Is Einstein Still Right?

by Clifford M. Will

  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.

  1 The attentive reader might ask if the deflection of the light ray adds to the distance traveled, and hence to the time. Indeed it does, but that effect is negligible compared to the effect we are describing.

  CHAPTER 4

  Does Gravity Do the Twist?

  Gravity Probe-B, the Relativity Gyroscope Experiment, may go down in the history of physics as one of the most difficult, most costly and longest physics experiments ever performed. From conception to completion it took almost half a century and cost $750 million, while the actual data taking took only sixteen months. The experiment was the brainchild of three naked men basking in the noonday California sun in the closing weeks of 1959. The three were all professors at Stanford University in Palo Alto. One of them was the eminent theoretical physicist Leonard I. Schiff, well known for his pioneering work in quantum theory and nuclear physics. In the late 1950s, however, he had become interested in gravitation theory. The second professor was William M. Fairbank, an authority on low-temperature physics and superconductivity, who had just arrived at Stanford in September of 1959, lured there from Duke University in North Carolina. The third was Robert H. Cannon, also a recent acquisition by Stanford, an expert in aeronautics and astronautics from MIT.

  But before we learn how these naked professors came to formulate this experiment, let us first answer the question, what does a gyroscope have to do with relativity? When we think of a gyroscope we imagine something like a spinning flywheel. If the flywheel spins rapidly enough, its axis of rotation always points in the same direction, no matter how we rotate the platform or laboratory in which it sits, as long as the gyroscope is mounted on the platform using gimbals that allow it to turn freely with minimum friction. In other words, the axis of the gyro always points in a fixed direction relative to inertial space or to the distant stars. The difficulty you have in turning a rapidly spinning bicycle wheel is an everyday example of this gyroscopic effect. This, of course, is the basic principle behind the use of gyroscopes in navigation of ships, airplanes, missiles and spacecraft (GPS has now taken over many aspects of such navigation, of course). When attached more rigidly to a platform, this gyroscopic action is what keeps personal transporters like Segways or hoverboards from toppling over. However, according to general relativity, a gyroscope moving through curved spacetime near a massive body such as the Earth will not necessarily point toward a fixed direction; instead, its axis of spin will change slightly, or precess. Two distinct general relativistic effects can cause such a precession.

  The first of these is called the “geodetic effect,” and is a consequence of curved spacetime. Our everyday experience with gyroscopes tells us that as a gyroscope moves through space, its spin axis should maintain the same direction, a direction parallel to its previous direction. However, in curved spacetime, parallel in the local sense does not necessarily mean parallel in the global sense, and so upon completing a closed path, the gyroscope axis can actually end up pointing in a different direction than the one it started with.

  A simple way to see how this can happen is to imagine a two-dimensional world, much like that of the nineteenth-century book Flatland by E. A. Abbott, but here confined to the surface of a sphere. Because the inhabitants of this “Sphereland” are only two-dimensional, they can’t really construct the right kind of gyroscopes; as an alternative, they can take a little pointer and slide it about on their sphere in a way that keeps it always parallel to its previous direction, veering neither to the right nor to the left (up and down are not options in Sphereland). The pointer’s tip then plays a role analogous to the spin axis of our gyroscope. To demonstrate what can happen, the Spherelanders consider the following closed route (see Figure 4.1): from a point at zero degrees longitude on the equator, move east along the equator to 90 degrees longitude, then go due north to the North Pole, make a 90 degree turn, and go due south back to the starting point on the equator. Suppose the Spherelanders start the pointer out parallel to the equator, pointing east. When they reach the first turn the pointer will still point east, and when they head north the pointer will now be perpendicular to the path. At the North Pole they make a 90 degree left turn, but now the pointer’s direction is to their rear, which is to the north as they head south. When they reach the equator once again, the pointer, which has been kept parallel to itself all the way, now points north, whereas it started out pointing east. This change in direction of the pointer is what we would call precession in the case of a gyroscope. The curvature of the two-dimensional surface of Sphereland accounts for this precession, and we understand it without much difficulty. The difference between this example and the geodetic effect on a moving gyroscope is that it is the curvature of spacetime and not just the curvature of space that is important.

  Figure 4.1 Precession in Sphereland. A pointer is carried parallel to itself from 0 degrees longitude to 90 degrees longitude. The path then turns north, but the pointer continues to point east, and maintains that direction up to the North Pole. A right-angle turn of the path leaves the pointer pointing to the rear. The pointer continues to point to the rear (north) back to the starting point. The result is a precession of the direction of the pointer from an easterly direction to a northerly direction.

  The geodetic effect has been known since the early days of general relativity. The first to calculate the effect was Willem de Sitter, the Dutch theorist who had played a pivotal role in bringing general relativity to the attention of Eddington and the British physics community. In a paper published in the Monthly Notices of the Royal Astronomical Society less than a year after Einstein’s November 1915 papers on general relativity, de Sitter showed that relativistic effects would cause the axis perpendicular to the orbital plane of the Earth–Moon system to precess at a rate of about 0.02 arcseconds per year. De Sitter was not thinking in terms of gyroscopes; instead, he had in mind how the combined relativistic gravitational fields of the Earth and Sun would perturb the Earth–Moon orbit. However, Eddington and others soon pointed out that the Earth–Moon system is really a kind of gyroscope, with the axis perpendicular to the orbital plane playing the role of the gyroscope’s rotation axis, so the de Sitter effect was effectively a precession of the Earth–Moon gyroscope. But if this is the case, then the Earth, as it spins about its own rotation axis, is also a gyroscope, so in fact both the Earth and the Earth–Moon system should precess in the same way. At the time, measuring such a small effect was hopeless. Only in recent years has a technique called “lunar laser ranging” (see Chapter 5 for a discussion) given such precise information about the Earth–Moon orbit that the de Sitter effect could be measured, to around half a percent.

  Instead of the Earth–Moon system, consider a more down-to-Earth situation: a laboratory-size gyroscope orbiting the Earth with its axis lying in the orbital plane, say pointing vertically (see the gyroscope labeled “start” in Figure 4.2). Without general relativity, the gyroscope would maintain its direction relative to distant stars as it orbits the Earth, and so after a complete orbit it
would again be pointing in the same vertical direction. But general relativity predicts that as the orbit carries the gyroscope around through the Earth’s curved spacetime, the gyroscope will experience a precession within the orbital plane at a rate of a little over a thousandth of an arcsecond per orbit. The direction of the precession is in the same sense as the motion of the gyroscope around its path, counterclockwise if looking down on the orbit from above (see Figure 4.2). Since the period of revolution of a low Earth orbit is about 1.5 hours, the net precession in one year will be around 6 arcseconds. This is the geodetic effect.

  Figure 4.2 Geodetic precession of a gyroscope in near-Earth orbit. After one orbit, the direction of the gyroscope axis has rotated relative to its initial direction in the same sense (counterclockwise) as that of the orbit. The net effect over one year (5,000 orbits) is 6 arcseconds.

  The other important relativistic effect on a gyroscope is known as the dragging of inertial frames, one of the most interesting and unusual of the predictions of general relativity (see Figure 4.3). The origin of this effect is the rotation of the body in whose gravitational field the gyroscope resides. According to general relativity, a rotating body attempts to “drag” the spacetime surrounding it into rotation. The simplest way to get a picture of the consequences of this dragging is to use a fluid analogy.

  Figure 4.3 Left: Swimmer and air mattresses in a pool with a whirlpool. All are tethered to the bottom of the pool so that they don’t move around the pool. The air mattress at the edge of the whirlpool rotates in a clockwise direction because water closer to the center moves faster, while the air mattress at the center rotates counterclockwise with the water. A vertical swimmer treading water does not rotate. Right: Dragging of inertial frames. Stationary gyroscopes near a rotating Earth can precess because of dragging of spacetime by rotation of the Earth. If the axis lies perpendicular to the rotation axis of the Earth, the precession will be opposite to the Earth’s rotation for a gyroscope at the equator, and with the Earth’s rotation for a gyroscope at the pole. If the axis is parallel to the Earth’s rotation axis then there is no precession. For other locations and other orientations of the gyroscope axis, the precession will be between these extremes.