, for short) do so from an extended region that can be as large as a degree in angular size. The discovery of quasistellar radio sources, or quasars as they are called, besides motivating applications of general relativity to astrophysics, provided the ideal source of radio waves to test the deflection of light. Because they are so distant, between one and twelve billion light years away, they appear much smaller in extent, making it possible to pinpoint their locations more accurately. Yet despite their distance, many of them are powerful radio sources and their light emission is constant enough to enable long-term observations.
Unfortunately, a powerful point source of radio waves is not the only ingredient for a successful light deflection experiment. We need at least two of them fairly close to each other on the sky, and they have to pass near the Sun as seen from Earth. We need at least two for the same reason as we needed a field of stars behind the eclipsed Sun in the optical deflection measurements: the stars whose images are far from the Sun are used to establish the scale because their light is relatively undeflected, and the movement of the star images close to the Sun is used to determine the deflection. Figure 3.6 illustrates how this would work. The Sun passes in front of a pair of quasars, one about 1 degree away, the other about 4 degrees away (top panel). Initially the angle between the two quasars as measured on Earth is the nominal, unperturbed angle (bottom panel). As the Sun approaches the lower quasar, the quasar’s image as seen from Earth is displaced toward the other quasar, causing the angle between them to decrease. Then, as the Sun passes the lower quasar, its image is displaced to the left, away from the other quasar, causing the angle between them to increase, although less dramatically. As the Sun moves away from the pair, the angle returns to its nominal value.
Figure 3.6 Testing the deflection of light using quasars. Top panel: As seen from Earth, the Sun moves across the sky passing near the sky position of two quasars. When the Sun is far to the left, the angle between the two quasars is the nominal, unperturbed angle. As the Sun approaches the position of the lower quasar, the quasar’s image is displaced toward the other quasar, causing the angle between them to decrease. Then, as the Sun continues past the lower quasar, its image is displaced to the left, away from the other quasar, causing the angle between them to increase. As the Sun moves far to the right, away from the pair, the angle returns to its nominal value. Bottom panel: The changes in angle between the quasars plotted against time.
Early measurements took advantage of the fact that groups of strong quasars annually pass very close to the Sun (as seen from the Earth), such as the group 3C273, 3C279 and 3C48 (the designation “3C” refers to the Third Cambridge Catalogue of radio sources). As the Earth moves in its orbit, changing the lines of sight of the quasars relative to the Sun, the angular separation between pairs of quasars varies. A number of measurements using radio interferometers over the period 1969–1975 yielded accurate determinations of the deflection, reaching levels of 1 percent.
In recent years, scientists interested in the Earth have made their own use of VLBI. The idea is to measure directions to hundreds of radio galaxies and quasars everywhere in the sky in order to monitor very precisely the Earth’s rotation rate and the orientation of its rotation axis. Variations in the rotation of the Earth can be caused by changes in ocean levels, variations in weather patterns, interactions between the Earth’s mantle and its core, and the gravitational tug of the Moon. A test of relativity is but a by-product of their work. But because the intrinsic accuracy of the measurements is now so high, they are sensitive to the deflection of light over almost the entire celestial sphere. While a ray grazing the Sun is deflected by 1.75 arcseconds, a ray approaching the Earth from 90° relative to the Sun is deflected by 0.004 arcseconds or 4 milliarcseconds. Even a ray coming from 175°, almost directly opposite the direction of the Sun, is deflected by just under a milliarcsecond. With accuracies of 10 to 100 microarcseconds, modern VLBI can detect these tiny deflections. Recent global analyses of several million VLBI observations of over 500 quasars and compact radio sources, made by telescopes spread around the globe, confirmed general relativity to about 0.01 percent, or one part in 10,000. The vast majority of the sources were more than 30 degrees from the Sun at all times. It is no longer necessary to look right at the Sun to detect Einstein’s light deflection effect!
At the time of the writing of this book, radio astronomers seem to have the upper hand in testing the deflection of light, but not for long. Optical astronomers may yet have the last laugh. The idea is to put Eddington into space, hypothetically speaking of course, to get above the effects of the Earth’s atmosphere. This was first demonstrated by the Hipparcos satellite, launched in 1989 by the European Space Agency (ESA) and operated until 1993. Hipparcos made precise measurements of the positions of over two million stars at optical wavelengths and was able to test the distorting effect of the deflection of light on the celestial sphere to about one part in 1,000, in agreement with general relativity, but not quite as precise as VLBI. But its follow-up mission, called Gaia, launched by ESA in 2013, is making even more precise position measurements of about a billion stars. This may permit a test of general relativity’s light bending to one part in a million.
Astronomers of the ancient world referred to the stars as residing on a “celestial sphere” that was fixed and immutable. This stellar realm had to be so perfect, because it was where the gods resided. We now know, thanks to Einstein, that it is more like a soap bubble; as the Sun wanders across the sky, the celestial sphere appears to warp and distend as light from those distant stars wends its way through the piece of curved spacetime that the Sun carries with it. Even after dark, with the Sun behind us, the night sky is warped. At fractions of a milliarcsecond, the effect is far too small for our eyes to sense, but astronomers now measure it routinely.
But this is not the only effect of gravity on light. Gravity also slows light down. As anyone knows who has ever tried and failed to spear a fish swimming in a river or a lake, there is a close relationship between the bending of light and changes in the speed of light. The speed of light in water is about 75 percent of its speed in air, because the light’s progress is impeded by its interactions with the atoms of the denser water, just as a person takes longer to get across a crowded room than an empty room. This means that as the waves of light reflect off the surface of the fish and cross the interface with the air, they begin to travel faster. This causes the wave fronts to be tilted more toward the vertical (Figure 3.7). To the observer on the shore, the apparent direction of the fish is defined by the verticals to the wavefronts, and thus the fish appears to be above its true location. This is why your spear misses the fish, unless you compensate for this effect, known as refraction.
Figure 3.7 Refraction of light. Because of the change of the speed of light in going from water to air, the spear fisher must compensate for the bending or refraction of the light rays from the fish (obviously not drawn to scale!).
Therefore, if the curved spacetime around the Sun causes light to bend, then there must be an associated change in its speed.
But wait a minute, this can’t be right! According to the equivalence principle, the speed of light as measured in any local freely falling frame is always the same. How then can we say that the light slows down near the Sun?
The problem here is the distinction between local effects, those that are observable in one very small, freely falling frame, and large-scale or global effects, which cover a range of space or an interval in time large enough that the effects of curvature of spacetime are important and cannot be described by a single freely falling frame. One indication of the global nature of an effect like the deflection of light was the fact that we could not detect it by looking at a single star or quasar; we always had to compare the light from one star or quasar with that from another that appeared to be farther from the Sun.
Similarly with the speed. An observer in a small, freely falling spaceship close to the Sun will find that the speed of light, given by the width of her ship divided by the time taken for the ray to cross it, is exactly the same as the speed obtained by a similarly freely falling observer far from the Sun. But the rates of the clocks of the two observers are not the same, because of the gravitational redshift discussed in Chapter 2, and the rulers they use to measure distances are not the same because of the warpage of space, as represented in our two-dimensional sheet of Figure 3.4. Thus, if we were to add up all the times taken for the ray to pass through a sequence of such frames laid side to side, we would find that the total travel time for the ray and frames that are close to the Sun is slightly longer than the time for a similar ray and frames that pass nowhere close to the Sun. Our rubber sheet of Figure 3.4 also suggests a delay, since the ray must “dip” down as it follows the rubber surface, thus taking longer to get across compared to the ray that passes far from the ball (remember that this is just an imperfect analogy!).
To make this slightly more concrete, consider Figure 3.8. An enormous circular rigid ring has been constructed with a diameter much larger than the solar system, with an emitter of light on one side of the ring and a receiver on the opposite side. The ring is so large that the Sun’s gravity has no measurable effect on it. The Sun is moving relative to the ring in such a way that it will pass through the center of the ring (clearly this is a gedanken experiment!). In (a), the Sun is initially on the left, very far from the center of the ring. The Sun’s gravity has a negligible effect on the propagation of the light ray, so that the time the ray takes to cross the ring is simply the diameter divided by the speed of light. In (b), the Sun is close to the center of the ring, and the light ray passes close to the Sun. According to our argument above, the ray takes a little longer to cross the ring.1 In (c), the Sun has passed to the other side, and the time to cross the ring returns to its normal value. Plot (d) shows schematically the increase in travel time as a function of time (the gap in the middle is where the light rays hit the Sun and never reach the other side).
Figure 3.8 Excess travel time of light. An emitter on an enormous rigid ring sends light signals to a receiver on the opposite side. (a) The Sun is far to the left, so the travel time is the diameter of the ring divided by the usual speed of light. (b) The Sun is about to pass through the center of the ring, so the signal suffers an excess delay caused by curved spacetime near the Sun. (c) The Sun is far to the right, so the travel time returns to normal. (d) Plot of the travel time vs. time.
Whether or not we use the words “light slows down near the Sun” is purely a question of semantics. Because the observer on the ring who receives the light ray never goes near the Sun to make the measurement, she can’t really make such a judgment; and if she had made such a measurement in a freely falling laboratory near the Sun, she would have found the same value for the speed of light as in a freely falling laboratory far from the Sun, and might have thoroughly confused herself. All the observer can say with no fear of contradiction is that she observed an excess time of travel that depended on how close the light ray came to the Sun. The only sense in which it can be said that the light slowed down is mathematical: in a particular mathematical representation of the equations that describe the motion of the light ray, what general relativists call a particular coordinate system, the light appears to have a variable speed. But in a different mathematical representation (a different coordinate system), this statement might be false. Nevertheless, the observable quantities, such as the net time of travel, are the same no matter what representation is used. This is one of those cases in relativity where the careless use of words or phrases that are not based on observable quantities can lead to confusion or contradiction. We have already seen an example of this in our discussion in Chapter 2 of what “really” changed in the gravitational redshift, the clock rates or the signal frequency.
Given this discussion, you might be tempted to assume that Einstein derived this delay effect and proposed that it be measured. But he didn’t. The effect was derived by the radio astronomer Irwin I. Shapiro in 1964.
After receiving a Ph.D. in physics from Harvard University in 1955, Shapiro had been at MIT’s Lincoln Laboratory working on the problem of “radar ranging.” The idea was to bounce radar signals off planets such as Venus and Mercury and measure the round-trip travel times. This new technique had already led to improved determinations of the astronomical unit, the mean radius of the Earth’s orbit, and promised significant improvements in determining the orbits of the planets.
Shapiro had only a passing acquaintance with general relativity, and might not ever have considered it relevant to radar ranging had it not been for a lecture he attended in 1961 on measurements of the speed of light. Purely in passing, the speaker mentioned that according to general relativity, the speed of light is not constant. This statement puzzled Shapiro, because he had always thought that according to relativity, the speed of light should be constant. He knew, of course, that general relativity predicts that light should be deflected by a gravitating body, and following the same logic as we presented for the fish in water, he asked if its speed would also be affected.
To be fair, Einstein had considered the possibility of speed variation. Once he understood, from the principle of equivalence, that gravity could have an effect on light (the gravitational redshift), he attempted to construct a theory of gravity in which the speed of light would vary in the vicinity of a gravitating body. It was the equations from this specific theory that he used in 1911 to calculate the (one-half) bending of light. As we discussed on page 42, we’re talking here about the speed in a specific coordinate system.
However, for some reason Einstein did not take the next step, the one that Shapiro took. Shapiro consulted the classic general relativity textbook by Eddington and found that, according to the equations of the full general theory, the “effective” speed of light should indeed vary, just as it did in Einstein’s earlier model (in the full theory of general relativity the effect was doubled, just as it was for the deflection of light). Shapiro then applied these equations to the problem of the round trip of a radar signal to a distant object and found, in agreement with our qualitative argument, that the radar signal should take slightly longer to make the round trip than one would have expected on the basis of Newtonian theory and a constant speed of light. The additional delay would increase if the signal passed closer to the Sun.
In the solar system, the effect would be most noticeable when the target was on the opposite side of the Sun from the Earth, so that the signal would pass very near the Sun on its trip, as in panel (b) of Figure 3.8. Such a configuration is called superior conjunction (when both planets are on the same side of the Sun, it is called inferior conjunction). For example, Shapiro found that a radar signal sent from Earth to Mars at superior conjunction that just grazes the surface of the Sun suffers a round-trip delay of 250 millionths of a second (250 microseconds). Don’t forget that the total round-trip travel time for such a signal is about 42 minutes! So the idea here would be to detect an additional delay in the round trip of 250 microseconds on a total travel time of three-quarters of an hour. This might seem to be a hopeless proposition until you realize that the distance that light travels in 250 microseconds is 75 kilometers. So the delay represents an apparent shift in the distance to the target of half of this, or about 38 kilometers. Since Shapiro saw that radar ranging could potentially achieve a precision in distance between Earth and planets corresponding to a few kilometers, then perhaps this effect could be observed. The problem was that no radio telescopes at the time had the capability of sending a powerful enough radar signal to any planet at superior conjunction and detecting the extremely weak return signal. So Shapiro’s calculation lay in his desk for two years.
In the fall of 1964, two events caused Shapiro to retrieve his superior conjunction calculation and take it more seriously. The first was the completion of the Haystack radar antenna in Westford, Massachusetts. The second was the birth of his son on 30 October. As often happens in creative endeavors, the event in his personal life may have elevated him to a higher level of awareness or of mental activity, for soon thereafter, while describing the time delay idea to a colleague at a party, he suddenly realized that Haystack might be able to range to Mercury at superior conjunction and provide a means to test the time delay prediction (Mars would be too far away at superior conjunction for Haystack to record a measurable signal). Shapiro decided then to write up his superior conjunction calculation for Physical Review Letters. The paper was submitted in the middle of November, and published under the title “Fourth Test of General Relativity” in late December, 1964. (The first three tests were the gravitational redshift, the light deflection and the perihelion advance of Mercury, the three proposed by Einstein.) In time, the effect would come to be called the Shapiro time delay.
The principle behind the measurement of the time delay is very much the same as the principle behind the measurement of the deflection of light. Just as we could not measure the deflection of a single star, we cannot detect the time delay in a single radar shot. The reason, of course, is that we cannot “turn off” the gravitational field of the Sun in order to see what the star’s “true” position is or to see what the “flat spacetime” round-trip travel time would have been. To get at the deflection, we had to compare the position of a star or quasar relative to other stars or quasars both when its light passed far from the Sun, and when its light passed very near the Sun. By the same token, to see the time delay, we must compare the round-trip travel time of a radar signal to the planet when the signal passes far from the Sun with that when the signal passes close to the Sun.
When the signal to the planet passes far from the Sun, the Shapiro time delay is relatively small, and the round-trip travel time is closer to being a measure of the “true” distance. This corresponds to the situation in Figure 3.4 where the signal traverses a portion of space that is virtually flat. As the planet moves into superior conjunction, however, and the signal passes closer and closer to the Sun, the Shapiro time delay becomes a larger contribution to the round-trip travel time.
However, even though the radar signal may go near the Sun, the planet itself never does. Its orbit is well away from the Sun, on the order of 230 million kilometers for Mars or 58 million kilometers for Mercury, for instance. Because of this, the planet always moves through a region of low spacetime warpage, and maintains a relatively low velocity; therefore, the relativistic effects on its orbit are small. To the accuracy desired for a time delay measurement, its orbit can be described quite adequately by standard Newtonian gravitational theory. Therefore, even though the planet moves during the experiment, its motion can be predicted accurately. Because of this circumstance, the time delay can be measured in four steps: (1) by ranging to the planet for a period of time when the signal stays far from the Sun, determine the parameters that describe its orbit at that time; (2) using the orbit equations of Newtonian theory, including the perturbations from all the other planets, make a prediction of its future orbit and that of the Earth, including especially the period of superior conjunction where the action will occur; (3) using the predicted orbit, calculate the round-trip travel times of signals to the planet assuming no Shapiro time delay; and (4) compare these predicted round-trip travel times with those actually observed during superior conjunction, attribute the difference to the Shapiro time delay, and see how well it agrees with the prediction of general relativity.
Within about a month of submitting his paper on the time-delay effect to Physical Review Letters, Shapiro’s colleagues at Lincoln Laboratory set out to upgrade the Laboratory’s Haystack radar by increasing its power fivefold and by making other electronic improvements. This would give them the capability to get a decent echo from Mercury and also Venus at superior conjunction, and to measure the round-trip travel times to within 10 microseconds. By late 1966 the improved system was ready, just in time for the 9 November superior conjunction of Venus. Unfortunately, Venus goes through superior conjunction only about once every year and a half, so after observing Venus they then turned the radar sights on Mercury. Because Mercury orbits the Sun almost three times faster than Venus, it has a superior conjunction more often, about three times per year, giving more opportunities to measure the time delay. Measurements were made during the 18 January, 11 May and 24 August 1967 conjunctions of Mercury. All told, over four hundred radar “observations” were used. Most of these measurements (the ones not taken near superior conjunction of either of the planets) were combined with existing optical observations of Mercury and Venus available through the US Naval Observatory to accomplish the first step of the method, namely, to establish accurate orbits for the two planets. The remaining radar measurements centered around the superior conjunctions were then used to compare the predicted time delays with the observed time delays (because of large amounts of noise, the Venus data turned out not to be very useful). The results using Mercury data agreed with general relativity to within 20 percent. The first new test of Einstein’s theory since 1915 was a reality.
But the story does not end there. During the summer of 1965, while Shapiro and his colleagues were busy working on the Haystack radar, a US spacecraft hurtled past Mars, the first man-made object to encounter the “red planet.” The spacecraft was Mariner 4, and on its way by the planet it took twenty-one pictures and examined the Martian atmosphere using radio waves. Buoyed by the success of Mariner 4, NASA in December 1965 authorized two more missions to Mars, Mariner 6 and 7 in 1969 (Mariner 5 was a Venus mission) and Mariner 8 and 9 in 1971, and planners began to think seriously about Martian landers. While these missions would bring planetary exploration to a zenith, at least temporarily, they would also have crucial consequences for general relativity.
At the Jet Propulsion Laboratory (JPL) in Pasadena, California, where the Mariner program was headquartered, the relativistic time delay was also on people’s minds, and they began to wonder if there was any way to make use of Mariner 6 and 7 to measure the time delay. In fact, two JPL scientists, Duane Muhleman and Paul Reichley, had calculated the delay effect of general relativity on radar propagation independently of Shapiro, although they only published the results in internal JPL reports. There was no reason in principle why a measurement of the delay should not be possible. Other than in size, there is no fundamental difference between a planet and a spacecraft. The orbit of the spacecraft can be determined by tracking, and its trajectory during superior conjunction can be predicted, just as for ranging to the planet, and the time delay of the radar ranging signal during superior conjunction can be measured and compared with the prediction of general relativity.
Mariners 6 and 7 were launched on 24 February and 27 March 1969, and reached Mars by the end of July. Both spacecraft performed their primary tasks of observing Mars’ surface and atmosphere beautifully, and then left the planet to go into orbit around the Sun. Between December 1969 and the end of 1970, several hundred range measurements were made to each spacecraft, with the heaviest concentration, involving almost daily measurements, around the time of each superior conjunction—on 29 April 1970 for Mariner 6 and on 10 May for Mariner 7. Neither spacecraft actually went behind the Sun. Because of the tilt of their post-Martian orbits, they both passed by the Sun slightly to the north, Mariner 6 about 1° away, Mariner 7 about 1.5° away, as seen from Earth. For Mariner 6, the distance of closest approach of the radar signal at superior conjunction was about 3.5 solar radii, corresponding to a Shapiro time delay of 200 microseconds out of a total round-trip travel time of 45 minutes. For Mariner 7, the radar signals came no closer than about 5.9 solar radii, giving a slightly smaller time delay of 180 microseconds. After feeding all the observations into the computer, they found that the measured delays agreed with the predictions of general relativity to within 3 percent. This was a dramatic improvement over the 20 percent figure from Venus and Mercury ranging.
Of course, the planetary radar ranging people at Lincoln Laboratory had not been idle since 1967. They had continued to make radar observations of Mercury and Venus using both the Haystack antenna and the Arecibo radio telescope in Puerto Rico. In fact, during late January and early February 1970, while the JPL rangers were busy getting distances to the Mariner spacecraft on their approach to superior conjunction, Venus passed through its own superior conjunction, bombarded almost twice a week by radar signals from Haystack and Arecibo. Data from that Venus conjunction, and from the numerous Mercury conjunctions between 1967 and the end of 1970, once again yielded relativistic time delays in agreement with general relativity, this time at the 5 percent level.
It was soon realized, however, that each method, planetary vs. spacecraft tracking, had advantages and disadvantages. One advantage of planets is that they are massive and therefore are completely unaffected by the constant bombardment of the solar wind and solar radiation pressure. Spacecraft, by contrast, are light and have large antennae and solar panels, and so they tend to get jostled around a lot on their way through the rough neighborhood of interplanetary space. This is important because of the need to predict the orbit accurately during the time of superior conjunction when range measurements are supposed to yield the Shapiro delay.
An advantage of spacecraft is that they receive the radar signal from Earth, pass it through a transponder (the same device we encountered in Chapter 2), which boosts the signal’s power and beams it right back to Earth, leading to very accurate round-trip travel times. By contrast, planets are poor reflectors of radar beams, and also have valleys and mountains that introduce uncertainties in the “true” round-trip travel time.
The way to combine the transponding capabilities of spacecraft with the imperturbable motions of planets was to anchor a spacecraft to a planet, by having the spacecraft orbit the planet, or even better by letting the spacecraft land on the planet.
The first anchored spacecraft was Mariner 9, the orbiter that reached Mars in November 1971 just in time to photograph the raging dust storm that obliterated most of the planetary surface for several weeks. The next Martian superior conjunction of 8 September 1972 gave a confirmation of the Einsteinian time delay to 2 percent, only a modest improvement over the previous results, but enough to prove the power of the anchoring idea.
And then came Viking. The Viking landers on Mars were a spectacular achievement for planetary exploration, with their close-up views of the Martian surface, their analyses of the atmosphere, and their search for signs of life in the Martian soil. But to the general relativist they were even more beautiful, for they were the perfect anchored spacecraft for the time delay experiment. After a ten-month voyage, the first Viking spacecraft reached Mars in mid June, 1976. After several weeks studying possible landing sites, Lander 1 was detached from the orbiter and descended to a plain called Chryse on 20 July. Eighteen days later, the second Viking reached Mars, and on 3 September, Lander 2 dropped to the surface in a region called Utopia Planitia.
While much of the world focused its attention on the remarkable photographs and scientific data radioed back to Earth by the landers and orbiters, Shapiro’s group at MIT and the JPL team, now working together, began to prepare for the 26 November superior conjunction. With two landers and two orbiters all providing ranging data, they had an excellent configuration of anchored spacecraft so as to avoid the errors of random orbit perturbations.
Figure 3.9 Paths of Mariner 9, Viking and Cassini during superior conjunctions, as seen from Earth. Although Cassini was almost 8.5 astronomical units away on the far side of the Sun, the tracking signal passed within 0.6 solar radii of the Sun’s surface. Even though the signal from the Viking landers passed closer to the Sun at superior conjunction, by the time of Cassini, improvements in radar tracking and atomic clocks, plus Cassini’s quiet orbit en route to Saturn, led to a much better test of the Shapiro delay.
Ranging measurements were made from the moment the landers reached the Martian surface, through the November superior conjunction, and on until September 1977, when the two groups felt they had enough data to measure the Shapiro delay. The final result was a measured time delay in complete agreement with the prediction of general relativity, with an accuracy of 0.1 percent, or one part in a thousand!
Another twenty-five years would pass before the next leap forward in testing the Shapiro delay: the Cassini-Huygens mission. And what a leap it was. Launched jointly by NASA and ESA on 15 October 1997 and terminating on 15 September 2017, the mission is best known to the general public for its extraordinary feats: detection of new features in the atmosphere of Jupiter during its flyby of that planet en route to Saturn; discovery of seven new moons of Saturn; close flybys of many Saturnian moons, including Titan, Phoebe and Enceladus; the first spacecraft to orbit Saturn; the successful landing of the Huygens probe on Titan; and the grand finale suicide dive of Cassini into Saturn’s atmosphere, sending back useful data until the very end.
What is less well known is that Cassini made possible a test of the Shapiro delay and general relativity to a precision of one part in 100,000, a hundred times better than Viking. Several things made such an accurate test possible. On 21 June 2002 Cassini was in “cruise” mode en route to Saturn, about 8.4 astronomical units from the Sun, and passed through superior conjunction. The alignment between the spacecraft, the Sun and the Earth was so good that the tracking signal passed within little more than half a solar radius of the Sun’s surface, leading to a Shapiro delay of over 260 microseconds. Tracking data was taken regularly for 15 days on either side of conjunction. Cassini was so far from the Sun that the buffeting effects of the solar wind and radiation pressure were negligible, so “anchoring” to a planet was not needed. Tracking the spacecraft was done using radar signals at two frequencies, X-band (7,175 megahertz) and Ka-band (34,316 megahertz), which made it possible to account for the small delay induced by the passage of the signals through the ionized corona of the Sun. This effect depends on the frequency of the signal, whereas the Shapiro delay does not. Twenty-five years of advances in transponders, atomic clocks and computational capabilities didn’t hurt.
This “perfect storm” of happy chance produced a test of Einstein’s delay effect that has not been surpassed. Since 2002 there have been numerous missions involving planetary orbiters, including Mars and Venus Express, Mars Reconnaissance Orbiter and Mercury MESSENGER, yet none has been able to improve upon Cassini’s result.
Gravity’s effect on light has by now been so thoroughly tested and confirmed that it has become useful to assume that general relativity is correct, at least with regard to this effect, and to use the bending and delay of light as a tool to explore other phenomena.
The classic example of using Einstein’s theory as a tool for something else is the “gravitational lens.” In 1979, astronomers Dennis Walsh, Robert Carswell and Ray Weymann, using telescopes of the University of Arizona and Kitt Peak National Observatory, discovered a system that they initially called the “double quasar.” This system, listed in astronomical catalogues as Q0957+561, was a pair of quasars separated in the sky by about 6 arcseconds. This by itself would not have been so unusual were it not for the fact that the two quasars were uncannily similar: their velocities of recession from the Earth were identical, within the precision of the measurement, and their spectra were almost identical. The only apparent difference was that one member of the pair was somewhat fainter than the other. The astronomers who discovered this system immediately proposed an explanation. They argued that there was actually only one quasar and that somewhere along the line of sight between us and it was a massive object that was deflecting the light from the quasar in such a way as to produce the multiple images (see Figure 3.10). The subsequent detection of a faint galaxy between the two quasar images along with a surrounding cluster of galaxies confirmed this interpretation. Since that time, the gravitational lens has become an important tool for astronomers and cosmologists.
Figure 3.10 Gravitational lens. Two light rays emitted by a source pass by a massive body. Each is deflected by the curved spacetime around the body, the ray passing closer deflected a little more than the ray passing farther away. The observer sees the two rays coming from the direction of image 1 and image 2.
The idea that a massive object could produce an image by gravitational lensing was not new. Ironically, Einstein was probably the first to consider the possibility of a gravitational lens, although he didn’t publish it. Indeed, the fact that he did this calculation was unearthed only in 1997. In the course of studying Einstein’s original notebooks, science historian Jürgen Renn and colleagues came across a notebook from around 1912 in which Einstein worked out the basic equations for gravitational lenses, including the possibility of double images for a lens consisting of a simple massive body, and he derived a formula for the magnification of each image. Everything he did was off by a factor of two, of course, because he was using his 1911 formula for the deflection of light based on the principle of equivalence. In his notebook, he remarked that the effects were too small and that the probability of two stars being so perfectly aligned one in front of the other to produce an astronomical lens was too low ever to be of interest, so he didn’t publish the calculations.
The physicist Oliver Lodge suggested the possibility of lenses shortly after the eclipse confirmations of the deflection of starlight in 1919, and in 1924 Orest Chwolson pointed out that if the source was perfectly aligned directly behind the lensing star, the image would be a perfect ring, today called the Einstein ring. In 1936, Einstein published a short note on gravitational lenses based on his earlier notes, but now with the correct factor. Apparently he did this primarily to get a retired engineer named Rudi Mandl to stop nagging him about it. He wrote to the editors of Science that “Mandl squeezed it out of me …, but it makes the poor guy happy.”
But in 1937 the astronomer Fritz Zwicky pointed out that galaxies or even clusters of galaxies could act as gravitational lenses, thus relaxing the need for such precise alignment between the source, the lens and the Earth. The large mass associated with a galaxy can provide plenty of warpage of spacetime to deflect light rays, but because galaxies are mostly empty space (and clusters of galaxies even more so), light can easily pass through them, just as light passes through a glass lens.
The actual discovery of gravitational lenses gave general relativity a new astronomical role. For example, the number of quasar images, their relative brightness and placement, and any distortion in their shape all depend in detail on the distribution of matter in the intervening galaxy or cluster of galaxies. This is especially important because it is now widely believed that galaxies and clusters are embedded in halos of “dark matter,” and that the mass of these dark halos can be anywhere between 10 and 100 times the mass of the visible galaxy or cluster. Even though this matter evidently does not produce light, its mass can warp spacetime and bend any light that goes through it. Thus, gravitational lensing is playing a major role in mapping the distribution of dark matter in the universe.
In 2003 a planetary system outside our own solar system was discovered using gravitational lensing, adding to the ever growing list of “exoplanet” systems discovered by other techniques, such as detecting the wobble of the star caused by its orbital motion relative to its planets. In this case, the combined lensing of a distant source by a star and its Jupiter-scale planet was measured and could be deconvolved to determine the ratio of the two masses and the approximate distance between the star and its planet. Additional systems were discovered subsequently, and gravitational lensing is proving to be a useful tool in the search for exoplanets.
Ilse Rosenthal-Schneider, one of Einstein’s students in 1919, was amazed at his remarkably serene reaction to the telegram from Eddington announcing the eclipse results. When she asked how he would have felt if the observations had not confirmed his prediction, he answered: “Then I would have been sorry for the dear Lord. The theory is correct.” Einstein was joking, of course. He understood full well that a theory stands or falls on the basis of its agreement with experimentation. Yet to his mind, general relativity was so beautiful, so elegant, so internally consistent that it had to be correct. The eclipse results merely justified his already supreme confidence. In their various ways, Don Bruns, VLBI radio astronomers and the Cassini spacecraft have shown that, so far, Einstein’s confidence was well placed.