Is Einstein Still Right?

by Clifford M. Will

  In fact, the distinction between clock and emitter/receiver of light that we have used is purely a semantic one. The term “clock” really means a device that performs some physical activity repetitively at a well-defined, constant rate. The activity could be the mechanical sweep of a second hand, the flashes of a strobe or the waves of an electromagnetic signal. Modern atomic clocks are based on the latter phenomenon—the emission of light with a constant, stable, well-defined frequency. The gravitational redshift affects all clock rates equally; this includes biological clocks, since, after all, biological processes fundamentally involve atoms and molecules, which are governed by the laws of physics. This can all be summed up in the simple statement that gravity warps time.

  Another thing that should be apparent is that we did not use general relativity itself anywhere in the discussion. The gravitational redshift depends only on the principle of equivalence. Even though the full version of the general theory predicts the redshift effect, and Einstein viewed the redshift as one of the three main tests of his theory, we now regard it as a test of the more fundamental equivalence principle. Any theory of gravity that is compatible with the equivalence principle (and there are many, including, for instance, the theory by Brans and Dicke) automatically predicts the same gravitational redshift as general relativity.

  A question that is often asked is: Do the intrinsic rates of the emitter and receiver or of the clocks change, or is it the light signal that changes frequency during its flight? The answer is simple: it doesn’t matter! Both descriptions are physically equivalent. Put differently, there is no way to carry out an experiment to distinguish between the two descriptions. Suppose that we tried to check whether the emitter and the receiver agreed in their rates by bringing the emitter down from the tower and setting it beside the receiver. We would find that indeed they agree. Similarly, if we were to transport the receiver to the top of the tower and set it beside the emitter, we would find that they also agree. But to get a gravitational redshift, we must separate the clocks in height; therefore, we must connect them by a signal that traverses the distance between them. But this makes it impossible to determine unambiguously whether the shift is due to the clocks or to the signal. The observable phenomenon is unambiguous: the received signal is blueshifted. To ask for more is to ask questions without observational meaning.

  This is a key aspect of relativity, and in fact of physics as a whole. We concentrate only on quantities that we can measure with physical devices, and avoid unanswerable questions.

  There is one way to see the effect of the gravitational redshift without an intervening signal, however, and that is to measure its effect on the elapsed time of two clocks. Begin with two clocks side by side, ticking at the same rate, and synchronized, so that at some chosen moment they read the same time and tick at the same rate. Take one clock slowly to the top of the tower and let it sit there for a while. Then, bring it back down slowly and compare it with the ground clock. While the rates at which they tick will once again be the same once the clocks are reunited, the tower clock will be ahead of the ground clock. The inference from this is that the tower clock ran faster while it was on the tower, but unless we connect the clocks by a light signal, we cannot see the difference in the ticking rate except after the fact, once we reunite them. This idea actually was the basis for a 1971 experiment using atomic clocks and jet aircraft, to be described shortly.

  Early attempts to measure the gravitational redshift focused on light from the Sun. When an atom undergoes a transition from one electronic level to another, it emits light at a frequency or wavelength that is a characteristic of the atom. In the laboratory, the frequencies of these “spectral lines” can be measured with high accuracy. The same atom on the surface of the Sun will emit light whose frequency is redshifted as seen from Earth because, in a thought experiment using a tower sitting on the surface of the Sun and stretching all the way to the Earth, the atom would be at the bottom of the tower and the receiver on Earth would be at the top of the tower (Figure 2.2). In this example we can ignore the effects of Earth’s gravity and of its orbital motion; these produce a correction of order 0.03 percent to the dominant effect due to the Sun. For a wavelength of 5,893 angstroms (an angstrom is ten billionths of a centimeter), corresponding to the bright-yellow emission line of the sodium atom, one of the most intense in the solar spectrum, the shift is 0.0125 angstroms toward longer wavelengths (lower frequencies), well within reach of standard measurement techniques.

  Figure 2.2 Gravitational redshift of light emitted by the Sun and received by an observer on Earth. The “tower” is used only to analyze the amount of the shift.

  In 1917, however, Charles E. St. John of the Mount Wilson Observatory in California reported that he had found no “Einstein” redshift of spectral lines from the Sun, and a 1918 report from an observatory in Kodiakanal in India was inconclusive. One can only imagine how Einstein must have felt! Science historians think that these results had a direct negative impact on Einstein’s candidacy for the 1918 Nobel Prize. The prize would not be awarded to him until 1921, and then only for his 1905 explanation of the photoelectric effect, and not for any of his relativistic theories. His theory of the photoelectric effect was verified definitively by experiment, whereas in 1921, verifying relativity still had a long way to go.

  Looking back, we see the results of St. John and others not as a failure of Einstein’s prediction but rather as a lack of understanding of the solar surface at the time. The gas at the surface of the Sun experiences violent and turbulent motions, with rising columns of hot gas and falling columns of cooler gas, which lead to Doppler shifts of the emitted frequencies both to the blue and to the red. The gas is also under high pressure, which causes shifts in the intrinsic frequencies emitted by certain atoms. These and other effects made it impossible in those early years to separate the gravitational shift clearly from other complex effects. It wasn’t until the 1960s, when these effects were better understood, that astronomers were able to measure the gravitational redshift of solar lines. A measurement in 1991 confirmed the prediction to about 2 percent.

  The problem with the Sun is that the relativistic shift is so tiny compared to other contaminating effects. But by 1920 astronomers had identified a few examples of a different kind of star, a white dwarf, which could be used to measure Einstein’s predicted redshift. A white dwarf is a star with a mass comparable to that of the Sun, but compressed into a ball the size of the Earth, a hundred times smaller than the Sun. The gravitational redshift is thus about a hundred times larger than that from the Sun, and accordingly easier to detect. But the prediction of the redshift depends on the mass and radius of the white dwarf, which are not as well known as those quantities are for the Sun.

  Fortunately, there was an exception to this even as far back as the 1920s. One of these unusual stars, called Sirius B, was actually in orbit around the “dog star,” Sirius (called Sirius A), the brightest star in the night sky. This allowed astronomers to determine that Sirius B has about the same mass as the Sun, as inferred from its orbital motion around Sirius A. In 1924, Arthur Stanley Eddington (1882–1944), who, in addition to his talents as an astronomer, was the world’s leading expert in stellar structure at the time, used his mathematical models to argue that the radius of Sirius B is about forty times smaller than that of the Sun. From that, he made a prediction of the gravitational redshift of spectral lines from Sirius B. At the same time, the noted spectroscopist Walter S. Adams of the Mount Wilson Observatory in California, who had first measured the spectrum of Sirius B in 1915, was engaged in making improved measurements with a view toward detecting the Einstein redshift. The spectrum was difficult to interpret, in part because of contamination of light from the much brighter Sirius A. But in 1925 Adams reported his results, in remarkably close agreement with Eddington’s prediction. The New York Times reported “New Test Supports Einstein’s Theory.”

  In time, however, it all began to unravel. First, it was realized t
hat the models used by Eddington to study white dwarfs were wrong. His Cambridge University colleague Ralph Fowler pointed out in 1926 that the white dwarf was an entirely new kind of astronomical beast, with an internal constitution radically different from normal stars, governed by the quantum mechanical principle that no two electrons can occupy the same state. In addition, as more white dwarfs were discovered and their unique spectral signatures identified, Adams’ interpretation of his spectra also came under heavy fire.

  The world would have to wait another forty years before the white dwarf test could be carried out correctly. It was not until 1961 that the orbit of Sirius B would bring it far enough away from Sirius A as seen from Earth to enable new and improved spectral measurements, now using the Mount Palomar 200 inch telescope. Meanwhile, modern theories of white dwarf structure had been developed that could make better predictions of the radius and of the redshift. Finally, in 1971 results were announced that confirmed Einstein, but showed that the redshift inferred from the spectra was four times larger than that claimed by Adams, and the predicted shift inferred from the theoretical models was four times larger than that predicted by Eddington. Adams and Eddington were both off by the same factor of four! Suggestions have been made of conscious or unconscious bias on Adams’ part, given that he and Eddington were in regular communication while he was making his measurements. But most science historians reject this claim, arguing that both the theory and the observations circa 1924 were so compromised that it was pure luck (good, bad or otherwise) that Adams and Eddington got apparent agreement. In 2005, measurements of Sirius B using the Hubble Space Telescope confirmed the Einstein redshift to about 6 percent.

  Einstein had proposed three crucial tests of his theory: the explanation of an anomalous advance of the orbit Mercury, which he had worked out in his 1915 papers (page 89), the deflection of light, as confirmed in 1919, and the gravitational redshift. In 1950, Einstein had to admit that evidence for the redshift effect was “not yet confirmed.” Within ten years, however, confirmation of the redshift would finally arrive, not from astronomy but from the physics laboratory.

  The first truly accurate and reliable test of the redshift was the Pound–Rebka experiment of 1960. This experiment is very close in concept to the one described in our thought experiment in Figure 2.1. In this case, the tower was the Jefferson Tower of the physics building at Harvard University. For the tower’s height of 74 feet the predicted frequency shift is only two parts in a thousand trillion, and thus an emitter and receiver of extremely well-defined frequency are required. Robert V. Pound and his student, Glen Rebka, Jr., used the unstable Fe57 isotope of iron, which has a lifetime or half-life of one ten-millionth of a second (one tenth of a microsecond). When this isotope decays it emits light in the form of gamma rays of wavelength 0.86 angstroms, within a very narrow range in wavelengths of only one part in a trillion of the basic wavelength. The same isotope can also absorb gamma rays of the same wavelength within the same narrow spread.

  However, this alone is not enough to measure the redshift. Inside any realistic sample containing Fe57, the iron nuclei are constantly in random motion because of the internal energy contained in any body at a finite temperature. This leads to Doppler shifts in the emitted gamma ray frequencies, whose result is to broaden or smear the range of wavelengths. In addition, upon emission or reception of a gamma ray, the iron nucleus “recoils,” just as a billiard ball will recoil slightly if struck by a ping-pong ball, and this recoil velocity also causes a Doppler shift in the frequency. These effects can broaden the range of frequencies emitted and absorbed by a realistic Fe57 sample so severely that a redshift measurement would have been impossible, had it not been for Rudolph Mössbauer.

  Working at the Max Planck Institute in Heidelberg, Germany, in the late 1950s, Mössbauer discovered that if an iron nucleus is implanted in the right kind of crystal, then the forces of the surrounding atoms not only reduce the heat-induced motions of the atom, but also transfer the recoil of the emitting atom to the crystal as a whole, thereby virtually eliminating it, because of the enormous mass of the crystal compared to the iron atom. For this discovery, Mössbauer was awarded the Nobel Prize in Physics in 1961. The Harvard gravitational redshift experiment was one of the many important applications of the effect cited at the awards ceremony by the Swedish Academy of Sciences, which is in charge of the Nobel Prize.

  Pound and Rebka carefully fabricated their Fe57 emitters and absorbers in order to take best advantage of the Mössbauer effect. But the range of frequencies emitted and absorbed was still a thousand times larger than the size of the expected gravitational shift, so they used a clever trick.

  They put the emitter on a movable platform that could be raised and lowered slowly using a hydraulic lift and a rack-and-pinion clock drive. If the emitter was at the top of the tower, so that the gamma rays would be blueshifted upon reaching the bottom, the platform was raised slowly, producing a Doppler shift toward the red. By adjusting the rate at which the emitter was raised, Pound and Rebka could produce a Doppler redshift that would cancel or compensate for the gravitational blueshift, thereby allowing the range of frequencies of gamma rays received at the bottom to match closely the range that could be absorbed by the receiver. The Doppler shift required to do this was then a measure of the gravitational blueshift. The needed velocity was about 2 millimeters per hour. In order to eliminate certain sources of error, the actual experiment was a symmetrical one. Half the measurements were made with an emitter at the top and an absorber at the bottom, to measure the blueshift, and half were made with an emitter at the bottom and an absorber at the top, to measure the equal and opposite redshift. The results of the 1960 experiment agreed with the prediction to 10 percent, and those of an improved 1965 experiment version by Pound and Joseph L. Snider agreed to 1 percent.

  As we mentioned earlier, another way to check the gravitational redshift is to compare the readings of two clocks that are separated temporarily. During October 1971, a remarkable experiment was performed that checked both these phenomena—gravitational redshift and special relativity’s time dilation—in their effects on traveling clocks. The idea behind the “jet-lagged clocks” experiment is this. Consider, for simplicity’s sake, a clock on Earth’s equator, and an identical clock on a jet plane flying overhead to the east at some altitude. Because of the gravitational blueshift, the flying clock will tick faster than the ground clock. What about special relativity’s time dilation? Here we must be a bit careful, because the Earth is also rotating about its axis, so both clocks are moving in circles around the center of the Earth, rather than in straight lines through empty space.

  According to special relativity, the rate of a moving clock must always be compared to a set of clocks that are in an inertial frame, in other words that are at rest or moving in straight lines at constant velocity. Therefore we can’t simply compare the flying clock directly with the ground clock. Let us instead compare the rates of both clocks to a set of fictitious clocks that are at rest with respect to the center of the Earth and not rotating with the Earth (Figure 2.3). The ground clock is moving at a speed determined by the rotation rate of the Earth, and thus ticks more slowly than the fictitious inertial clocks (as represented by a master inertial clock in Figure 2.3); when the flying clock is moving in the samedirection as Earth’s rotation (eastbound), then it is moving even more quickly than the ground clock relative to the inertial clocks, so it is ticking even more slowly. Thus time dilation makes the flying clock run slowly relative to the ground clock.

  Figure 2.3 Jet-lagged clocks. (a) Eastbound. At the start, the flying clock is directly above the ground clock (in gray); after some time the ground clock has moved, while the flying clock is farther to the east. The flying clock has traveled more quickly than the ground clock relative to a stationary master clock, and therefore ticks more slowly relative to it than does the ground clock because of the time dilation of special relativity. Thus, the flying clock ticks more slowly than th
e ground clock. On the other hand, the gravitational blueshift makes the flying clock tick more quickly than the ground clock. The two effects can offset each other. (b) Westbound. At the start, the flying clock is again directly above the ground clock; after some time the ground clock has moved, but the flying clock has not moved as far because a typical commercial jet cannot overtake the Earth’s rotation. It has traveled more slowly relative to the inertial clock than has the ground clock. Thus, the flying clock ticks more quickly than the ground clock relative to the master clock because of time dilation. The gravitational blueshift also causes the flying clock to tick more quickly, so the two effects add to each other.

  In this thought experiment, the two effects, gravitational blueshift and time dilation, tend to offset one another, and whether the net effect is that the flying clock ticks more quickly or ticks more slowly than the ground clock will depend on the height of the flight, which determines the amount of gravitational blueshift or speed-up, and the ground speed of the flight, which determines the amount of time dilation slow-down. Consider now a westbound flying clock at the same altitude. The gravitational blueshift is the same, but now the flying clock is traveling more slowly relative to the inertial clocks than is the ground clock, and therefore it is the ground clock that ticks more slowly relative to the inertial clocks. Therefore, the flying clock ticks more quickly than the ground clock, and in this case both the gravitational and time dilation effects work together, causing the flying clock to tick more quickly. If we were to start with three identical, synchronized clocks, and we were to leave one at home while sending one around the world to the east and the other around the world to the west, we would expect the westward clock to return having gained time, or aged more quickly, while the eastward clock would have gained or lost time depending on the altitude and speed of the flight.