Is Einstein Still Right?

by Clifford M. Will

  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 t
han 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).